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Analyze one and two-sample Z-test in R
In this article, you will learn how to perform three types of Z tests (one sample, two samples, and paired Z test) in R. If you would like to learn more about Z test background, hypotheses, formula,
and assumptions, you can refer to this detailed article on how to perform Z-test in Python.
1. One Sample Z-test
One sample Z-test compares the sample mean with a specific hypothesized mean of the population. Unlike t-test, the mean and standard deviation of the population should be known in the Z-test.
Example: A factory produces balls of a diameter of 5 cm. But due to manufacturing conditions, every ball may not have the exactly same diameter. The standard deviation of the diameters of balls is
0.4. Now, the quality officer would like to test whether the ball diameter is significantly different from 5 cm in a sample of 50 balls randomly taken from the manufacturing line.
Import dataset for testing ball diameter using Z-test,
Read more here how to import CSV dataset in R
df = read.csv('https://reneshbedre.github.io/assets/posts/ztest/z_one_samp.csv')
head(df, 2)
# output
1 4.819289
2 3.569358
# determine the size of data frame
# output
[1] 50 1
Now, perform one sample Z-test using the z.test() function available in the BSDA package. z.test() function takes the following required arguments,
x: Numeric vector for one sample data (use this in case of one sample Z-test)
mu: hypothesized or known population mean (specified in null hypothesis)
alternative: Type of test to calculate p value (two-sided, greater, or less). By default, it performs two-sided test.
sigma.x: population standard deviation for x
# perform two-sided test
z.test(x = df, mu = 5, sigma.x = 0.4)
# output
One-sample z-Test
data: df
z = 0.31746, p-value = 0.7509
alternative hypothesis: true mean is not equal to 5
95 percent confidence interval:
4.907086 5.128831
sample estimates:
mean of x
If you want to perform one-sided test, then add an option alternative: 'greater' or alternative: 'less' based on the hypothesis.
As the p value of the one-sample Z-test is not significant [Z value = 0.3174, p = 0.7509], we conclude that the mean diameter of balls in a random sample is equal to the population mean of 5 cm. It
is, therefore, a sample drawn from an identical population.
2.2 Two sample Z-test (unpaired or independent Z-test)
The two-sample (unpaired or independent) Z-test calculates if the means of two independent groups are equal or significantly different from each other. Unlike the t-test, Z-test is performed when the
population means and standard deviation are known.
Calculate Two sample Z-test in R
Example: a factory in A and B cities produces balls of a diameter of 5 cm. The defect is reported in the B factory which is known to cause a change in ball size. The standard deviation of the
diameters of the balls is 0.1. The factory quality officer randomly selects 50 balls from both factories to test whether the ball diameter is significantly different in factory B than that produced
in factory A.
Import dataset for testing ball diameter using two sample Z-test,
Read more here how to import CSV dataset in R
df = read.csv('https://reneshbedre.github.io/assets/posts/ztest/z_two_samp.csv')
head(df, 2)
# output
fact_A fact_B
1 4.977904 5.887947
2 5.166254 5.990616
# determine the size of data frame
# output
[1] 50 2
Now, perform one sample Z-test using the z.test() function available in the BSDA package. z.test() function takes the following required arguments,
x: Numeric vector for x group
y: Numeric vector for y group
sigma.x: population standard deviation for x group
sigma.y: population standard deviation for y group
alternative: Type of test to calculate p value (two-sided, greater, or less). By default, it performs two-sided test.
# perform two-sided test
z.test(x = df1$fact_A, y = df1$fact_B, sigma.x = 0.1, sigma.y = 0.1)
# output
Two-sample z-Test
data: df1$fact_A and df1$fact_B
z = -48.866, p-value < 2.2e-16
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-1.0165115 -0.9381129
sample estimates:
mean of x mean of y
5.012839 5.990151
Z-test results for the two samples show a highly significant p value [Z = -48.866, p < 0.001]. This indicates that the balls produced in factories A and B are significantly different in size. As a
result, factory B’s ball production machine has a defect.
2.3 Paired Z-test (dependent Z-test)
• Paired Z-test is used for checking whether there is difference between the two paired samples or not. For example, we have plant variety A and would like to compare the yield of variety A before
and after the application of fertilizer.
Calculate paired Z-test in R
Example: a researcher wants to test the effect of fertilizer on plant growth. The researcher measures the height of the plants before and after the application of fertilizer. The standard deviations
for two plant height differences (before and after the application of fertilizer) is 1.2. A researcher wants to test if there is an increase in plant heights after the application of fertilizers.
Import dataset for paired Z-test,
Read more here how to import CSV dataset in R
df = read.csv('https://reneshbedre.github.io/assets/posts/ztest/paired_z.csv')
head(df, 2)
# output
before after
1 75.73154 83.15463
2 75.27301 83.32258
# output
[1] 60 2
If the assumed population difference is zero (as stated in the null hypothesis), the paired Z-test reduces to the one sample Z-test. Hence, we will perform one sample Z-test on paired
# get difference
df$diff = df$after - df$before
head(df, 2)
# output
before after diff
1 75.73154 83.15463 7.423096
2 75.27301 83.32258 8.049578
# perform one sample Z-test on differences (two-sided test)
z.test(x = df$diff, mu = 0, sigma.x = 1.2)
# output
One-sample z-Test
data: df$diff
z = 67.278, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
10.11903 10.72630
sample estimates:
mean of x
Based on the paired Z-test, the p value is highly significant [Z = 67.278, p < 0.001]. The plant height increased significantly after fertilizer was applied. We conclude that the application of
fertilizer significantly contributes to the height increase of plants.
Enhance your skills with courses on Statistics and R
• Heumann C, Shalabh MS. Introduction to statistics and data analysis. Springer; 2016.
If you have any questions, comments or recommendations, please email me at reneshbe@gmail.com
This work is licensed under a Creative Commons Attribution 4.0 International License
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Pyramid Volume Calculator
A pyramid is one of the most complicated figures in geometrical Mathematics. It requires proficiency in this skill to perform different calculations including area and volume calculations. As it is a
three-dimensional figure, that’s why it involves an extra step to find its volume. If you are not proficient in this field, you should use this Pyramid Volume Calculator.
This online handy tool has been designed for this calculation particularly. With the help of this maths calculator, you can easily find the volume of the Pyramid within seconds and with 100% accuracy
. You can solve simple as well as complex problems using this advanced calculator.
What is Pyramid Volume?
A pyramid is a three-dimensional figure that has triangular dimensions from all sides. The volume of a pyramid is the region or area that comes under its boundaries. As it belongs to three
dimensions, so, it covers length, width, and height in that region.
We can also say that the volume of a pyramid is the region that comes under its boundaries in all three dimensions. To find this measurement, we need to find the base and height of the figure. Like
other volume measurements, it must be calculated in cubic units. For example, if we are given lengths in meters, the volume of that pyramid must be in cubic meters
How to calculate Pyramid Volume?
To learn the method of pyramid volume calculation, you first need to learn the formula used for it. The formula for pyramid volume calculation keeps varying with the sides involved or the shape of
the figure. In simple words, the formula for Triangular pyramid volume is different from square pyramid volume.
For the sake of your understanding, we have written the formula related to triangular pyramid volume here and solved an example too.
Triangular Pyramid Volume = √3/12 a^2 h
• “a” represents the base measurement of the pyramid
• “h” represents the height of the pyramid that is measured from base to top corner perpendicular.
Here we have solved an example related to this problem that you can follow for learning how to find the volume of a pyramid.
Example 1:
Find the triangular pyramid volume having a base length equal to 7m and a height equal to 12m.
As we know, By dividing, we get the following factors,
Volume of a Pyramid = √3/12 a^2 h
= √3/12 (7)^2 (12)
= 84.9 sq. meter
How to use Pyramid Volume Calculator?
Using the pyramid volume calculator isn’t difficult when you have this online tool by Calculator’s Bag. You can use this tool using the simple steps mentioned below:
• Choose the type of Pyramid first
• Insert the length of the base and choose its unit
• Insert the measurement of height and choose its unit
• This calculator will automatically show you a measurement of the volume of that pyramid.
FAQ | Pyramid Volume Calculator
How does the pyramid volume calculator work?
This pyramid volume calculator has an advanced and pre-programmed algorithm using which it performs the calculation using the formula.
What are the different types of pyramids that can be used with the pyramid volume calculator?
You can solve problems related to triangular pyramids, square pyramids, pentagon pyramids, and all others
Can the pyramid volume calculator be used for other 3-dimensional shapes?
Yes, it has been designed for three-dimensional shapes.
How accurate is the pyramid volume calculator?
This pyramid volume calculator has an advanced algorithm with which it will provide you 100% accurate answers.
Is the pyramid volume calculator suitable for use in engineering and construction projects?
Yes, if you have a problem related to volume calculation, you can use this calculator. It will make your work easier and more accurate at the same time.
Why is 1/3 used to find the volume of a pyramid?
Because a pyramid is a three-dimensional shape and the volume is also a three-dimensional quantity, that’s why 1/3 is used to find its volume. | {"url":"https://calculatorsbag.com/calculators/math/pyramid-volume","timestamp":"2024-11-12T19:15:16Z","content_type":"text/html","content_length":"59697","record_id":"<urn:uuid:bcdeda2b-be5c-4c9c-a306-3778e390cad1>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00659.warc.gz"} |
0.7 Kilometers to Centimeters
0.7 km to cm conversion result above is displayed in three different forms: as a decimal (which could be rounded), in scientific notation (scientific form, standard index form or standard form in the
United Kingdom) and as a fraction (exact result). Every display form has its own advantages and in different situations particular form is more convenient than another. For example usage of
scientific notation when working with big numbers is recommended due to easier reading and comprehension. Usage of fractions is recommended when more precision is needed.
If we want to calculate how many Centimeters are 0.7 Kilometers we have to multiply 0.7 by 100000 and divide the product by 1. So for 0.7 we have: (0.7 × 100000) ÷ 1 = 70000 ÷ 1 = 70000 Centimeters
So finally 0.7 km = 70000 cm | {"url":"https://unitchefs.com/kilometers/centimeters/0.7/","timestamp":"2024-11-04T05:05:54Z","content_type":"text/html","content_length":"22942","record_id":"<urn:uuid:57cf7f62-fc22-4ad4-b254-e3fea415a8cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00399.warc.gz"} |
How to Understand the Statistical Information Definitions of the Question Basic Analysis Report
The Question Basic Analysis Report displays a significant amount of statistical information that can be useful in analyzing the effectiveness of assessment material, offering guidance towards
Below are detailed explanations of what each term means on the Question Basic Analysis.
Statistical Information
Item Description
Total The total number of respondents that answered the question.
Blank The total number of respondents that did not provide an answer to the question.
Response The mean (average) of the numeric position of the answer choices within the question. This is useful under certain circumstances such as when there is a linear progression of the answer
Mean choices.
Standard The standard deviation of the numeric position of the answer choices within the question. This is useful under certain circumstances, such as when there is a linear progression of the
Deviation answer choices.
Answer The average amount of time respondents spend on this question.
The calculated average of all the Recap times for respondents answering the question.
Recap Recap time is the amount of time that could not be attributed to a single question on a page of questions. More specifically, the Recap Time is the amount of time between a respondent
Time selecting an answer to the last question on a page of questions (or selects an answer to the only question on a page), and navigating to another page of questions. If there is more than one
question displayed on a page, the total Recap Time is divided evenly between all the questions on the page when calculating the Average Recap Time.
Item Analysis
Item Description
Mean Score When Item The mean score value for respondents answering the question correctly
Correct (Mp)
Mean Score When Item The mean score value for respondents answering the question incorrectly
Incorrect (Mq)
Overall Standard Measures the range of deviation across the entire group of respondents for the item across the assessment, showing how the respondent group scores are spread out from the
Deviation (St) average.
Proportion Answering The question Item Difficulty Index. The p-value, expressed as a decimal value between 0.0 and 1, displays the proportion of respondents answering the question item correctly. A
Correctly (p-value) higher p-value indicates a greater proportion of respondents answered the item correctly, meaning it was an easier question item. A lower p-value indicates a lower proportion
of respondents answered the item correctly, meaning it was a more difficult question item.
Proportion Answering The False Discovery Rate or the proportion of false positives you can expect to get from a question item, typically used when working with smaller samples.
Incorrectly (q-value)
Point-Biserial The correlation between respondent scores on a question item and the respondents overall total assessment score. The correlation between respondent scores on a question item is
Correlation measured as 1 if a respondent answers the question item correctly, and 0 if the respondent answers a question item incorrectly.
Coefficient (rpbi) | {"url":"https://support.brillium.com/en-us/knowledgebase/article/KA-01028","timestamp":"2024-11-10T17:44:18Z","content_type":"text/html","content_length":"51872","record_id":"<urn:uuid:5269a7a5-bc2f-4ef2-88d0-c3329a780926>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00565.warc.gz"} |
Modeling the Relative Size of the Solar System | BFSU Community
Modeling the Relative Size of the Solar System
• Posts
From artists renditions of the solar system, kids readily gain a “picture” of the relative positions of the sun, earth, moon, other planets, etc. What is more difficult to appreciate is the
distances between these objects. The following modeling exercise, known as the cosmic distance ladder, may help:
In this model, the sun is the size of a grape (one half inch in diameter). The earth and other planets are grains of sand. Now for distances between:
☆ Student representing the sun (grape): stands at the edge of the area
☆ Mercury (tiny grain of sand) = 1 step from sun
☆ Venus (grain of sand) = 2 steps from sun
☆ Earth (grain of sand) = 2.5 steps from sun
☆ Mars (grain of sand) = 4 steps from sun
☆ Asteroid belt (dust particles) = 8 steps from sun
☆ Jupiter (big grain of sand) = 13 steps from sun
☆ Saturn (big grain of sand) = 24 steps from sun
☆ Uranus (big grain of sand) = 49 steps from sun
☆ Neptune (big grain of sand) = 76 steps from sun
☆ Kuiper belt = 100 steps from sun
Remind students that these are relative dimensions within the solar system. Lets leave the solar system and go on to the nearest star, about 4 light years away. On the same scale, the student
representing this distance would have to take close to 7000 steps, roughly 1.3 miles.
Return again, to the fact that this relative distance scale is based on the size of the sun being the size of a grape (one half inch in diameter) and the Earth bring a tiny grain of sand!
Then the nearest star (represented by another grape) is over a mile away!!!! Impress on students how much of outer space is exactly that–vast stretches of empty space.
Measurement of Cosmic Distances
Giving these relative distances should lead students to ask: How are such distances measured? The following video provides a historical sketch of how such distances have been estimated.
See also, type into your browser: cosmic ladder
• Posts
• You must be logged in to reply to this topic. | {"url":"https://www.bfsucommunity.com/forums/topic/modeling-relative-size-solar-system/","timestamp":"2024-11-11T20:35:42Z","content_type":"text/html","content_length":"38518","record_id":"<urn:uuid:1aca0bfd-a579-4cf6-afdc-8b6063bafca0>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00022.warc.gz"} |
The fundamental gap of simplices
Journal article, 2013
The gap function of a domain Ω⊂Rn is ξ(Ω):=d2(λ2−λ1) , where d is the diameter of Ω, and λ1 and λ2 are the first two positive Dirichlet eigenvalues of the Euclidean Laplacian on Ω. It was recently
shown by Andrews and Clutterbuck (J Amer Math Soc 24:899–916, 2011) that for any convex Ω⊂Rn , ξ(Ω)≥3π2 , where the infimum occurs for n = 1. On the other hand, the gap function on the moduli space
of n-simplices behaves differently. Our first theorem is a compactness result for the gap function on the moduli space of n-simplices. Next, specializing to n = 2, our second main result proves the
recent conjecture of Antunes-Freitas (J Phys A: Math Theor 41(5):055201, 2008) for any triangle T⊂R2 , ξ(T)≥64π29 , with equality if and only if T is equilateral.
Communications in Mathematical Physics
0010-3616 (ISSN) 1432-0916 (eISSN)
Vol. 319 1 111--145-
Subject Categories | {"url":"https://research.chalmers.se/en/publication/214309","timestamp":"2024-11-03T19:35:51Z","content_type":"text/html","content_length":"30466","record_id":"<urn:uuid:ff00ef68-8cfa-4b09-af30-327e892cf54b>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00801.warc.gz"} |
Statistics Translated: Second Edition: A Step-by-Step Guide to Analyzing and Interpreting Data
Statistics Translated
Second Edition
A Step-by-Step Guide to Analyzing and Interpreting Data
HardcoverPaperbacke-bookprint + e-book
February 17, 2021
ISBN 9781462545414
Price: $89.00
433 Pages
Size: 7" x 10"
March 8, 2021
ISBN 9781462545407
Price: $59.00
433 Pages
Size: 7" x 10"
January 22, 2021
Price: $59.00
433 Pages
print + e-book
Paperback + e-Book (PDF)
Price: [S:$118.00:S] $70.80
433 Pages
- Introduction: You Do Not Need to Be a Statistician to Understand Statistics!
A Little Background
Many Students Do Not Know What They’re Getting Into
A Few Simple Steps
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
So, What’s New in This Edition?
Do You Understand These Key Words and Phrases?
1. Identifying a Research Problem and Stating Hypotheses
Identify the problem
- Characteristics of a Good Problem Statement
- Finding a Good Research Problem
- The Problem Is Interesting to the Researcher
- The Scope of the Problem Is Manageable by the Researcher
- The Researcher Has the Knowledge, Time, and Resources Needed to Investigate the Problem
- The Problem Can Be Researched through the Collection and Analysis of Numeric Data
- Investigating the Problem Has Theoretical or Practical Significance
- It Is Ethical to Investigate the Problem
- Writing the Problem Statement
- Problem Statements Must Be Clear and Concise
- The Problem Statement Must Include All Variables to Be Considered
- The Problem Statement Should Not Interject the Researcher’s Bias
- Summary of Step 1: Identify the Problem
State a Hypothesis
- An Example of Stating Our Hypothesis
- A Little More Detail
- The Direction of Hypotheses
- Using Directional Hypotheses to Test a “Greater Than” Relationship
- Using Directional Hypotheses to Test a “Less Than” Relationship
- Nondirectional Hypotheses
- Hypotheses Must Be Testable via the Collection and Analysis of Data
- Research versus Null Hypotheses
- Stating Null Hypotheses for Directional Hypotheses
- Issues Underlying the Null Hypothesis for Directional Research Hypotheses
- Stating Null Hypotheses for Nondirectional Hypotheses
- A Preview of Testing the Null Hypothesis
- Where Does That Leave Us?
- Statistical Words of Wisdom
- Summary of Step 2: State a Hypothesis
Do You Understand These Key Words and Phrases?
Quiz Time!
Problem Statements
Case Studies:
The Case of Distance Therapy
- The Case of the New Teacher
- The Case of Being Exactly Right
- The Case of “Does It Really Work?”
- The Case of Advertising
- The Case of Learning to Speak
- The Case of Kids on Cruises
2. Identifying the Independent and Dependent Variables in a Hypothesis
Identify the Independent Variable
- Nonmanipulated Independent Variables
- Another Way of Thinking about Nonmanipulated Independent Variables
- Manipulated or Experimental Independent Variables
- Levels of the Independent Variable
- Summary of Step 3: Identify the Independent Variable
Identify and Describe the Dependent Variable
- Identifying Your Dependent Variable
- What Type of Data Are We Collecting?
- Interval Data
- Data Types—What Is the Good News?
- Summary of the Dependent Variable and Data Types
- Measures of Central Tendency
- The Mean, Median, and Mode—Measures of Central Tendency
- The Mode
- Using Statistical Software to Analyze Our Data
- Summary of the First Part of Step 4: Identify and Describe the Dependent Variable
Do You Understand These Key Words and Phrases?
Do You Understand These Formulas?
Quiz Time!
3. Measures of Dispersion and Measures of Relative Standing
Measures of Dispersion
- The Range
- The Standard Deviation
- The Variance
Measures of Relative Standing
- Percentiles
- Computing and Interpreting T-Scores
- Stanines
- Putting It All Together
- Using SPSS for T-Scores and Stanines—Not So Fast!
Do You Understand These Key Words and Phrases?
Do You Understand These Formulas?
Quiz Time!
4. Graphically Describing the Dependent Variable
Graphical Descriptive Statistics
- Graphically Describing Nominal Data
- Pie Charts
- Bar Charts
- Graphically Describing Quantitative Data
- Scatterplots
- Histograms
- Don’t Let a Picture Tell You the Wrong Story!
- Summary of Graphical Descriptive Statistics
The Normal Distribution
- Things That Can Affect the Shape of a Distribution of Quantitative Data
Summary of the Normal Distribution
Do You Understand These Key Words and Phrases?
Quiz Time!
5. Choosing the Right Statistical Test
The Very Basics
- The Central Limit Theorem
- The Sampling Distribution of the Means
- Summary of the Central Limit Theorem and the Sampling Distribution of the Means
How Are We Doing So Far?
Estimating Population Parameters Using Confidence Intervals
- The Alpha Value
- Type I and Type II Errors
Predicting a Population Parameter Based on a Sample Statistic Using Confidence Intervals
- Pay Close Attention Here
- Confidence Intervals for Alpha = .01 and Alpha = .10
- Another Way to Think about z Scores in Confidence Intervals
- Tying This All Together
- Be Careful When Changing Your Alpha Values
- Do We Understand Everything We Need to Know about Confidence Intervals?
Testing Hypotheses about a Population Parameter Based on a Sample Statistic
- Making a Decision about the Certification Examination Scores
- We Are Finally Going to Test Our Hypothesis!
- Testing a One-Tailed Hypothesis
Testing a One-Tailed “Less Than” Hypothesis
Summarizing What We Just Said
Be Careful When Changing Your Alpha Values
The Heart of Inferential Statistics
- Probability Values
- A Few More Examples
- Great News—We Will Always Use Software to Compute Our p Value
Choose the Right Statistical Test
- You Already Know a Few Things
- A Couple of Notes about the Table
- Summary of Step 5: Choose the Right Statistical Test
Do You Understand These Key Words and Phrases?
Do You Understand These Formulas and Symbols?
Quiz Time!
6. The One-Sample t-Test
Welcome to the Guinness Breweries
The t Distribution
- Putting This Together
- Determining the Critical Value of t
- Degrees of Freedom
- Be Careful Computing Degrees of Freedom
- Let’s Get Back to Our Anxiety Hypothesis
- Plotting Our Critical Value of t
- The Statistical Effect Size of Our Example
Let’s Look at a Directional Hypothesis
- Using the p Value
- Check Your Mean Scores!
One More Time
- Important Note about Software Packages
Let’s Use the Six-Step Model!
- The Case of Slow Response Time
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Case of Stopping Sneezing
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Case of Growing Tomatoes
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
Do You Understand These Key Words and Phrases?
7. The Independent-Sample t-Test
If We Have Samples from Two Independent Populations, How Do We Know If They Are Significantly Different from One Another?
- The Sampling Distribution of Mean Differences
- Calculating the t Value for the Independent-Sample t-Test
- Pay Attention Here
- Testing Our Hypothesis
- The p Value
- Note on Variance and the t-Test
- The Statistical Effect Size of Our Example
- Let’s Try Another Example
- Remember the Effect Size
- How Does This Work for a Directional Hypothesis?
- Reminder—Always Pay Attention to the Direction of the Means!
Putting the Independent-Sample t-Test to Work
- The Case of the Cavernous Lab
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Case of the Report Cards
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Case of the Anxious Athletes
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
Do You Understand These Key Words and Phrases?
Quiz Time!
The Case of the Homesick Blues
The Case of the Cold Call
The Case of the Prima Donnas
The Case of the Wrong Side of the Road
The Case of Workplace Satisfaction
The Case of the Flower Show
8. The Dependent-Sample t-Test
That’s Great, But How Do We Test Our Hypotheses?
Independence versus Dependence
- Computing the t Value for a Dependent-Sample t-Test
- Testing a One-Tailed “Greater Than” Hypothesis
- The Effect Size for a Dependent-Sample t-Test
- Testing a One-Tailed “Less Than” Hypothesis
- Testing a Two-Tailed Hypothesis
- Let’s Move Forward and Use Our Six-Step Model
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
The Case of the Unexcused Students
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- Just in Case—A Nonparametric Alternative
Do You Understand These Key Words and Phrases?
Quiz Time!
The Case of Technology and Achievement
The Case of Worrying about Our Neighbors
The Case of SPAM
The Case of “We Can’t Get No Satisfaction”
The Case of “Winning at the Lottery”
9. Analysis of Variance and Multivariate Analysis of Variance
Understanding the ANOVA
The Different Types of ANOVAs
- One-Way ANOVA
- Factorial ANOVA
- Multivariate ANOVA (MANOVA)
- Assumptions of the ANOVA
- Random Samples
- Independence of Scores
- Normal Distribution of Data
- Homogeneity of Variance
Calculating the ANOVA
- Descriptive Statistics
- The Total Variance
- The Total Sum of Squares
- The Between Sum of Squares
- The Within Sum of Squares
- Computing the Degrees of Freedom
- Computing the Mean Square
- Computing the F Value
- The F Distribution
- Determining the Area under the Curve for F Distributions
- The p Value for an ANOVA
- Effect Size for the ANOVA
Testing a Hypothesis Using the ANOVA
- The Case of Multiple Means of Math Mastery
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Post-Hoc Comparisons
- Multiple-Comparison Tests
- Always Observe the Means!
- The Case of Seniors Skipping School
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Case of Quality Time
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Case of Regional Discrepancies
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Factorial ANOVA
- The Case of Age Affecting Ability
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- Interpreting the Interaction p value
- The Case of the Coach
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Multivariate ANOVA (MANOVA)
- Assumptions of the MANOVA
- Using the MANOVA
- The Case of Balancing Time
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
Do You Understand These Key Words and Phrases?
Quiz Time!
The Case of Degree Completion
The Case of Seasonal Depression
The Case of Driving Away
The Case of Climbing
The Case of Employee Productivity
10. The Chi-Square Tests
The One-Way Chi-Square Test
The Factorial Chi-Square Test (the Chi-Square Test of Independence)
- Computing the Chi-Square Statistic
- The Chi-Square Distribution
- What about the Post-Hoc Test?
- Working with an Even Number of Expected Values
- The Case of the Belligerent Bus Drivers
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Case of the Irate Parents
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Chi-Square Test of Independence
- Computing Chi-Square for the Test of Independence
- Computing Expected Values for the Test of Independence
- Computing the Chi-Square Value for the Test of Independence
- Determining the Degrees of Freedom for the Test of Independence
- We Are Finally Going to Test Our Hypothesis
- Using SPSS to Check What We Just Computed
- The Corporal Punishment Conundrum
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
Post-Hoc Tests Following the Chi-Square
- The Case of Type of Instruction and Learning Style
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
Do You Understand These Key Words and Phrases?
Quiz Time!
The Case of Prerequisites and Performance
The Case of Getting What You Asked For
The Case of Money Meaning Nothing
The Case of Equal Opportunity
11. The Correlational Procedures
Understanding the Idea of Correlations
Interpreting Pearson’s r
- A Word of Caution
- An Even More Important Word of Caution!
A Nonparametric Correlational Procedure
- The p Value of a Correlation
- The Case of the Absent Students
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- Another Example: The Case against Sleep
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Case of Height versus Weight
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- The Case of Different Tastes
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
- Once We Have a Linear Relationship, What Can We Do with It?
Linear Regression
- The Regression Equation
- Computing the Slope
- Computing the Intercept
- Why Wasn’t It Exactly Right?
- Using the Six-Step Model: The Case of Age and Driving
- Identify the Problem
- State a Hypothesis
- Identify the Independent Variable
- Identify and Describe the Dependent Variable
- Choose the Right Statistical Test
- Use Data Analysis Software to Test the Hypothesis
Do You Understand These Key Words and Phrases?
Quiz Time!
The Case of “Like Father, Like Son”
The Case of “Can’t We All Just Get Along?”
The Case of More Is Better
The Case of More Is Better Still
- Conclusion: Have We Accomplished What We Set Out to Do?
Statistics in a New Light
A Limited Set of Statistical Techniques
The Use of Statistical Software packages
A Straightforward Approach
At Long Last
- Appendix A. Area under the Normal Curve Table (Critical Values of z)
- Appendix B. Critical Values of t
- Appendix C. Critical Values of F When Alpha = .01
- Appendix D. Critical Values of F When Alpha = .05
- Appendix E. Critical Values of F When Alpha = .10
- Appendix F. Critical Values of Chi-Square
- Appendix G. Selecting the Right Statistical Test
- Glossary
- Answers to Quiz Time!
- Index | {"url":"https://www.guilford.com/books/Statistics-Translated/Steven-Terrell/9781462545407/contents","timestamp":"2024-11-03T17:15:39Z","content_type":"text/html","content_length":"90118","record_id":"<urn:uuid:635e2e32-ce90-4c7b-91a7-f817ac134e01>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00787.warc.gz"} |
L notes
expression = runif(20)
expression = expression[order(expression,decreasing = T)]
types = c(rep(1,times=10),rep(2,times=10))
data = data.frame(expression=expression,types=types)
names = paste(‘exon_’,1:20,sep=””)
barplot(data$expression,col = c(“deepskyblue”,”gray”)[as.factor(data$types)],horiz=F,border = F,las=2,names=rownames(data),xlab=”Splicing genes”,ylab=”conservation scores”,cex.names = 0.6,cex.lab =
An example script of creating a heatmap using the heatmap.2 function in R
# Just copy all below into R to test
# Notes: the styles of “double quotes” when copied may change and make the script not run normally, but you can manually change the “double quotes”.
breaks = 20 # number of colors
my_palette <- colorRampPalette(c(“green”,”red”))(n =breaks-1) # self-define and discretize color maps
heatmap.2(X,Colv = NA,col=my_palette,breaks = breaks,density.info=”none”,trace = “none”,scale=”none”)
Venn diagram in R
# personal notes
data=list(A=c(1,2,3,4,5,6),B=c(3,4,5,6,7),C=c(5,6,7,8,9,10)) # demo data
venn.diagram(data,”vennFigure.tiff”,fill = c(“cornflowerblue”, “green”, “yellow”)) # venn diagram saved to a file
File handle arrays in PERL
# for personal notes
# generate files and file handles
foreach $i (@group){
$train[$i]=”train_$i.txt”; # file name
$FILE[$i]=”FILE”.”$i”; # file handle
open $FILE[$i],”>$train[$i]” or die;
# write to file
foreach $i (@group){
print {$FILE[$i]} “test\n”;
close $FILE[$i];
convert between RefSeq, Entrez and Ensembl gene IDs using R package biomaRt
# R code for convert gene/transcripts names
# personal notes
# settings
# human
hmart=useMart(“ENSEMBL_MART_ENSEMBL”, dataset=”hsapiens_gene_ensembl”,host=host)
# mouse
mmart=useMart(“ENSEMBL_MART_ENSEMBL”, dataset=”mmusculus_gene_ensembl”,host=host)
Partial Least Squares (PLS) code and basics
There are various ways to implement PLS, including the NIPALS, SIMPLS and the bi-diagonalizaton method of Rolf Manne. Here I provide a code based Wold’s 2001 paper: Chemometr. Intell. Lab. 58(2001)
X: data matrix of size n x p
Y: response variable of size n x 1
A: the number of PLS components to extract, which is usually optimized by cross validation.
B: a p-dimensional regression vector, where p equals the number of columns in X. If you want to add an intercept in your model, just add an additional column of ones to X.
T: PLS component or score matrix of size n x A. Can be thought of dimension-reduced representation of X. Similar to principal components in PCA but obtained in a different way.
Wstar: [Wstar1, Wstar2,…,WstarA], weight matrix to calculate T from original input X. Mathematically, T=XWstar.
W: [W1, W2,…,WA], weight matrix to calculate T from the residual-X at each iteration. Note that W is different from Wstar in addition to W1=Wstar1.
P: Loading matrix. X=TP’+E
R2X: a A-dimensional vector, records the explained variance of X by each PLS component
R2Y: a A-dimensional vector, records the explained variance of Y by each PLS component
Code: copy the whole below and save as a function.
function [B,Wstar,T,P,Q,W,R2X,R2Y]=pls_basic(X,Y,A)
%+++ The NIPALS algorithm for both PLS-1 (a single y) and PLS-2 (multiple Y)
%+++ The model is assumed to be: Y=XB+E,where E is random errors.
%+++ X: n x p matrix
%+++ Y: n x m matrix
%+++ A: number of latent variables
%+++ Code: Hongdong Li, lhdcsu@gmail.com, Feb, 2014
%+++ reference: Wold, S., M. Sj?str?m, and L. Eriksson, 2001. PLS-regression: a basic tool of chemometrics,
% Chemometr. Intell. Lab. 58(2001)109-130.
for i=1:A
while (error>1e-8 && niter<1000) % for convergence test
q=Y’*t/(t’*t); % regress Y against t;
%+++ store
%+++ calculate explained variance
Random Frog and results interpretation
Random Frog is a variable selection algorithm that computationally assign a probability to each variable as a measure of its IMPORTANCE in a predictive model.
CARS, Competitive Adaptive Reweighted Sampling, is a variable or feature selection methods, especially suitable for p>>n setting and data with high correlation (coupled with PLS). Here I will give an
example to detail how to use CARS and the meaning of its output.
Suppose that you have MATLAB installed (I am using 2010a version on Mac), downloaded the libPLS_1.95 and have added libPLS_1.95 in the search path.
First, let’s get our data. Copy the codes below to the command window and press ENTER:
%+++ load data
Keep in mind that this data has n=80 samples which were measured on p=700 wavelengths. The target variable to model is stored in vector y. In this p>>n setting, ordinary least squares (OLS) can not
be used to build a model y=Xb. Why? because b is calculated as b=inv(X’*X)X’y (inv indicates inverse, ‘ indicates transpose) in OLS. Due to p > n, X’*X is not full rank and hence do NOT have its
inverse and therefore OLS fails in this situation. One type of solution is to use regularized methods such as ridge regression or latent variable-based methods such as principal component regression
(PCR) and partial least squares (PLS). Why not simply reduce p? A good idea! This motivates another type of solution which is the well known branch of statistics: variable selection or called feature
Then, we split the data into a training and a test set, which will be used for model training and validation, respectively. There are different ways to split data, such as the KS (Kennard-Stone)
method and random partition. Here, I will randomly split the data using the codes below:
%+++ split data into a training and test set
ktrain=perm(1:60); % 60 samples as training
ktest=perm(61:80); % 20 samples as test
At this point, data is ready! We can start to select a subset of variables using the CARS method. One may ask why not use the full spectral data? A good question to think about. Run the codes below
to select variables (wavelengths):
%+++ perform variable selection using only the training data
A=10; % maximum number of PLS components
fold=5; % fold for cross validation
method=’center’; % for internal pretreatment
num=50; % number of iterations of CARS, 50 is default
To see the output of CARS, type “CARS” in command window and press ENTER and you will see:
>> CARS
CARS =
W: [700×50 double]
time: 2.8165
RMSECV: [1×50 double]
RMSECV_min: 3.0894e-04
Q2_max: 1.0000
iterOPT: 49
optLV: 2
ratio: [1×50 double]
vsel: [405 505]
Technically, CARS is a structural data with different components like optLV as shown above. Your can use CARS.RMSECV to fetch the value of RMSECV. The meaning of each component of CARS is given here:
W: [700×50 double] % weight of each variable at each iteration.
time: 2.8165
RMSECV: [1×50 double] % Root Mean Squares Errors of Cross Validation, one for each iteration
RMSECV_min: 3.0894e-04 % the minimum of RMSECV
Q2_max: 1.0000 % Q2 is the cross validated R2.
iterOPT: 49 % the optimal iteration number (corresponding to the RMSECV_min)
optLV: 2 % the optimal number of Latent Variables of PLS model at iterOPT.
ratio: [1×50 double]
vsel: [405 505] % The selected variables. THIS IS WHAT WE WANT.
Recall that this data contains p=700 variables. Here only two out of the 700 were selected to be useful by CARS: the 405th and the 505th. To understand the behavior of CARS, let’s issue the
following command:
which gives you a plot:
In Figure 1, the upper panel shows the number of variables kept by CARS at each iteration; the middle panel plots the the RMSECV at each iteration, which shows that RMSECV decreases with reduced
number of variables (interesting? why?); the bottom panel provides the regression coefficients of each of the 700 variables at each iteration, called Regression Coefficient Paths (RCP) where each
line denotes the RCP of one variable. Eliminated variables will have RCP ending with zero (thus you can not see it); some variables will have their RCP first increase and then decrease and finally
drop to zero (notice iterations around 30); some variables survived along the whole 50 iterations of CARS and are winners and are selected by CARS. Notice that there two lines (just the 405th and
505th variables) clearly stand out!
We have selected 2 variables. Then let’s check out how only these two will perform. The variable-reduced data is:
Xtrain_sub=Xtrain(:,[405 505]);
Xtest_sub=Xtest(:,[405 505]);
The we build a model and test it:
An intuitive way to check model performance is to plot predicted value against the experimental value:
plot(ytest, ypred,’b.’);
This gives you a plot:
You want to give a quantitative measurement of model performance. Usually, you calculate RMSEP (Root Mean Squared Errors of Prediction):
which is 3.0285e-04 in this case, very small (this data is very special).
So far, we have selected a subset of variables (405 and 505) using CARS and tested the performance of this subset on an independent test set. You may ask whether using a subset will improve model
performance over the full-spectral model? It will as long as the full-model contains information that is irrelevant to the target variable you are predicting. As a test case, you can build a model
using the full training data and tested it on the full test set and compares it to the simpler model with only two selected variables. | {"url":"http://libpls.net/fundata/?author=1","timestamp":"2024-11-07T06:37:26Z","content_type":"text/html","content_length":"41726","record_id":"<urn:uuid:b3b4fd24-9956-493f-88c5-f64a3327958e>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00311.warc.gz"} |
For each one, first the puzzle is posed for you to solve within a recommended time. A detailed systematic solution then follows.
Matchstick puzzles, riddles, math puzzles, logic puzzles, river crossing puzzles, ball weighing puzzles are some of the categories.
The full list of all brain teasers with solutions is available in the link,
This will always be the most up-to-date list with the brain teasers classified into categories that can be browsed separately.
Number lock puzzle 9285: Can you crack the 4 digit number lock code from the five clues? Each 4 digit clue hints on correct digits and placements.
Twin birth statistics puzzle: In a year, 3% of births give rise to twins. What percentage of population is a twin: 3%, less than 3% or, more than 3%?
A customer ordered fifteen Zingers that are to be placed in packages of four, three or one. In how many ways can this order be fulfilled?
Find all two digit sets of two or more consecutive positive integers that can be added to obtain a sum of 100. Follow a systematic approach.
Analyze the one condition and four clues each a 4 digit code with hints on right digit and right place to find the 4 digit code to open the number lock.
A thrifty shop-owner made a collection of 10 rupee, 5 rupee and 2 rupee coins. One night the shop-owner had an idea of dividing his coins into...read on...
Duorp was born in planet Urp on the first day of the fourth month on a Fourday. Each week has 5 days, each month 10 days, a year 180 days except..read on...
How to find the total when a series of fractions of the total and fractions of remainders were given in a will? The key is to use portions in the fractions.
What will be the least number of exposed cube faces when 500 cubes of side length 1 cm each are glued together in the form of a prism?
Four persons to cross a bridge in a dark night with one torch in 15 minutes. Two persons max can cross at a time. One crosses in 1 min, second in 2 mins... | {"url":"https://suresolv.com/brain-teaser?page=4","timestamp":"2024-11-04T04:45:26Z","content_type":"text/html","content_length":"48655","record_id":"<urn:uuid:c3e16d81-b0b5-4193-b0a0-867e6cdfb7fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00145.warc.gz"} |
Understanding Mathematical Functions: What Is The Range Of The Linear
Understanding Linear Parent Functions in Mathematics
In mathematics, linear parent functions play a fundamental role in understanding algebraic concepts and solving various mathematical problems. In this chapter, we will explore the definition of a
linear parent function, its significance in algebra, an overview of function terminology, and the importance of understanding the range in various mathematical contexts.
A Definition of a linear parent function and its significance in algebra
A linear parent function is a function that can be represented by a straight line when graphed on a coordinate plane. It is in the form of y = mx + b, where m represents the slope of the line, and b
represents the y-intercept. The significance of the linear parent function in algebra lies in its ability to provide a simple and foundational understanding of the relationship between two variables,
making it a crucial concept in introductory mathematics.
Overview of function terminology: domain, range, slope, and intercept
When dealing with linear parent functions, it is essential to understand the key terminology associated with functions. The domain of a function refers to the set of all possible input values, while
the range represents the set of all possible output values. The slope of a linear function determines the steepness of the line, and the y-intercept is the point where the line intersects the y-axis.
Importance of understanding the range in various mathematical contexts
Understanding the range of a linear parent function is crucial in various mathematical contexts. It allows us to determine the possible output values of the function, providing insights into the
behavior of the function and its limitations. Additionally, the range helps us identify the maximum and minimum values that the function can attain, which is essential in applications such as
optimization and problem-solving.
Key Takeaways
• Linear parent function has a constant rate of change.
• Range is all real numbers.
• Graph is a straight line.
• Equation is y = mx + b.
• Useful for modeling simple relationships.
Basic Properties of the Linear Parent Function
The linear parent function is the simplest form of a linear equation, and it serves as the foundation for understanding more complex linear functions. Let's explore the basic properties of the linear
parent function y = x.
A The formula of the linear parent function y = x
The formula of the linear parent function is y = x, where x represents the independent variable and y represents the dependent variable. This means that the value of y is directly proportional to the
value of x, with a constant rate of change.
B Characteristics: constant rate of change, straight line graph
One of the key characteristics of the linear parent function is its constant rate of change. This means that for every unit increase in the value of x, there is a corresponding unit increase in the
value of y. This results in a straight line graph when the function is plotted on a coordinate plane.
C Visual representation through graphing
Graphing the linear parent function y = x results in a straight line that passes through the origin (0,0) on the coordinate plane. This visual representation helps to illustrate the constant rate of
change, as well as the direct proportionality between the values of x and y.
When graphed, the linear parent function forms a 45-degree angle with the x-axis, indicating that the slope of the line is 1. This further emphasizes the constant rate of change and the linear
relationship between the variables.
Understanding the visual representation of the linear parent function through graphing is essential for grasping its fundamental properties and how it differs from other types of functions.
Diving into the Range Concept
When it comes to understanding mathematical functions, one of the key concepts to grasp is the range of a function. The range of a function refers to the set of all possible output values that the
function can produce. In simpler terms, it is the collection of all the y-values that the function can generate when x-values are inputted into the function.
A Definition and explanation of the range of a function
The range of a function can be defined as the set of all possible output values of the function. It is denoted as Range(f) or f(x). In other words, if we have a function f(x), the range of the
function is the set of all possible values that f(x) can take as x varies throughout the domain of the function.
For example, if we have a function f(x) = 2x + 3, the range of this function would be all real numbers, as for any real number input for x, the function will produce a real number output.
Determining the range of a linear parent function
When it comes to linear parent functions, such as f(x) = x, the range is also all real numbers. This is because for any real number input for x, the function will produce a real number output. In the
case of the linear parent function, the range is the same as the domain, which brings us to the next point of comparison.
Comparison with the domain of a linear parent function
While the range of a linear parent function is all real numbers, the domain of the function is also all real numbers. The domain of a function refers to the set of all possible input values that the
function can accept. In the case of the linear parent function, since it can accept any real number as input, its domain is also all real numbers.
It is important to note that while the range and domain of the linear parent function are the same, this is not always the case for other types of functions. Understanding the range and domain of a
function is crucial in analyzing and graphing functions, as it provides insight into the behavior and limitations of the function.
The Range of a Linear Parent Function: Theoretical Overview
When it comes to understanding the range of a linear parent function, it is important to grasp the fundamental concept that the range of the linear parent function is all real numbers. This concept
is based on the unbounded nature of the line on a graph and has significant implications in mathematical analysis.
Explanation that the range of the linear parent function is all real numbers
The range of a linear parent function, represented by the equation y = x, is all real numbers. This means that for any real number 'y', there exists a corresponding input 'x' such that y = x. In
other words, the output (y) can take on any real number value, making the range of the linear parent function infinite.
Justification based on the unbounded nature of the line on a graph
Graphically, the linear parent function y = x represents a straight line that extends infinitely in both the positive and negative directions on the coordinate plane. This unbounded nature of the
line illustrates that there are no restrictions on the y-values that the function can take. As a result, the range of the linear parent function encompasses all real numbers.
Implications of having no restrictions on the y-values for the range
The fact that the range of the linear parent function includes all real numbers has significant implications in mathematical analysis. It means that the function can produce any real number as an
output, and there are no limitations on the values that the function can attain. This unbounded nature of the range has practical applications in various fields, including physics, engineering, and
economics, where linear relationships are prevalent.
Understanding Mathematical Functions: What is the range of the linear parent function
When it comes to understanding mathematical functions, the range of the linear parent function is an important concept to grasp. In this chapter, we will explore the graphical and analytical
interpretation of the range of a linear function, as well as real-world examples where linear functions and their ranges are applied.
Graphical and Analytical Interpretation
Graphical and analytical interpretation of the range of a linear function involves visually determining the range using a graph and analyzing the slope and y-intercept to understand the function's
A. Using a graph to visually determine the range
Graphs are powerful tools for visually understanding the behavior of mathematical functions. When it comes to determining the range of a linear function, the graph can provide valuable insights. The
range of a linear function is the set of all possible output values (y-values) that the function can produce for any given input value (x-value).
By examining the graph of a linear function, we can visually identify the range by looking at the vertical spread of the function. The range will be the set of all y-values covered by the function as
it extends vertically along the y-axis. This visual representation can help us understand the possible output values of the function.
B. Analyzing the slope and y-intercept to understand the function's behavior
Another way to understand the range of a linear function is by analyzing its slope and y-intercept. The slope of a linear function determines the rate at which the function's output values change
with respect to its input values. The y-intercept, on the other hand, represents the value of the function when the input is zero.
By considering the slope and y-intercept, we can gain insights into the behavior of the linear function and infer the range of possible output values. For example, if the slope is positive, the
function will have a range of positive y-values, and if the slope is negative, the function will have a range of negative y-values.
C. Real-world examples where linear functions and their ranges are applied
Linear functions and their ranges are applied in various real-world scenarios. For instance, in economics, linear functions are used to model relationships between variables such as supply and
demand, cost and revenue, and profit and quantity. Understanding the range of these linear functions is crucial for making informed decisions in business and economics.
In physics, linear functions are used to describe the motion of objects, such as the position of an object over time. By understanding the range of these linear functions, physicists can predict the
possible positions of objects at different points in time.
Overall, understanding the range of linear functions is essential for interpreting their behavior graphically and analytically, as well as for applying them to real-world situations.
Troubleshooting Common Misconceptions and Calculation Errors
When dealing with mathematical functions, it is important to be aware of common misconceptions and calculation errors that can arise when determining the range of a function. By understanding these
potential pitfalls, you can ensure that you accurately identify the range of the linear parent function.
Misinterpreting the range when graph restrictions are present
One common mistake when determining the range of a linear parent function is misinterpreting the range when graph restrictions are present. It is important to remember that the range of a function is
the set of all possible output values, or y-values, that the function can produce. When graph restrictions are present, such as a limited domain or a specific portion of the graph being considered,
it is crucial to take these restrictions into account when identifying the range. Failure to do so can lead to an inaccurate determination of the range.
Distinguishing between the range of the parent function and transformations of it
Another potential source of confusion is distinguishing between the range of the parent function and transformations of it. When applying transformations, such as shifts, reflections, or stretches,
to the linear parent function, it is important to understand how these transformations affect the range. It is not uncommon for errors to occur when identifying the range of a transformed function,
as the transformations can alter the range in unexpected ways. Therefore, it is essential to carefully consider how each transformation impacts the range of the function.
Avoiding errors in identifying ranges for non-linear functions
Finally, when working with non-linear functions, it is important to avoid errors in identifying ranges. Non-linear functions can exhibit a wide range of behaviors, including asymptotes,
discontinuities, and complex curves. These complexities can make it challenging to accurately determine the range of a non-linear function. It is crucial to carefully analyze the behavior of the
function and consider any special cases that may arise, such as vertical or horizontal asymptotes, in order to accurately identify the range.
Conclusion & Best Practices: Mastering the Concept of Range
Understanding the concept of range in mathematical functions, particularly in the context of linear parent functions, is essential for mastering the fundamentals of algebra and calculus. In this
final section, we will recapitulate the key points discussed in the blog post, highlight best practices for identifying the range in linear parent functions and their transformations, and encourage
the readers to practice graphing and analyzing different linear functions for a better grasp of the range concept.
A Recapitulation of key points from the blog post
• Definition of Range: The range of a function refers to the set of all possible output values that the function can produce.
• Linear Parent Function: The linear parent function, represented by f(x) = x, has a range that extends from negative infinity to positive infinity, encompassing all real numbers.
• Transformation of Linear Functions: When linear functions undergo transformations such as shifts, reflections, or stretches, the range may be affected based on the nature of the transformation.
Best practices for identifying the range in linear parent functions and their transformations
When dealing with linear parent functions and their transformations, it is important to follow certain best practices to accurately determine the range:
• Understand the Basic Function: Gain a thorough understanding of the range of the linear parent function f(x) = x, which includes all real numbers.
• Analyze Transformations: When applying transformations to the linear function, carefully analyze how the transformations impact the range. For example, a vertical stretch or compression may alter
the range of the function.
• Use Algebraic Techniques: Utilize algebraic techniques such as solving inequalities and manipulating equations to determine the range of transformed linear functions.
• Graphical Analysis: Graph the linear functions and their transformations to visually observe the changes in the range as the functions are modified.
Encouragement to practice graphing and analyzing different linear functions for a better grasp of the range concept
Mastering the concept of range in linear functions requires practice and application. We encourage readers to engage in the following activities to enhance their understanding:
• Graphing Exercises: Practice graphing various linear functions and their transformations to observe how the range changes with different modifications.
• Real-World Applications: Explore real-world scenarios where linear functions are utilized and analyze the range in the context of these applications.
• Problem-Solving: Solve problems involving linear functions and determine the range based on the given parameters and transformations.
By actively engaging in these activities, individuals can develop a deeper comprehension of the range concept in linear functions and strengthen their overall proficiency in mathematical analysis. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-range-linear-parent-function","timestamp":"2024-11-14T19:04:41Z","content_type":"text/html","content_length":"225325","record_id":"<urn:uuid:406ccfd0-9aa8-492f-99e7-2fad6a2d445d>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00658.warc.gz"} |
Semisymmetric Graph -- from Wolfram MathWorld
A regular graph that is edge-transitive but not vertex-transitive is called a semisymmetric graph (Marušič and Potočnik 2001). In contrast, any graph that is both edge-transitive and
vertex-transitive is called a symmetric graph.
Note that it is possible for a graph to be edge-transitive yet neither vertex-transitive not regular. An example is the Pasch graph, which is therefore not semisymmetric. Other examples include the
star graphs, rhombic dodecahedral graph, rhombic triacontahedral graph, and Schläfli double sixes graph.
Every semisymmetric graph is necessarily bipartite, with the two parts having equal size and the automorphism group acting transitively on each of these parts. The numbers of regular bipartite graphs
on A087114).
Folkman (1967) proved that there are no semisymmetric graphs of order prime number and constructed some semisymmetric graphs of order Folkman graph. Folkman (1967) also asked if there exists a
semisymmetric graph of order 30, which was subsequently answered in the negative by Ivanov (1987).
There are no semisymmetric graphs on fewer than 20 vertices (Skiena 1990, p. 186). Examples of semisymmetric graphs are illustrated above and summarized in the following table. | {"url":"https://mathworld.wolfram.com/SemisymmetricGraph.html","timestamp":"2024-11-13T14:56:12Z","content_type":"text/html","content_length":"62535","record_id":"<urn:uuid:714ba33e-1367-4a13-bb66-0679778d423e>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00172.warc.gz"} |
Bitrate Calculation for KB and Time Transfer
27 Sep 2024
Popularity: ⭐⭐⭐
Bitrate Calculator
This calculator helps you find the bitrate (bps) by providing the data size in kilobytes (kb) and the time taken to transfer the data in seconds (s).
Understanding Bitrate: Bitrate, often measured in bits per second (bps), is a measure of the amount of data transferred per unit of time. It’s a crucial factor in determining the speed and efficiency
of data transmission.
Related Questions
Q: What factors affect bitrate?
A: Several factors can impact bitrate, including the size of the data being transferred, the available bandwidth, and the efficiency of the transmission protocol.
Q: How can I improve my bitrate?
A: To enhance your bitrate, consider upgrading your internet connection, using a wired connection instead of Wi-Fi, and closing any unnecessary programs that may be consuming bandwidth.
Symbol Name Unit
kb Kilobytes kb
t Time s
Calculation Expression
Bitrate Calculation: To calculate the bitrate in bits per second, use the formula: bps = kb * 8 / t
Calculated values
Considering these as variable values: t=20.0, kb=500.0, the calculated value(s) are given in table below
Derived Variable Value
Bitrate Calculation 200.0
Calculator Apps | {"url":"https://blog.truegeometry.com/calculators/kbeapp_com_calculation.html","timestamp":"2024-11-11T21:14:52Z","content_type":"text/html","content_length":"13660","record_id":"<urn:uuid:8e2e55c5-8575-4453-bf82-60c0904a23d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00116.warc.gz"} |
Chapter 3: Data Preprocessing and Feature Engineering
3.1 Data Cleaning and Handling Missing Data
Data preprocessing stands as the cornerstone of any robust machine learning pipeline, serving as the critical initial step that can make or break the success of your model. In the complex landscape
of real-world data science, practitioners often encounter raw data that is far from ideal - it may be riddled with inconsistencies, plagued by missing values, or lack the structure necessary for
immediate analysis.
Attempting to feed such unrefined data directly into a machine learning algorithm is a recipe for suboptimal performance and unreliable results. This is precisely where the twin pillars of data
preprocessing and feature engineering come into play, offering a systematic approach to data refinement.
These essential processes encompass a wide range of techniques aimed at cleaning, transforming, and optimizing your dataset. By meticulously preparing your data, you create a solid foundation that
enables machine learning algorithms to uncover meaningful patterns and generate accurate predictions. The goal is to present your model with a dataset that is not only clean and complete but also
structured in a way that highlights the most relevant features and relationships within the data.
Throughout this chapter, we will delve deep into the crucial steps that comprise effective data preprocessing. We'll explore the intricacies of data cleaning, a fundamental process that involves
identifying and rectifying errors, inconsistencies, and anomalies in your dataset. We'll tackle the challenge of handling missing data, discussing various strategies to address gaps in your
information without compromising the integrity of your analysis. The chapter will also cover scaling and normalization techniques, essential for ensuring that all features contribute proportionally
to the model's decision-making process.
Furthermore, we'll examine methods for encoding categorical variables, transforming non-numeric data into a format that machine learning algorithms can interpret and utilize effectively. Lastly,
we'll dive into the art and science of feature engineering, where domain knowledge and creativity converge to craft new, informative features that can significantly enhance your model's predictive
By mastering these preprocessing steps, you'll be equipped to lay a rock-solid foundation for your machine learning projects. This meticulous preparation of your data is what separates mediocre
models from those that truly excel, maximizing performance and ensuring that your algorithms can extract the most valuable insights from the information at hand.
We'll kick off our journey into data preprocessing with an in-depth look at data cleaning. This critical process serves as the first line of defense against the myriad issues that can plague raw
datasets. By ensuring that your data is accurate, complete, and primed for analysis, data cleaning sets the stage for all subsequent preprocessing steps and ultimately contributes to the overall
success of your machine learning endeavors.
Data cleaning is a crucial step in the data preprocessing pipeline, involving the systematic identification and rectification of issues within datasets. This process encompasses a wide range of
activities, including:
Detecting corrupt data
This crucial step involves a comprehensive and meticulous examination of the dataset to identify any data points that have been compromised or altered during various stages of the data lifecycle.
This includes, but is not limited to, the collection phase, where errors might occur due to faulty sensors or human input mistakes; the transmission phase, where data corruption can happen due to
network issues or interference; and the storage phase, where data might be corrupted due to hardware failures or software glitches.
The process of detecting corrupt data often involves multiple techniques:
• Statistical analysis: Using statistical methods to identify outliers or values that deviate significantly from expected patterns.
• Data validation rules: Implementing specific rules based on domain knowledge to flag potentially corrupt entries.
• Consistency checks: Comparing data across different fields or time periods to ensure logical consistency.
• Format verification: Ensuring that data adheres to expected formats, such as date structures or numerical ranges.
By pinpointing these corrupted elements through such rigorous methods, data scientists can take appropriate actions such as removing, correcting, or flagging the corrupt data. This process is
fundamental in ensuring the integrity and reliability of the dataset, which is crucial for any subsequent analysis or machine learning model development. Without this step, corrupt data could lead to
skewed results, incorrect conclusions, or poorly performing models, potentially undermining the entire data science project.
Example: Detecting Corrupt Data
import pandas as pd
import numpy as np
# Create a sample DataFrame with potentially corrupt data
data = {
'ID': [1, 2, 3, 4, 5],
'Value': [10, 20, 'error', 40, 50],
'Date': ['2023-01-01', '2023-02-30', '2023-03-15', '2023-04-01', '2023-05-01']
df = pd.DataFrame(data)
# Function to detect corrupt data
def detect_corrupt_data(df):
corrupt_rows = []
# Check for non-numeric values in 'Value' column
numeric_errors = pd.to_numeric(df['Value'], errors='coerce').isna()
# Check for invalid dates
df['Date'] = pd.to_datetime(df['Date'], errors='coerce')
date_errors = df['Date'].isna()
return list(set(corrupt_rows)) # Remove duplicates
# Detect corrupt data
corrupt_indices = detect_corrupt_data(df)
print("Corrupt data found at indices:", corrupt_indices)
print("\nCorrupt rows:")
This code demonstrates how to detect corrupt data in a pandas DataFrame. Here's a breakdown of its functionality:
• It creates a sample DataFrame with potentially corrupt data, including non-numeric values in the 'Value' column and invalid dates in the 'Date' column.
• The detect_corrupt_data() function is defined to identify corrupt rows. It checks for:
• Non-numeric values in the 'Value' column using pd.to_numeric() with errors='coerce'.
• Invalid dates in the 'Date' column using pd.to_datetime() with errors='coerce'.
• The function returns a list of unique indices where corrupt data was found.
• Finally, it prints the indices of corrupt rows and displays the corrupt data.
This code is an example of how to implement data cleaning techniques, specifically for detecting corrupt data, which is a crucial step in the data preprocessing pipeline.
Correcting incomplete data
This process involves a comprehensive and meticulous examination of the dataset to identify and address any instances of incomplete or missing information. The approach to handling such gaps depends
on several factors, including the nature of the data, the extent of incompleteness, and the potential impact on subsequent analyses.
When dealing with missing data, data scientists employ a range of sophisticated techniques:
• Imputation methods: These involve estimating and filling in missing values based on patterns observed in the existing data. Techniques can range from simple mean or median imputation to more
advanced methods like regression imputation or multiple imputation.
• Machine learning-based approaches: Algorithms such as K-Nearest Neighbors (KNN) or Random Forest can be used to predict missing values based on the relationships between variables in the dataset.
• Time series-specific methods: For temporal data, techniques like interpolation or forecasting models may be employed to estimate missing values based on trends and seasonality.
However, in cases where the gaps in the data are too significant or the missing information is deemed crucial, careful consideration must be given to the removal of incomplete records. This decision
is not taken lightly, as it involves balancing the need for data quality with the potential loss of valuable information.
Factors influencing the decision to remove incomplete records include:
• The proportion of missing data: If a large percentage of a record or variable is missing, removal might be more appropriate than imputation.
• The mechanism of missingness: Understanding whether data is missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR) can inform the decision-making process.
• The importance of the missing information: If the missing data is critical to the analysis or model, removal might be necessary to maintain the integrity of the results.
Ultimately, the goal is to strike a balance between preserving as much valuable information as possible while ensuring the overall quality and reliability of the dataset for subsequent analysis and
modeling tasks.
Example: Correcting Incomplete Data
import pandas as pd
import numpy as np
from sklearn.impute import SimpleImputer
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
# Create a sample DataFrame with incomplete data
data = {
'Age': [25, np.nan, 30, np.nan, 40],
'Income': [50000, 60000, np.nan, 75000, 80000],
'Education': ['Bachelor', 'Master', np.nan, 'PhD', 'Bachelor']
df = pd.DataFrame(data)
print("Original DataFrame:")
# Method 1: Simple Imputation (Mean for numerical, Most frequent for categorical)
imputer_mean = SimpleImputer(strategy='mean')
imputer_most_frequent = SimpleImputer(strategy='most_frequent')
df_imputed_simple = df.copy()
df_imputed_simple[['Age', 'Income']] = imputer_mean.fit_transform(df[['Age', 'Income']])
df_imputed_simple[['Education']] = imputer_most_frequent.fit_transform(df[['Education']])
print("\nDataFrame after Simple Imputation:")
# Method 2: Iterative Imputation (uses the IterativeImputer, aka MICE)
imputer_iterative = IterativeImputer(random_state=0)
df_imputed_iterative = df.copy()
df_imputed_iterative.iloc[:, :] = imputer_iterative.fit_transform(df)
print("\nDataFrame after Iterative Imputation:")
# Method 3: Custom logic (e.g., filling Age based on median of similar Education levels)
df_custom = df.copy()
df_custom['Age'] = df_custom.groupby('Education')['Age'].transform(lambda x: x.fillna(x.median()))
df_custom['Income'].fillna(df_custom['Income'].mean(), inplace=True)
df_custom['Education'].fillna(df_custom['Education'].mode()[0], inplace=True)
print("\nDataFrame after Custom Imputation:")
This example demonstrates three different methods for correcting incomplete data:
• 1. Simple Imputation: Uses Scikit-learn's SimpleImputer to fill missing values with the mean for numerical columns (Age and Income) and the most frequent value for categorical columns
• 2. Iterative Imputation: Employs Scikit-learn's IterativeImputer (also known as MICE - Multivariate Imputation by Chained Equations) to estimate missing values based on the relationships between
• 3. Custom Logic: Implements a tailored approach where Age is imputed based on the median age of similar education levels, Income is filled with the mean, and Education uses the mode (most
frequent value).
Breakdown of the code:
1. We start by importing necessary libraries and creating a sample DataFrame with missing values.
2. For Simple Imputation, we use SimpleImputer with different strategies for numerical and categorical data.
3. Iterative Imputation uses the IterativeImputer, which estimates each feature from all the others iteratively.
4. The custom logic demonstrates how domain knowledge can be applied to impute data more accurately, such as using education level to estimate age.
This example showcases the flexibility and power of different imputation techniques. The choice of method depends on the nature of your data and the specific requirements of your analysis. Simple
imputation is quick and easy but may not capture complex relationships in the data. Iterative imputation can be more accurate but is computationally intensive. Custom logic allows for the
incorporation of domain expertise but requires more manual effort and understanding of the data.
Addressing inaccurate data
This crucial step in the data cleaning process involves a comprehensive and meticulous approach to identifying and rectifying errors that may have infiltrated the dataset during various stages of
data collection and management. These errors can arise from multiple sources:
• Data Entry Errors: Human mistakes during manual data input, such as typos, transposed digits, or incorrect categorizations.
• Measurement Errors: Inaccuracies stemming from faulty equipment, miscalibrated instruments, or inconsistent measurement techniques.
• Recording Errors: Issues that occur during the data recording process, including system glitches, software bugs, or data transmission failures.
To address these challenges, data scientists employ a range of sophisticated validation techniques:
• Statistical Outlier Detection: Utilizing statistical methods to identify data points that deviate significantly from the expected patterns or distributions.
• Domain-Specific Rule Validation: Implementing checks based on expert knowledge of the field to flag logically inconsistent or impossible values.
• Cross-Referencing: Comparing data against reliable external sources or internal databases to verify accuracy and consistency.
• Machine Learning-Based Anomaly Detection: Leveraging advanced algorithms to detect subtle patterns of inaccuracy that might escape traditional validation methods.
By rigorously applying these validation techniques and diligently cross-referencing with trusted sources, data scientists can substantially enhance the accuracy and reliability of their datasets.
This meticulous process not only improves the quality of the data but also bolsters the credibility of subsequent analyses and machine learning models built upon this foundation. Ultimately,
addressing inaccurate data is a critical investment in ensuring the integrity and trustworthiness of data-driven insights and decision-making processes.
Example: Addressing Inaccurate Data
import pandas as pd
import numpy as np
from scipy import stats
# Create a sample DataFrame with potentially inaccurate data
data = {
'ID': range(1, 11),
'Age': [25, 30, 35, 40, 45, 50, 55, 60, 65, 1000],
'Income': [50000, 60000, 70000, 80000, 90000, 100000, 110000, 120000, 130000, 10000000],
'Height': [170, 175, 180, 185, 190, 195, 200, 205, 210, 150]
df = pd.DataFrame(data)
print("Original DataFrame:")
def detect_and_correct_outliers(df, column, method='zscore', threshold=3):
if method == 'zscore':
z_scores = np.abs(stats.zscore(df[column]))
outliers = df[z_scores > threshold]
df.loc[z_scores > threshold, column] = df[column].median()
elif method == 'iqr':
Q1 = df[column].quantile(0.25)
Q3 = df[column].quantile(0.75)
IQR = Q3 - Q1
lower_bound = Q1 - 1.5 * IQR
upper_bound = Q3 + 1.5 * IQR
outliers = df[(df[column] < lower_bound) | (df[column] > upper_bound)]
df.loc[(df[column] < lower_bound) | (df[column] > upper_bound), column] = df[column].median()
return outliers
# Detect and correct outliers in 'Age' column using Z-score method
age_outliers = detect_and_correct_outliers(df, 'Age', method='zscore')
# Detect and correct outliers in 'Income' column using IQR method
income_outliers = detect_and_correct_outliers(df, 'Income', method='iqr')
# Custom logic for 'Height' column
height_outliers = df[(df['Height'] < 150) | (df['Height'] > 220)]
df.loc[(df['Height'] < 150) | (df['Height'] > 220), 'Height'] = df['Height'].median()
print("\nOutliers detected:")
print("Age outliers:", age_outliers['Age'].tolist())
print("Income outliers:", income_outliers['Income'].tolist())
print("Height outliers:", height_outliers['Height'].tolist())
print("\nCorrected DataFrame:")
This example demonstrates a comprehensive approach to addressing inaccurate data, specifically focusing on outlier detection and correction.
Here's a breakdown of the code and its functionality:
1. Data Creation: We start by creating a sample DataFrame with potentially inaccurate data, including extreme values in the 'Age', 'Income', and 'Height' columns.
2. Outlier Detection and Correction Function: The detect_and_correct_outliers() function is defined to handle outliers using two common methods:
□ Z-score method: Identifies outliers based on the number of standard deviations from the mean.
□ IQR (Interquartile Range) method: Detects outliers using the concept of quartiles.
3. Applying Outlier Detection:
□ For the 'Age' column, we use the Z-score method with a threshold of 3 standard deviations.
□ For the 'Income' column, we apply the IQR method to account for potential skewness in income distribution.
□ For the 'Height' column, we implement a custom logic to flag values below 150 cm or above 220 cm as outliers.
4. Outlier Correction: Once outliers are detected, they are replaced with the median value of the respective column. This approach helps maintain data integrity while reducing the impact of extreme
5. Reporting: The code prints out the detected outliers for each column and displays the corrected DataFrame.
This example showcases different strategies for addressing inaccurate data:
• Statistical methods (Z-score and IQR) for automated outlier detection
• Custom logic for domain-specific outlier identification
• Median imputation for correcting outliers, which is more robust to extreme values than mean imputation
By employing these techniques, data scientists can significantly improve the quality of their datasets, leading to more reliable analyses and machine learning models. It's important to note that
while this example uses median imputation for simplicity, in practice, the choice of correction method should be carefully considered based on the specific characteristics of the data and the
requirements of the analysis.
Removing irrelevant data
This final step in the data cleaning process, known as data relevance assessment, involves a meticulous evaluation of each data point to determine its significance and applicability to the specific
analysis or problem at hand. This crucial phase requires data scientists to critically examine the dataset through multiple lenses:
1. Contextual Relevance: Assessing whether each variable or feature directly contributes to answering the research questions or achieving the project goals.
2. Temporal Relevance: Determining if the data is current enough to be meaningful for the analysis, especially in rapidly changing domains.
3. Granularity: Evaluating if the level of detail in the data is appropriate for the intended analysis, neither too broad nor too specific.
4. Redundancy: Identifying and removing duplicate or highly correlated variables that don't provide additional informational value.
5. Signal-to-Noise Ratio: Distinguishing between data that carries meaningful information (signal) and data that introduces unnecessary complexity or variability (noise).
By meticulously eliminating extraneous or irrelevant information through this process, data scientists can significantly enhance the quality and focus of their dataset. This refinement yields several
critical benefits:
• Improved Model Performance: A streamlined dataset with only relevant features often leads to more accurate and robust machine learning models.
• Enhanced Computational Efficiency: Reducing the dataset's dimensionality can dramatically decrease processing time and resource requirements, especially crucial when dealing with large-scale data.
• Clearer Insights: By removing noise and focusing on pertinent data, analysts can derive more meaningful and actionable insights from their analyses.
• Reduced Overfitting Risk: Eliminating irrelevant features helps prevent models from learning spurious patterns, thus improving generalization to new, unseen data.
• Simplified Interpretability: A more focused dataset often results in models and analyses that are easier to interpret and explain to stakeholders.
In essence, this careful curation of relevant data serves as a critical foundation, significantly enhancing the efficiency, effectiveness, and reliability of subsequent analyses and machine learning
models. It ensures that the final insights and decisions are based on the most pertinent and high-quality information available.
Example: Removing Irrelevant Data
import pandas as pd
import numpy as np
from sklearn.feature_selection import VarianceThreshold
from sklearn.feature_selection import mutual_info_regression
# Create a sample DataFrame with potentially irrelevant features
data = {
'ID': range(1, 101),
'Age': np.random.randint(18, 80, 100),
'Income': np.random.randint(20000, 150000, 100),
'Education': np.random.choice(['High School', 'Bachelor', 'Master', 'PhD'], 100),
'Constant_Feature': [5] * 100,
'Random_Feature': np.random.random(100),
'Target': np.random.randint(0, 2, 100)
df = pd.DataFrame(data)
print("Original DataFrame shape:", df.shape)
# Step 1: Remove constant features
constant_filter = VarianceThreshold(threshold=0)
constant_columns = df.columns[~constant_filter.get_support()]
df = df.drop(columns=constant_columns)
print("After removing constant features:", df.shape)
# Step 2: Remove features with low variance
variance_filter = VarianceThreshold(threshold=0.1)
low_variance_columns = df.select_dtypes(include=[np.number]).columns[~variance_filter.get_support()]
df = df.drop(columns=low_variance_columns)
print("After removing low variance features:", df.shape)
# Step 3: Feature importance based on mutual information
numerical_features = df.select_dtypes(include=[np.number]).columns.drop('Target')
mi_scores = mutual_info_regression(df[numerical_features], df['Target'])
mi_scores = pd.Series(mi_scores, index=numerical_features)
important_features = mi_scores[mi_scores > 0.01].index
df = df[important_features.tolist() + ['Education', 'Target']]
print("After removing less important features:", df.shape)
print("\nFinal DataFrame columns:", df.columns.tolist())
This code example demonstrates various techniques for removing irrelevant data from a dataset.
Let's break down the code and explain each step:
1. Data Creation: We start by creating a sample DataFrame with potentially irrelevant features, including a constant feature and a random feature.
2. Removing Constant Features:
□ We use VarianceThreshold with a threshold of 0 to identify and remove features that have the same value in all samples.
□ This step eliminates features that provide no discriminative information for the model.
3. Removing Low Variance Features:
□ We apply VarianceThreshold again, this time with a threshold of 0.1, to remove features with very low variance.
□ Features with low variance often contain little information and may not contribute significantly to the model's predictive power.
4. Feature Importance based on Mutual Information:
□ We use mutual_info_regression to calculate the mutual information between each feature and the target variable.
□ Features with mutual information scores below a certain threshold (0.01 in this example) are considered less important and are removed.
□ This step helps in identifying features that have a strong relationship with the target variable.
5. Retaining Categorical Features: We manually include the 'Education' column to demonstrate how you might retain important categorical features that weren't part of the numerical analysis.
This example showcases a multi-faceted approach to removing irrelevant data:
• It addresses constant features that provide no discriminative information.
• It removes features with very low variance, which often contribute little to model performance.
• It uses a statistical measure (mutual information) to identify features most relevant to the target variable.
By applying these techniques, we significantly reduce the dimensionality of the dataset, focusing on the most relevant features. This can lead to improved model performance, reduced overfitting, and
increased computational efficiency. However, it's crucial to validate the impact of feature removal on your specific problem and adjust thresholds as necessary.
The importance of data cleaning cannot be overstated, as it directly impacts the quality and reliability of machine learning models. Clean, high-quality data is essential for accurate predictions and
meaningful insights.
Missing values are a common challenge in real-world datasets, often arising from various sources such as equipment malfunctions, human error, or intentional non-responses. Handling these missing
values appropriately is critical, as they can significantly affect model performance and lead to biased or incorrect conclusions if not addressed properly.
The approach to dealing with missing data is not one-size-fits-all and depends on several factors:
1. The nature and characteristics of your dataset: The specific type of data you're working with (such as numerical, categorical, or time series) and its underlying distribution patterns play a
crucial role in determining the most appropriate technique for handling missing data. For instance, certain imputation methods may be more suitable for continuous numerical data, while others
might be better suited for categorical variables or time-dependent information.
2. The quantity and distribution pattern of missing data: The extent of missing information and the underlying mechanism causing the data gaps significantly influence the choice of handling
strategy. It's essential to distinguish between data that is missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR), as each scenario may require a
different approach to maintain the integrity and representativeness of your dataset.
3. The selected machine learning algorithm and its inherent properties: Different machine learning models exhibit varying degrees of sensitivity to missing data, which can substantially impact their
performance and the reliability of their predictions. Some algorithms, like decision trees, can handle missing values intrinsically, while others, such as support vector machines, may require
more extensive preprocessing to address data gaps effectively. Understanding these model-specific characteristics is crucial in selecting an appropriate missing data handling technique that
aligns with your chosen algorithm.
By understanding these concepts and techniques, data scientists can make informed decisions about how to preprocess their data effectively, ensuring the development of robust and accurate machine
learning models.
3.1.1 Types of Missing Data
Before delving deeper into the intricacies of handling missing data, it is crucial to grasp the three primary categories of missing data, each with its own unique characteristics and implications for
data analysis:
1. Missing Completely at Random (MCAR)
This type of missing data represents a scenario where the absence of information follows no discernible pattern or relationship with any variables in the dataset, whether observed or unobserved. MCAR
is characterized by an equal probability of data being missing across all cases, effectively creating an unbiased subset of the complete dataset.
The key features of MCAR include:
• Randomness: The missingness is entirely random and not influenced by any factors within or outside the dataset.
• Unbiased representation: The remaining data can be considered a random sample of the full dataset, maintaining its statistical properties.
• Statistical implications: Analyses conducted on the complete cases (after removing missing data) remain unbiased, although there may be a loss in statistical power due to reduced sample size.
To illustrate MCAR, consider a comprehensive survey scenario:
Imagine a large-scale health survey where participants are required to fill out a lengthy questionnaire. Some respondents might inadvertently skip certain questions due to factors entirely unrelated
to the survey content or their personal characteristics. For instance:
• A respondent might be momentarily distracted by an external noise and accidentally skip a question.
• Technical glitches in the survey platform could randomly fail to record some responses.
• A participant might unintentionally turn two pages at once, missing a set of questions.
In these cases, the missing data would be considered MCAR because the likelihood of a response being missing is not related to the question itself, the respondent's characteristics, or any other
variables in the study. This randomness ensures that the remaining data still provides an unbiased, albeit smaller, representation of the population under study.
While MCAR is often considered the "best-case scenario" for missing data, it's important to note that it's relatively rare in real-world datasets. Researchers and data scientists must carefully
examine their data and the data collection process to determine if the MCAR assumption truly holds before proceeding with analyses or imputation methods based on this assumption.
2. Missing at Random (MAR):
In this scenario, known as Missing at Random (MAR), the missing data exhibits a systematic relationship with the observed data, but crucially, not with the missing data itself. This means that the
probability of data being missing can be explained by other observed variables in the dataset, but is not directly related to the unobserved values.
To better understand MAR, let's break it down further:
• Systematic relationship: The pattern of missingness is not completely random, but follows a discernible pattern based on other observed variables.
• Observed data dependency: The likelihood of a value being missing depends on other variables that we can observe and measure in the dataset.
• Independence from unobserved values: Importantly, the probability of missingness is not related to the actual value that would have been observed, had it not been missing.
Let's consider an expanded illustration to clarify this concept:
Imagine a comprehensive health survey where participants are asked about their age, exercise habits, and overall health satisfaction. In this scenario:
• Younger participants (ages 18-30) might be less likely to respond to questions about their exercise habits, regardless of how much they actually exercise.
• This lower response rate among younger participants is observable and can be accounted for in the analysis.
• Crucially, their tendency to not respond is not directly related to their actual exercise habits (which would be the missing data), but rather to their age group (which is observed).
In this MAR scenario, we can use the observed data (age) to make informed decisions about handling the missing data (exercise habits). This characteristic of MAR allows for more sophisticated
imputation methods that can leverage the relationships between variables to estimate missing values more accurately.
Understanding that data is MAR is vital for choosing appropriate missing data handling techniques. Unlike Missing Completely at Random (MCAR), where simple techniques like listwise deletion might
suffice, MAR often requires more advanced methods such as multiple imputation or maximum likelihood estimation to avoid bias in analyses.
3. Missing Not at Random (MNAR)
This category represents the most complex type of missing data, where the missingness is directly related to the unobserved values themselves. In MNAR situations, the very reason for the data being
missing is intrinsically linked to the information that would have been collected. This creates a significant challenge for data analysis and imputation methods, as the missing data mechanism cannot
be ignored without potentially introducing bias.
To better understand MNAR, let's break it down further:
• Direct relationship: The probability of a value being missing depends on the value itself, which is unobserved.
• Systematic bias: The missingness creates a systematic bias in the dataset that cannot be fully accounted for using only the observed data.
• Complexity in analysis: MNAR scenarios often require specialized statistical techniques to handle properly, as simple imputation methods may lead to incorrect conclusions.
A prime example of MNAR is when patients with severe health conditions are less inclined to disclose their health status. This leads to systematic gaps in health-related data that are directly
correlated with the severity of their conditions. Let's explore this example in more depth:
• Self-selection bias: Patients with more severe conditions might avoid participating in health surveys or medical studies due to physical limitations or psychological factors.
• Privacy concerns: Those with serious health issues might be more reluctant to share their medical information, fearing stigma or discrimination.
• Incomplete medical records: Patients with complex health conditions might have incomplete medical records if they frequently switch healthcare providers or avoid certain types of care.
The implications of MNAR data in this health-related scenario are significant:
• Underestimation of disease prevalence: If those with severe conditions are systematically missing from the data, the true prevalence of the disease might be underestimated.
• Biased treatment efficacy assessments: In clinical trials, if patients with severe side effects are more likely to drop out, the remaining data might overestimate the treatment's effectiveness.
• Skewed health policy decisions: Policymakers relying on this data might allocate resources based on an incomplete picture of public health needs.
Handling MNAR data requires careful consideration and often involves advanced statistical methods such as selection models or pattern-mixture models. These approaches attempt to model the missing
data mechanism explicitly, allowing for more accurate inferences from incomplete datasets. However, they often rely on untestable assumptions about the nature of the missingness, highlighting the
complexity and challenges associated with MNAR scenarios in data analysis.
Understanding these distinct types of missing data is paramount, as each category necessitates a unique approach in data handling and analysis. The choice of method for addressing missing
data—whether it involves imputation, deletion, or more advanced techniques—should be carefully tailored to the specific type of missingness encountered in the dataset.
This nuanced understanding ensures that the subsequent data analysis and modeling efforts are built on a foundation that accurately reflects the underlying data structure and minimizes potential
biases introduced by missing information.
3.1.2 Detecting and Visualizing Missing Data
The first step in handling missing data is detecting where the missing values are within your dataset. This crucial initial phase sets the foundation for all subsequent data preprocessing and
analysis tasks. Pandas, a powerful data manipulation library in Python, provides an efficient and user-friendly way to check for missing values in a dataset.
To begin this process, you typically load your data into a Pandas DataFrame, which is a two-dimensional labeled data structure. Once your data is in this format, Pandas offers several built-in
functions to identify missing values:
• The isnull() or isna() methods: These functions return a boolean mask of the same shape as your DataFrame, where True indicates a missing value and False indicates a non-missing value.
• The notnull() method: This is the inverse of isnull(), returning True for non-missing values.
• The info() method: This provides a concise summary of your DataFrame, including the number of non-null values in each column.
By combining these functions with other Pandas operations, you can gain a comprehensive understanding of the missing data in your dataset. For example, you can use df.isnull().sum() to count the
number of missing values in each column, or df.isnull().any() to check if any column contains missing values.
Understanding the pattern and extent of missing data is crucial as it informs your strategy for handling these gaps. It helps you decide whether to remove rows or columns with missing data, impute
the missing values, or employ more advanced techniques like multiple imputation or machine learning models designed to handle missing data.
Example: Detecting Missing Data with Pandas
import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer, KNNImputer
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
# Create a sample DataFrame with missing data
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve', 'Frank'],
'Age': [25, None, 35, 40, None, 50],
'Salary': [50000, 60000, None, 80000, 55000, None],
'Department': ['HR', 'IT', 'Finance', 'IT', None, 'HR']
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
# Check for missing data
print("Missing Data in Each Column:")
# Calculate percentage of missing data
print("Percentage of Missing Data in Each Column:")
print(df.isnull().sum() / len(df) * 100)
# Visualize missing data with a heatmap
plt.figure(figsize=(10, 6))
sns.heatmap(df.isnull(), cbar=False, cmap='viridis', yticklabels=False)
plt.title("Missing Data Heatmap")
# Handling missing data
# 1. Removing rows with missing data
df_dropna = df.dropna()
print("DataFrame after dropping rows with missing data:")
# 2. Simple imputation methods
# Mean imputation for numerical columns
df_mean_imputed = df.copy()
df_mean_imputed['Age'].fillna(df_mean_imputed['Age'].mean(), inplace=True)
df_mean_imputed['Salary'].fillna(df_mean_imputed['Salary'].mean(), inplace=True)
# Mode imputation for categorical column
df_mean_imputed['Department'].fillna(df_mean_imputed['Department'].mode()[0], inplace=True)
print("DataFrame after mean/mode imputation:")
# 3. KNN Imputation
imputer_knn = KNNImputer(n_neighbors=2)
df_knn_imputed = pd.DataFrame(imputer_knn.fit_transform(df.drop('Name', axis=1)),
df_knn_imputed.insert(0, 'Name', df['Name']) # Add back the 'Name' column
print("DataFrame after KNN imputation:")
# 4. Multiple Imputation by Chained Equations (MICE)
imputer_mice = IterativeImputer(random_state=0)
df_mice_imputed = pd.DataFrame(imputer_mice.fit_transform(df.drop('Name', axis=1)),
df_mice_imputed.insert(0, 'Name', df['Name']) # Add back the 'Name' column
print("DataFrame after MICE imputation:")
This code example provides a comprehensive demonstration of detecting, visualizing, and handling missing data in Python using pandas, numpy, seaborn, matplotlib, and scikit-learn.
Let's break down the code and explain each section:
1. Data Creation and Exploration:
• We start by creating a sample DataFrame with missing values in different columns.
• The original DataFrame is displayed to show the initial state of the data.
• We use df.isnull().sum() to count the number of missing values in each column.
• The percentage of missing data in each column is calculated to give a better perspective on the extent of missing data.
2. Visualization:
• A heatmap is created using seaborn to visualize the pattern of missing data. This provides an intuitive way to identify which columns and rows contain missing values.
3. Handling Missing Data:
The code demonstrates four different approaches to handling missing data:
• a. Removing rows with missing data: Using df.dropna(), we remove all rows that contain any missing values. This method is simple but can lead to significant data loss if many rows contain missing
• b. Simple imputation methods:
□ For numerical columns ('Age' and 'Salary'), we use mean imputation with fillna(df['column'].mean()).
□ For the categorical column ('Department'), we use mode imputation with fillna(df['Department'].mode()[0]).
□ These methods are straightforward but don't consider the relationships between variables.
• c. K-Nearest Neighbors (KNN) Imputation: Using KNNImputer from scikit-learn, we impute missing values based on the values of the k-nearest neighbors. This method can capture some of the
relationships between variables but may not work well with categorical data.
• d. Multiple Imputation by Chained Equations (MICE): Using IterativeImputer from scikit-learn, we perform multiple imputations. This method models each feature with missing values as a function of
other features and uses that estimate for imputation. It's more sophisticated and can handle both numerical and categorical data.
4. Output and Comparison:
• After each imputation method, the resulting DataFrame is printed, allowing for easy comparison between different approaches.
• This enables the user to assess the impact of each method on the data and choose the most appropriate one for their specific use case.
This example showcases multiple imputation techniques, provides a step-by-step breakdown, and offers a comprehensive look at handling missing data in Python. It demonstrates the progression from
simple techniques (like deletion and mean imputation) to more advanced methods (KNN and MICE). This approach allows users to understand and compare different strategies for missing data imputation.
The isnull() function in Pandas detects missing values (represented as NaN), and by using .sum(), you can get the total number of missing values in each column. Additionally, the Seaborn heatmap
provides a quick visual representation of where the missing data is located.
3.1.3 Techniques for Handling Missing Data
After identifying missing values in your dataset, the crucial next step involves determining the most appropriate strategy for addressing these gaps. The approach you choose can significantly impact
your analysis and model performance. There are multiple techniques available for handling missing data, each with its own strengths and limitations.
The selection of the most suitable method depends on various factors, including the volume of missing data, the pattern of missingness (whether it's missing completely at random, missing at random,
or missing not at random), and the relative importance of the features containing missing values. It's essential to carefully consider these aspects to ensure that your chosen method aligns with your
specific data characteristics and analytical goals.
1. Removing Missing Data
If the amount of missing data is small (typically less than 5% of the total dataset) and the missingness pattern is random (MCAR - Missing Completely At Random), you can consider removing rows or
columns with missing values. This method, known as listwise deletion or complete case analysis, is straightforward and easy to implement.
However, this approach should be used cautiously for several reasons:
• Loss of Information: Removing entire rows or columns can lead to a significant loss of potentially valuable information, especially if the missing data is in different rows across multiple
• Reduced Statistical Power: A smaller sample size due to data removal can decrease the statistical power of your analyses, potentially making it harder to detect significant effects.
• Bias Introduction: If the data is not MCAR, removing rows with missing values can introduce bias into your dataset, potentially skewing your results and leading to incorrect conclusions.
• Inefficiency: In cases where multiple variables have missing values, you might end up discarding a large portion of your dataset, which is inefficient and can lead to unstable estimates.
Before opting for this method, it's crucial to thoroughly analyze the pattern and extent of missing data in your dataset. Consider alternative approaches like various imputation techniques if the
proportion of missing data is substantial or if the missingness pattern suggests that the data is not MCAR.
Example: Removing Rows with Missing Data
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
# Create a sample DataFrame with missing values
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve'],
'Age': [25, np.nan, 35, 40, np.nan],
'Salary': [50000, 60000, np.nan, 80000, 55000],
'Department': ['HR', 'IT', 'Finance', 'IT', np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
# Check for missing values
print("Missing values in each column:")
# Remove rows with any missing values
df_clean = df.dropna()
print("DataFrame after removing rows with missing data:")
# Remove rows with missing values in specific columns
df_clean_specific = df.dropna(subset=['Age', 'Salary'])
print("DataFrame after removing rows with missing data in 'Age' and 'Salary':")
# Remove columns with missing values
df_clean_columns = df.dropna(axis=1)
print("DataFrame after removing columns with missing data:")
# Visualize the impact of removing missing data
plt.figure(figsize=(10, 6))
plt.bar(['Original', 'After row removal', 'After column removal'],
[len(df), len(df_clean), len(df_clean_columns)],
color=['blue', 'green', 'red'])
plt.title('Impact of Removing Missing Data')
plt.ylabel('Number of rows')
This code example demonstrates various aspects of handling missing data using the dropna() method in pandas.
Here's a comprehensive breakdown of the code:
1. Data Creation:
□ We start by creating a sample DataFrame with missing values (represented as np.nan) in different columns.
□ This simulates a real-world scenario where data might be incomplete.
2. Displaying Original Data:
□ The original DataFrame is printed to show the initial state of the data, including the missing values.
3. Checking for Missing Values:
□ We use df.isnull().sum() to count the number of missing values in each column.
□ This step is crucial for understanding the extent of missing data before deciding on a removal strategy.
4. Removing Rows with Any Missing Values:
□ df.dropna() is used without any parameters to remove all rows that contain any missing values.
□ This is the most stringent approach and can lead to significant data loss if many rows have missing values.
5. Removing Rows with Missing Values in Specific Columns:
□ df.dropna(subset=['Age', 'Salary']) removes rows only if there are missing values in the 'Age' or 'Salary' columns.
□ This approach is more targeted and preserves more data compared to removing all rows with any missing values.
6. Removing Columns with Missing Values:
□ df.dropna(axis=1) removes any column that contains missing values.
□ This approach is useful when certain features are deemed unreliable due to missing data.
7. Visualizing the Impact:
□ A bar chart is created to visually compare the number of rows in the original DataFrame versus the DataFrames after row and column removal.
□ This visualization helps in understanding the trade-off between data completeness and data loss.
This comprehensive example illustrates different strategies for handling missing data through removal, allowing for a comparison of their impacts on the dataset. It's important to choose the
appropriate method based on the specific requirements of your analysis and the nature of your data.
In this example, the dropna() function removes any rows that contain missing values. You can also specify whether to drop rows or columns depending on your use case.
2. Imputing Missing Data
If you have a significant amount of missing data, removing rows may not be a viable option as it could lead to substantial loss of information. In such cases, imputation becomes a crucial technique.
Imputation involves filling in the missing values with estimated data, allowing you to preserve the overall structure and size of your dataset.
There are several common imputation methods, each with its own strengths and use cases:
a. Mean Imputation
Mean imputation is a widely used method for handling missing numeric data. This technique involves replacing missing values in a column with the arithmetic mean (average) of all non-missing values in
that same column. For instance, if a dataset has missing age values, the average age of all individuals with recorded ages would be calculated and used to fill in the gaps.
The popularity of mean imputation stems from its simplicity and ease of implementation. It requires minimal computational resources and can be quickly applied to large datasets. This makes it an
attractive option for data scientists and analysts working with time constraints or limited processing power.
However, while mean imputation is straightforward, it comes with several important caveats:
1. Distribution Distortion: By replacing missing values with the mean, this method can alter the overall distribution of the data. It artificially increases the frequency of the mean value,
potentially creating a spike in the distribution around this point. This can lead to a reduction in the data's variance and standard deviation, which may impact statistical analyses that rely on
these measures.
2. Relationship Alteration: Mean imputation doesn't account for relationships between variables. In reality, missing values might be correlated with other features in the dataset. By using the
overall mean, these potential relationships are ignored, which could lead to biased results in subsequent analyses.
3. Uncertainty Misrepresentation: This method doesn't capture the uncertainty associated with the missing data. It treats imputed values with the same confidence as observed values, which may not be
appropriate, especially if the proportion of missing data is substantial.
4. Impact on Statistical Tests: The artificially reduced variability can lead to narrower confidence intervals and potentially inflated t-statistics, which might result in false positives in
hypothesis testing.
5. Bias in Multivariate Analyses: In analyses involving multiple variables, such as regression or clustering, mean imputation can introduce bias by weakening the relationships between variables.
Given these limitations, while mean imputation remains a useful tool in certain scenarios, it's crucial for data scientists to carefully consider its appropriateness for their specific dataset and
analysis goals. In many cases, more sophisticated imputation methods that preserve the data's statistical properties and relationships might be preferable, especially for complex analyses or when
dealing with a significant amount of missing data.
Example: Imputing Missing Data with the Mean
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve'],
'Age': [25, np.nan, 35, 40, np.nan],
'Salary': [50000, 60000, np.nan, 80000, 55000],
'Department': ['HR', 'IT', 'Finance', 'IT', np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
print("\nMissing values in each column:")
# Impute missing values in the 'Age' and 'Salary' columns with the mean
df['Age'] = df['Age'].fillna(df['Age'].mean())
df['Salary'] = df['Salary'].fillna(df['Salary'].mean())
print("\nDataFrame After Mean Imputation:")
# Using SimpleImputer for comparison
imputer = SimpleImputer(strategy='mean')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After SimpleImputer Mean Imputation:")
# Visualize the impact of imputation
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))
ax1.bar(df['Name'], df['Age'], color='blue', alpha=0.7)
ax1.set_title('Age Distribution After Imputation')
ax1.tick_params(axis='x', rotation=45)
ax2.bar(df['Name'], df['Salary'], color='green', alpha=0.7)
ax2.set_title('Salary Distribution After Imputation')
ax2.tick_params(axis='x', rotation=45)
# Calculate and print statistics
print("\nStatistics After Imputation:")
print(df[['Age', 'Salary']].describe())
This code example provides a more comprehensive approach to mean imputation and includes visualization and statistical analysis.
Here's a breakdown of the code:
• Data Creation and Inspection:
□ We create a sample DataFrame with missing values in different columns.
□ The original DataFrame is displayed along with a count of missing values in each column.
• Mean Imputation:
□ We use the fillna() method with df['column'].mean() to impute missing values in the 'Age' and 'Salary' columns.
□ The DataFrame after imputation is displayed to show the changes.
• SimpleImputer Comparison:
□ We use sklearn's SimpleImputer with 'mean' strategy to perform imputation.
□ This demonstrates an alternative method for mean imputation, which can be useful for larger datasets or when working with scikit-learn pipelines.
• Visualization:
□ Two bar plots are created to visualize the Age and Salary distributions after imputation.
□ This helps in understanding the impact of imputation on the data distribution.
• Statistical Analysis:
□ We calculate and display descriptive statistics for the 'Age' and 'Salary' columns after imputation.
□ This provides insights into how imputation has affected the central tendencies and spread of the data.
This code example not only demonstrates how to perform mean imputation but also shows how to assess its impact through visualization and statistical analysis. It's important to note that while mean
imputation is simple and often effective, it can reduce the variance in your data and may not be suitable for all situations, especially when data is not missing at random.
b. Median Imputation
Median imputation is a robust alternative to mean imputation for handling missing data. This method uses the median value of the non-missing data to fill in gaps. The median is the middle value when
a dataset is ordered from least to greatest, effectively separating the higher half from the lower half of a data sample.
Median imputation is particularly valuable when dealing with skewed distributions or datasets containing outliers. In these scenarios, the median proves to be more resilient and representative than
the mean. This is because outliers can significantly pull the mean towards extreme values, whereas the median remains stable.
For instance, consider a dataset of salaries where most employees earn between $40,000 and $60,000, but there are a few executives with salaries over $1,000,000. The mean salary would be heavily
influenced by these high earners, potentially leading to overestimation when imputing missing values. The median, however, would provide a more accurate representation of the typical salary.
Furthermore, median imputation helps maintain the overall shape of the data distribution better than mean imputation in cases of skewed data. This is crucial for preserving important characteristics
of the dataset, which can be essential for subsequent analyses or modeling tasks.
It's worth noting that while median imputation is often superior to mean imputation for skewed data, it still has limitations. Like mean imputation, it doesn't account for relationships between
variables and may not be suitable for datasets where missing values are not randomly distributed. In such cases, more advanced imputation techniques might be necessary.
Example: Median Imputation
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values and outliers
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve', 'Frank', 'Grace', 'Henry', 'Ivy', 'Jack'],
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Salary': [50000, 60000, np.nan, 80000, 55000, 75000, np.nan, 70000, 1000000, np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
print("\nMissing values in each column:")
# Perform median imputation
df_median_imputed = df.copy()
df_median_imputed['Age'] = df_median_imputed['Age'].fillna(df_median_imputed['Age'].median())
df_median_imputed['Salary'] = df_median_imputed['Salary'].fillna(df_median_imputed['Salary'].median())
print("\nDataFrame After Median Imputation:")
# Using SimpleImputer for comparison
imputer = SimpleImputer(strategy='median')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After SimpleImputer Median Imputation:")
# Visualize the impact of imputation
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))
ax1.boxplot([df['Salary'].dropna(), df_median_imputed['Salary']], labels=['Original', 'Imputed'])
ax1.set_title('Salary Distribution: Original vs Imputed')
ax2.scatter(df['Age'], df['Salary'], label='Original', alpha=0.7)
ax2.scatter(df_median_imputed['Age'], df_median_imputed['Salary'], label='Imputed', alpha=0.7)
ax2.set_title('Age vs Salary: Original and Imputed Data')
# Calculate and print statistics
print("\nStatistics After Imputation:")
print(df_median_imputed[['Age', 'Salary']].describe())
This comprehensive example demonstrates median imputation and includes visualization and statistical analysis. Here's a breakdown of the code:
1. Data Creation and Inspection:
□ We create a sample DataFrame with missing values in the 'Age' and 'Salary' columns, including an outlier in the 'Salary' column.
□ The original DataFrame is displayed along with a count of missing values in each column.
2. Median Imputation:
□ We use the fillna() method with df['column'].median() to impute missing values in the 'Age' and 'Salary' columns.
□ The DataFrame after imputation is displayed to show the changes.
3. SimpleImputer Comparison:
□ We use sklearn's SimpleImputer with 'median' strategy to perform imputation.
□ This demonstrates an alternative method for median imputation, which can be useful for larger datasets or when working with scikit-learn pipelines.
4. Visualization:
□ A box plot is created to compare the original and imputed salary distributions, highlighting the impact of median imputation on the outlier.
□ A scatter plot shows the relationship between Age and Salary, comparing original and imputed data.
5. Statistical Analysis:
□ We calculate and display descriptive statistics for the 'Age' and 'Salary' columns after imputation.
□ This provides insights into how imputation has affected the central tendencies and spread of the data.
This example illustrates how median imputation handles outliers better than mean imputation. The salary outlier of 1,000,000 doesn't significantly affect the imputed values, as it would with mean
imputation. The visualization helps to understand the impact of imputation on the data distribution and relationships between variables.
Median imputation is particularly useful when dealing with skewed data or datasets with outliers, as it provides a more robust measure of central tendency compared to the mean. However, like other
simple imputation methods, it doesn't account for relationships between variables and may not be suitable for all types of missing data mechanisms.
c. Mode Imputation
Mode imputation is a technique used to handle missing data by replacing missing values with the most frequently occurring value (mode) in the column. This method is particularly useful for
categorical data where numerical concepts like mean or median are not applicable.
Here's a more detailed explanation:
Application in Categorical Data: Mode imputation is primarily used for categorical variables, such as 'color', 'gender', or 'product type'. For instance, if in a 'favorite color' column, most
responses are 'blue', missing values would be filled with 'blue'.
Effectiveness for Nominal Variables: Mode imputation can be quite effective for nominal categorical variables, where categories have no inherent order. Examples include variables like 'blood type' or
'country of origin'. In these cases, using the most frequent category as a replacement is often a reasonable assumption.
Limitations with Ordinal Data: However, mode imputation may not be suitable for ordinal data, where the order of categories matters. For example, in a variable like 'education level' (high school,
bachelor's, master's, PhD), simply using the most frequent category could disrupt the inherent order and potentially introduce bias in subsequent analyses.
Preserving Data Distribution: One advantage of mode imputation is that it preserves the original distribution of the data more closely than methods like mean imputation, especially for categorical
variables with a clear majority category.
Potential Drawbacks: It's important to note that mode imputation can oversimplify the data, especially if there's no clear mode or if the variable has multiple modes. It also doesn't account for
relationships between variables, which could lead to loss of important information or introduction of bias.
Alternative Approaches: For more complex scenarios, especially with ordinal data or when preserving relationships between variables is crucial, more sophisticated methods like multiple imputation or
machine learning-based imputation techniques might be more appropriate.
Example: Mode Imputation
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve', 'Frank', 'Grace', 'Henry', 'Ivy', 'Jack'],
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Category': ['A', 'B', np.nan, 'A', 'C', 'B', np.nan, 'A', 'C', np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
print("\nMissing values in each column:")
# Perform mode imputation
df_mode_imputed = df.copy()
df_mode_imputed['Category'] = df_mode_imputed['Category'].fillna(df_mode_imputed['Category'].mode()[0])
print("\nDataFrame After Mode Imputation:")
# Using SimpleImputer for comparison
imputer = SimpleImputer(strategy='most_frequent')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After SimpleImputer Mode Imputation:")
# Visualize the impact of imputation
fig, ax = plt.subplots(figsize=(10, 6))
category_counts = df_mode_imputed['Category'].value_counts()
ax.bar(category_counts.index, category_counts.values)
ax.set_title('Category Distribution After Mode Imputation')
# Calculate and print statistics
print("\nCategory Distribution After Imputation:")
This comprehensive example demonstrates mode imputation and includes visualization and statistical analysis. Here's a breakdown of the code:
1. Data Creation and Inspection:
□ We create a sample DataFrame with missing values in the 'Age' and 'Category' columns.
□ The original DataFrame is displayed along with a count of missing values in each column.
2. Mode Imputation:
□ We use the fillna() method with df['column'].mode()[0] to impute missing values in the 'Category' column.
□ The DataFrame after imputation is displayed to show the changes.
3. SimpleImputer Comparison:
□ We use sklearn's SimpleImputer with 'most_frequent' strategy to perform imputation.
□ This demonstrates an alternative method for mode imputation, which can be useful for larger datasets or when working with scikit-learn pipelines.
4. Visualization:
□ A bar plot is created to show the distribution of categories after imputation.
□ This helps in understanding the impact of mode imputation on the categorical data distribution.
5. Statistical Analysis:
□ We calculate and display the proportion of each category after imputation.
□ This provides insights into how imputation has affected the distribution of the categorical variable.
This example illustrates how mode imputation works for categorical data. It fills in missing values with the most frequent category, which in this case is 'A'. The visualization helps to understand
the impact of imputation on the distribution of categories.
Mode imputation is particularly useful for nominal categorical data where concepts like mean or median don't apply. However, it's important to note that this method can potentially amplify the bias
towards the most common category, especially if there's a significant imbalance in the original data.
While mode imputation is simple and often effective for categorical data, it doesn't account for relationships between variables and may not be suitable for ordinal categorical data or when the
missingness mechanism is not completely at random. In such cases, more advanced techniques like multiple imputation or machine learning-based approaches might be more appropriate.
While these methods are commonly used due to their simplicity and ease of implementation, it's crucial to consider their limitations. They don't account for relationships between variables and can
introduce bias if the data is not missing completely at random. More advanced techniques like multiple imputation or machine learning-based imputation methods may be necessary for complex datasets or
when the missingness mechanism is not random.
d. Advanced Imputation Methods
In some cases, simple mean or median imputation might not be sufficient for handling missing data effectively. More sophisticated methods such as K-nearest neighbors (KNN) imputation or regression
imputation can be applied to achieve better results. These advanced techniques go beyond simple statistical measures and take into account the complex relationships between variables to predict
missing values more accurately.
K-nearest neighbors (KNN) imputation works by identifying the K most similar data points (neighbors) to the one with missing values, based on other available features. It then uses the values from
these neighbors to estimate the missing value, often by taking their average. This method is particularly useful when there are strong correlations between features in the dataset.
Regression imputation, on the other hand, involves building a regression model using the available data to predict the missing values. This method can capture more complex relationships between
variables and can be especially effective when there are clear patterns or trends in the data that can be leveraged for prediction.
These advanced imputation methods offer several advantages over simple imputation:
• They preserve the relationships between variables, which can be crucial for maintaining the integrity of the dataset.
• They can handle both numerical and categorical data more effectively.
• They often provide more accurate estimates of missing values, leading to better model performance downstream.
Fortunately, popular machine learning libraries like Scikit-learn provide easy-to-use implementations of these advanced imputation techniques. This accessibility allows data scientists and analysts
to quickly experiment with and apply these sophisticated methods in their preprocessing pipelines, potentially improving the overall quality of their data and the performance of their models.
Example: K-Nearest Neighbors (KNN) Imputation
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import KNNImputer
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
# Create a sample DataFrame with missing values
data = {
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Salary': [50000, 60000, np.nan, 75000, 65000, np.nan, 70000, 80000, np.nan, 90000],
'Experience': [2, 3, 5, np.nan, 4, 8, np.nan, 7, 6, 10]
df = pd.DataFrame(data)
print("Original DataFrame:")
print("\nMissing values in each column:")
# Initialize the KNN Imputer
imputer = KNNImputer(n_neighbors=2)
# Fit and transform the data
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After KNN Imputation:")
# Visualize the imputation results
fig, axes = plt.subplots(1, 3, figsize=(15, 5))
for i, column in enumerate(df.columns):
axes[i].scatter(df.index, df[column], label='Original', alpha=0.5)
axes[i].scatter(df_imputed.index, df_imputed[column], label='Imputed', alpha=0.5)
axes[i].set_title(f'{column} - Before and After Imputation')
# Evaluate the impact of imputation on a simple model
X = df_imputed[['Age', 'Experience']]
y = df_imputed['Salary']
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
print(f"\nMean Squared Error after imputation: {mse:.2f}")
This code example demonstrates a more comprehensive approach to KNN imputation and its evaluation.
Here's a breakdown of the code:
• Data Preparation:
□ We create a sample DataFrame with missing values in 'Age', 'Salary', and 'Experience' columns.
□ The original DataFrame and the count of missing values are displayed.
• KNN Imputation:
□ We initialize a KNNImputer with 2 neighbors.
□ The imputer is applied to the DataFrame, filling in missing values based on the K-nearest neighbors.
• Visualization:
□ We create scatter plots for each column, comparing the original data with missing values to the imputed data.
□ This visual representation helps in understanding how KNN imputation affects the data distribution.
• Model Evaluation:
□ We use the imputed data to train a simple Linear Regression model.
□ The model predicts 'Salary' based on 'Age' and 'Experience'.
□ We calculate the Mean Squared Error to evaluate the model's performance after imputation.
This comprehensive example showcases not only how to perform KNN imputation but also how to visualize its effects and evaluate its impact on a subsequent machine learning task. It provides a more
holistic view of the imputation process and its consequences in a data science workflow.
In this example, the KNN Imputer fills in missing values by finding the nearest neighbors in the dataset and using their values to estimate the missing ones. This method is often more accurate than
simple mean imputation when the data has strong relationships between features.
3.1.4 Evaluating the Impact of Missing Data
Handling missing data is not merely a matter of filling in gaps—it's crucial to thoroughly evaluate how missing data impacts your model's performance. This evaluation process is multifaceted and
requires careful consideration. When certain features in your dataset contain an excessive number of missing values, they may prove to be unreliable predictors. In such cases, it might be more
beneficial to remove these features entirely rather than attempting to impute the missing values.
Furthermore, it's essential to rigorously test imputed data to ensure its validity and reliability. This testing process should focus on two key aspects: first, verifying that the imputation method
hasn't inadvertently distorted the underlying relationships within the data, and second, confirming that it hasn't introduced any bias into the model. Both of these factors can significantly affect
the accuracy and generalizability of your machine learning model.
To gain a comprehensive understanding of how your chosen method for handling missing data affects your model, it's advisable to assess the model's performance both before and after implementing your
missing data strategy. This comparative analysis can be conducted using robust validation techniques such as cross-validation or holdout validation.
These methods provide valuable insights into how your model's predictive capabilities have been influenced by your approach to missing data, allowing you to make informed decisions about the most
effective preprocessing strategies for your specific dataset and modeling objectives.
Example: Model Evaluation Before and After Handling Missing Data
import pandas as pd
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values
data = {
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Salary': [50000, 60000, np.nan, 75000, 65000, np.nan, 70000, 80000, np.nan, 90000],
'Experience': [2, 3, 5, np.nan, 4, 8, np.nan, 7, 6, 10]
df = pd.DataFrame(data)
print("Original DataFrame:")
print("\nMissing values in each column:")
# Function to evaluate model performance
def evaluate_model(X, y, model_name):
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)
print(f"\n{model_name} - Mean Squared Error: {mse:.2f}")
print(f"{model_name} - R-squared Score: {r2:.2f}")
# Evaluate model with missing data
X_missing = df[['Age', 'Experience']]
y_missing = df['Salary']
evaluate_model(X_missing.dropna(), y_missing.dropna(), "Model with Missing Data")
# Simple Imputation
imputer = SimpleImputer(strategy='mean')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After Mean Imputation:")
# Evaluate model after imputation
X_imputed = df_imputed[['Age', 'Experience']]
y_imputed = df_imputed['Salary']
evaluate_model(X_imputed, y_imputed, "Model After Imputation")
# Advanced: Multiple models comparison
from sklearn.ensemble import RandomForestRegressor
from sklearn.svm import SVR
models = {
'Linear Regression': LinearRegression(),
'Random Forest': RandomForestRegressor(n_estimators=100, random_state=42),
'Support Vector Regression': SVR()
for name, model in models.items():
X_train, X_test, y_train, y_test = train_test_split(X_imputed, y_imputed, test_size=0.2, random_state=42)
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)
print(f"\n{name} - Mean Squared Error: {mse:.2f}")
print(f"{name} - R-squared Score: {r2:.2f}")
This code example provides a comprehensive approach to evaluating the impact of missing data and imputation on model performance.
Here's a detailed breakdown of the code:
1. Data Preparation:
□ We create a sample DataFrame with missing values in 'Age', 'Salary', and 'Experience' columns.
□ The original DataFrame and the count of missing values are displayed.
2. Model Evaluation Function:
□ A function evaluate_model() is defined to assess model performance using Mean Squared Error (MSE) and R-squared score.
□ This function will be used to compare model performance before and after imputation.
3. Evaluation with Missing Data:
□ We first evaluate the model's performance using only the complete cases (rows without missing values).
□ This serves as a baseline for comparison.
4. Simple Imputation:
□ Mean imputation is performed using sklearn's SimpleImputer.
□ The imputed DataFrame is displayed to show the changes.
5. Evaluation After Imputation:
□ We evaluate the model's performance again using the imputed data.
□ This allows us to compare the impact of imputation on model performance.
6. Advanced Model Comparison:
□ We introduce two additional models: Random Forest and Support Vector Regression.
□ All three models (including Linear Regression) are trained and evaluated on the imputed data.
□ This comparison helps in understanding if the choice of model affects the impact of imputation.
This example demonstrates how to handle missing data, perform imputation, and evaluate its impact on different models. It provides insights into:
• The effect of missing data on model performance
• The impact of mean imputation on data distribution and model accuracy
• How different models perform on the imputed data
By comparing the results, data scientists can make informed decisions about the most appropriate imputation method and model selection for their specific dataset and problem.
Handling missing data is one of the most critical steps in data preprocessing. Whether you choose to remove or impute missing values, understanding the nature of the missing data and selecting the
appropriate method is essential for building a reliable machine learning model. In this section, we covered several strategies, ranging from simple mean imputation to more advanced techniques like
KNN imputation, and demonstrated how to evaluate their impact on your model's performance.
3.1 Data Cleaning and Handling Missing Data
Data preprocessing stands as the cornerstone of any robust machine learning pipeline, serving as the critical initial step that can make or break the success of your model. In the complex landscape
of real-world data science, practitioners often encounter raw data that is far from ideal - it may be riddled with inconsistencies, plagued by missing values, or lack the structure necessary for
immediate analysis.
Attempting to feed such unrefined data directly into a machine learning algorithm is a recipe for suboptimal performance and unreliable results. This is precisely where the twin pillars of data
preprocessing and feature engineering come into play, offering a systematic approach to data refinement.
These essential processes encompass a wide range of techniques aimed at cleaning, transforming, and optimizing your dataset. By meticulously preparing your data, you create a solid foundation that
enables machine learning algorithms to uncover meaningful patterns and generate accurate predictions. The goal is to present your model with a dataset that is not only clean and complete but also
structured in a way that highlights the most relevant features and relationships within the data.
Throughout this chapter, we will delve deep into the crucial steps that comprise effective data preprocessing. We'll explore the intricacies of data cleaning, a fundamental process that involves
identifying and rectifying errors, inconsistencies, and anomalies in your dataset. We'll tackle the challenge of handling missing data, discussing various strategies to address gaps in your
information without compromising the integrity of your analysis. The chapter will also cover scaling and normalization techniques, essential for ensuring that all features contribute proportionally
to the model's decision-making process.
Furthermore, we'll examine methods for encoding categorical variables, transforming non-numeric data into a format that machine learning algorithms can interpret and utilize effectively. Lastly,
we'll dive into the art and science of feature engineering, where domain knowledge and creativity converge to craft new, informative features that can significantly enhance your model's predictive
By mastering these preprocessing steps, you'll be equipped to lay a rock-solid foundation for your machine learning projects. This meticulous preparation of your data is what separates mediocre
models from those that truly excel, maximizing performance and ensuring that your algorithms can extract the most valuable insights from the information at hand.
We'll kick off our journey into data preprocessing with an in-depth look at data cleaning. This critical process serves as the first line of defense against the myriad issues that can plague raw
datasets. By ensuring that your data is accurate, complete, and primed for analysis, data cleaning sets the stage for all subsequent preprocessing steps and ultimately contributes to the overall
success of your machine learning endeavors.
Data cleaning is a crucial step in the data preprocessing pipeline, involving the systematic identification and rectification of issues within datasets. This process encompasses a wide range of
activities, including:
Detecting corrupt data
This crucial step involves a comprehensive and meticulous examination of the dataset to identify any data points that have been compromised or altered during various stages of the data lifecycle.
This includes, but is not limited to, the collection phase, where errors might occur due to faulty sensors or human input mistakes; the transmission phase, where data corruption can happen due to
network issues or interference; and the storage phase, where data might be corrupted due to hardware failures or software glitches.
The process of detecting corrupt data often involves multiple techniques:
• Statistical analysis: Using statistical methods to identify outliers or values that deviate significantly from expected patterns.
• Data validation rules: Implementing specific rules based on domain knowledge to flag potentially corrupt entries.
• Consistency checks: Comparing data across different fields or time periods to ensure logical consistency.
• Format verification: Ensuring that data adheres to expected formats, such as date structures or numerical ranges.
By pinpointing these corrupted elements through such rigorous methods, data scientists can take appropriate actions such as removing, correcting, or flagging the corrupt data. This process is
fundamental in ensuring the integrity and reliability of the dataset, which is crucial for any subsequent analysis or machine learning model development. Without this step, corrupt data could lead to
skewed results, incorrect conclusions, or poorly performing models, potentially undermining the entire data science project.
Example: Detecting Corrupt Data
import pandas as pd
import numpy as np
# Create a sample DataFrame with potentially corrupt data
data = {
'ID': [1, 2, 3, 4, 5],
'Value': [10, 20, 'error', 40, 50],
'Date': ['2023-01-01', '2023-02-30', '2023-03-15', '2023-04-01', '2023-05-01']
df = pd.DataFrame(data)
# Function to detect corrupt data
def detect_corrupt_data(df):
corrupt_rows = []
# Check for non-numeric values in 'Value' column
numeric_errors = pd.to_numeric(df['Value'], errors='coerce').isna()
# Check for invalid dates
df['Date'] = pd.to_datetime(df['Date'], errors='coerce')
date_errors = df['Date'].isna()
return list(set(corrupt_rows)) # Remove duplicates
# Detect corrupt data
corrupt_indices = detect_corrupt_data(df)
print("Corrupt data found at indices:", corrupt_indices)
print("\nCorrupt rows:")
This code demonstrates how to detect corrupt data in a pandas DataFrame. Here's a breakdown of its functionality:
• It creates a sample DataFrame with potentially corrupt data, including non-numeric values in the 'Value' column and invalid dates in the 'Date' column.
• The detect_corrupt_data() function is defined to identify corrupt rows. It checks for:
• Non-numeric values in the 'Value' column using pd.to_numeric() with errors='coerce'.
• Invalid dates in the 'Date' column using pd.to_datetime() with errors='coerce'.
• The function returns a list of unique indices where corrupt data was found.
• Finally, it prints the indices of corrupt rows and displays the corrupt data.
This code is an example of how to implement data cleaning techniques, specifically for detecting corrupt data, which is a crucial step in the data preprocessing pipeline.
Correcting incomplete data
This process involves a comprehensive and meticulous examination of the dataset to identify and address any instances of incomplete or missing information. The approach to handling such gaps depends
on several factors, including the nature of the data, the extent of incompleteness, and the potential impact on subsequent analyses.
When dealing with missing data, data scientists employ a range of sophisticated techniques:
• Imputation methods: These involve estimating and filling in missing values based on patterns observed in the existing data. Techniques can range from simple mean or median imputation to more
advanced methods like regression imputation or multiple imputation.
• Machine learning-based approaches: Algorithms such as K-Nearest Neighbors (KNN) or Random Forest can be used to predict missing values based on the relationships between variables in the dataset.
• Time series-specific methods: For temporal data, techniques like interpolation or forecasting models may be employed to estimate missing values based on trends and seasonality.
However, in cases where the gaps in the data are too significant or the missing information is deemed crucial, careful consideration must be given to the removal of incomplete records. This decision
is not taken lightly, as it involves balancing the need for data quality with the potential loss of valuable information.
Factors influencing the decision to remove incomplete records include:
• The proportion of missing data: If a large percentage of a record or variable is missing, removal might be more appropriate than imputation.
• The mechanism of missingness: Understanding whether data is missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR) can inform the decision-making process.
• The importance of the missing information: If the missing data is critical to the analysis or model, removal might be necessary to maintain the integrity of the results.
Ultimately, the goal is to strike a balance between preserving as much valuable information as possible while ensuring the overall quality and reliability of the dataset for subsequent analysis and
modeling tasks.
Example: Correcting Incomplete Data
import pandas as pd
import numpy as np
from sklearn.impute import SimpleImputer
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
# Create a sample DataFrame with incomplete data
data = {
'Age': [25, np.nan, 30, np.nan, 40],
'Income': [50000, 60000, np.nan, 75000, 80000],
'Education': ['Bachelor', 'Master', np.nan, 'PhD', 'Bachelor']
df = pd.DataFrame(data)
print("Original DataFrame:")
# Method 1: Simple Imputation (Mean for numerical, Most frequent for categorical)
imputer_mean = SimpleImputer(strategy='mean')
imputer_most_frequent = SimpleImputer(strategy='most_frequent')
df_imputed_simple = df.copy()
df_imputed_simple[['Age', 'Income']] = imputer_mean.fit_transform(df[['Age', 'Income']])
df_imputed_simple[['Education']] = imputer_most_frequent.fit_transform(df[['Education']])
print("\nDataFrame after Simple Imputation:")
# Method 2: Iterative Imputation (uses the IterativeImputer, aka MICE)
imputer_iterative = IterativeImputer(random_state=0)
df_imputed_iterative = df.copy()
df_imputed_iterative.iloc[:, :] = imputer_iterative.fit_transform(df)
print("\nDataFrame after Iterative Imputation:")
# Method 3: Custom logic (e.g., filling Age based on median of similar Education levels)
df_custom = df.copy()
df_custom['Age'] = df_custom.groupby('Education')['Age'].transform(lambda x: x.fillna(x.median()))
df_custom['Income'].fillna(df_custom['Income'].mean(), inplace=True)
df_custom['Education'].fillna(df_custom['Education'].mode()[0], inplace=True)
print("\nDataFrame after Custom Imputation:")
This example demonstrates three different methods for correcting incomplete data:
• 1. Simple Imputation: Uses Scikit-learn's SimpleImputer to fill missing values with the mean for numerical columns (Age and Income) and the most frequent value for categorical columns
• 2. Iterative Imputation: Employs Scikit-learn's IterativeImputer (also known as MICE - Multivariate Imputation by Chained Equations) to estimate missing values based on the relationships between
• 3. Custom Logic: Implements a tailored approach where Age is imputed based on the median age of similar education levels, Income is filled with the mean, and Education uses the mode (most
frequent value).
Breakdown of the code:
1. We start by importing necessary libraries and creating a sample DataFrame with missing values.
2. For Simple Imputation, we use SimpleImputer with different strategies for numerical and categorical data.
3. Iterative Imputation uses the IterativeImputer, which estimates each feature from all the others iteratively.
4. The custom logic demonstrates how domain knowledge can be applied to impute data more accurately, such as using education level to estimate age.
This example showcases the flexibility and power of different imputation techniques. The choice of method depends on the nature of your data and the specific requirements of your analysis. Simple
imputation is quick and easy but may not capture complex relationships in the data. Iterative imputation can be more accurate but is computationally intensive. Custom logic allows for the
incorporation of domain expertise but requires more manual effort and understanding of the data.
Addressing inaccurate data
This crucial step in the data cleaning process involves a comprehensive and meticulous approach to identifying and rectifying errors that may have infiltrated the dataset during various stages of
data collection and management. These errors can arise from multiple sources:
• Data Entry Errors: Human mistakes during manual data input, such as typos, transposed digits, or incorrect categorizations.
• Measurement Errors: Inaccuracies stemming from faulty equipment, miscalibrated instruments, or inconsistent measurement techniques.
• Recording Errors: Issues that occur during the data recording process, including system glitches, software bugs, or data transmission failures.
To address these challenges, data scientists employ a range of sophisticated validation techniques:
• Statistical Outlier Detection: Utilizing statistical methods to identify data points that deviate significantly from the expected patterns or distributions.
• Domain-Specific Rule Validation: Implementing checks based on expert knowledge of the field to flag logically inconsistent or impossible values.
• Cross-Referencing: Comparing data against reliable external sources or internal databases to verify accuracy and consistency.
• Machine Learning-Based Anomaly Detection: Leveraging advanced algorithms to detect subtle patterns of inaccuracy that might escape traditional validation methods.
By rigorously applying these validation techniques and diligently cross-referencing with trusted sources, data scientists can substantially enhance the accuracy and reliability of their datasets.
This meticulous process not only improves the quality of the data but also bolsters the credibility of subsequent analyses and machine learning models built upon this foundation. Ultimately,
addressing inaccurate data is a critical investment in ensuring the integrity and trustworthiness of data-driven insights and decision-making processes.
Example: Addressing Inaccurate Data
import pandas as pd
import numpy as np
from scipy import stats
# Create a sample DataFrame with potentially inaccurate data
data = {
'ID': range(1, 11),
'Age': [25, 30, 35, 40, 45, 50, 55, 60, 65, 1000],
'Income': [50000, 60000, 70000, 80000, 90000, 100000, 110000, 120000, 130000, 10000000],
'Height': [170, 175, 180, 185, 190, 195, 200, 205, 210, 150]
df = pd.DataFrame(data)
print("Original DataFrame:")
def detect_and_correct_outliers(df, column, method='zscore', threshold=3):
if method == 'zscore':
z_scores = np.abs(stats.zscore(df[column]))
outliers = df[z_scores > threshold]
df.loc[z_scores > threshold, column] = df[column].median()
elif method == 'iqr':
Q1 = df[column].quantile(0.25)
Q3 = df[column].quantile(0.75)
IQR = Q3 - Q1
lower_bound = Q1 - 1.5 * IQR
upper_bound = Q3 + 1.5 * IQR
outliers = df[(df[column] < lower_bound) | (df[column] > upper_bound)]
df.loc[(df[column] < lower_bound) | (df[column] > upper_bound), column] = df[column].median()
return outliers
# Detect and correct outliers in 'Age' column using Z-score method
age_outliers = detect_and_correct_outliers(df, 'Age', method='zscore')
# Detect and correct outliers in 'Income' column using IQR method
income_outliers = detect_and_correct_outliers(df, 'Income', method='iqr')
# Custom logic for 'Height' column
height_outliers = df[(df['Height'] < 150) | (df['Height'] > 220)]
df.loc[(df['Height'] < 150) | (df['Height'] > 220), 'Height'] = df['Height'].median()
print("\nOutliers detected:")
print("Age outliers:", age_outliers['Age'].tolist())
print("Income outliers:", income_outliers['Income'].tolist())
print("Height outliers:", height_outliers['Height'].tolist())
print("\nCorrected DataFrame:")
This example demonstrates a comprehensive approach to addressing inaccurate data, specifically focusing on outlier detection and correction.
Here's a breakdown of the code and its functionality:
1. Data Creation: We start by creating a sample DataFrame with potentially inaccurate data, including extreme values in the 'Age', 'Income', and 'Height' columns.
2. Outlier Detection and Correction Function: The detect_and_correct_outliers() function is defined to handle outliers using two common methods:
□ Z-score method: Identifies outliers based on the number of standard deviations from the mean.
□ IQR (Interquartile Range) method: Detects outliers using the concept of quartiles.
3. Applying Outlier Detection:
□ For the 'Age' column, we use the Z-score method with a threshold of 3 standard deviations.
□ For the 'Income' column, we apply the IQR method to account for potential skewness in income distribution.
□ For the 'Height' column, we implement a custom logic to flag values below 150 cm or above 220 cm as outliers.
4. Outlier Correction: Once outliers are detected, they are replaced with the median value of the respective column. This approach helps maintain data integrity while reducing the impact of extreme
5. Reporting: The code prints out the detected outliers for each column and displays the corrected DataFrame.
This example showcases different strategies for addressing inaccurate data:
• Statistical methods (Z-score and IQR) for automated outlier detection
• Custom logic for domain-specific outlier identification
• Median imputation for correcting outliers, which is more robust to extreme values than mean imputation
By employing these techniques, data scientists can significantly improve the quality of their datasets, leading to more reliable analyses and machine learning models. It's important to note that
while this example uses median imputation for simplicity, in practice, the choice of correction method should be carefully considered based on the specific characteristics of the data and the
requirements of the analysis.
Removing irrelevant data
This final step in the data cleaning process, known as data relevance assessment, involves a meticulous evaluation of each data point to determine its significance and applicability to the specific
analysis or problem at hand. This crucial phase requires data scientists to critically examine the dataset through multiple lenses:
1. Contextual Relevance: Assessing whether each variable or feature directly contributes to answering the research questions or achieving the project goals.
2. Temporal Relevance: Determining if the data is current enough to be meaningful for the analysis, especially in rapidly changing domains.
3. Granularity: Evaluating if the level of detail in the data is appropriate for the intended analysis, neither too broad nor too specific.
4. Redundancy: Identifying and removing duplicate or highly correlated variables that don't provide additional informational value.
5. Signal-to-Noise Ratio: Distinguishing between data that carries meaningful information (signal) and data that introduces unnecessary complexity or variability (noise).
By meticulously eliminating extraneous or irrelevant information through this process, data scientists can significantly enhance the quality and focus of their dataset. This refinement yields several
critical benefits:
• Improved Model Performance: A streamlined dataset with only relevant features often leads to more accurate and robust machine learning models.
• Enhanced Computational Efficiency: Reducing the dataset's dimensionality can dramatically decrease processing time and resource requirements, especially crucial when dealing with large-scale data.
• Clearer Insights: By removing noise and focusing on pertinent data, analysts can derive more meaningful and actionable insights from their analyses.
• Reduced Overfitting Risk: Eliminating irrelevant features helps prevent models from learning spurious patterns, thus improving generalization to new, unseen data.
• Simplified Interpretability: A more focused dataset often results in models and analyses that are easier to interpret and explain to stakeholders.
In essence, this careful curation of relevant data serves as a critical foundation, significantly enhancing the efficiency, effectiveness, and reliability of subsequent analyses and machine learning
models. It ensures that the final insights and decisions are based on the most pertinent and high-quality information available.
Example: Removing Irrelevant Data
import pandas as pd
import numpy as np
from sklearn.feature_selection import VarianceThreshold
from sklearn.feature_selection import mutual_info_regression
# Create a sample DataFrame with potentially irrelevant features
data = {
'ID': range(1, 101),
'Age': np.random.randint(18, 80, 100),
'Income': np.random.randint(20000, 150000, 100),
'Education': np.random.choice(['High School', 'Bachelor', 'Master', 'PhD'], 100),
'Constant_Feature': [5] * 100,
'Random_Feature': np.random.random(100),
'Target': np.random.randint(0, 2, 100)
df = pd.DataFrame(data)
print("Original DataFrame shape:", df.shape)
# Step 1: Remove constant features
constant_filter = VarianceThreshold(threshold=0)
constant_columns = df.columns[~constant_filter.get_support()]
df = df.drop(columns=constant_columns)
print("After removing constant features:", df.shape)
# Step 2: Remove features with low variance
variance_filter = VarianceThreshold(threshold=0.1)
low_variance_columns = df.select_dtypes(include=[np.number]).columns[~variance_filter.get_support()]
df = df.drop(columns=low_variance_columns)
print("After removing low variance features:", df.shape)
# Step 3: Feature importance based on mutual information
numerical_features = df.select_dtypes(include=[np.number]).columns.drop('Target')
mi_scores = mutual_info_regression(df[numerical_features], df['Target'])
mi_scores = pd.Series(mi_scores, index=numerical_features)
important_features = mi_scores[mi_scores > 0.01].index
df = df[important_features.tolist() + ['Education', 'Target']]
print("After removing less important features:", df.shape)
print("\nFinal DataFrame columns:", df.columns.tolist())
This code example demonstrates various techniques for removing irrelevant data from a dataset.
Let's break down the code and explain each step:
1. Data Creation: We start by creating a sample DataFrame with potentially irrelevant features, including a constant feature and a random feature.
2. Removing Constant Features:
□ We use VarianceThreshold with a threshold of 0 to identify and remove features that have the same value in all samples.
□ This step eliminates features that provide no discriminative information for the model.
3. Removing Low Variance Features:
□ We apply VarianceThreshold again, this time with a threshold of 0.1, to remove features with very low variance.
□ Features with low variance often contain little information and may not contribute significantly to the model's predictive power.
4. Feature Importance based on Mutual Information:
□ We use mutual_info_regression to calculate the mutual information between each feature and the target variable.
□ Features with mutual information scores below a certain threshold (0.01 in this example) are considered less important and are removed.
□ This step helps in identifying features that have a strong relationship with the target variable.
5. Retaining Categorical Features: We manually include the 'Education' column to demonstrate how you might retain important categorical features that weren't part of the numerical analysis.
This example showcases a multi-faceted approach to removing irrelevant data:
• It addresses constant features that provide no discriminative information.
• It removes features with very low variance, which often contribute little to model performance.
• It uses a statistical measure (mutual information) to identify features most relevant to the target variable.
By applying these techniques, we significantly reduce the dimensionality of the dataset, focusing on the most relevant features. This can lead to improved model performance, reduced overfitting, and
increased computational efficiency. However, it's crucial to validate the impact of feature removal on your specific problem and adjust thresholds as necessary.
The importance of data cleaning cannot be overstated, as it directly impacts the quality and reliability of machine learning models. Clean, high-quality data is essential for accurate predictions and
meaningful insights.
Missing values are a common challenge in real-world datasets, often arising from various sources such as equipment malfunctions, human error, or intentional non-responses. Handling these missing
values appropriately is critical, as they can significantly affect model performance and lead to biased or incorrect conclusions if not addressed properly.
The approach to dealing with missing data is not one-size-fits-all and depends on several factors:
1. The nature and characteristics of your dataset: The specific type of data you're working with (such as numerical, categorical, or time series) and its underlying distribution patterns play a
crucial role in determining the most appropriate technique for handling missing data. For instance, certain imputation methods may be more suitable for continuous numerical data, while others
might be better suited for categorical variables or time-dependent information.
2. The quantity and distribution pattern of missing data: The extent of missing information and the underlying mechanism causing the data gaps significantly influence the choice of handling
strategy. It's essential to distinguish between data that is missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR), as each scenario may require a
different approach to maintain the integrity and representativeness of your dataset.
3. The selected machine learning algorithm and its inherent properties: Different machine learning models exhibit varying degrees of sensitivity to missing data, which can substantially impact their
performance and the reliability of their predictions. Some algorithms, like decision trees, can handle missing values intrinsically, while others, such as support vector machines, may require
more extensive preprocessing to address data gaps effectively. Understanding these model-specific characteristics is crucial in selecting an appropriate missing data handling technique that
aligns with your chosen algorithm.
By understanding these concepts and techniques, data scientists can make informed decisions about how to preprocess their data effectively, ensuring the development of robust and accurate machine
learning models.
3.1.1 Types of Missing Data
Before delving deeper into the intricacies of handling missing data, it is crucial to grasp the three primary categories of missing data, each with its own unique characteristics and implications for
data analysis:
1. Missing Completely at Random (MCAR)
This type of missing data represents a scenario where the absence of information follows no discernible pattern or relationship with any variables in the dataset, whether observed or unobserved. MCAR
is characterized by an equal probability of data being missing across all cases, effectively creating an unbiased subset of the complete dataset.
The key features of MCAR include:
• Randomness: The missingness is entirely random and not influenced by any factors within or outside the dataset.
• Unbiased representation: The remaining data can be considered a random sample of the full dataset, maintaining its statistical properties.
• Statistical implications: Analyses conducted on the complete cases (after removing missing data) remain unbiased, although there may be a loss in statistical power due to reduced sample size.
To illustrate MCAR, consider a comprehensive survey scenario:
Imagine a large-scale health survey where participants are required to fill out a lengthy questionnaire. Some respondents might inadvertently skip certain questions due to factors entirely unrelated
to the survey content or their personal characteristics. For instance:
• A respondent might be momentarily distracted by an external noise and accidentally skip a question.
• Technical glitches in the survey platform could randomly fail to record some responses.
• A participant might unintentionally turn two pages at once, missing a set of questions.
In these cases, the missing data would be considered MCAR because the likelihood of a response being missing is not related to the question itself, the respondent's characteristics, or any other
variables in the study. This randomness ensures that the remaining data still provides an unbiased, albeit smaller, representation of the population under study.
While MCAR is often considered the "best-case scenario" for missing data, it's important to note that it's relatively rare in real-world datasets. Researchers and data scientists must carefully
examine their data and the data collection process to determine if the MCAR assumption truly holds before proceeding with analyses or imputation methods based on this assumption.
2. Missing at Random (MAR):
In this scenario, known as Missing at Random (MAR), the missing data exhibits a systematic relationship with the observed data, but crucially, not with the missing data itself. This means that the
probability of data being missing can be explained by other observed variables in the dataset, but is not directly related to the unobserved values.
To better understand MAR, let's break it down further:
• Systematic relationship: The pattern of missingness is not completely random, but follows a discernible pattern based on other observed variables.
• Observed data dependency: The likelihood of a value being missing depends on other variables that we can observe and measure in the dataset.
• Independence from unobserved values: Importantly, the probability of missingness is not related to the actual value that would have been observed, had it not been missing.
Let's consider an expanded illustration to clarify this concept:
Imagine a comprehensive health survey where participants are asked about their age, exercise habits, and overall health satisfaction. In this scenario:
• Younger participants (ages 18-30) might be less likely to respond to questions about their exercise habits, regardless of how much they actually exercise.
• This lower response rate among younger participants is observable and can be accounted for in the analysis.
• Crucially, their tendency to not respond is not directly related to their actual exercise habits (which would be the missing data), but rather to their age group (which is observed).
In this MAR scenario, we can use the observed data (age) to make informed decisions about handling the missing data (exercise habits). This characteristic of MAR allows for more sophisticated
imputation methods that can leverage the relationships between variables to estimate missing values more accurately.
Understanding that data is MAR is vital for choosing appropriate missing data handling techniques. Unlike Missing Completely at Random (MCAR), where simple techniques like listwise deletion might
suffice, MAR often requires more advanced methods such as multiple imputation or maximum likelihood estimation to avoid bias in analyses.
3. Missing Not at Random (MNAR)
This category represents the most complex type of missing data, where the missingness is directly related to the unobserved values themselves. In MNAR situations, the very reason for the data being
missing is intrinsically linked to the information that would have been collected. This creates a significant challenge for data analysis and imputation methods, as the missing data mechanism cannot
be ignored without potentially introducing bias.
To better understand MNAR, let's break it down further:
• Direct relationship: The probability of a value being missing depends on the value itself, which is unobserved.
• Systematic bias: The missingness creates a systematic bias in the dataset that cannot be fully accounted for using only the observed data.
• Complexity in analysis: MNAR scenarios often require specialized statistical techniques to handle properly, as simple imputation methods may lead to incorrect conclusions.
A prime example of MNAR is when patients with severe health conditions are less inclined to disclose their health status. This leads to systematic gaps in health-related data that are directly
correlated with the severity of their conditions. Let's explore this example in more depth:
• Self-selection bias: Patients with more severe conditions might avoid participating in health surveys or medical studies due to physical limitations or psychological factors.
• Privacy concerns: Those with serious health issues might be more reluctant to share their medical information, fearing stigma or discrimination.
• Incomplete medical records: Patients with complex health conditions might have incomplete medical records if they frequently switch healthcare providers or avoid certain types of care.
The implications of MNAR data in this health-related scenario are significant:
• Underestimation of disease prevalence: If those with severe conditions are systematically missing from the data, the true prevalence of the disease might be underestimated.
• Biased treatment efficacy assessments: In clinical trials, if patients with severe side effects are more likely to drop out, the remaining data might overestimate the treatment's effectiveness.
• Skewed health policy decisions: Policymakers relying on this data might allocate resources based on an incomplete picture of public health needs.
Handling MNAR data requires careful consideration and often involves advanced statistical methods such as selection models or pattern-mixture models. These approaches attempt to model the missing
data mechanism explicitly, allowing for more accurate inferences from incomplete datasets. However, they often rely on untestable assumptions about the nature of the missingness, highlighting the
complexity and challenges associated with MNAR scenarios in data analysis.
Understanding these distinct types of missing data is paramount, as each category necessitates a unique approach in data handling and analysis. The choice of method for addressing missing
data—whether it involves imputation, deletion, or more advanced techniques—should be carefully tailored to the specific type of missingness encountered in the dataset.
This nuanced understanding ensures that the subsequent data analysis and modeling efforts are built on a foundation that accurately reflects the underlying data structure and minimizes potential
biases introduced by missing information.
3.1.2 Detecting and Visualizing Missing Data
The first step in handling missing data is detecting where the missing values are within your dataset. This crucial initial phase sets the foundation for all subsequent data preprocessing and
analysis tasks. Pandas, a powerful data manipulation library in Python, provides an efficient and user-friendly way to check for missing values in a dataset.
To begin this process, you typically load your data into a Pandas DataFrame, which is a two-dimensional labeled data structure. Once your data is in this format, Pandas offers several built-in
functions to identify missing values:
• The isnull() or isna() methods: These functions return a boolean mask of the same shape as your DataFrame, where True indicates a missing value and False indicates a non-missing value.
• The notnull() method: This is the inverse of isnull(), returning True for non-missing values.
• The info() method: This provides a concise summary of your DataFrame, including the number of non-null values in each column.
By combining these functions with other Pandas operations, you can gain a comprehensive understanding of the missing data in your dataset. For example, you can use df.isnull().sum() to count the
number of missing values in each column, or df.isnull().any() to check if any column contains missing values.
Understanding the pattern and extent of missing data is crucial as it informs your strategy for handling these gaps. It helps you decide whether to remove rows or columns with missing data, impute
the missing values, or employ more advanced techniques like multiple imputation or machine learning models designed to handle missing data.
Example: Detecting Missing Data with Pandas
import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer, KNNImputer
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
# Create a sample DataFrame with missing data
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve', 'Frank'],
'Age': [25, None, 35, 40, None, 50],
'Salary': [50000, 60000, None, 80000, 55000, None],
'Department': ['HR', 'IT', 'Finance', 'IT', None, 'HR']
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
# Check for missing data
print("Missing Data in Each Column:")
# Calculate percentage of missing data
print("Percentage of Missing Data in Each Column:")
print(df.isnull().sum() / len(df) * 100)
# Visualize missing data with a heatmap
plt.figure(figsize=(10, 6))
sns.heatmap(df.isnull(), cbar=False, cmap='viridis', yticklabels=False)
plt.title("Missing Data Heatmap")
# Handling missing data
# 1. Removing rows with missing data
df_dropna = df.dropna()
print("DataFrame after dropping rows with missing data:")
# 2. Simple imputation methods
# Mean imputation for numerical columns
df_mean_imputed = df.copy()
df_mean_imputed['Age'].fillna(df_mean_imputed['Age'].mean(), inplace=True)
df_mean_imputed['Salary'].fillna(df_mean_imputed['Salary'].mean(), inplace=True)
# Mode imputation for categorical column
df_mean_imputed['Department'].fillna(df_mean_imputed['Department'].mode()[0], inplace=True)
print("DataFrame after mean/mode imputation:")
# 3. KNN Imputation
imputer_knn = KNNImputer(n_neighbors=2)
df_knn_imputed = pd.DataFrame(imputer_knn.fit_transform(df.drop('Name', axis=1)),
df_knn_imputed.insert(0, 'Name', df['Name']) # Add back the 'Name' column
print("DataFrame after KNN imputation:")
# 4. Multiple Imputation by Chained Equations (MICE)
imputer_mice = IterativeImputer(random_state=0)
df_mice_imputed = pd.DataFrame(imputer_mice.fit_transform(df.drop('Name', axis=1)),
df_mice_imputed.insert(0, 'Name', df['Name']) # Add back the 'Name' column
print("DataFrame after MICE imputation:")
This code example provides a comprehensive demonstration of detecting, visualizing, and handling missing data in Python using pandas, numpy, seaborn, matplotlib, and scikit-learn.
Let's break down the code and explain each section:
1. Data Creation and Exploration:
• We start by creating a sample DataFrame with missing values in different columns.
• The original DataFrame is displayed to show the initial state of the data.
• We use df.isnull().sum() to count the number of missing values in each column.
• The percentage of missing data in each column is calculated to give a better perspective on the extent of missing data.
2. Visualization:
• A heatmap is created using seaborn to visualize the pattern of missing data. This provides an intuitive way to identify which columns and rows contain missing values.
3. Handling Missing Data:
The code demonstrates four different approaches to handling missing data:
• a. Removing rows with missing data: Using df.dropna(), we remove all rows that contain any missing values. This method is simple but can lead to significant data loss if many rows contain missing
• b. Simple imputation methods:
□ For numerical columns ('Age' and 'Salary'), we use mean imputation with fillna(df['column'].mean()).
□ For the categorical column ('Department'), we use mode imputation with fillna(df['Department'].mode()[0]).
□ These methods are straightforward but don't consider the relationships between variables.
• c. K-Nearest Neighbors (KNN) Imputation: Using KNNImputer from scikit-learn, we impute missing values based on the values of the k-nearest neighbors. This method can capture some of the
relationships between variables but may not work well with categorical data.
• d. Multiple Imputation by Chained Equations (MICE): Using IterativeImputer from scikit-learn, we perform multiple imputations. This method models each feature with missing values as a function of
other features and uses that estimate for imputation. It's more sophisticated and can handle both numerical and categorical data.
4. Output and Comparison:
• After each imputation method, the resulting DataFrame is printed, allowing for easy comparison between different approaches.
• This enables the user to assess the impact of each method on the data and choose the most appropriate one for their specific use case.
This example showcases multiple imputation techniques, provides a step-by-step breakdown, and offers a comprehensive look at handling missing data in Python. It demonstrates the progression from
simple techniques (like deletion and mean imputation) to more advanced methods (KNN and MICE). This approach allows users to understand and compare different strategies for missing data imputation.
The isnull() function in Pandas detects missing values (represented as NaN), and by using .sum(), you can get the total number of missing values in each column. Additionally, the Seaborn heatmap
provides a quick visual representation of where the missing data is located.
3.1.3 Techniques for Handling Missing Data
After identifying missing values in your dataset, the crucial next step involves determining the most appropriate strategy for addressing these gaps. The approach you choose can significantly impact
your analysis and model performance. There are multiple techniques available for handling missing data, each with its own strengths and limitations.
The selection of the most suitable method depends on various factors, including the volume of missing data, the pattern of missingness (whether it's missing completely at random, missing at random,
or missing not at random), and the relative importance of the features containing missing values. It's essential to carefully consider these aspects to ensure that your chosen method aligns with your
specific data characteristics and analytical goals.
1. Removing Missing Data
If the amount of missing data is small (typically less than 5% of the total dataset) and the missingness pattern is random (MCAR - Missing Completely At Random), you can consider removing rows or
columns with missing values. This method, known as listwise deletion or complete case analysis, is straightforward and easy to implement.
However, this approach should be used cautiously for several reasons:
• Loss of Information: Removing entire rows or columns can lead to a significant loss of potentially valuable information, especially if the missing data is in different rows across multiple
• Reduced Statistical Power: A smaller sample size due to data removal can decrease the statistical power of your analyses, potentially making it harder to detect significant effects.
• Bias Introduction: If the data is not MCAR, removing rows with missing values can introduce bias into your dataset, potentially skewing your results and leading to incorrect conclusions.
• Inefficiency: In cases where multiple variables have missing values, you might end up discarding a large portion of your dataset, which is inefficient and can lead to unstable estimates.
Before opting for this method, it's crucial to thoroughly analyze the pattern and extent of missing data in your dataset. Consider alternative approaches like various imputation techniques if the
proportion of missing data is substantial or if the missingness pattern suggests that the data is not MCAR.
Example: Removing Rows with Missing Data
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
# Create a sample DataFrame with missing values
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve'],
'Age': [25, np.nan, 35, 40, np.nan],
'Salary': [50000, 60000, np.nan, 80000, 55000],
'Department': ['HR', 'IT', 'Finance', 'IT', np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
# Check for missing values
print("Missing values in each column:")
# Remove rows with any missing values
df_clean = df.dropna()
print("DataFrame after removing rows with missing data:")
# Remove rows with missing values in specific columns
df_clean_specific = df.dropna(subset=['Age', 'Salary'])
print("DataFrame after removing rows with missing data in 'Age' and 'Salary':")
# Remove columns with missing values
df_clean_columns = df.dropna(axis=1)
print("DataFrame after removing columns with missing data:")
# Visualize the impact of removing missing data
plt.figure(figsize=(10, 6))
plt.bar(['Original', 'After row removal', 'After column removal'],
[len(df), len(df_clean), len(df_clean_columns)],
color=['blue', 'green', 'red'])
plt.title('Impact of Removing Missing Data')
plt.ylabel('Number of rows')
This code example demonstrates various aspects of handling missing data using the dropna() method in pandas.
Here's a comprehensive breakdown of the code:
1. Data Creation:
□ We start by creating a sample DataFrame with missing values (represented as np.nan) in different columns.
□ This simulates a real-world scenario where data might be incomplete.
2. Displaying Original Data:
□ The original DataFrame is printed to show the initial state of the data, including the missing values.
3. Checking for Missing Values:
□ We use df.isnull().sum() to count the number of missing values in each column.
□ This step is crucial for understanding the extent of missing data before deciding on a removal strategy.
4. Removing Rows with Any Missing Values:
□ df.dropna() is used without any parameters to remove all rows that contain any missing values.
□ This is the most stringent approach and can lead to significant data loss if many rows have missing values.
5. Removing Rows with Missing Values in Specific Columns:
□ df.dropna(subset=['Age', 'Salary']) removes rows only if there are missing values in the 'Age' or 'Salary' columns.
□ This approach is more targeted and preserves more data compared to removing all rows with any missing values.
6. Removing Columns with Missing Values:
□ df.dropna(axis=1) removes any column that contains missing values.
□ This approach is useful when certain features are deemed unreliable due to missing data.
7. Visualizing the Impact:
□ A bar chart is created to visually compare the number of rows in the original DataFrame versus the DataFrames after row and column removal.
□ This visualization helps in understanding the trade-off between data completeness and data loss.
This comprehensive example illustrates different strategies for handling missing data through removal, allowing for a comparison of their impacts on the dataset. It's important to choose the
appropriate method based on the specific requirements of your analysis and the nature of your data.
In this example, the dropna() function removes any rows that contain missing values. You can also specify whether to drop rows or columns depending on your use case.
2. Imputing Missing Data
If you have a significant amount of missing data, removing rows may not be a viable option as it could lead to substantial loss of information. In such cases, imputation becomes a crucial technique.
Imputation involves filling in the missing values with estimated data, allowing you to preserve the overall structure and size of your dataset.
There are several common imputation methods, each with its own strengths and use cases:
a. Mean Imputation
Mean imputation is a widely used method for handling missing numeric data. This technique involves replacing missing values in a column with the arithmetic mean (average) of all non-missing values in
that same column. For instance, if a dataset has missing age values, the average age of all individuals with recorded ages would be calculated and used to fill in the gaps.
The popularity of mean imputation stems from its simplicity and ease of implementation. It requires minimal computational resources and can be quickly applied to large datasets. This makes it an
attractive option for data scientists and analysts working with time constraints or limited processing power.
However, while mean imputation is straightforward, it comes with several important caveats:
1. Distribution Distortion: By replacing missing values with the mean, this method can alter the overall distribution of the data. It artificially increases the frequency of the mean value,
potentially creating a spike in the distribution around this point. This can lead to a reduction in the data's variance and standard deviation, which may impact statistical analyses that rely on
these measures.
2. Relationship Alteration: Mean imputation doesn't account for relationships between variables. In reality, missing values might be correlated with other features in the dataset. By using the
overall mean, these potential relationships are ignored, which could lead to biased results in subsequent analyses.
3. Uncertainty Misrepresentation: This method doesn't capture the uncertainty associated with the missing data. It treats imputed values with the same confidence as observed values, which may not be
appropriate, especially if the proportion of missing data is substantial.
4. Impact on Statistical Tests: The artificially reduced variability can lead to narrower confidence intervals and potentially inflated t-statistics, which might result in false positives in
hypothesis testing.
5. Bias in Multivariate Analyses: In analyses involving multiple variables, such as regression or clustering, mean imputation can introduce bias by weakening the relationships between variables.
Given these limitations, while mean imputation remains a useful tool in certain scenarios, it's crucial for data scientists to carefully consider its appropriateness for their specific dataset and
analysis goals. In many cases, more sophisticated imputation methods that preserve the data's statistical properties and relationships might be preferable, especially for complex analyses or when
dealing with a significant amount of missing data.
Example: Imputing Missing Data with the Mean
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve'],
'Age': [25, np.nan, 35, 40, np.nan],
'Salary': [50000, 60000, np.nan, 80000, 55000],
'Department': ['HR', 'IT', 'Finance', 'IT', np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
print("\nMissing values in each column:")
# Impute missing values in the 'Age' and 'Salary' columns with the mean
df['Age'] = df['Age'].fillna(df['Age'].mean())
df['Salary'] = df['Salary'].fillna(df['Salary'].mean())
print("\nDataFrame After Mean Imputation:")
# Using SimpleImputer for comparison
imputer = SimpleImputer(strategy='mean')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After SimpleImputer Mean Imputation:")
# Visualize the impact of imputation
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))
ax1.bar(df['Name'], df['Age'], color='blue', alpha=0.7)
ax1.set_title('Age Distribution After Imputation')
ax1.tick_params(axis='x', rotation=45)
ax2.bar(df['Name'], df['Salary'], color='green', alpha=0.7)
ax2.set_title('Salary Distribution After Imputation')
ax2.tick_params(axis='x', rotation=45)
# Calculate and print statistics
print("\nStatistics After Imputation:")
print(df[['Age', 'Salary']].describe())
This code example provides a more comprehensive approach to mean imputation and includes visualization and statistical analysis.
Here's a breakdown of the code:
• Data Creation and Inspection:
□ We create a sample DataFrame with missing values in different columns.
□ The original DataFrame is displayed along with a count of missing values in each column.
• Mean Imputation:
□ We use the fillna() method with df['column'].mean() to impute missing values in the 'Age' and 'Salary' columns.
□ The DataFrame after imputation is displayed to show the changes.
• SimpleImputer Comparison:
□ We use sklearn's SimpleImputer with 'mean' strategy to perform imputation.
□ This demonstrates an alternative method for mean imputation, which can be useful for larger datasets or when working with scikit-learn pipelines.
• Visualization:
□ Two bar plots are created to visualize the Age and Salary distributions after imputation.
□ This helps in understanding the impact of imputation on the data distribution.
• Statistical Analysis:
□ We calculate and display descriptive statistics for the 'Age' and 'Salary' columns after imputation.
□ This provides insights into how imputation has affected the central tendencies and spread of the data.
This code example not only demonstrates how to perform mean imputation but also shows how to assess its impact through visualization and statistical analysis. It's important to note that while mean
imputation is simple and often effective, it can reduce the variance in your data and may not be suitable for all situations, especially when data is not missing at random.
b. Median Imputation
Median imputation is a robust alternative to mean imputation for handling missing data. This method uses the median value of the non-missing data to fill in gaps. The median is the middle value when
a dataset is ordered from least to greatest, effectively separating the higher half from the lower half of a data sample.
Median imputation is particularly valuable when dealing with skewed distributions or datasets containing outliers. In these scenarios, the median proves to be more resilient and representative than
the mean. This is because outliers can significantly pull the mean towards extreme values, whereas the median remains stable.
For instance, consider a dataset of salaries where most employees earn between $40,000 and $60,000, but there are a few executives with salaries over $1,000,000. The mean salary would be heavily
influenced by these high earners, potentially leading to overestimation when imputing missing values. The median, however, would provide a more accurate representation of the typical salary.
Furthermore, median imputation helps maintain the overall shape of the data distribution better than mean imputation in cases of skewed data. This is crucial for preserving important characteristics
of the dataset, which can be essential for subsequent analyses or modeling tasks.
It's worth noting that while median imputation is often superior to mean imputation for skewed data, it still has limitations. Like mean imputation, it doesn't account for relationships between
variables and may not be suitable for datasets where missing values are not randomly distributed. In such cases, more advanced imputation techniques might be necessary.
Example: Median Imputation
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values and outliers
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve', 'Frank', 'Grace', 'Henry', 'Ivy', 'Jack'],
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Salary': [50000, 60000, np.nan, 80000, 55000, 75000, np.nan, 70000, 1000000, np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
print("\nMissing values in each column:")
# Perform median imputation
df_median_imputed = df.copy()
df_median_imputed['Age'] = df_median_imputed['Age'].fillna(df_median_imputed['Age'].median())
df_median_imputed['Salary'] = df_median_imputed['Salary'].fillna(df_median_imputed['Salary'].median())
print("\nDataFrame After Median Imputation:")
# Using SimpleImputer for comparison
imputer = SimpleImputer(strategy='median')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After SimpleImputer Median Imputation:")
# Visualize the impact of imputation
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))
ax1.boxplot([df['Salary'].dropna(), df_median_imputed['Salary']], labels=['Original', 'Imputed'])
ax1.set_title('Salary Distribution: Original vs Imputed')
ax2.scatter(df['Age'], df['Salary'], label='Original', alpha=0.7)
ax2.scatter(df_median_imputed['Age'], df_median_imputed['Salary'], label='Imputed', alpha=0.7)
ax2.set_title('Age vs Salary: Original and Imputed Data')
# Calculate and print statistics
print("\nStatistics After Imputation:")
print(df_median_imputed[['Age', 'Salary']].describe())
This comprehensive example demonstrates median imputation and includes visualization and statistical analysis. Here's a breakdown of the code:
1. Data Creation and Inspection:
□ We create a sample DataFrame with missing values in the 'Age' and 'Salary' columns, including an outlier in the 'Salary' column.
□ The original DataFrame is displayed along with a count of missing values in each column.
2. Median Imputation:
□ We use the fillna() method with df['column'].median() to impute missing values in the 'Age' and 'Salary' columns.
□ The DataFrame after imputation is displayed to show the changes.
3. SimpleImputer Comparison:
□ We use sklearn's SimpleImputer with 'median' strategy to perform imputation.
□ This demonstrates an alternative method for median imputation, which can be useful for larger datasets or when working with scikit-learn pipelines.
4. Visualization:
□ A box plot is created to compare the original and imputed salary distributions, highlighting the impact of median imputation on the outlier.
□ A scatter plot shows the relationship between Age and Salary, comparing original and imputed data.
5. Statistical Analysis:
□ We calculate and display descriptive statistics for the 'Age' and 'Salary' columns after imputation.
□ This provides insights into how imputation has affected the central tendencies and spread of the data.
This example illustrates how median imputation handles outliers better than mean imputation. The salary outlier of 1,000,000 doesn't significantly affect the imputed values, as it would with mean
imputation. The visualization helps to understand the impact of imputation on the data distribution and relationships between variables.
Median imputation is particularly useful when dealing with skewed data or datasets with outliers, as it provides a more robust measure of central tendency compared to the mean. However, like other
simple imputation methods, it doesn't account for relationships between variables and may not be suitable for all types of missing data mechanisms.
c. Mode Imputation
Mode imputation is a technique used to handle missing data by replacing missing values with the most frequently occurring value (mode) in the column. This method is particularly useful for
categorical data where numerical concepts like mean or median are not applicable.
Here's a more detailed explanation:
Application in Categorical Data: Mode imputation is primarily used for categorical variables, such as 'color', 'gender', or 'product type'. For instance, if in a 'favorite color' column, most
responses are 'blue', missing values would be filled with 'blue'.
Effectiveness for Nominal Variables: Mode imputation can be quite effective for nominal categorical variables, where categories have no inherent order. Examples include variables like 'blood type' or
'country of origin'. In these cases, using the most frequent category as a replacement is often a reasonable assumption.
Limitations with Ordinal Data: However, mode imputation may not be suitable for ordinal data, where the order of categories matters. For example, in a variable like 'education level' (high school,
bachelor's, master's, PhD), simply using the most frequent category could disrupt the inherent order and potentially introduce bias in subsequent analyses.
Preserving Data Distribution: One advantage of mode imputation is that it preserves the original distribution of the data more closely than methods like mean imputation, especially for categorical
variables with a clear majority category.
Potential Drawbacks: It's important to note that mode imputation can oversimplify the data, especially if there's no clear mode or if the variable has multiple modes. It also doesn't account for
relationships between variables, which could lead to loss of important information or introduction of bias.
Alternative Approaches: For more complex scenarios, especially with ordinal data or when preserving relationships between variables is crucial, more sophisticated methods like multiple imputation or
machine learning-based imputation techniques might be more appropriate.
Example: Mode Imputation
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve', 'Frank', 'Grace', 'Henry', 'Ivy', 'Jack'],
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Category': ['A', 'B', np.nan, 'A', 'C', 'B', np.nan, 'A', 'C', np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
print("\nMissing values in each column:")
# Perform mode imputation
df_mode_imputed = df.copy()
df_mode_imputed['Category'] = df_mode_imputed['Category'].fillna(df_mode_imputed['Category'].mode()[0])
print("\nDataFrame After Mode Imputation:")
# Using SimpleImputer for comparison
imputer = SimpleImputer(strategy='most_frequent')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After SimpleImputer Mode Imputation:")
# Visualize the impact of imputation
fig, ax = plt.subplots(figsize=(10, 6))
category_counts = df_mode_imputed['Category'].value_counts()
ax.bar(category_counts.index, category_counts.values)
ax.set_title('Category Distribution After Mode Imputation')
# Calculate and print statistics
print("\nCategory Distribution After Imputation:")
This comprehensive example demonstrates mode imputation and includes visualization and statistical analysis. Here's a breakdown of the code:
1. Data Creation and Inspection:
□ We create a sample DataFrame with missing values in the 'Age' and 'Category' columns.
□ The original DataFrame is displayed along with a count of missing values in each column.
2. Mode Imputation:
□ We use the fillna() method with df['column'].mode()[0] to impute missing values in the 'Category' column.
□ The DataFrame after imputation is displayed to show the changes.
3. SimpleImputer Comparison:
□ We use sklearn's SimpleImputer with 'most_frequent' strategy to perform imputation.
□ This demonstrates an alternative method for mode imputation, which can be useful for larger datasets or when working with scikit-learn pipelines.
4. Visualization:
□ A bar plot is created to show the distribution of categories after imputation.
□ This helps in understanding the impact of mode imputation on the categorical data distribution.
5. Statistical Analysis:
□ We calculate and display the proportion of each category after imputation.
□ This provides insights into how imputation has affected the distribution of the categorical variable.
This example illustrates how mode imputation works for categorical data. It fills in missing values with the most frequent category, which in this case is 'A'. The visualization helps to understand
the impact of imputation on the distribution of categories.
Mode imputation is particularly useful for nominal categorical data where concepts like mean or median don't apply. However, it's important to note that this method can potentially amplify the bias
towards the most common category, especially if there's a significant imbalance in the original data.
While mode imputation is simple and often effective for categorical data, it doesn't account for relationships between variables and may not be suitable for ordinal categorical data or when the
missingness mechanism is not completely at random. In such cases, more advanced techniques like multiple imputation or machine learning-based approaches might be more appropriate.
While these methods are commonly used due to their simplicity and ease of implementation, it's crucial to consider their limitations. They don't account for relationships between variables and can
introduce bias if the data is not missing completely at random. More advanced techniques like multiple imputation or machine learning-based imputation methods may be necessary for complex datasets or
when the missingness mechanism is not random.
d. Advanced Imputation Methods
In some cases, simple mean or median imputation might not be sufficient for handling missing data effectively. More sophisticated methods such as K-nearest neighbors (KNN) imputation or regression
imputation can be applied to achieve better results. These advanced techniques go beyond simple statistical measures and take into account the complex relationships between variables to predict
missing values more accurately.
K-nearest neighbors (KNN) imputation works by identifying the K most similar data points (neighbors) to the one with missing values, based on other available features. It then uses the values from
these neighbors to estimate the missing value, often by taking their average. This method is particularly useful when there are strong correlations between features in the dataset.
Regression imputation, on the other hand, involves building a regression model using the available data to predict the missing values. This method can capture more complex relationships between
variables and can be especially effective when there are clear patterns or trends in the data that can be leveraged for prediction.
These advanced imputation methods offer several advantages over simple imputation:
• They preserve the relationships between variables, which can be crucial for maintaining the integrity of the dataset.
• They can handle both numerical and categorical data more effectively.
• They often provide more accurate estimates of missing values, leading to better model performance downstream.
Fortunately, popular machine learning libraries like Scikit-learn provide easy-to-use implementations of these advanced imputation techniques. This accessibility allows data scientists and analysts
to quickly experiment with and apply these sophisticated methods in their preprocessing pipelines, potentially improving the overall quality of their data and the performance of their models.
Example: K-Nearest Neighbors (KNN) Imputation
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import KNNImputer
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
# Create a sample DataFrame with missing values
data = {
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Salary': [50000, 60000, np.nan, 75000, 65000, np.nan, 70000, 80000, np.nan, 90000],
'Experience': [2, 3, 5, np.nan, 4, 8, np.nan, 7, 6, 10]
df = pd.DataFrame(data)
print("Original DataFrame:")
print("\nMissing values in each column:")
# Initialize the KNN Imputer
imputer = KNNImputer(n_neighbors=2)
# Fit and transform the data
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After KNN Imputation:")
# Visualize the imputation results
fig, axes = plt.subplots(1, 3, figsize=(15, 5))
for i, column in enumerate(df.columns):
axes[i].scatter(df.index, df[column], label='Original', alpha=0.5)
axes[i].scatter(df_imputed.index, df_imputed[column], label='Imputed', alpha=0.5)
axes[i].set_title(f'{column} - Before and After Imputation')
# Evaluate the impact of imputation on a simple model
X = df_imputed[['Age', 'Experience']]
y = df_imputed['Salary']
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
print(f"\nMean Squared Error after imputation: {mse:.2f}")
This code example demonstrates a more comprehensive approach to KNN imputation and its evaluation.
Here's a breakdown of the code:
• Data Preparation:
□ We create a sample DataFrame with missing values in 'Age', 'Salary', and 'Experience' columns.
□ The original DataFrame and the count of missing values are displayed.
• KNN Imputation:
□ We initialize a KNNImputer with 2 neighbors.
□ The imputer is applied to the DataFrame, filling in missing values based on the K-nearest neighbors.
• Visualization:
□ We create scatter plots for each column, comparing the original data with missing values to the imputed data.
□ This visual representation helps in understanding how KNN imputation affects the data distribution.
• Model Evaluation:
□ We use the imputed data to train a simple Linear Regression model.
□ The model predicts 'Salary' based on 'Age' and 'Experience'.
□ We calculate the Mean Squared Error to evaluate the model's performance after imputation.
This comprehensive example showcases not only how to perform KNN imputation but also how to visualize its effects and evaluate its impact on a subsequent machine learning task. It provides a more
holistic view of the imputation process and its consequences in a data science workflow.
In this example, the KNN Imputer fills in missing values by finding the nearest neighbors in the dataset and using their values to estimate the missing ones. This method is often more accurate than
simple mean imputation when the data has strong relationships between features.
3.1.4 Evaluating the Impact of Missing Data
Handling missing data is not merely a matter of filling in gaps—it's crucial to thoroughly evaluate how missing data impacts your model's performance. This evaluation process is multifaceted and
requires careful consideration. When certain features in your dataset contain an excessive number of missing values, they may prove to be unreliable predictors. In such cases, it might be more
beneficial to remove these features entirely rather than attempting to impute the missing values.
Furthermore, it's essential to rigorously test imputed data to ensure its validity and reliability. This testing process should focus on two key aspects: first, verifying that the imputation method
hasn't inadvertently distorted the underlying relationships within the data, and second, confirming that it hasn't introduced any bias into the model. Both of these factors can significantly affect
the accuracy and generalizability of your machine learning model.
To gain a comprehensive understanding of how your chosen method for handling missing data affects your model, it's advisable to assess the model's performance both before and after implementing your
missing data strategy. This comparative analysis can be conducted using robust validation techniques such as cross-validation or holdout validation.
These methods provide valuable insights into how your model's predictive capabilities have been influenced by your approach to missing data, allowing you to make informed decisions about the most
effective preprocessing strategies for your specific dataset and modeling objectives.
Example: Model Evaluation Before and After Handling Missing Data
import pandas as pd
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values
data = {
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Salary': [50000, 60000, np.nan, 75000, 65000, np.nan, 70000, 80000, np.nan, 90000],
'Experience': [2, 3, 5, np.nan, 4, 8, np.nan, 7, 6, 10]
df = pd.DataFrame(data)
print("Original DataFrame:")
print("\nMissing values in each column:")
# Function to evaluate model performance
def evaluate_model(X, y, model_name):
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)
print(f"\n{model_name} - Mean Squared Error: {mse:.2f}")
print(f"{model_name} - R-squared Score: {r2:.2f}")
# Evaluate model with missing data
X_missing = df[['Age', 'Experience']]
y_missing = df['Salary']
evaluate_model(X_missing.dropna(), y_missing.dropna(), "Model with Missing Data")
# Simple Imputation
imputer = SimpleImputer(strategy='mean')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After Mean Imputation:")
# Evaluate model after imputation
X_imputed = df_imputed[['Age', 'Experience']]
y_imputed = df_imputed['Salary']
evaluate_model(X_imputed, y_imputed, "Model After Imputation")
# Advanced: Multiple models comparison
from sklearn.ensemble import RandomForestRegressor
from sklearn.svm import SVR
models = {
'Linear Regression': LinearRegression(),
'Random Forest': RandomForestRegressor(n_estimators=100, random_state=42),
'Support Vector Regression': SVR()
for name, model in models.items():
X_train, X_test, y_train, y_test = train_test_split(X_imputed, y_imputed, test_size=0.2, random_state=42)
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)
print(f"\n{name} - Mean Squared Error: {mse:.2f}")
print(f"{name} - R-squared Score: {r2:.2f}")
This code example provides a comprehensive approach to evaluating the impact of missing data and imputation on model performance.
Here's a detailed breakdown of the code:
1. Data Preparation:
□ We create a sample DataFrame with missing values in 'Age', 'Salary', and 'Experience' columns.
□ The original DataFrame and the count of missing values are displayed.
2. Model Evaluation Function:
□ A function evaluate_model() is defined to assess model performance using Mean Squared Error (MSE) and R-squared score.
□ This function will be used to compare model performance before and after imputation.
3. Evaluation with Missing Data:
□ We first evaluate the model's performance using only the complete cases (rows without missing values).
□ This serves as a baseline for comparison.
4. Simple Imputation:
□ Mean imputation is performed using sklearn's SimpleImputer.
□ The imputed DataFrame is displayed to show the changes.
5. Evaluation After Imputation:
□ We evaluate the model's performance again using the imputed data.
□ This allows us to compare the impact of imputation on model performance.
6. Advanced Model Comparison:
□ We introduce two additional models: Random Forest and Support Vector Regression.
□ All three models (including Linear Regression) are trained and evaluated on the imputed data.
□ This comparison helps in understanding if the choice of model affects the impact of imputation.
This example demonstrates how to handle missing data, perform imputation, and evaluate its impact on different models. It provides insights into:
• The effect of missing data on model performance
• The impact of mean imputation on data distribution and model accuracy
• How different models perform on the imputed data
By comparing the results, data scientists can make informed decisions about the most appropriate imputation method and model selection for their specific dataset and problem.
Handling missing data is one of the most critical steps in data preprocessing. Whether you choose to remove or impute missing values, understanding the nature of the missing data and selecting the
appropriate method is essential for building a reliable machine learning model. In this section, we covered several strategies, ranging from simple mean imputation to more advanced techniques like
KNN imputation, and demonstrated how to evaluate their impact on your model's performance.
3.1 Data Cleaning and Handling Missing Data
Data preprocessing stands as the cornerstone of any robust machine learning pipeline, serving as the critical initial step that can make or break the success of your model. In the complex landscape
of real-world data science, practitioners often encounter raw data that is far from ideal - it may be riddled with inconsistencies, plagued by missing values, or lack the structure necessary for
immediate analysis.
Attempting to feed such unrefined data directly into a machine learning algorithm is a recipe for suboptimal performance and unreliable results. This is precisely where the twin pillars of data
preprocessing and feature engineering come into play, offering a systematic approach to data refinement.
These essential processes encompass a wide range of techniques aimed at cleaning, transforming, and optimizing your dataset. By meticulously preparing your data, you create a solid foundation that
enables machine learning algorithms to uncover meaningful patterns and generate accurate predictions. The goal is to present your model with a dataset that is not only clean and complete but also
structured in a way that highlights the most relevant features and relationships within the data.
Throughout this chapter, we will delve deep into the crucial steps that comprise effective data preprocessing. We'll explore the intricacies of data cleaning, a fundamental process that involves
identifying and rectifying errors, inconsistencies, and anomalies in your dataset. We'll tackle the challenge of handling missing data, discussing various strategies to address gaps in your
information without compromising the integrity of your analysis. The chapter will also cover scaling and normalization techniques, essential for ensuring that all features contribute proportionally
to the model's decision-making process.
Furthermore, we'll examine methods for encoding categorical variables, transforming non-numeric data into a format that machine learning algorithms can interpret and utilize effectively. Lastly,
we'll dive into the art and science of feature engineering, where domain knowledge and creativity converge to craft new, informative features that can significantly enhance your model's predictive
By mastering these preprocessing steps, you'll be equipped to lay a rock-solid foundation for your machine learning projects. This meticulous preparation of your data is what separates mediocre
models from those that truly excel, maximizing performance and ensuring that your algorithms can extract the most valuable insights from the information at hand.
We'll kick off our journey into data preprocessing with an in-depth look at data cleaning. This critical process serves as the first line of defense against the myriad issues that can plague raw
datasets. By ensuring that your data is accurate, complete, and primed for analysis, data cleaning sets the stage for all subsequent preprocessing steps and ultimately contributes to the overall
success of your machine learning endeavors.
Data cleaning is a crucial step in the data preprocessing pipeline, involving the systematic identification and rectification of issues within datasets. This process encompasses a wide range of
activities, including:
Detecting corrupt data
This crucial step involves a comprehensive and meticulous examination of the dataset to identify any data points that have been compromised or altered during various stages of the data lifecycle.
This includes, but is not limited to, the collection phase, where errors might occur due to faulty sensors or human input mistakes; the transmission phase, where data corruption can happen due to
network issues or interference; and the storage phase, where data might be corrupted due to hardware failures or software glitches.
The process of detecting corrupt data often involves multiple techniques:
• Statistical analysis: Using statistical methods to identify outliers or values that deviate significantly from expected patterns.
• Data validation rules: Implementing specific rules based on domain knowledge to flag potentially corrupt entries.
• Consistency checks: Comparing data across different fields or time periods to ensure logical consistency.
• Format verification: Ensuring that data adheres to expected formats, such as date structures or numerical ranges.
By pinpointing these corrupted elements through such rigorous methods, data scientists can take appropriate actions such as removing, correcting, or flagging the corrupt data. This process is
fundamental in ensuring the integrity and reliability of the dataset, which is crucial for any subsequent analysis or machine learning model development. Without this step, corrupt data could lead to
skewed results, incorrect conclusions, or poorly performing models, potentially undermining the entire data science project.
Example: Detecting Corrupt Data
import pandas as pd
import numpy as np
# Create a sample DataFrame with potentially corrupt data
data = {
'ID': [1, 2, 3, 4, 5],
'Value': [10, 20, 'error', 40, 50],
'Date': ['2023-01-01', '2023-02-30', '2023-03-15', '2023-04-01', '2023-05-01']
df = pd.DataFrame(data)
# Function to detect corrupt data
def detect_corrupt_data(df):
corrupt_rows = []
# Check for non-numeric values in 'Value' column
numeric_errors = pd.to_numeric(df['Value'], errors='coerce').isna()
# Check for invalid dates
df['Date'] = pd.to_datetime(df['Date'], errors='coerce')
date_errors = df['Date'].isna()
return list(set(corrupt_rows)) # Remove duplicates
# Detect corrupt data
corrupt_indices = detect_corrupt_data(df)
print("Corrupt data found at indices:", corrupt_indices)
print("\nCorrupt rows:")
This code demonstrates how to detect corrupt data in a pandas DataFrame. Here's a breakdown of its functionality:
• It creates a sample DataFrame with potentially corrupt data, including non-numeric values in the 'Value' column and invalid dates in the 'Date' column.
• The detect_corrupt_data() function is defined to identify corrupt rows. It checks for:
• Non-numeric values in the 'Value' column using pd.to_numeric() with errors='coerce'.
• Invalid dates in the 'Date' column using pd.to_datetime() with errors='coerce'.
• The function returns a list of unique indices where corrupt data was found.
• Finally, it prints the indices of corrupt rows and displays the corrupt data.
This code is an example of how to implement data cleaning techniques, specifically for detecting corrupt data, which is a crucial step in the data preprocessing pipeline.
Correcting incomplete data
This process involves a comprehensive and meticulous examination of the dataset to identify and address any instances of incomplete or missing information. The approach to handling such gaps depends
on several factors, including the nature of the data, the extent of incompleteness, and the potential impact on subsequent analyses.
When dealing with missing data, data scientists employ a range of sophisticated techniques:
• Imputation methods: These involve estimating and filling in missing values based on patterns observed in the existing data. Techniques can range from simple mean or median imputation to more
advanced methods like regression imputation or multiple imputation.
• Machine learning-based approaches: Algorithms such as K-Nearest Neighbors (KNN) or Random Forest can be used to predict missing values based on the relationships between variables in the dataset.
• Time series-specific methods: For temporal data, techniques like interpolation or forecasting models may be employed to estimate missing values based on trends and seasonality.
However, in cases where the gaps in the data are too significant or the missing information is deemed crucial, careful consideration must be given to the removal of incomplete records. This decision
is not taken lightly, as it involves balancing the need for data quality with the potential loss of valuable information.
Factors influencing the decision to remove incomplete records include:
• The proportion of missing data: If a large percentage of a record or variable is missing, removal might be more appropriate than imputation.
• The mechanism of missingness: Understanding whether data is missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR) can inform the decision-making process.
• The importance of the missing information: If the missing data is critical to the analysis or model, removal might be necessary to maintain the integrity of the results.
Ultimately, the goal is to strike a balance between preserving as much valuable information as possible while ensuring the overall quality and reliability of the dataset for subsequent analysis and
modeling tasks.
Example: Correcting Incomplete Data
import pandas as pd
import numpy as np
from sklearn.impute import SimpleImputer
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
# Create a sample DataFrame with incomplete data
data = {
'Age': [25, np.nan, 30, np.nan, 40],
'Income': [50000, 60000, np.nan, 75000, 80000],
'Education': ['Bachelor', 'Master', np.nan, 'PhD', 'Bachelor']
df = pd.DataFrame(data)
print("Original DataFrame:")
# Method 1: Simple Imputation (Mean for numerical, Most frequent for categorical)
imputer_mean = SimpleImputer(strategy='mean')
imputer_most_frequent = SimpleImputer(strategy='most_frequent')
df_imputed_simple = df.copy()
df_imputed_simple[['Age', 'Income']] = imputer_mean.fit_transform(df[['Age', 'Income']])
df_imputed_simple[['Education']] = imputer_most_frequent.fit_transform(df[['Education']])
print("\nDataFrame after Simple Imputation:")
# Method 2: Iterative Imputation (uses the IterativeImputer, aka MICE)
imputer_iterative = IterativeImputer(random_state=0)
df_imputed_iterative = df.copy()
df_imputed_iterative.iloc[:, :] = imputer_iterative.fit_transform(df)
print("\nDataFrame after Iterative Imputation:")
# Method 3: Custom logic (e.g., filling Age based on median of similar Education levels)
df_custom = df.copy()
df_custom['Age'] = df_custom.groupby('Education')['Age'].transform(lambda x: x.fillna(x.median()))
df_custom['Income'].fillna(df_custom['Income'].mean(), inplace=True)
df_custom['Education'].fillna(df_custom['Education'].mode()[0], inplace=True)
print("\nDataFrame after Custom Imputation:")
This example demonstrates three different methods for correcting incomplete data:
• 1. Simple Imputation: Uses Scikit-learn's SimpleImputer to fill missing values with the mean for numerical columns (Age and Income) and the most frequent value for categorical columns
• 2. Iterative Imputation: Employs Scikit-learn's IterativeImputer (also known as MICE - Multivariate Imputation by Chained Equations) to estimate missing values based on the relationships between
• 3. Custom Logic: Implements a tailored approach where Age is imputed based on the median age of similar education levels, Income is filled with the mean, and Education uses the mode (most
frequent value).
Breakdown of the code:
1. We start by importing necessary libraries and creating a sample DataFrame with missing values.
2. For Simple Imputation, we use SimpleImputer with different strategies for numerical and categorical data.
3. Iterative Imputation uses the IterativeImputer, which estimates each feature from all the others iteratively.
4. The custom logic demonstrates how domain knowledge can be applied to impute data more accurately, such as using education level to estimate age.
This example showcases the flexibility and power of different imputation techniques. The choice of method depends on the nature of your data and the specific requirements of your analysis. Simple
imputation is quick and easy but may not capture complex relationships in the data. Iterative imputation can be more accurate but is computationally intensive. Custom logic allows for the
incorporation of domain expertise but requires more manual effort and understanding of the data.
Addressing inaccurate data
This crucial step in the data cleaning process involves a comprehensive and meticulous approach to identifying and rectifying errors that may have infiltrated the dataset during various stages of
data collection and management. These errors can arise from multiple sources:
• Data Entry Errors: Human mistakes during manual data input, such as typos, transposed digits, or incorrect categorizations.
• Measurement Errors: Inaccuracies stemming from faulty equipment, miscalibrated instruments, or inconsistent measurement techniques.
• Recording Errors: Issues that occur during the data recording process, including system glitches, software bugs, or data transmission failures.
To address these challenges, data scientists employ a range of sophisticated validation techniques:
• Statistical Outlier Detection: Utilizing statistical methods to identify data points that deviate significantly from the expected patterns or distributions.
• Domain-Specific Rule Validation: Implementing checks based on expert knowledge of the field to flag logically inconsistent or impossible values.
• Cross-Referencing: Comparing data against reliable external sources or internal databases to verify accuracy and consistency.
• Machine Learning-Based Anomaly Detection: Leveraging advanced algorithms to detect subtle patterns of inaccuracy that might escape traditional validation methods.
By rigorously applying these validation techniques and diligently cross-referencing with trusted sources, data scientists can substantially enhance the accuracy and reliability of their datasets.
This meticulous process not only improves the quality of the data but also bolsters the credibility of subsequent analyses and machine learning models built upon this foundation. Ultimately,
addressing inaccurate data is a critical investment in ensuring the integrity and trustworthiness of data-driven insights and decision-making processes.
Example: Addressing Inaccurate Data
import pandas as pd
import numpy as np
from scipy import stats
# Create a sample DataFrame with potentially inaccurate data
data = {
'ID': range(1, 11),
'Age': [25, 30, 35, 40, 45, 50, 55, 60, 65, 1000],
'Income': [50000, 60000, 70000, 80000, 90000, 100000, 110000, 120000, 130000, 10000000],
'Height': [170, 175, 180, 185, 190, 195, 200, 205, 210, 150]
df = pd.DataFrame(data)
print("Original DataFrame:")
def detect_and_correct_outliers(df, column, method='zscore', threshold=3):
if method == 'zscore':
z_scores = np.abs(stats.zscore(df[column]))
outliers = df[z_scores > threshold]
df.loc[z_scores > threshold, column] = df[column].median()
elif method == 'iqr':
Q1 = df[column].quantile(0.25)
Q3 = df[column].quantile(0.75)
IQR = Q3 - Q1
lower_bound = Q1 - 1.5 * IQR
upper_bound = Q3 + 1.5 * IQR
outliers = df[(df[column] < lower_bound) | (df[column] > upper_bound)]
df.loc[(df[column] < lower_bound) | (df[column] > upper_bound), column] = df[column].median()
return outliers
# Detect and correct outliers in 'Age' column using Z-score method
age_outliers = detect_and_correct_outliers(df, 'Age', method='zscore')
# Detect and correct outliers in 'Income' column using IQR method
income_outliers = detect_and_correct_outliers(df, 'Income', method='iqr')
# Custom logic for 'Height' column
height_outliers = df[(df['Height'] < 150) | (df['Height'] > 220)]
df.loc[(df['Height'] < 150) | (df['Height'] > 220), 'Height'] = df['Height'].median()
print("\nOutliers detected:")
print("Age outliers:", age_outliers['Age'].tolist())
print("Income outliers:", income_outliers['Income'].tolist())
print("Height outliers:", height_outliers['Height'].tolist())
print("\nCorrected DataFrame:")
This example demonstrates a comprehensive approach to addressing inaccurate data, specifically focusing on outlier detection and correction.
Here's a breakdown of the code and its functionality:
1. Data Creation: We start by creating a sample DataFrame with potentially inaccurate data, including extreme values in the 'Age', 'Income', and 'Height' columns.
2. Outlier Detection and Correction Function: The detect_and_correct_outliers() function is defined to handle outliers using two common methods:
□ Z-score method: Identifies outliers based on the number of standard deviations from the mean.
□ IQR (Interquartile Range) method: Detects outliers using the concept of quartiles.
3. Applying Outlier Detection:
□ For the 'Age' column, we use the Z-score method with a threshold of 3 standard deviations.
□ For the 'Income' column, we apply the IQR method to account for potential skewness in income distribution.
□ For the 'Height' column, we implement a custom logic to flag values below 150 cm or above 220 cm as outliers.
4. Outlier Correction: Once outliers are detected, they are replaced with the median value of the respective column. This approach helps maintain data integrity while reducing the impact of extreme
5. Reporting: The code prints out the detected outliers for each column and displays the corrected DataFrame.
This example showcases different strategies for addressing inaccurate data:
• Statistical methods (Z-score and IQR) for automated outlier detection
• Custom logic for domain-specific outlier identification
• Median imputation for correcting outliers, which is more robust to extreme values than mean imputation
By employing these techniques, data scientists can significantly improve the quality of their datasets, leading to more reliable analyses and machine learning models. It's important to note that
while this example uses median imputation for simplicity, in practice, the choice of correction method should be carefully considered based on the specific characteristics of the data and the
requirements of the analysis.
Removing irrelevant data
This final step in the data cleaning process, known as data relevance assessment, involves a meticulous evaluation of each data point to determine its significance and applicability to the specific
analysis or problem at hand. This crucial phase requires data scientists to critically examine the dataset through multiple lenses:
1. Contextual Relevance: Assessing whether each variable or feature directly contributes to answering the research questions or achieving the project goals.
2. Temporal Relevance: Determining if the data is current enough to be meaningful for the analysis, especially in rapidly changing domains.
3. Granularity: Evaluating if the level of detail in the data is appropriate for the intended analysis, neither too broad nor too specific.
4. Redundancy: Identifying and removing duplicate or highly correlated variables that don't provide additional informational value.
5. Signal-to-Noise Ratio: Distinguishing between data that carries meaningful information (signal) and data that introduces unnecessary complexity or variability (noise).
By meticulously eliminating extraneous or irrelevant information through this process, data scientists can significantly enhance the quality and focus of their dataset. This refinement yields several
critical benefits:
• Improved Model Performance: A streamlined dataset with only relevant features often leads to more accurate and robust machine learning models.
• Enhanced Computational Efficiency: Reducing the dataset's dimensionality can dramatically decrease processing time and resource requirements, especially crucial when dealing with large-scale data.
• Clearer Insights: By removing noise and focusing on pertinent data, analysts can derive more meaningful and actionable insights from their analyses.
• Reduced Overfitting Risk: Eliminating irrelevant features helps prevent models from learning spurious patterns, thus improving generalization to new, unseen data.
• Simplified Interpretability: A more focused dataset often results in models and analyses that are easier to interpret and explain to stakeholders.
In essence, this careful curation of relevant data serves as a critical foundation, significantly enhancing the efficiency, effectiveness, and reliability of subsequent analyses and machine learning
models. It ensures that the final insights and decisions are based on the most pertinent and high-quality information available.
Example: Removing Irrelevant Data
import pandas as pd
import numpy as np
from sklearn.feature_selection import VarianceThreshold
from sklearn.feature_selection import mutual_info_regression
# Create a sample DataFrame with potentially irrelevant features
data = {
'ID': range(1, 101),
'Age': np.random.randint(18, 80, 100),
'Income': np.random.randint(20000, 150000, 100),
'Education': np.random.choice(['High School', 'Bachelor', 'Master', 'PhD'], 100),
'Constant_Feature': [5] * 100,
'Random_Feature': np.random.random(100),
'Target': np.random.randint(0, 2, 100)
df = pd.DataFrame(data)
print("Original DataFrame shape:", df.shape)
# Step 1: Remove constant features
constant_filter = VarianceThreshold(threshold=0)
constant_columns = df.columns[~constant_filter.get_support()]
df = df.drop(columns=constant_columns)
print("After removing constant features:", df.shape)
# Step 2: Remove features with low variance
variance_filter = VarianceThreshold(threshold=0.1)
low_variance_columns = df.select_dtypes(include=[np.number]).columns[~variance_filter.get_support()]
df = df.drop(columns=low_variance_columns)
print("After removing low variance features:", df.shape)
# Step 3: Feature importance based on mutual information
numerical_features = df.select_dtypes(include=[np.number]).columns.drop('Target')
mi_scores = mutual_info_regression(df[numerical_features], df['Target'])
mi_scores = pd.Series(mi_scores, index=numerical_features)
important_features = mi_scores[mi_scores > 0.01].index
df = df[important_features.tolist() + ['Education', 'Target']]
print("After removing less important features:", df.shape)
print("\nFinal DataFrame columns:", df.columns.tolist())
This code example demonstrates various techniques for removing irrelevant data from a dataset.
Let's break down the code and explain each step:
1. Data Creation: We start by creating a sample DataFrame with potentially irrelevant features, including a constant feature and a random feature.
2. Removing Constant Features:
□ We use VarianceThreshold with a threshold of 0 to identify and remove features that have the same value in all samples.
□ This step eliminates features that provide no discriminative information for the model.
3. Removing Low Variance Features:
□ We apply VarianceThreshold again, this time with a threshold of 0.1, to remove features with very low variance.
□ Features with low variance often contain little information and may not contribute significantly to the model's predictive power.
4. Feature Importance based on Mutual Information:
□ We use mutual_info_regression to calculate the mutual information between each feature and the target variable.
□ Features with mutual information scores below a certain threshold (0.01 in this example) are considered less important and are removed.
□ This step helps in identifying features that have a strong relationship with the target variable.
5. Retaining Categorical Features: We manually include the 'Education' column to demonstrate how you might retain important categorical features that weren't part of the numerical analysis.
This example showcases a multi-faceted approach to removing irrelevant data:
• It addresses constant features that provide no discriminative information.
• It removes features with very low variance, which often contribute little to model performance.
• It uses a statistical measure (mutual information) to identify features most relevant to the target variable.
By applying these techniques, we significantly reduce the dimensionality of the dataset, focusing on the most relevant features. This can lead to improved model performance, reduced overfitting, and
increased computational efficiency. However, it's crucial to validate the impact of feature removal on your specific problem and adjust thresholds as necessary.
The importance of data cleaning cannot be overstated, as it directly impacts the quality and reliability of machine learning models. Clean, high-quality data is essential for accurate predictions and
meaningful insights.
Missing values are a common challenge in real-world datasets, often arising from various sources such as equipment malfunctions, human error, or intentional non-responses. Handling these missing
values appropriately is critical, as they can significantly affect model performance and lead to biased or incorrect conclusions if not addressed properly.
The approach to dealing with missing data is not one-size-fits-all and depends on several factors:
1. The nature and characteristics of your dataset: The specific type of data you're working with (such as numerical, categorical, or time series) and its underlying distribution patterns play a
crucial role in determining the most appropriate technique for handling missing data. For instance, certain imputation methods may be more suitable for continuous numerical data, while others
might be better suited for categorical variables or time-dependent information.
2. The quantity and distribution pattern of missing data: The extent of missing information and the underlying mechanism causing the data gaps significantly influence the choice of handling
strategy. It's essential to distinguish between data that is missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR), as each scenario may require a
different approach to maintain the integrity and representativeness of your dataset.
3. The selected machine learning algorithm and its inherent properties: Different machine learning models exhibit varying degrees of sensitivity to missing data, which can substantially impact their
performance and the reliability of their predictions. Some algorithms, like decision trees, can handle missing values intrinsically, while others, such as support vector machines, may require
more extensive preprocessing to address data gaps effectively. Understanding these model-specific characteristics is crucial in selecting an appropriate missing data handling technique that
aligns with your chosen algorithm.
By understanding these concepts and techniques, data scientists can make informed decisions about how to preprocess their data effectively, ensuring the development of robust and accurate machine
learning models.
3.1.1 Types of Missing Data
Before delving deeper into the intricacies of handling missing data, it is crucial to grasp the three primary categories of missing data, each with its own unique characteristics and implications for
data analysis:
1. Missing Completely at Random (MCAR)
This type of missing data represents a scenario where the absence of information follows no discernible pattern or relationship with any variables in the dataset, whether observed or unobserved. MCAR
is characterized by an equal probability of data being missing across all cases, effectively creating an unbiased subset of the complete dataset.
The key features of MCAR include:
• Randomness: The missingness is entirely random and not influenced by any factors within or outside the dataset.
• Unbiased representation: The remaining data can be considered a random sample of the full dataset, maintaining its statistical properties.
• Statistical implications: Analyses conducted on the complete cases (after removing missing data) remain unbiased, although there may be a loss in statistical power due to reduced sample size.
To illustrate MCAR, consider a comprehensive survey scenario:
Imagine a large-scale health survey where participants are required to fill out a lengthy questionnaire. Some respondents might inadvertently skip certain questions due to factors entirely unrelated
to the survey content or their personal characteristics. For instance:
• A respondent might be momentarily distracted by an external noise and accidentally skip a question.
• Technical glitches in the survey platform could randomly fail to record some responses.
• A participant might unintentionally turn two pages at once, missing a set of questions.
In these cases, the missing data would be considered MCAR because the likelihood of a response being missing is not related to the question itself, the respondent's characteristics, or any other
variables in the study. This randomness ensures that the remaining data still provides an unbiased, albeit smaller, representation of the population under study.
While MCAR is often considered the "best-case scenario" for missing data, it's important to note that it's relatively rare in real-world datasets. Researchers and data scientists must carefully
examine their data and the data collection process to determine if the MCAR assumption truly holds before proceeding with analyses or imputation methods based on this assumption.
2. Missing at Random (MAR):
In this scenario, known as Missing at Random (MAR), the missing data exhibits a systematic relationship with the observed data, but crucially, not with the missing data itself. This means that the
probability of data being missing can be explained by other observed variables in the dataset, but is not directly related to the unobserved values.
To better understand MAR, let's break it down further:
• Systematic relationship: The pattern of missingness is not completely random, but follows a discernible pattern based on other observed variables.
• Observed data dependency: The likelihood of a value being missing depends on other variables that we can observe and measure in the dataset.
• Independence from unobserved values: Importantly, the probability of missingness is not related to the actual value that would have been observed, had it not been missing.
Let's consider an expanded illustration to clarify this concept:
Imagine a comprehensive health survey where participants are asked about their age, exercise habits, and overall health satisfaction. In this scenario:
• Younger participants (ages 18-30) might be less likely to respond to questions about their exercise habits, regardless of how much they actually exercise.
• This lower response rate among younger participants is observable and can be accounted for in the analysis.
• Crucially, their tendency to not respond is not directly related to their actual exercise habits (which would be the missing data), but rather to their age group (which is observed).
In this MAR scenario, we can use the observed data (age) to make informed decisions about handling the missing data (exercise habits). This characteristic of MAR allows for more sophisticated
imputation methods that can leverage the relationships between variables to estimate missing values more accurately.
Understanding that data is MAR is vital for choosing appropriate missing data handling techniques. Unlike Missing Completely at Random (MCAR), where simple techniques like listwise deletion might
suffice, MAR often requires more advanced methods such as multiple imputation or maximum likelihood estimation to avoid bias in analyses.
3. Missing Not at Random (MNAR)
This category represents the most complex type of missing data, where the missingness is directly related to the unobserved values themselves. In MNAR situations, the very reason for the data being
missing is intrinsically linked to the information that would have been collected. This creates a significant challenge for data analysis and imputation methods, as the missing data mechanism cannot
be ignored without potentially introducing bias.
To better understand MNAR, let's break it down further:
• Direct relationship: The probability of a value being missing depends on the value itself, which is unobserved.
• Systematic bias: The missingness creates a systematic bias in the dataset that cannot be fully accounted for using only the observed data.
• Complexity in analysis: MNAR scenarios often require specialized statistical techniques to handle properly, as simple imputation methods may lead to incorrect conclusions.
A prime example of MNAR is when patients with severe health conditions are less inclined to disclose their health status. This leads to systematic gaps in health-related data that are directly
correlated with the severity of their conditions. Let's explore this example in more depth:
• Self-selection bias: Patients with more severe conditions might avoid participating in health surveys or medical studies due to physical limitations or psychological factors.
• Privacy concerns: Those with serious health issues might be more reluctant to share their medical information, fearing stigma or discrimination.
• Incomplete medical records: Patients with complex health conditions might have incomplete medical records if they frequently switch healthcare providers or avoid certain types of care.
The implications of MNAR data in this health-related scenario are significant:
• Underestimation of disease prevalence: If those with severe conditions are systematically missing from the data, the true prevalence of the disease might be underestimated.
• Biased treatment efficacy assessments: In clinical trials, if patients with severe side effects are more likely to drop out, the remaining data might overestimate the treatment's effectiveness.
• Skewed health policy decisions: Policymakers relying on this data might allocate resources based on an incomplete picture of public health needs.
Handling MNAR data requires careful consideration and often involves advanced statistical methods such as selection models or pattern-mixture models. These approaches attempt to model the missing
data mechanism explicitly, allowing for more accurate inferences from incomplete datasets. However, they often rely on untestable assumptions about the nature of the missingness, highlighting the
complexity and challenges associated with MNAR scenarios in data analysis.
Understanding these distinct types of missing data is paramount, as each category necessitates a unique approach in data handling and analysis. The choice of method for addressing missing
data—whether it involves imputation, deletion, or more advanced techniques—should be carefully tailored to the specific type of missingness encountered in the dataset.
This nuanced understanding ensures that the subsequent data analysis and modeling efforts are built on a foundation that accurately reflects the underlying data structure and minimizes potential
biases introduced by missing information.
3.1.2 Detecting and Visualizing Missing Data
The first step in handling missing data is detecting where the missing values are within your dataset. This crucial initial phase sets the foundation for all subsequent data preprocessing and
analysis tasks. Pandas, a powerful data manipulation library in Python, provides an efficient and user-friendly way to check for missing values in a dataset.
To begin this process, you typically load your data into a Pandas DataFrame, which is a two-dimensional labeled data structure. Once your data is in this format, Pandas offers several built-in
functions to identify missing values:
• The isnull() or isna() methods: These functions return a boolean mask of the same shape as your DataFrame, where True indicates a missing value and False indicates a non-missing value.
• The notnull() method: This is the inverse of isnull(), returning True for non-missing values.
• The info() method: This provides a concise summary of your DataFrame, including the number of non-null values in each column.
By combining these functions with other Pandas operations, you can gain a comprehensive understanding of the missing data in your dataset. For example, you can use df.isnull().sum() to count the
number of missing values in each column, or df.isnull().any() to check if any column contains missing values.
Understanding the pattern and extent of missing data is crucial as it informs your strategy for handling these gaps. It helps you decide whether to remove rows or columns with missing data, impute
the missing values, or employ more advanced techniques like multiple imputation or machine learning models designed to handle missing data.
Example: Detecting Missing Data with Pandas
import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer, KNNImputer
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
# Create a sample DataFrame with missing data
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve', 'Frank'],
'Age': [25, None, 35, 40, None, 50],
'Salary': [50000, 60000, None, 80000, 55000, None],
'Department': ['HR', 'IT', 'Finance', 'IT', None, 'HR']
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
# Check for missing data
print("Missing Data in Each Column:")
# Calculate percentage of missing data
print("Percentage of Missing Data in Each Column:")
print(df.isnull().sum() / len(df) * 100)
# Visualize missing data with a heatmap
plt.figure(figsize=(10, 6))
sns.heatmap(df.isnull(), cbar=False, cmap='viridis', yticklabels=False)
plt.title("Missing Data Heatmap")
# Handling missing data
# 1. Removing rows with missing data
df_dropna = df.dropna()
print("DataFrame after dropping rows with missing data:")
# 2. Simple imputation methods
# Mean imputation for numerical columns
df_mean_imputed = df.copy()
df_mean_imputed['Age'].fillna(df_mean_imputed['Age'].mean(), inplace=True)
df_mean_imputed['Salary'].fillna(df_mean_imputed['Salary'].mean(), inplace=True)
# Mode imputation for categorical column
df_mean_imputed['Department'].fillna(df_mean_imputed['Department'].mode()[0], inplace=True)
print("DataFrame after mean/mode imputation:")
# 3. KNN Imputation
imputer_knn = KNNImputer(n_neighbors=2)
df_knn_imputed = pd.DataFrame(imputer_knn.fit_transform(df.drop('Name', axis=1)),
df_knn_imputed.insert(0, 'Name', df['Name']) # Add back the 'Name' column
print("DataFrame after KNN imputation:")
# 4. Multiple Imputation by Chained Equations (MICE)
imputer_mice = IterativeImputer(random_state=0)
df_mice_imputed = pd.DataFrame(imputer_mice.fit_transform(df.drop('Name', axis=1)),
df_mice_imputed.insert(0, 'Name', df['Name']) # Add back the 'Name' column
print("DataFrame after MICE imputation:")
This code example provides a comprehensive demonstration of detecting, visualizing, and handling missing data in Python using pandas, numpy, seaborn, matplotlib, and scikit-learn.
Let's break down the code and explain each section:
1. Data Creation and Exploration:
• We start by creating a sample DataFrame with missing values in different columns.
• The original DataFrame is displayed to show the initial state of the data.
• We use df.isnull().sum() to count the number of missing values in each column.
• The percentage of missing data in each column is calculated to give a better perspective on the extent of missing data.
2. Visualization:
• A heatmap is created using seaborn to visualize the pattern of missing data. This provides an intuitive way to identify which columns and rows contain missing values.
3. Handling Missing Data:
The code demonstrates four different approaches to handling missing data:
• a. Removing rows with missing data: Using df.dropna(), we remove all rows that contain any missing values. This method is simple but can lead to significant data loss if many rows contain missing
• b. Simple imputation methods:
□ For numerical columns ('Age' and 'Salary'), we use mean imputation with fillna(df['column'].mean()).
□ For the categorical column ('Department'), we use mode imputation with fillna(df['Department'].mode()[0]).
□ These methods are straightforward but don't consider the relationships between variables.
• c. K-Nearest Neighbors (KNN) Imputation: Using KNNImputer from scikit-learn, we impute missing values based on the values of the k-nearest neighbors. This method can capture some of the
relationships between variables but may not work well with categorical data.
• d. Multiple Imputation by Chained Equations (MICE): Using IterativeImputer from scikit-learn, we perform multiple imputations. This method models each feature with missing values as a function of
other features and uses that estimate for imputation. It's more sophisticated and can handle both numerical and categorical data.
4. Output and Comparison:
• After each imputation method, the resulting DataFrame is printed, allowing for easy comparison between different approaches.
• This enables the user to assess the impact of each method on the data and choose the most appropriate one for their specific use case.
This example showcases multiple imputation techniques, provides a step-by-step breakdown, and offers a comprehensive look at handling missing data in Python. It demonstrates the progression from
simple techniques (like deletion and mean imputation) to more advanced methods (KNN and MICE). This approach allows users to understand and compare different strategies for missing data imputation.
The isnull() function in Pandas detects missing values (represented as NaN), and by using .sum(), you can get the total number of missing values in each column. Additionally, the Seaborn heatmap
provides a quick visual representation of where the missing data is located.
3.1.3 Techniques for Handling Missing Data
After identifying missing values in your dataset, the crucial next step involves determining the most appropriate strategy for addressing these gaps. The approach you choose can significantly impact
your analysis and model performance. There are multiple techniques available for handling missing data, each with its own strengths and limitations.
The selection of the most suitable method depends on various factors, including the volume of missing data, the pattern of missingness (whether it's missing completely at random, missing at random,
or missing not at random), and the relative importance of the features containing missing values. It's essential to carefully consider these aspects to ensure that your chosen method aligns with your
specific data characteristics and analytical goals.
1. Removing Missing Data
If the amount of missing data is small (typically less than 5% of the total dataset) and the missingness pattern is random (MCAR - Missing Completely At Random), you can consider removing rows or
columns with missing values. This method, known as listwise deletion or complete case analysis, is straightforward and easy to implement.
However, this approach should be used cautiously for several reasons:
• Loss of Information: Removing entire rows or columns can lead to a significant loss of potentially valuable information, especially if the missing data is in different rows across multiple
• Reduced Statistical Power: A smaller sample size due to data removal can decrease the statistical power of your analyses, potentially making it harder to detect significant effects.
• Bias Introduction: If the data is not MCAR, removing rows with missing values can introduce bias into your dataset, potentially skewing your results and leading to incorrect conclusions.
• Inefficiency: In cases where multiple variables have missing values, you might end up discarding a large portion of your dataset, which is inefficient and can lead to unstable estimates.
Before opting for this method, it's crucial to thoroughly analyze the pattern and extent of missing data in your dataset. Consider alternative approaches like various imputation techniques if the
proportion of missing data is substantial or if the missingness pattern suggests that the data is not MCAR.
Example: Removing Rows with Missing Data
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
# Create a sample DataFrame with missing values
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve'],
'Age': [25, np.nan, 35, 40, np.nan],
'Salary': [50000, 60000, np.nan, 80000, 55000],
'Department': ['HR', 'IT', 'Finance', 'IT', np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
# Check for missing values
print("Missing values in each column:")
# Remove rows with any missing values
df_clean = df.dropna()
print("DataFrame after removing rows with missing data:")
# Remove rows with missing values in specific columns
df_clean_specific = df.dropna(subset=['Age', 'Salary'])
print("DataFrame after removing rows with missing data in 'Age' and 'Salary':")
# Remove columns with missing values
df_clean_columns = df.dropna(axis=1)
print("DataFrame after removing columns with missing data:")
# Visualize the impact of removing missing data
plt.figure(figsize=(10, 6))
plt.bar(['Original', 'After row removal', 'After column removal'],
[len(df), len(df_clean), len(df_clean_columns)],
color=['blue', 'green', 'red'])
plt.title('Impact of Removing Missing Data')
plt.ylabel('Number of rows')
This code example demonstrates various aspects of handling missing data using the dropna() method in pandas.
Here's a comprehensive breakdown of the code:
1. Data Creation:
□ We start by creating a sample DataFrame with missing values (represented as np.nan) in different columns.
□ This simulates a real-world scenario where data might be incomplete.
2. Displaying Original Data:
□ The original DataFrame is printed to show the initial state of the data, including the missing values.
3. Checking for Missing Values:
□ We use df.isnull().sum() to count the number of missing values in each column.
□ This step is crucial for understanding the extent of missing data before deciding on a removal strategy.
4. Removing Rows with Any Missing Values:
□ df.dropna() is used without any parameters to remove all rows that contain any missing values.
□ This is the most stringent approach and can lead to significant data loss if many rows have missing values.
5. Removing Rows with Missing Values in Specific Columns:
□ df.dropna(subset=['Age', 'Salary']) removes rows only if there are missing values in the 'Age' or 'Salary' columns.
□ This approach is more targeted and preserves more data compared to removing all rows with any missing values.
6. Removing Columns with Missing Values:
□ df.dropna(axis=1) removes any column that contains missing values.
□ This approach is useful when certain features are deemed unreliable due to missing data.
7. Visualizing the Impact:
□ A bar chart is created to visually compare the number of rows in the original DataFrame versus the DataFrames after row and column removal.
□ This visualization helps in understanding the trade-off between data completeness and data loss.
This comprehensive example illustrates different strategies for handling missing data through removal, allowing for a comparison of their impacts on the dataset. It's important to choose the
appropriate method based on the specific requirements of your analysis and the nature of your data.
In this example, the dropna() function removes any rows that contain missing values. You can also specify whether to drop rows or columns depending on your use case.
2. Imputing Missing Data
If you have a significant amount of missing data, removing rows may not be a viable option as it could lead to substantial loss of information. In such cases, imputation becomes a crucial technique.
Imputation involves filling in the missing values with estimated data, allowing you to preserve the overall structure and size of your dataset.
There are several common imputation methods, each with its own strengths and use cases:
a. Mean Imputation
Mean imputation is a widely used method for handling missing numeric data. This technique involves replacing missing values in a column with the arithmetic mean (average) of all non-missing values in
that same column. For instance, if a dataset has missing age values, the average age of all individuals with recorded ages would be calculated and used to fill in the gaps.
The popularity of mean imputation stems from its simplicity and ease of implementation. It requires minimal computational resources and can be quickly applied to large datasets. This makes it an
attractive option for data scientists and analysts working with time constraints or limited processing power.
However, while mean imputation is straightforward, it comes with several important caveats:
1. Distribution Distortion: By replacing missing values with the mean, this method can alter the overall distribution of the data. It artificially increases the frequency of the mean value,
potentially creating a spike in the distribution around this point. This can lead to a reduction in the data's variance and standard deviation, which may impact statistical analyses that rely on
these measures.
2. Relationship Alteration: Mean imputation doesn't account for relationships between variables. In reality, missing values might be correlated with other features in the dataset. By using the
overall mean, these potential relationships are ignored, which could lead to biased results in subsequent analyses.
3. Uncertainty Misrepresentation: This method doesn't capture the uncertainty associated with the missing data. It treats imputed values with the same confidence as observed values, which may not be
appropriate, especially if the proportion of missing data is substantial.
4. Impact on Statistical Tests: The artificially reduced variability can lead to narrower confidence intervals and potentially inflated t-statistics, which might result in false positives in
hypothesis testing.
5. Bias in Multivariate Analyses: In analyses involving multiple variables, such as regression or clustering, mean imputation can introduce bias by weakening the relationships between variables.
Given these limitations, while mean imputation remains a useful tool in certain scenarios, it's crucial for data scientists to carefully consider its appropriateness for their specific dataset and
analysis goals. In many cases, more sophisticated imputation methods that preserve the data's statistical properties and relationships might be preferable, especially for complex analyses or when
dealing with a significant amount of missing data.
Example: Imputing Missing Data with the Mean
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve'],
'Age': [25, np.nan, 35, 40, np.nan],
'Salary': [50000, 60000, np.nan, 80000, 55000],
'Department': ['HR', 'IT', 'Finance', 'IT', np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
print("\nMissing values in each column:")
# Impute missing values in the 'Age' and 'Salary' columns with the mean
df['Age'] = df['Age'].fillna(df['Age'].mean())
df['Salary'] = df['Salary'].fillna(df['Salary'].mean())
print("\nDataFrame After Mean Imputation:")
# Using SimpleImputer for comparison
imputer = SimpleImputer(strategy='mean')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After SimpleImputer Mean Imputation:")
# Visualize the impact of imputation
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))
ax1.bar(df['Name'], df['Age'], color='blue', alpha=0.7)
ax1.set_title('Age Distribution After Imputation')
ax1.tick_params(axis='x', rotation=45)
ax2.bar(df['Name'], df['Salary'], color='green', alpha=0.7)
ax2.set_title('Salary Distribution After Imputation')
ax2.tick_params(axis='x', rotation=45)
# Calculate and print statistics
print("\nStatistics After Imputation:")
print(df[['Age', 'Salary']].describe())
This code example provides a more comprehensive approach to mean imputation and includes visualization and statistical analysis.
Here's a breakdown of the code:
• Data Creation and Inspection:
□ We create a sample DataFrame with missing values in different columns.
□ The original DataFrame is displayed along with a count of missing values in each column.
• Mean Imputation:
□ We use the fillna() method with df['column'].mean() to impute missing values in the 'Age' and 'Salary' columns.
□ The DataFrame after imputation is displayed to show the changes.
• SimpleImputer Comparison:
□ We use sklearn's SimpleImputer with 'mean' strategy to perform imputation.
□ This demonstrates an alternative method for mean imputation, which can be useful for larger datasets or when working with scikit-learn pipelines.
• Visualization:
□ Two bar plots are created to visualize the Age and Salary distributions after imputation.
□ This helps in understanding the impact of imputation on the data distribution.
• Statistical Analysis:
□ We calculate and display descriptive statistics for the 'Age' and 'Salary' columns after imputation.
□ This provides insights into how imputation has affected the central tendencies and spread of the data.
This code example not only demonstrates how to perform mean imputation but also shows how to assess its impact through visualization and statistical analysis. It's important to note that while mean
imputation is simple and often effective, it can reduce the variance in your data and may not be suitable for all situations, especially when data is not missing at random.
b. Median Imputation
Median imputation is a robust alternative to mean imputation for handling missing data. This method uses the median value of the non-missing data to fill in gaps. The median is the middle value when
a dataset is ordered from least to greatest, effectively separating the higher half from the lower half of a data sample.
Median imputation is particularly valuable when dealing with skewed distributions or datasets containing outliers. In these scenarios, the median proves to be more resilient and representative than
the mean. This is because outliers can significantly pull the mean towards extreme values, whereas the median remains stable.
For instance, consider a dataset of salaries where most employees earn between $40,000 and $60,000, but there are a few executives with salaries over $1,000,000. The mean salary would be heavily
influenced by these high earners, potentially leading to overestimation when imputing missing values. The median, however, would provide a more accurate representation of the typical salary.
Furthermore, median imputation helps maintain the overall shape of the data distribution better than mean imputation in cases of skewed data. This is crucial for preserving important characteristics
of the dataset, which can be essential for subsequent analyses or modeling tasks.
It's worth noting that while median imputation is often superior to mean imputation for skewed data, it still has limitations. Like mean imputation, it doesn't account for relationships between
variables and may not be suitable for datasets where missing values are not randomly distributed. In such cases, more advanced imputation techniques might be necessary.
Example: Median Imputation
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values and outliers
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve', 'Frank', 'Grace', 'Henry', 'Ivy', 'Jack'],
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Salary': [50000, 60000, np.nan, 80000, 55000, 75000, np.nan, 70000, 1000000, np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
print("\nMissing values in each column:")
# Perform median imputation
df_median_imputed = df.copy()
df_median_imputed['Age'] = df_median_imputed['Age'].fillna(df_median_imputed['Age'].median())
df_median_imputed['Salary'] = df_median_imputed['Salary'].fillna(df_median_imputed['Salary'].median())
print("\nDataFrame After Median Imputation:")
# Using SimpleImputer for comparison
imputer = SimpleImputer(strategy='median')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After SimpleImputer Median Imputation:")
# Visualize the impact of imputation
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))
ax1.boxplot([df['Salary'].dropna(), df_median_imputed['Salary']], labels=['Original', 'Imputed'])
ax1.set_title('Salary Distribution: Original vs Imputed')
ax2.scatter(df['Age'], df['Salary'], label='Original', alpha=0.7)
ax2.scatter(df_median_imputed['Age'], df_median_imputed['Salary'], label='Imputed', alpha=0.7)
ax2.set_title('Age vs Salary: Original and Imputed Data')
# Calculate and print statistics
print("\nStatistics After Imputation:")
print(df_median_imputed[['Age', 'Salary']].describe())
This comprehensive example demonstrates median imputation and includes visualization and statistical analysis. Here's a breakdown of the code:
1. Data Creation and Inspection:
□ We create a sample DataFrame with missing values in the 'Age' and 'Salary' columns, including an outlier in the 'Salary' column.
□ The original DataFrame is displayed along with a count of missing values in each column.
2. Median Imputation:
□ We use the fillna() method with df['column'].median() to impute missing values in the 'Age' and 'Salary' columns.
□ The DataFrame after imputation is displayed to show the changes.
3. SimpleImputer Comparison:
□ We use sklearn's SimpleImputer with 'median' strategy to perform imputation.
□ This demonstrates an alternative method for median imputation, which can be useful for larger datasets or when working with scikit-learn pipelines.
4. Visualization:
□ A box plot is created to compare the original and imputed salary distributions, highlighting the impact of median imputation on the outlier.
□ A scatter plot shows the relationship between Age and Salary, comparing original and imputed data.
5. Statistical Analysis:
□ We calculate and display descriptive statistics for the 'Age' and 'Salary' columns after imputation.
□ This provides insights into how imputation has affected the central tendencies and spread of the data.
This example illustrates how median imputation handles outliers better than mean imputation. The salary outlier of 1,000,000 doesn't significantly affect the imputed values, as it would with mean
imputation. The visualization helps to understand the impact of imputation on the data distribution and relationships between variables.
Median imputation is particularly useful when dealing with skewed data or datasets with outliers, as it provides a more robust measure of central tendency compared to the mean. However, like other
simple imputation methods, it doesn't account for relationships between variables and may not be suitable for all types of missing data mechanisms.
c. Mode Imputation
Mode imputation is a technique used to handle missing data by replacing missing values with the most frequently occurring value (mode) in the column. This method is particularly useful for
categorical data where numerical concepts like mean or median are not applicable.
Here's a more detailed explanation:
Application in Categorical Data: Mode imputation is primarily used for categorical variables, such as 'color', 'gender', or 'product type'. For instance, if in a 'favorite color' column, most
responses are 'blue', missing values would be filled with 'blue'.
Effectiveness for Nominal Variables: Mode imputation can be quite effective for nominal categorical variables, where categories have no inherent order. Examples include variables like 'blood type' or
'country of origin'. In these cases, using the most frequent category as a replacement is often a reasonable assumption.
Limitations with Ordinal Data: However, mode imputation may not be suitable for ordinal data, where the order of categories matters. For example, in a variable like 'education level' (high school,
bachelor's, master's, PhD), simply using the most frequent category could disrupt the inherent order and potentially introduce bias in subsequent analyses.
Preserving Data Distribution: One advantage of mode imputation is that it preserves the original distribution of the data more closely than methods like mean imputation, especially for categorical
variables with a clear majority category.
Potential Drawbacks: It's important to note that mode imputation can oversimplify the data, especially if there's no clear mode or if the variable has multiple modes. It also doesn't account for
relationships between variables, which could lead to loss of important information or introduction of bias.
Alternative Approaches: For more complex scenarios, especially with ordinal data or when preserving relationships between variables is crucial, more sophisticated methods like multiple imputation or
machine learning-based imputation techniques might be more appropriate.
Example: Mode Imputation
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve', 'Frank', 'Grace', 'Henry', 'Ivy', 'Jack'],
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Category': ['A', 'B', np.nan, 'A', 'C', 'B', np.nan, 'A', 'C', np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
print("\nMissing values in each column:")
# Perform mode imputation
df_mode_imputed = df.copy()
df_mode_imputed['Category'] = df_mode_imputed['Category'].fillna(df_mode_imputed['Category'].mode()[0])
print("\nDataFrame After Mode Imputation:")
# Using SimpleImputer for comparison
imputer = SimpleImputer(strategy='most_frequent')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After SimpleImputer Mode Imputation:")
# Visualize the impact of imputation
fig, ax = plt.subplots(figsize=(10, 6))
category_counts = df_mode_imputed['Category'].value_counts()
ax.bar(category_counts.index, category_counts.values)
ax.set_title('Category Distribution After Mode Imputation')
# Calculate and print statistics
print("\nCategory Distribution After Imputation:")
This comprehensive example demonstrates mode imputation and includes visualization and statistical analysis. Here's a breakdown of the code:
1. Data Creation and Inspection:
□ We create a sample DataFrame with missing values in the 'Age' and 'Category' columns.
□ The original DataFrame is displayed along with a count of missing values in each column.
2. Mode Imputation:
□ We use the fillna() method with df['column'].mode()[0] to impute missing values in the 'Category' column.
□ The DataFrame after imputation is displayed to show the changes.
3. SimpleImputer Comparison:
□ We use sklearn's SimpleImputer with 'most_frequent' strategy to perform imputation.
□ This demonstrates an alternative method for mode imputation, which can be useful for larger datasets or when working with scikit-learn pipelines.
4. Visualization:
□ A bar plot is created to show the distribution of categories after imputation.
□ This helps in understanding the impact of mode imputation on the categorical data distribution.
5. Statistical Analysis:
□ We calculate and display the proportion of each category after imputation.
□ This provides insights into how imputation has affected the distribution of the categorical variable.
This example illustrates how mode imputation works for categorical data. It fills in missing values with the most frequent category, which in this case is 'A'. The visualization helps to understand
the impact of imputation on the distribution of categories.
Mode imputation is particularly useful for nominal categorical data where concepts like mean or median don't apply. However, it's important to note that this method can potentially amplify the bias
towards the most common category, especially if there's a significant imbalance in the original data.
While mode imputation is simple and often effective for categorical data, it doesn't account for relationships between variables and may not be suitable for ordinal categorical data or when the
missingness mechanism is not completely at random. In such cases, more advanced techniques like multiple imputation or machine learning-based approaches might be more appropriate.
While these methods are commonly used due to their simplicity and ease of implementation, it's crucial to consider their limitations. They don't account for relationships between variables and can
introduce bias if the data is not missing completely at random. More advanced techniques like multiple imputation or machine learning-based imputation methods may be necessary for complex datasets or
when the missingness mechanism is not random.
d. Advanced Imputation Methods
In some cases, simple mean or median imputation might not be sufficient for handling missing data effectively. More sophisticated methods such as K-nearest neighbors (KNN) imputation or regression
imputation can be applied to achieve better results. These advanced techniques go beyond simple statistical measures and take into account the complex relationships between variables to predict
missing values more accurately.
K-nearest neighbors (KNN) imputation works by identifying the K most similar data points (neighbors) to the one with missing values, based on other available features. It then uses the values from
these neighbors to estimate the missing value, often by taking their average. This method is particularly useful when there are strong correlations between features in the dataset.
Regression imputation, on the other hand, involves building a regression model using the available data to predict the missing values. This method can capture more complex relationships between
variables and can be especially effective when there are clear patterns or trends in the data that can be leveraged for prediction.
These advanced imputation methods offer several advantages over simple imputation:
• They preserve the relationships between variables, which can be crucial for maintaining the integrity of the dataset.
• They can handle both numerical and categorical data more effectively.
• They often provide more accurate estimates of missing values, leading to better model performance downstream.
Fortunately, popular machine learning libraries like Scikit-learn provide easy-to-use implementations of these advanced imputation techniques. This accessibility allows data scientists and analysts
to quickly experiment with and apply these sophisticated methods in their preprocessing pipelines, potentially improving the overall quality of their data and the performance of their models.
Example: K-Nearest Neighbors (KNN) Imputation
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import KNNImputer
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
# Create a sample DataFrame with missing values
data = {
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Salary': [50000, 60000, np.nan, 75000, 65000, np.nan, 70000, 80000, np.nan, 90000],
'Experience': [2, 3, 5, np.nan, 4, 8, np.nan, 7, 6, 10]
df = pd.DataFrame(data)
print("Original DataFrame:")
print("\nMissing values in each column:")
# Initialize the KNN Imputer
imputer = KNNImputer(n_neighbors=2)
# Fit and transform the data
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After KNN Imputation:")
# Visualize the imputation results
fig, axes = plt.subplots(1, 3, figsize=(15, 5))
for i, column in enumerate(df.columns):
axes[i].scatter(df.index, df[column], label='Original', alpha=0.5)
axes[i].scatter(df_imputed.index, df_imputed[column], label='Imputed', alpha=0.5)
axes[i].set_title(f'{column} - Before and After Imputation')
# Evaluate the impact of imputation on a simple model
X = df_imputed[['Age', 'Experience']]
y = df_imputed['Salary']
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
print(f"\nMean Squared Error after imputation: {mse:.2f}")
This code example demonstrates a more comprehensive approach to KNN imputation and its evaluation.
Here's a breakdown of the code:
• Data Preparation:
□ We create a sample DataFrame with missing values in 'Age', 'Salary', and 'Experience' columns.
□ The original DataFrame and the count of missing values are displayed.
• KNN Imputation:
□ We initialize a KNNImputer with 2 neighbors.
□ The imputer is applied to the DataFrame, filling in missing values based on the K-nearest neighbors.
• Visualization:
□ We create scatter plots for each column, comparing the original data with missing values to the imputed data.
□ This visual representation helps in understanding how KNN imputation affects the data distribution.
• Model Evaluation:
□ We use the imputed data to train a simple Linear Regression model.
□ The model predicts 'Salary' based on 'Age' and 'Experience'.
□ We calculate the Mean Squared Error to evaluate the model's performance after imputation.
This comprehensive example showcases not only how to perform KNN imputation but also how to visualize its effects and evaluate its impact on a subsequent machine learning task. It provides a more
holistic view of the imputation process and its consequences in a data science workflow.
In this example, the KNN Imputer fills in missing values by finding the nearest neighbors in the dataset and using their values to estimate the missing ones. This method is often more accurate than
simple mean imputation when the data has strong relationships between features.
3.1.4 Evaluating the Impact of Missing Data
Handling missing data is not merely a matter of filling in gaps—it's crucial to thoroughly evaluate how missing data impacts your model's performance. This evaluation process is multifaceted and
requires careful consideration. When certain features in your dataset contain an excessive number of missing values, they may prove to be unreliable predictors. In such cases, it might be more
beneficial to remove these features entirely rather than attempting to impute the missing values.
Furthermore, it's essential to rigorously test imputed data to ensure its validity and reliability. This testing process should focus on two key aspects: first, verifying that the imputation method
hasn't inadvertently distorted the underlying relationships within the data, and second, confirming that it hasn't introduced any bias into the model. Both of these factors can significantly affect
the accuracy and generalizability of your machine learning model.
To gain a comprehensive understanding of how your chosen method for handling missing data affects your model, it's advisable to assess the model's performance both before and after implementing your
missing data strategy. This comparative analysis can be conducted using robust validation techniques such as cross-validation or holdout validation.
These methods provide valuable insights into how your model's predictive capabilities have been influenced by your approach to missing data, allowing you to make informed decisions about the most
effective preprocessing strategies for your specific dataset and modeling objectives.
Example: Model Evaluation Before and After Handling Missing Data
import pandas as pd
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values
data = {
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Salary': [50000, 60000, np.nan, 75000, 65000, np.nan, 70000, 80000, np.nan, 90000],
'Experience': [2, 3, 5, np.nan, 4, 8, np.nan, 7, 6, 10]
df = pd.DataFrame(data)
print("Original DataFrame:")
print("\nMissing values in each column:")
# Function to evaluate model performance
def evaluate_model(X, y, model_name):
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)
print(f"\n{model_name} - Mean Squared Error: {mse:.2f}")
print(f"{model_name} - R-squared Score: {r2:.2f}")
# Evaluate model with missing data
X_missing = df[['Age', 'Experience']]
y_missing = df['Salary']
evaluate_model(X_missing.dropna(), y_missing.dropna(), "Model with Missing Data")
# Simple Imputation
imputer = SimpleImputer(strategy='mean')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After Mean Imputation:")
# Evaluate model after imputation
X_imputed = df_imputed[['Age', 'Experience']]
y_imputed = df_imputed['Salary']
evaluate_model(X_imputed, y_imputed, "Model After Imputation")
# Advanced: Multiple models comparison
from sklearn.ensemble import RandomForestRegressor
from sklearn.svm import SVR
models = {
'Linear Regression': LinearRegression(),
'Random Forest': RandomForestRegressor(n_estimators=100, random_state=42),
'Support Vector Regression': SVR()
for name, model in models.items():
X_train, X_test, y_train, y_test = train_test_split(X_imputed, y_imputed, test_size=0.2, random_state=42)
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)
print(f"\n{name} - Mean Squared Error: {mse:.2f}")
print(f"{name} - R-squared Score: {r2:.2f}")
This code example provides a comprehensive approach to evaluating the impact of missing data and imputation on model performance.
Here's a detailed breakdown of the code:
1. Data Preparation:
□ We create a sample DataFrame with missing values in 'Age', 'Salary', and 'Experience' columns.
□ The original DataFrame and the count of missing values are displayed.
2. Model Evaluation Function:
□ A function evaluate_model() is defined to assess model performance using Mean Squared Error (MSE) and R-squared score.
□ This function will be used to compare model performance before and after imputation.
3. Evaluation with Missing Data:
□ We first evaluate the model's performance using only the complete cases (rows without missing values).
□ This serves as a baseline for comparison.
4. Simple Imputation:
□ Mean imputation is performed using sklearn's SimpleImputer.
□ The imputed DataFrame is displayed to show the changes.
5. Evaluation After Imputation:
□ We evaluate the model's performance again using the imputed data.
□ This allows us to compare the impact of imputation on model performance.
6. Advanced Model Comparison:
□ We introduce two additional models: Random Forest and Support Vector Regression.
□ All three models (including Linear Regression) are trained and evaluated on the imputed data.
□ This comparison helps in understanding if the choice of model affects the impact of imputation.
This example demonstrates how to handle missing data, perform imputation, and evaluate its impact on different models. It provides insights into:
• The effect of missing data on model performance
• The impact of mean imputation on data distribution and model accuracy
• How different models perform on the imputed data
By comparing the results, data scientists can make informed decisions about the most appropriate imputation method and model selection for their specific dataset and problem.
Handling missing data is one of the most critical steps in data preprocessing. Whether you choose to remove or impute missing values, understanding the nature of the missing data and selecting the
appropriate method is essential for building a reliable machine learning model. In this section, we covered several strategies, ranging from simple mean imputation to more advanced techniques like
KNN imputation, and demonstrated how to evaluate their impact on your model's performance.
3.1 Data Cleaning and Handling Missing Data
Data preprocessing stands as the cornerstone of any robust machine learning pipeline, serving as the critical initial step that can make or break the success of your model. In the complex landscape
of real-world data science, practitioners often encounter raw data that is far from ideal - it may be riddled with inconsistencies, plagued by missing values, or lack the structure necessary for
immediate analysis.
Attempting to feed such unrefined data directly into a machine learning algorithm is a recipe for suboptimal performance and unreliable results. This is precisely where the twin pillars of data
preprocessing and feature engineering come into play, offering a systematic approach to data refinement.
These essential processes encompass a wide range of techniques aimed at cleaning, transforming, and optimizing your dataset. By meticulously preparing your data, you create a solid foundation that
enables machine learning algorithms to uncover meaningful patterns and generate accurate predictions. The goal is to present your model with a dataset that is not only clean and complete but also
structured in a way that highlights the most relevant features and relationships within the data.
Throughout this chapter, we will delve deep into the crucial steps that comprise effective data preprocessing. We'll explore the intricacies of data cleaning, a fundamental process that involves
identifying and rectifying errors, inconsistencies, and anomalies in your dataset. We'll tackle the challenge of handling missing data, discussing various strategies to address gaps in your
information without compromising the integrity of your analysis. The chapter will also cover scaling and normalization techniques, essential for ensuring that all features contribute proportionally
to the model's decision-making process.
Furthermore, we'll examine methods for encoding categorical variables, transforming non-numeric data into a format that machine learning algorithms can interpret and utilize effectively. Lastly,
we'll dive into the art and science of feature engineering, where domain knowledge and creativity converge to craft new, informative features that can significantly enhance your model's predictive
By mastering these preprocessing steps, you'll be equipped to lay a rock-solid foundation for your machine learning projects. This meticulous preparation of your data is what separates mediocre
models from those that truly excel, maximizing performance and ensuring that your algorithms can extract the most valuable insights from the information at hand.
We'll kick off our journey into data preprocessing with an in-depth look at data cleaning. This critical process serves as the first line of defense against the myriad issues that can plague raw
datasets. By ensuring that your data is accurate, complete, and primed for analysis, data cleaning sets the stage for all subsequent preprocessing steps and ultimately contributes to the overall
success of your machine learning endeavors.
Data cleaning is a crucial step in the data preprocessing pipeline, involving the systematic identification and rectification of issues within datasets. This process encompasses a wide range of
activities, including:
Detecting corrupt data
This crucial step involves a comprehensive and meticulous examination of the dataset to identify any data points that have been compromised or altered during various stages of the data lifecycle.
This includes, but is not limited to, the collection phase, where errors might occur due to faulty sensors or human input mistakes; the transmission phase, where data corruption can happen due to
network issues or interference; and the storage phase, where data might be corrupted due to hardware failures or software glitches.
The process of detecting corrupt data often involves multiple techniques:
• Statistical analysis: Using statistical methods to identify outliers or values that deviate significantly from expected patterns.
• Data validation rules: Implementing specific rules based on domain knowledge to flag potentially corrupt entries.
• Consistency checks: Comparing data across different fields or time periods to ensure logical consistency.
• Format verification: Ensuring that data adheres to expected formats, such as date structures or numerical ranges.
By pinpointing these corrupted elements through such rigorous methods, data scientists can take appropriate actions such as removing, correcting, or flagging the corrupt data. This process is
fundamental in ensuring the integrity and reliability of the dataset, which is crucial for any subsequent analysis or machine learning model development. Without this step, corrupt data could lead to
skewed results, incorrect conclusions, or poorly performing models, potentially undermining the entire data science project.
Example: Detecting Corrupt Data
import pandas as pd
import numpy as np
# Create a sample DataFrame with potentially corrupt data
data = {
'ID': [1, 2, 3, 4, 5],
'Value': [10, 20, 'error', 40, 50],
'Date': ['2023-01-01', '2023-02-30', '2023-03-15', '2023-04-01', '2023-05-01']
df = pd.DataFrame(data)
# Function to detect corrupt data
def detect_corrupt_data(df):
corrupt_rows = []
# Check for non-numeric values in 'Value' column
numeric_errors = pd.to_numeric(df['Value'], errors='coerce').isna()
# Check for invalid dates
df['Date'] = pd.to_datetime(df['Date'], errors='coerce')
date_errors = df['Date'].isna()
return list(set(corrupt_rows)) # Remove duplicates
# Detect corrupt data
corrupt_indices = detect_corrupt_data(df)
print("Corrupt data found at indices:", corrupt_indices)
print("\nCorrupt rows:")
This code demonstrates how to detect corrupt data in a pandas DataFrame. Here's a breakdown of its functionality:
• It creates a sample DataFrame with potentially corrupt data, including non-numeric values in the 'Value' column and invalid dates in the 'Date' column.
• The detect_corrupt_data() function is defined to identify corrupt rows. It checks for:
• Non-numeric values in the 'Value' column using pd.to_numeric() with errors='coerce'.
• Invalid dates in the 'Date' column using pd.to_datetime() with errors='coerce'.
• The function returns a list of unique indices where corrupt data was found.
• Finally, it prints the indices of corrupt rows and displays the corrupt data.
This code is an example of how to implement data cleaning techniques, specifically for detecting corrupt data, which is a crucial step in the data preprocessing pipeline.
Correcting incomplete data
This process involves a comprehensive and meticulous examination of the dataset to identify and address any instances of incomplete or missing information. The approach to handling such gaps depends
on several factors, including the nature of the data, the extent of incompleteness, and the potential impact on subsequent analyses.
When dealing with missing data, data scientists employ a range of sophisticated techniques:
• Imputation methods: These involve estimating and filling in missing values based on patterns observed in the existing data. Techniques can range from simple mean or median imputation to more
advanced methods like regression imputation or multiple imputation.
• Machine learning-based approaches: Algorithms such as K-Nearest Neighbors (KNN) or Random Forest can be used to predict missing values based on the relationships between variables in the dataset.
• Time series-specific methods: For temporal data, techniques like interpolation or forecasting models may be employed to estimate missing values based on trends and seasonality.
However, in cases where the gaps in the data are too significant or the missing information is deemed crucial, careful consideration must be given to the removal of incomplete records. This decision
is not taken lightly, as it involves balancing the need for data quality with the potential loss of valuable information.
Factors influencing the decision to remove incomplete records include:
• The proportion of missing data: If a large percentage of a record or variable is missing, removal might be more appropriate than imputation.
• The mechanism of missingness: Understanding whether data is missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR) can inform the decision-making process.
• The importance of the missing information: If the missing data is critical to the analysis or model, removal might be necessary to maintain the integrity of the results.
Ultimately, the goal is to strike a balance between preserving as much valuable information as possible while ensuring the overall quality and reliability of the dataset for subsequent analysis and
modeling tasks.
Example: Correcting Incomplete Data
import pandas as pd
import numpy as np
from sklearn.impute import SimpleImputer
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
# Create a sample DataFrame with incomplete data
data = {
'Age': [25, np.nan, 30, np.nan, 40],
'Income': [50000, 60000, np.nan, 75000, 80000],
'Education': ['Bachelor', 'Master', np.nan, 'PhD', 'Bachelor']
df = pd.DataFrame(data)
print("Original DataFrame:")
# Method 1: Simple Imputation (Mean for numerical, Most frequent for categorical)
imputer_mean = SimpleImputer(strategy='mean')
imputer_most_frequent = SimpleImputer(strategy='most_frequent')
df_imputed_simple = df.copy()
df_imputed_simple[['Age', 'Income']] = imputer_mean.fit_transform(df[['Age', 'Income']])
df_imputed_simple[['Education']] = imputer_most_frequent.fit_transform(df[['Education']])
print("\nDataFrame after Simple Imputation:")
# Method 2: Iterative Imputation (uses the IterativeImputer, aka MICE)
imputer_iterative = IterativeImputer(random_state=0)
df_imputed_iterative = df.copy()
df_imputed_iterative.iloc[:, :] = imputer_iterative.fit_transform(df)
print("\nDataFrame after Iterative Imputation:")
# Method 3: Custom logic (e.g., filling Age based on median of similar Education levels)
df_custom = df.copy()
df_custom['Age'] = df_custom.groupby('Education')['Age'].transform(lambda x: x.fillna(x.median()))
df_custom['Income'].fillna(df_custom['Income'].mean(), inplace=True)
df_custom['Education'].fillna(df_custom['Education'].mode()[0], inplace=True)
print("\nDataFrame after Custom Imputation:")
This example demonstrates three different methods for correcting incomplete data:
• 1. Simple Imputation: Uses Scikit-learn's SimpleImputer to fill missing values with the mean for numerical columns (Age and Income) and the most frequent value for categorical columns
• 2. Iterative Imputation: Employs Scikit-learn's IterativeImputer (also known as MICE - Multivariate Imputation by Chained Equations) to estimate missing values based on the relationships between
• 3. Custom Logic: Implements a tailored approach where Age is imputed based on the median age of similar education levels, Income is filled with the mean, and Education uses the mode (most
frequent value).
Breakdown of the code:
1. We start by importing necessary libraries and creating a sample DataFrame with missing values.
2. For Simple Imputation, we use SimpleImputer with different strategies for numerical and categorical data.
3. Iterative Imputation uses the IterativeImputer, which estimates each feature from all the others iteratively.
4. The custom logic demonstrates how domain knowledge can be applied to impute data more accurately, such as using education level to estimate age.
This example showcases the flexibility and power of different imputation techniques. The choice of method depends on the nature of your data and the specific requirements of your analysis. Simple
imputation is quick and easy but may not capture complex relationships in the data. Iterative imputation can be more accurate but is computationally intensive. Custom logic allows for the
incorporation of domain expertise but requires more manual effort and understanding of the data.
Addressing inaccurate data
This crucial step in the data cleaning process involves a comprehensive and meticulous approach to identifying and rectifying errors that may have infiltrated the dataset during various stages of
data collection and management. These errors can arise from multiple sources:
• Data Entry Errors: Human mistakes during manual data input, such as typos, transposed digits, or incorrect categorizations.
• Measurement Errors: Inaccuracies stemming from faulty equipment, miscalibrated instruments, or inconsistent measurement techniques.
• Recording Errors: Issues that occur during the data recording process, including system glitches, software bugs, or data transmission failures.
To address these challenges, data scientists employ a range of sophisticated validation techniques:
• Statistical Outlier Detection: Utilizing statistical methods to identify data points that deviate significantly from the expected patterns or distributions.
• Domain-Specific Rule Validation: Implementing checks based on expert knowledge of the field to flag logically inconsistent or impossible values.
• Cross-Referencing: Comparing data against reliable external sources or internal databases to verify accuracy and consistency.
• Machine Learning-Based Anomaly Detection: Leveraging advanced algorithms to detect subtle patterns of inaccuracy that might escape traditional validation methods.
By rigorously applying these validation techniques and diligently cross-referencing with trusted sources, data scientists can substantially enhance the accuracy and reliability of their datasets.
This meticulous process not only improves the quality of the data but also bolsters the credibility of subsequent analyses and machine learning models built upon this foundation. Ultimately,
addressing inaccurate data is a critical investment in ensuring the integrity and trustworthiness of data-driven insights and decision-making processes.
Example: Addressing Inaccurate Data
import pandas as pd
import numpy as np
from scipy import stats
# Create a sample DataFrame with potentially inaccurate data
data = {
'ID': range(1, 11),
'Age': [25, 30, 35, 40, 45, 50, 55, 60, 65, 1000],
'Income': [50000, 60000, 70000, 80000, 90000, 100000, 110000, 120000, 130000, 10000000],
'Height': [170, 175, 180, 185, 190, 195, 200, 205, 210, 150]
df = pd.DataFrame(data)
print("Original DataFrame:")
def detect_and_correct_outliers(df, column, method='zscore', threshold=3):
if method == 'zscore':
z_scores = np.abs(stats.zscore(df[column]))
outliers = df[z_scores > threshold]
df.loc[z_scores > threshold, column] = df[column].median()
elif method == 'iqr':
Q1 = df[column].quantile(0.25)
Q3 = df[column].quantile(0.75)
IQR = Q3 - Q1
lower_bound = Q1 - 1.5 * IQR
upper_bound = Q3 + 1.5 * IQR
outliers = df[(df[column] < lower_bound) | (df[column] > upper_bound)]
df.loc[(df[column] < lower_bound) | (df[column] > upper_bound), column] = df[column].median()
return outliers
# Detect and correct outliers in 'Age' column using Z-score method
age_outliers = detect_and_correct_outliers(df, 'Age', method='zscore')
# Detect and correct outliers in 'Income' column using IQR method
income_outliers = detect_and_correct_outliers(df, 'Income', method='iqr')
# Custom logic for 'Height' column
height_outliers = df[(df['Height'] < 150) | (df['Height'] > 220)]
df.loc[(df['Height'] < 150) | (df['Height'] > 220), 'Height'] = df['Height'].median()
print("\nOutliers detected:")
print("Age outliers:", age_outliers['Age'].tolist())
print("Income outliers:", income_outliers['Income'].tolist())
print("Height outliers:", height_outliers['Height'].tolist())
print("\nCorrected DataFrame:")
This example demonstrates a comprehensive approach to addressing inaccurate data, specifically focusing on outlier detection and correction.
Here's a breakdown of the code and its functionality:
1. Data Creation: We start by creating a sample DataFrame with potentially inaccurate data, including extreme values in the 'Age', 'Income', and 'Height' columns.
2. Outlier Detection and Correction Function: The detect_and_correct_outliers() function is defined to handle outliers using two common methods:
□ Z-score method: Identifies outliers based on the number of standard deviations from the mean.
□ IQR (Interquartile Range) method: Detects outliers using the concept of quartiles.
3. Applying Outlier Detection:
□ For the 'Age' column, we use the Z-score method with a threshold of 3 standard deviations.
□ For the 'Income' column, we apply the IQR method to account for potential skewness in income distribution.
□ For the 'Height' column, we implement a custom logic to flag values below 150 cm or above 220 cm as outliers.
4. Outlier Correction: Once outliers are detected, they are replaced with the median value of the respective column. This approach helps maintain data integrity while reducing the impact of extreme
5. Reporting: The code prints out the detected outliers for each column and displays the corrected DataFrame.
This example showcases different strategies for addressing inaccurate data:
• Statistical methods (Z-score and IQR) for automated outlier detection
• Custom logic for domain-specific outlier identification
• Median imputation for correcting outliers, which is more robust to extreme values than mean imputation
By employing these techniques, data scientists can significantly improve the quality of their datasets, leading to more reliable analyses and machine learning models. It's important to note that
while this example uses median imputation for simplicity, in practice, the choice of correction method should be carefully considered based on the specific characteristics of the data and the
requirements of the analysis.
Removing irrelevant data
This final step in the data cleaning process, known as data relevance assessment, involves a meticulous evaluation of each data point to determine its significance and applicability to the specific
analysis or problem at hand. This crucial phase requires data scientists to critically examine the dataset through multiple lenses:
1. Contextual Relevance: Assessing whether each variable or feature directly contributes to answering the research questions or achieving the project goals.
2. Temporal Relevance: Determining if the data is current enough to be meaningful for the analysis, especially in rapidly changing domains.
3. Granularity: Evaluating if the level of detail in the data is appropriate for the intended analysis, neither too broad nor too specific.
4. Redundancy: Identifying and removing duplicate or highly correlated variables that don't provide additional informational value.
5. Signal-to-Noise Ratio: Distinguishing between data that carries meaningful information (signal) and data that introduces unnecessary complexity or variability (noise).
By meticulously eliminating extraneous or irrelevant information through this process, data scientists can significantly enhance the quality and focus of their dataset. This refinement yields several
critical benefits:
• Improved Model Performance: A streamlined dataset with only relevant features often leads to more accurate and robust machine learning models.
• Enhanced Computational Efficiency: Reducing the dataset's dimensionality can dramatically decrease processing time and resource requirements, especially crucial when dealing with large-scale data.
• Clearer Insights: By removing noise and focusing on pertinent data, analysts can derive more meaningful and actionable insights from their analyses.
• Reduced Overfitting Risk: Eliminating irrelevant features helps prevent models from learning spurious patterns, thus improving generalization to new, unseen data.
• Simplified Interpretability: A more focused dataset often results in models and analyses that are easier to interpret and explain to stakeholders.
In essence, this careful curation of relevant data serves as a critical foundation, significantly enhancing the efficiency, effectiveness, and reliability of subsequent analyses and machine learning
models. It ensures that the final insights and decisions are based on the most pertinent and high-quality information available.
Example: Removing Irrelevant Data
import pandas as pd
import numpy as np
from sklearn.feature_selection import VarianceThreshold
from sklearn.feature_selection import mutual_info_regression
# Create a sample DataFrame with potentially irrelevant features
data = {
'ID': range(1, 101),
'Age': np.random.randint(18, 80, 100),
'Income': np.random.randint(20000, 150000, 100),
'Education': np.random.choice(['High School', 'Bachelor', 'Master', 'PhD'], 100),
'Constant_Feature': [5] * 100,
'Random_Feature': np.random.random(100),
'Target': np.random.randint(0, 2, 100)
df = pd.DataFrame(data)
print("Original DataFrame shape:", df.shape)
# Step 1: Remove constant features
constant_filter = VarianceThreshold(threshold=0)
constant_columns = df.columns[~constant_filter.get_support()]
df = df.drop(columns=constant_columns)
print("After removing constant features:", df.shape)
# Step 2: Remove features with low variance
variance_filter = VarianceThreshold(threshold=0.1)
low_variance_columns = df.select_dtypes(include=[np.number]).columns[~variance_filter.get_support()]
df = df.drop(columns=low_variance_columns)
print("After removing low variance features:", df.shape)
# Step 3: Feature importance based on mutual information
numerical_features = df.select_dtypes(include=[np.number]).columns.drop('Target')
mi_scores = mutual_info_regression(df[numerical_features], df['Target'])
mi_scores = pd.Series(mi_scores, index=numerical_features)
important_features = mi_scores[mi_scores > 0.01].index
df = df[important_features.tolist() + ['Education', 'Target']]
print("After removing less important features:", df.shape)
print("\nFinal DataFrame columns:", df.columns.tolist())
This code example demonstrates various techniques for removing irrelevant data from a dataset.
Let's break down the code and explain each step:
1. Data Creation: We start by creating a sample DataFrame with potentially irrelevant features, including a constant feature and a random feature.
2. Removing Constant Features:
□ We use VarianceThreshold with a threshold of 0 to identify and remove features that have the same value in all samples.
□ This step eliminates features that provide no discriminative information for the model.
3. Removing Low Variance Features:
□ We apply VarianceThreshold again, this time with a threshold of 0.1, to remove features with very low variance.
□ Features with low variance often contain little information and may not contribute significantly to the model's predictive power.
4. Feature Importance based on Mutual Information:
□ We use mutual_info_regression to calculate the mutual information between each feature and the target variable.
□ Features with mutual information scores below a certain threshold (0.01 in this example) are considered less important and are removed.
□ This step helps in identifying features that have a strong relationship with the target variable.
5. Retaining Categorical Features: We manually include the 'Education' column to demonstrate how you might retain important categorical features that weren't part of the numerical analysis.
This example showcases a multi-faceted approach to removing irrelevant data:
• It addresses constant features that provide no discriminative information.
• It removes features with very low variance, which often contribute little to model performance.
• It uses a statistical measure (mutual information) to identify features most relevant to the target variable.
By applying these techniques, we significantly reduce the dimensionality of the dataset, focusing on the most relevant features. This can lead to improved model performance, reduced overfitting, and
increased computational efficiency. However, it's crucial to validate the impact of feature removal on your specific problem and adjust thresholds as necessary.
The importance of data cleaning cannot be overstated, as it directly impacts the quality and reliability of machine learning models. Clean, high-quality data is essential for accurate predictions and
meaningful insights.
Missing values are a common challenge in real-world datasets, often arising from various sources such as equipment malfunctions, human error, or intentional non-responses. Handling these missing
values appropriately is critical, as they can significantly affect model performance and lead to biased or incorrect conclusions if not addressed properly.
The approach to dealing with missing data is not one-size-fits-all and depends on several factors:
1. The nature and characteristics of your dataset: The specific type of data you're working with (such as numerical, categorical, or time series) and its underlying distribution patterns play a
crucial role in determining the most appropriate technique for handling missing data. For instance, certain imputation methods may be more suitable for continuous numerical data, while others
might be better suited for categorical variables or time-dependent information.
2. The quantity and distribution pattern of missing data: The extent of missing information and the underlying mechanism causing the data gaps significantly influence the choice of handling
strategy. It's essential to distinguish between data that is missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR), as each scenario may require a
different approach to maintain the integrity and representativeness of your dataset.
3. The selected machine learning algorithm and its inherent properties: Different machine learning models exhibit varying degrees of sensitivity to missing data, which can substantially impact their
performance and the reliability of their predictions. Some algorithms, like decision trees, can handle missing values intrinsically, while others, such as support vector machines, may require
more extensive preprocessing to address data gaps effectively. Understanding these model-specific characteristics is crucial in selecting an appropriate missing data handling technique that
aligns with your chosen algorithm.
By understanding these concepts and techniques, data scientists can make informed decisions about how to preprocess their data effectively, ensuring the development of robust and accurate machine
learning models.
3.1.1 Types of Missing Data
Before delving deeper into the intricacies of handling missing data, it is crucial to grasp the three primary categories of missing data, each with its own unique characteristics and implications for
data analysis:
1. Missing Completely at Random (MCAR)
This type of missing data represents a scenario where the absence of information follows no discernible pattern or relationship with any variables in the dataset, whether observed or unobserved. MCAR
is characterized by an equal probability of data being missing across all cases, effectively creating an unbiased subset of the complete dataset.
The key features of MCAR include:
• Randomness: The missingness is entirely random and not influenced by any factors within or outside the dataset.
• Unbiased representation: The remaining data can be considered a random sample of the full dataset, maintaining its statistical properties.
• Statistical implications: Analyses conducted on the complete cases (after removing missing data) remain unbiased, although there may be a loss in statistical power due to reduced sample size.
To illustrate MCAR, consider a comprehensive survey scenario:
Imagine a large-scale health survey where participants are required to fill out a lengthy questionnaire. Some respondents might inadvertently skip certain questions due to factors entirely unrelated
to the survey content or their personal characteristics. For instance:
• A respondent might be momentarily distracted by an external noise and accidentally skip a question.
• Technical glitches in the survey platform could randomly fail to record some responses.
• A participant might unintentionally turn two pages at once, missing a set of questions.
In these cases, the missing data would be considered MCAR because the likelihood of a response being missing is not related to the question itself, the respondent's characteristics, or any other
variables in the study. This randomness ensures that the remaining data still provides an unbiased, albeit smaller, representation of the population under study.
While MCAR is often considered the "best-case scenario" for missing data, it's important to note that it's relatively rare in real-world datasets. Researchers and data scientists must carefully
examine their data and the data collection process to determine if the MCAR assumption truly holds before proceeding with analyses or imputation methods based on this assumption.
2. Missing at Random (MAR):
In this scenario, known as Missing at Random (MAR), the missing data exhibits a systematic relationship with the observed data, but crucially, not with the missing data itself. This means that the
probability of data being missing can be explained by other observed variables in the dataset, but is not directly related to the unobserved values.
To better understand MAR, let's break it down further:
• Systematic relationship: The pattern of missingness is not completely random, but follows a discernible pattern based on other observed variables.
• Observed data dependency: The likelihood of a value being missing depends on other variables that we can observe and measure in the dataset.
• Independence from unobserved values: Importantly, the probability of missingness is not related to the actual value that would have been observed, had it not been missing.
Let's consider an expanded illustration to clarify this concept:
Imagine a comprehensive health survey where participants are asked about their age, exercise habits, and overall health satisfaction. In this scenario:
• Younger participants (ages 18-30) might be less likely to respond to questions about their exercise habits, regardless of how much they actually exercise.
• This lower response rate among younger participants is observable and can be accounted for in the analysis.
• Crucially, their tendency to not respond is not directly related to their actual exercise habits (which would be the missing data), but rather to their age group (which is observed).
In this MAR scenario, we can use the observed data (age) to make informed decisions about handling the missing data (exercise habits). This characteristic of MAR allows for more sophisticated
imputation methods that can leverage the relationships between variables to estimate missing values more accurately.
Understanding that data is MAR is vital for choosing appropriate missing data handling techniques. Unlike Missing Completely at Random (MCAR), where simple techniques like listwise deletion might
suffice, MAR often requires more advanced methods such as multiple imputation or maximum likelihood estimation to avoid bias in analyses.
3. Missing Not at Random (MNAR)
This category represents the most complex type of missing data, where the missingness is directly related to the unobserved values themselves. In MNAR situations, the very reason for the data being
missing is intrinsically linked to the information that would have been collected. This creates a significant challenge for data analysis and imputation methods, as the missing data mechanism cannot
be ignored without potentially introducing bias.
To better understand MNAR, let's break it down further:
• Direct relationship: The probability of a value being missing depends on the value itself, which is unobserved.
• Systematic bias: The missingness creates a systematic bias in the dataset that cannot be fully accounted for using only the observed data.
• Complexity in analysis: MNAR scenarios often require specialized statistical techniques to handle properly, as simple imputation methods may lead to incorrect conclusions.
A prime example of MNAR is when patients with severe health conditions are less inclined to disclose their health status. This leads to systematic gaps in health-related data that are directly
correlated with the severity of their conditions. Let's explore this example in more depth:
• Self-selection bias: Patients with more severe conditions might avoid participating in health surveys or medical studies due to physical limitations or psychological factors.
• Privacy concerns: Those with serious health issues might be more reluctant to share their medical information, fearing stigma or discrimination.
• Incomplete medical records: Patients with complex health conditions might have incomplete medical records if they frequently switch healthcare providers or avoid certain types of care.
The implications of MNAR data in this health-related scenario are significant:
• Underestimation of disease prevalence: If those with severe conditions are systematically missing from the data, the true prevalence of the disease might be underestimated.
• Biased treatment efficacy assessments: In clinical trials, if patients with severe side effects are more likely to drop out, the remaining data might overestimate the treatment's effectiveness.
• Skewed health policy decisions: Policymakers relying on this data might allocate resources based on an incomplete picture of public health needs.
Handling MNAR data requires careful consideration and often involves advanced statistical methods such as selection models or pattern-mixture models. These approaches attempt to model the missing
data mechanism explicitly, allowing for more accurate inferences from incomplete datasets. However, they often rely on untestable assumptions about the nature of the missingness, highlighting the
complexity and challenges associated with MNAR scenarios in data analysis.
Understanding these distinct types of missing data is paramount, as each category necessitates a unique approach in data handling and analysis. The choice of method for addressing missing
data—whether it involves imputation, deletion, or more advanced techniques—should be carefully tailored to the specific type of missingness encountered in the dataset.
This nuanced understanding ensures that the subsequent data analysis and modeling efforts are built on a foundation that accurately reflects the underlying data structure and minimizes potential
biases introduced by missing information.
3.1.2 Detecting and Visualizing Missing Data
The first step in handling missing data is detecting where the missing values are within your dataset. This crucial initial phase sets the foundation for all subsequent data preprocessing and
analysis tasks. Pandas, a powerful data manipulation library in Python, provides an efficient and user-friendly way to check for missing values in a dataset.
To begin this process, you typically load your data into a Pandas DataFrame, which is a two-dimensional labeled data structure. Once your data is in this format, Pandas offers several built-in
functions to identify missing values:
• The isnull() or isna() methods: These functions return a boolean mask of the same shape as your DataFrame, where True indicates a missing value and False indicates a non-missing value.
• The notnull() method: This is the inverse of isnull(), returning True for non-missing values.
• The info() method: This provides a concise summary of your DataFrame, including the number of non-null values in each column.
By combining these functions with other Pandas operations, you can gain a comprehensive understanding of the missing data in your dataset. For example, you can use df.isnull().sum() to count the
number of missing values in each column, or df.isnull().any() to check if any column contains missing values.
Understanding the pattern and extent of missing data is crucial as it informs your strategy for handling these gaps. It helps you decide whether to remove rows or columns with missing data, impute
the missing values, or employ more advanced techniques like multiple imputation or machine learning models designed to handle missing data.
Example: Detecting Missing Data with Pandas
import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer, KNNImputer
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
# Create a sample DataFrame with missing data
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve', 'Frank'],
'Age': [25, None, 35, 40, None, 50],
'Salary': [50000, 60000, None, 80000, 55000, None],
'Department': ['HR', 'IT', 'Finance', 'IT', None, 'HR']
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
# Check for missing data
print("Missing Data in Each Column:")
# Calculate percentage of missing data
print("Percentage of Missing Data in Each Column:")
print(df.isnull().sum() / len(df) * 100)
# Visualize missing data with a heatmap
plt.figure(figsize=(10, 6))
sns.heatmap(df.isnull(), cbar=False, cmap='viridis', yticklabels=False)
plt.title("Missing Data Heatmap")
# Handling missing data
# 1. Removing rows with missing data
df_dropna = df.dropna()
print("DataFrame after dropping rows with missing data:")
# 2. Simple imputation methods
# Mean imputation for numerical columns
df_mean_imputed = df.copy()
df_mean_imputed['Age'].fillna(df_mean_imputed['Age'].mean(), inplace=True)
df_mean_imputed['Salary'].fillna(df_mean_imputed['Salary'].mean(), inplace=True)
# Mode imputation for categorical column
df_mean_imputed['Department'].fillna(df_mean_imputed['Department'].mode()[0], inplace=True)
print("DataFrame after mean/mode imputation:")
# 3. KNN Imputation
imputer_knn = KNNImputer(n_neighbors=2)
df_knn_imputed = pd.DataFrame(imputer_knn.fit_transform(df.drop('Name', axis=1)),
df_knn_imputed.insert(0, 'Name', df['Name']) # Add back the 'Name' column
print("DataFrame after KNN imputation:")
# 4. Multiple Imputation by Chained Equations (MICE)
imputer_mice = IterativeImputer(random_state=0)
df_mice_imputed = pd.DataFrame(imputer_mice.fit_transform(df.drop('Name', axis=1)),
df_mice_imputed.insert(0, 'Name', df['Name']) # Add back the 'Name' column
print("DataFrame after MICE imputation:")
This code example provides a comprehensive demonstration of detecting, visualizing, and handling missing data in Python using pandas, numpy, seaborn, matplotlib, and scikit-learn.
Let's break down the code and explain each section:
1. Data Creation and Exploration:
• We start by creating a sample DataFrame with missing values in different columns.
• The original DataFrame is displayed to show the initial state of the data.
• We use df.isnull().sum() to count the number of missing values in each column.
• The percentage of missing data in each column is calculated to give a better perspective on the extent of missing data.
2. Visualization:
• A heatmap is created using seaborn to visualize the pattern of missing data. This provides an intuitive way to identify which columns and rows contain missing values.
3. Handling Missing Data:
The code demonstrates four different approaches to handling missing data:
• a. Removing rows with missing data: Using df.dropna(), we remove all rows that contain any missing values. This method is simple but can lead to significant data loss if many rows contain missing
• b. Simple imputation methods:
□ For numerical columns ('Age' and 'Salary'), we use mean imputation with fillna(df['column'].mean()).
□ For the categorical column ('Department'), we use mode imputation with fillna(df['Department'].mode()[0]).
□ These methods are straightforward but don't consider the relationships between variables.
• c. K-Nearest Neighbors (KNN) Imputation: Using KNNImputer from scikit-learn, we impute missing values based on the values of the k-nearest neighbors. This method can capture some of the
relationships between variables but may not work well with categorical data.
• d. Multiple Imputation by Chained Equations (MICE): Using IterativeImputer from scikit-learn, we perform multiple imputations. This method models each feature with missing values as a function of
other features and uses that estimate for imputation. It's more sophisticated and can handle both numerical and categorical data.
4. Output and Comparison:
• After each imputation method, the resulting DataFrame is printed, allowing for easy comparison between different approaches.
• This enables the user to assess the impact of each method on the data and choose the most appropriate one for their specific use case.
This example showcases multiple imputation techniques, provides a step-by-step breakdown, and offers a comprehensive look at handling missing data in Python. It demonstrates the progression from
simple techniques (like deletion and mean imputation) to more advanced methods (KNN and MICE). This approach allows users to understand and compare different strategies for missing data imputation.
The isnull() function in Pandas detects missing values (represented as NaN), and by using .sum(), you can get the total number of missing values in each column. Additionally, the Seaborn heatmap
provides a quick visual representation of where the missing data is located.
3.1.3 Techniques for Handling Missing Data
After identifying missing values in your dataset, the crucial next step involves determining the most appropriate strategy for addressing these gaps. The approach you choose can significantly impact
your analysis and model performance. There are multiple techniques available for handling missing data, each with its own strengths and limitations.
The selection of the most suitable method depends on various factors, including the volume of missing data, the pattern of missingness (whether it's missing completely at random, missing at random,
or missing not at random), and the relative importance of the features containing missing values. It's essential to carefully consider these aspects to ensure that your chosen method aligns with your
specific data characteristics and analytical goals.
1. Removing Missing Data
If the amount of missing data is small (typically less than 5% of the total dataset) and the missingness pattern is random (MCAR - Missing Completely At Random), you can consider removing rows or
columns with missing values. This method, known as listwise deletion or complete case analysis, is straightforward and easy to implement.
However, this approach should be used cautiously for several reasons:
• Loss of Information: Removing entire rows or columns can lead to a significant loss of potentially valuable information, especially if the missing data is in different rows across multiple
• Reduced Statistical Power: A smaller sample size due to data removal can decrease the statistical power of your analyses, potentially making it harder to detect significant effects.
• Bias Introduction: If the data is not MCAR, removing rows with missing values can introduce bias into your dataset, potentially skewing your results and leading to incorrect conclusions.
• Inefficiency: In cases where multiple variables have missing values, you might end up discarding a large portion of your dataset, which is inefficient and can lead to unstable estimates.
Before opting for this method, it's crucial to thoroughly analyze the pattern and extent of missing data in your dataset. Consider alternative approaches like various imputation techniques if the
proportion of missing data is substantial or if the missingness pattern suggests that the data is not MCAR.
Example: Removing Rows with Missing Data
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
# Create a sample DataFrame with missing values
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve'],
'Age': [25, np.nan, 35, 40, np.nan],
'Salary': [50000, 60000, np.nan, 80000, 55000],
'Department': ['HR', 'IT', 'Finance', 'IT', np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
# Check for missing values
print("Missing values in each column:")
# Remove rows with any missing values
df_clean = df.dropna()
print("DataFrame after removing rows with missing data:")
# Remove rows with missing values in specific columns
df_clean_specific = df.dropna(subset=['Age', 'Salary'])
print("DataFrame after removing rows with missing data in 'Age' and 'Salary':")
# Remove columns with missing values
df_clean_columns = df.dropna(axis=1)
print("DataFrame after removing columns with missing data:")
# Visualize the impact of removing missing data
plt.figure(figsize=(10, 6))
plt.bar(['Original', 'After row removal', 'After column removal'],
[len(df), len(df_clean), len(df_clean_columns)],
color=['blue', 'green', 'red'])
plt.title('Impact of Removing Missing Data')
plt.ylabel('Number of rows')
This code example demonstrates various aspects of handling missing data using the dropna() method in pandas.
Here's a comprehensive breakdown of the code:
1. Data Creation:
□ We start by creating a sample DataFrame with missing values (represented as np.nan) in different columns.
□ This simulates a real-world scenario where data might be incomplete.
2. Displaying Original Data:
□ The original DataFrame is printed to show the initial state of the data, including the missing values.
3. Checking for Missing Values:
□ We use df.isnull().sum() to count the number of missing values in each column.
□ This step is crucial for understanding the extent of missing data before deciding on a removal strategy.
4. Removing Rows with Any Missing Values:
□ df.dropna() is used without any parameters to remove all rows that contain any missing values.
□ This is the most stringent approach and can lead to significant data loss if many rows have missing values.
5. Removing Rows with Missing Values in Specific Columns:
□ df.dropna(subset=['Age', 'Salary']) removes rows only if there are missing values in the 'Age' or 'Salary' columns.
□ This approach is more targeted and preserves more data compared to removing all rows with any missing values.
6. Removing Columns with Missing Values:
□ df.dropna(axis=1) removes any column that contains missing values.
□ This approach is useful when certain features are deemed unreliable due to missing data.
7. Visualizing the Impact:
□ A bar chart is created to visually compare the number of rows in the original DataFrame versus the DataFrames after row and column removal.
□ This visualization helps in understanding the trade-off between data completeness and data loss.
This comprehensive example illustrates different strategies for handling missing data through removal, allowing for a comparison of their impacts on the dataset. It's important to choose the
appropriate method based on the specific requirements of your analysis and the nature of your data.
In this example, the dropna() function removes any rows that contain missing values. You can also specify whether to drop rows or columns depending on your use case.
2. Imputing Missing Data
If you have a significant amount of missing data, removing rows may not be a viable option as it could lead to substantial loss of information. In such cases, imputation becomes a crucial technique.
Imputation involves filling in the missing values with estimated data, allowing you to preserve the overall structure and size of your dataset.
There are several common imputation methods, each with its own strengths and use cases:
a. Mean Imputation
Mean imputation is a widely used method for handling missing numeric data. This technique involves replacing missing values in a column with the arithmetic mean (average) of all non-missing values in
that same column. For instance, if a dataset has missing age values, the average age of all individuals with recorded ages would be calculated and used to fill in the gaps.
The popularity of mean imputation stems from its simplicity and ease of implementation. It requires minimal computational resources and can be quickly applied to large datasets. This makes it an
attractive option for data scientists and analysts working with time constraints or limited processing power.
However, while mean imputation is straightforward, it comes with several important caveats:
1. Distribution Distortion: By replacing missing values with the mean, this method can alter the overall distribution of the data. It artificially increases the frequency of the mean value,
potentially creating a spike in the distribution around this point. This can lead to a reduction in the data's variance and standard deviation, which may impact statistical analyses that rely on
these measures.
2. Relationship Alteration: Mean imputation doesn't account for relationships between variables. In reality, missing values might be correlated with other features in the dataset. By using the
overall mean, these potential relationships are ignored, which could lead to biased results in subsequent analyses.
3. Uncertainty Misrepresentation: This method doesn't capture the uncertainty associated with the missing data. It treats imputed values with the same confidence as observed values, which may not be
appropriate, especially if the proportion of missing data is substantial.
4. Impact on Statistical Tests: The artificially reduced variability can lead to narrower confidence intervals and potentially inflated t-statistics, which might result in false positives in
hypothesis testing.
5. Bias in Multivariate Analyses: In analyses involving multiple variables, such as regression or clustering, mean imputation can introduce bias by weakening the relationships between variables.
Given these limitations, while mean imputation remains a useful tool in certain scenarios, it's crucial for data scientists to carefully consider its appropriateness for their specific dataset and
analysis goals. In many cases, more sophisticated imputation methods that preserve the data's statistical properties and relationships might be preferable, especially for complex analyses or when
dealing with a significant amount of missing data.
Example: Imputing Missing Data with the Mean
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve'],
'Age': [25, np.nan, 35, 40, np.nan],
'Salary': [50000, 60000, np.nan, 80000, 55000],
'Department': ['HR', 'IT', 'Finance', 'IT', np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
print("\nMissing values in each column:")
# Impute missing values in the 'Age' and 'Salary' columns with the mean
df['Age'] = df['Age'].fillna(df['Age'].mean())
df['Salary'] = df['Salary'].fillna(df['Salary'].mean())
print("\nDataFrame After Mean Imputation:")
# Using SimpleImputer for comparison
imputer = SimpleImputer(strategy='mean')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After SimpleImputer Mean Imputation:")
# Visualize the impact of imputation
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))
ax1.bar(df['Name'], df['Age'], color='blue', alpha=0.7)
ax1.set_title('Age Distribution After Imputation')
ax1.tick_params(axis='x', rotation=45)
ax2.bar(df['Name'], df['Salary'], color='green', alpha=0.7)
ax2.set_title('Salary Distribution After Imputation')
ax2.tick_params(axis='x', rotation=45)
# Calculate and print statistics
print("\nStatistics After Imputation:")
print(df[['Age', 'Salary']].describe())
This code example provides a more comprehensive approach to mean imputation and includes visualization and statistical analysis.
Here's a breakdown of the code:
• Data Creation and Inspection:
□ We create a sample DataFrame with missing values in different columns.
□ The original DataFrame is displayed along with a count of missing values in each column.
• Mean Imputation:
□ We use the fillna() method with df['column'].mean() to impute missing values in the 'Age' and 'Salary' columns.
□ The DataFrame after imputation is displayed to show the changes.
• SimpleImputer Comparison:
□ We use sklearn's SimpleImputer with 'mean' strategy to perform imputation.
□ This demonstrates an alternative method for mean imputation, which can be useful for larger datasets or when working with scikit-learn pipelines.
• Visualization:
□ Two bar plots are created to visualize the Age and Salary distributions after imputation.
□ This helps in understanding the impact of imputation on the data distribution.
• Statistical Analysis:
□ We calculate and display descriptive statistics for the 'Age' and 'Salary' columns after imputation.
□ This provides insights into how imputation has affected the central tendencies and spread of the data.
This code example not only demonstrates how to perform mean imputation but also shows how to assess its impact through visualization and statistical analysis. It's important to note that while mean
imputation is simple and often effective, it can reduce the variance in your data and may not be suitable for all situations, especially when data is not missing at random.
b. Median Imputation
Median imputation is a robust alternative to mean imputation for handling missing data. This method uses the median value of the non-missing data to fill in gaps. The median is the middle value when
a dataset is ordered from least to greatest, effectively separating the higher half from the lower half of a data sample.
Median imputation is particularly valuable when dealing with skewed distributions or datasets containing outliers. In these scenarios, the median proves to be more resilient and representative than
the mean. This is because outliers can significantly pull the mean towards extreme values, whereas the median remains stable.
For instance, consider a dataset of salaries where most employees earn between $40,000 and $60,000, but there are a few executives with salaries over $1,000,000. The mean salary would be heavily
influenced by these high earners, potentially leading to overestimation when imputing missing values. The median, however, would provide a more accurate representation of the typical salary.
Furthermore, median imputation helps maintain the overall shape of the data distribution better than mean imputation in cases of skewed data. This is crucial for preserving important characteristics
of the dataset, which can be essential for subsequent analyses or modeling tasks.
It's worth noting that while median imputation is often superior to mean imputation for skewed data, it still has limitations. Like mean imputation, it doesn't account for relationships between
variables and may not be suitable for datasets where missing values are not randomly distributed. In such cases, more advanced imputation techniques might be necessary.
Example: Median Imputation
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values and outliers
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve', 'Frank', 'Grace', 'Henry', 'Ivy', 'Jack'],
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Salary': [50000, 60000, np.nan, 80000, 55000, 75000, np.nan, 70000, 1000000, np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
print("\nMissing values in each column:")
# Perform median imputation
df_median_imputed = df.copy()
df_median_imputed['Age'] = df_median_imputed['Age'].fillna(df_median_imputed['Age'].median())
df_median_imputed['Salary'] = df_median_imputed['Salary'].fillna(df_median_imputed['Salary'].median())
print("\nDataFrame After Median Imputation:")
# Using SimpleImputer for comparison
imputer = SimpleImputer(strategy='median')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After SimpleImputer Median Imputation:")
# Visualize the impact of imputation
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))
ax1.boxplot([df['Salary'].dropna(), df_median_imputed['Salary']], labels=['Original', 'Imputed'])
ax1.set_title('Salary Distribution: Original vs Imputed')
ax2.scatter(df['Age'], df['Salary'], label='Original', alpha=0.7)
ax2.scatter(df_median_imputed['Age'], df_median_imputed['Salary'], label='Imputed', alpha=0.7)
ax2.set_title('Age vs Salary: Original and Imputed Data')
# Calculate and print statistics
print("\nStatistics After Imputation:")
print(df_median_imputed[['Age', 'Salary']].describe())
This comprehensive example demonstrates median imputation and includes visualization and statistical analysis. Here's a breakdown of the code:
1. Data Creation and Inspection:
□ We create a sample DataFrame with missing values in the 'Age' and 'Salary' columns, including an outlier in the 'Salary' column.
□ The original DataFrame is displayed along with a count of missing values in each column.
2. Median Imputation:
□ We use the fillna() method with df['column'].median() to impute missing values in the 'Age' and 'Salary' columns.
□ The DataFrame after imputation is displayed to show the changes.
3. SimpleImputer Comparison:
□ We use sklearn's SimpleImputer with 'median' strategy to perform imputation.
□ This demonstrates an alternative method for median imputation, which can be useful for larger datasets or when working with scikit-learn pipelines.
4. Visualization:
□ A box plot is created to compare the original and imputed salary distributions, highlighting the impact of median imputation on the outlier.
□ A scatter plot shows the relationship between Age and Salary, comparing original and imputed data.
5. Statistical Analysis:
□ We calculate and display descriptive statistics for the 'Age' and 'Salary' columns after imputation.
□ This provides insights into how imputation has affected the central tendencies and spread of the data.
This example illustrates how median imputation handles outliers better than mean imputation. The salary outlier of 1,000,000 doesn't significantly affect the imputed values, as it would with mean
imputation. The visualization helps to understand the impact of imputation on the data distribution and relationships between variables.
Median imputation is particularly useful when dealing with skewed data or datasets with outliers, as it provides a more robust measure of central tendency compared to the mean. However, like other
simple imputation methods, it doesn't account for relationships between variables and may not be suitable for all types of missing data mechanisms.
c. Mode Imputation
Mode imputation is a technique used to handle missing data by replacing missing values with the most frequently occurring value (mode) in the column. This method is particularly useful for
categorical data where numerical concepts like mean or median are not applicable.
Here's a more detailed explanation:
Application in Categorical Data: Mode imputation is primarily used for categorical variables, such as 'color', 'gender', or 'product type'. For instance, if in a 'favorite color' column, most
responses are 'blue', missing values would be filled with 'blue'.
Effectiveness for Nominal Variables: Mode imputation can be quite effective for nominal categorical variables, where categories have no inherent order. Examples include variables like 'blood type' or
'country of origin'. In these cases, using the most frequent category as a replacement is often a reasonable assumption.
Limitations with Ordinal Data: However, mode imputation may not be suitable for ordinal data, where the order of categories matters. For example, in a variable like 'education level' (high school,
bachelor's, master's, PhD), simply using the most frequent category could disrupt the inherent order and potentially introduce bias in subsequent analyses.
Preserving Data Distribution: One advantage of mode imputation is that it preserves the original distribution of the data more closely than methods like mean imputation, especially for categorical
variables with a clear majority category.
Potential Drawbacks: It's important to note that mode imputation can oversimplify the data, especially if there's no clear mode or if the variable has multiple modes. It also doesn't account for
relationships between variables, which could lead to loss of important information or introduction of bias.
Alternative Approaches: For more complex scenarios, especially with ordinal data or when preserving relationships between variables is crucial, more sophisticated methods like multiple imputation or
machine learning-based imputation techniques might be more appropriate.
Example: Mode Imputation
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values
data = {
'Name': ['Alice', 'Bob', 'Charlie', 'David', 'Eve', 'Frank', 'Grace', 'Henry', 'Ivy', 'Jack'],
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Category': ['A', 'B', np.nan, 'A', 'C', 'B', np.nan, 'A', 'C', np.nan]
df = pd.DataFrame(data)
# Display the original DataFrame
print("Original DataFrame:")
print("\nMissing values in each column:")
# Perform mode imputation
df_mode_imputed = df.copy()
df_mode_imputed['Category'] = df_mode_imputed['Category'].fillna(df_mode_imputed['Category'].mode()[0])
print("\nDataFrame After Mode Imputation:")
# Using SimpleImputer for comparison
imputer = SimpleImputer(strategy='most_frequent')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After SimpleImputer Mode Imputation:")
# Visualize the impact of imputation
fig, ax = plt.subplots(figsize=(10, 6))
category_counts = df_mode_imputed['Category'].value_counts()
ax.bar(category_counts.index, category_counts.values)
ax.set_title('Category Distribution After Mode Imputation')
# Calculate and print statistics
print("\nCategory Distribution After Imputation:")
This comprehensive example demonstrates mode imputation and includes visualization and statistical analysis. Here's a breakdown of the code:
1. Data Creation and Inspection:
□ We create a sample DataFrame with missing values in the 'Age' and 'Category' columns.
□ The original DataFrame is displayed along with a count of missing values in each column.
2. Mode Imputation:
□ We use the fillna() method with df['column'].mode()[0] to impute missing values in the 'Category' column.
□ The DataFrame after imputation is displayed to show the changes.
3. SimpleImputer Comparison:
□ We use sklearn's SimpleImputer with 'most_frequent' strategy to perform imputation.
□ This demonstrates an alternative method for mode imputation, which can be useful for larger datasets or when working with scikit-learn pipelines.
4. Visualization:
□ A bar plot is created to show the distribution of categories after imputation.
□ This helps in understanding the impact of mode imputation on the categorical data distribution.
5. Statistical Analysis:
□ We calculate and display the proportion of each category after imputation.
□ This provides insights into how imputation has affected the distribution of the categorical variable.
This example illustrates how mode imputation works for categorical data. It fills in missing values with the most frequent category, which in this case is 'A'. The visualization helps to understand
the impact of imputation on the distribution of categories.
Mode imputation is particularly useful for nominal categorical data where concepts like mean or median don't apply. However, it's important to note that this method can potentially amplify the bias
towards the most common category, especially if there's a significant imbalance in the original data.
While mode imputation is simple and often effective for categorical data, it doesn't account for relationships between variables and may not be suitable for ordinal categorical data or when the
missingness mechanism is not completely at random. In such cases, more advanced techniques like multiple imputation or machine learning-based approaches might be more appropriate.
While these methods are commonly used due to their simplicity and ease of implementation, it's crucial to consider their limitations. They don't account for relationships between variables and can
introduce bias if the data is not missing completely at random. More advanced techniques like multiple imputation or machine learning-based imputation methods may be necessary for complex datasets or
when the missingness mechanism is not random.
d. Advanced Imputation Methods
In some cases, simple mean or median imputation might not be sufficient for handling missing data effectively. More sophisticated methods such as K-nearest neighbors (KNN) imputation or regression
imputation can be applied to achieve better results. These advanced techniques go beyond simple statistical measures and take into account the complex relationships between variables to predict
missing values more accurately.
K-nearest neighbors (KNN) imputation works by identifying the K most similar data points (neighbors) to the one with missing values, based on other available features. It then uses the values from
these neighbors to estimate the missing value, often by taking their average. This method is particularly useful when there are strong correlations between features in the dataset.
Regression imputation, on the other hand, involves building a regression model using the available data to predict the missing values. This method can capture more complex relationships between
variables and can be especially effective when there are clear patterns or trends in the data that can be leveraged for prediction.
These advanced imputation methods offer several advantages over simple imputation:
• They preserve the relationships between variables, which can be crucial for maintaining the integrity of the dataset.
• They can handle both numerical and categorical data more effectively.
• They often provide more accurate estimates of missing values, leading to better model performance downstream.
Fortunately, popular machine learning libraries like Scikit-learn provide easy-to-use implementations of these advanced imputation techniques. This accessibility allows data scientists and analysts
to quickly experiment with and apply these sophisticated methods in their preprocessing pipelines, potentially improving the overall quality of their data and the performance of their models.
Example: K-Nearest Neighbors (KNN) Imputation
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.impute import KNNImputer
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
# Create a sample DataFrame with missing values
data = {
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Salary': [50000, 60000, np.nan, 75000, 65000, np.nan, 70000, 80000, np.nan, 90000],
'Experience': [2, 3, 5, np.nan, 4, 8, np.nan, 7, 6, 10]
df = pd.DataFrame(data)
print("Original DataFrame:")
print("\nMissing values in each column:")
# Initialize the KNN Imputer
imputer = KNNImputer(n_neighbors=2)
# Fit and transform the data
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After KNN Imputation:")
# Visualize the imputation results
fig, axes = plt.subplots(1, 3, figsize=(15, 5))
for i, column in enumerate(df.columns):
axes[i].scatter(df.index, df[column], label='Original', alpha=0.5)
axes[i].scatter(df_imputed.index, df_imputed[column], label='Imputed', alpha=0.5)
axes[i].set_title(f'{column} - Before and After Imputation')
# Evaluate the impact of imputation on a simple model
X = df_imputed[['Age', 'Experience']]
y = df_imputed['Salary']
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
print(f"\nMean Squared Error after imputation: {mse:.2f}")
This code example demonstrates a more comprehensive approach to KNN imputation and its evaluation.
Here's a breakdown of the code:
• Data Preparation:
□ We create a sample DataFrame with missing values in 'Age', 'Salary', and 'Experience' columns.
□ The original DataFrame and the count of missing values are displayed.
• KNN Imputation:
□ We initialize a KNNImputer with 2 neighbors.
□ The imputer is applied to the DataFrame, filling in missing values based on the K-nearest neighbors.
• Visualization:
□ We create scatter plots for each column, comparing the original data with missing values to the imputed data.
□ This visual representation helps in understanding how KNN imputation affects the data distribution.
• Model Evaluation:
□ We use the imputed data to train a simple Linear Regression model.
□ The model predicts 'Salary' based on 'Age' and 'Experience'.
□ We calculate the Mean Squared Error to evaluate the model's performance after imputation.
This comprehensive example showcases not only how to perform KNN imputation but also how to visualize its effects and evaluate its impact on a subsequent machine learning task. It provides a more
holistic view of the imputation process and its consequences in a data science workflow.
In this example, the KNN Imputer fills in missing values by finding the nearest neighbors in the dataset and using their values to estimate the missing ones. This method is often more accurate than
simple mean imputation when the data has strong relationships between features.
3.1.4 Evaluating the Impact of Missing Data
Handling missing data is not merely a matter of filling in gaps—it's crucial to thoroughly evaluate how missing data impacts your model's performance. This evaluation process is multifaceted and
requires careful consideration. When certain features in your dataset contain an excessive number of missing values, they may prove to be unreliable predictors. In such cases, it might be more
beneficial to remove these features entirely rather than attempting to impute the missing values.
Furthermore, it's essential to rigorously test imputed data to ensure its validity and reliability. This testing process should focus on two key aspects: first, verifying that the imputation method
hasn't inadvertently distorted the underlying relationships within the data, and second, confirming that it hasn't introduced any bias into the model. Both of these factors can significantly affect
the accuracy and generalizability of your machine learning model.
To gain a comprehensive understanding of how your chosen method for handling missing data affects your model, it's advisable to assess the model's performance both before and after implementing your
missing data strategy. This comparative analysis can be conducted using robust validation techniques such as cross-validation or holdout validation.
These methods provide valuable insights into how your model's predictive capabilities have been influenced by your approach to missing data, allowing you to make informed decisions about the most
effective preprocessing strategies for your specific dataset and modeling objectives.
Example: Model Evaluation Before and After Handling Missing Data
import pandas as pd
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score
from sklearn.impute import SimpleImputer
# Create a sample DataFrame with missing values
data = {
'Age': [25, np.nan, 35, 40, np.nan, 55, 30, np.nan, 45, 50],
'Salary': [50000, 60000, np.nan, 75000, 65000, np.nan, 70000, 80000, np.nan, 90000],
'Experience': [2, 3, 5, np.nan, 4, 8, np.nan, 7, 6, 10]
df = pd.DataFrame(data)
print("Original DataFrame:")
print("\nMissing values in each column:")
# Function to evaluate model performance
def evaluate_model(X, y, model_name):
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)
print(f"\n{model_name} - Mean Squared Error: {mse:.2f}")
print(f"{model_name} - R-squared Score: {r2:.2f}")
# Evaluate model with missing data
X_missing = df[['Age', 'Experience']]
y_missing = df['Salary']
evaluate_model(X_missing.dropna(), y_missing.dropna(), "Model with Missing Data")
# Simple Imputation
imputer = SimpleImputer(strategy='mean')
df_imputed = pd.DataFrame(imputer.fit_transform(df), columns=df.columns)
print("\nDataFrame After Mean Imputation:")
# Evaluate model after imputation
X_imputed = df_imputed[['Age', 'Experience']]
y_imputed = df_imputed['Salary']
evaluate_model(X_imputed, y_imputed, "Model After Imputation")
# Advanced: Multiple models comparison
from sklearn.ensemble import RandomForestRegressor
from sklearn.svm import SVR
models = {
'Linear Regression': LinearRegression(),
'Random Forest': RandomForestRegressor(n_estimators=100, random_state=42),
'Support Vector Regression': SVR()
for name, model in models.items():
X_train, X_test, y_train, y_test = train_test_split(X_imputed, y_imputed, test_size=0.2, random_state=42)
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)
print(f"\n{name} - Mean Squared Error: {mse:.2f}")
print(f"{name} - R-squared Score: {r2:.2f}")
This code example provides a comprehensive approach to evaluating the impact of missing data and imputation on model performance.
Here's a detailed breakdown of the code:
1. Data Preparation:
□ We create a sample DataFrame with missing values in 'Age', 'Salary', and 'Experience' columns.
□ The original DataFrame and the count of missing values are displayed.
2. Model Evaluation Function:
□ A function evaluate_model() is defined to assess model performance using Mean Squared Error (MSE) and R-squared score.
□ This function will be used to compare model performance before and after imputation.
3. Evaluation with Missing Data:
□ We first evaluate the model's performance using only the complete cases (rows without missing values).
□ This serves as a baseline for comparison.
4. Simple Imputation:
□ Mean imputation is performed using sklearn's SimpleImputer.
□ The imputed DataFrame is displayed to show the changes.
5. Evaluation After Imputation:
□ We evaluate the model's performance again using the imputed data.
□ This allows us to compare the impact of imputation on model performance.
6. Advanced Model Comparison:
□ We introduce two additional models: Random Forest and Support Vector Regression.
□ All three models (including Linear Regression) are trained and evaluated on the imputed data.
□ This comparison helps in understanding if the choice of model affects the impact of imputation.
This example demonstrates how to handle missing data, perform imputation, and evaluate its impact on different models. It provides insights into:
• The effect of missing data on model performance
• The impact of mean imputation on data distribution and model accuracy
• How different models perform on the imputed data
By comparing the results, data scientists can make informed decisions about the most appropriate imputation method and model selection for their specific dataset and problem.
Handling missing data is one of the most critical steps in data preprocessing. Whether you choose to remove or impute missing values, understanding the nature of the missing data and selecting the
appropriate method is essential for building a reliable machine learning model. In this section, we covered several strategies, ranging from simple mean imputation to more advanced techniques like
KNN imputation, and demonstrated how to evaluate their impact on your model's performance. | {"url":"https://www.cuantum.tech/app/section/31-data-cleaning-and-handling-missing-data-119ba44c15f481fc8455c9080daa6961","timestamp":"2024-11-04T19:56:52Z","content_type":"text/html","content_length":"424600","record_id":"<urn:uuid:127c602b-0a0b-43d2-8686-2d654da6cbd1>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00416.warc.gz"} |
The Balance of Trade, Terms of Trade, and Real Exchange Rate An Intertemporal Optimizing Framework
An intertemporal optimizing model of a small open economy is used to analyze how terms of trade changes affect real exchange rates and the trade balance. Temporary current, (expected) future, and
permanent changes in the terms of trade are considered. The results suggest that the relationship between the terms of trade and the current account (the so-called Harberger-Laursen-Metzler effect)
is sensitive to whether the model incorporates nontradable goods. Thus, the real exchange rate may be an important variable through which terms of trade shocks are transmitted to the current account.
Several developed and developing countries have experienced—in the course of the past fifteen years—what are by historical standards large disturbances in their terms of trade. Such changes in the
terms of trade have taken many forms: changes in the relative price of intermediate inputs, in the relative price of final manufactured goods, or in the relative price of some final primary
commodity. Along with these developments in the world economy has arisen a renewed theoretical interest among economists about the likely effects of terms of trade shifts on various macroeconomic
variables such as spending, saving, investment, and, in particular, the current account balance. Part of this interest is no doubt due to recent historical experience, but it is also due to
dissatisfaction with certain aspects of the more traditional approaches to the analysis of the relationship between the terms of trade and the current account that view the latter as being determined
in a nonoptimizing, static setting.
Of course the current account—being the difference between saving and investment—is inherently a forward-looking economic variable; hence the question of how various terms of trade changes affect the
current account balance requires an explicit intertemporal model. Only in such a setting can the important distinctions about the different effects of current versus future shocks, or of temporary
versus permanent shocks, be made clear.
The analysis of the effects of changes in the terms of trade on spending, saving, and the current account has a long history in the open-economy literature. The early papers, which include Harberger
(1950) and Laursen and Metzler (1950), were based on nonoptimizing models. Their argument was essentially that a deterioration in the terms of trade would lower real income and hence reduce saving
out of any given level of nominal income, both measured in terms of exportables. If investment, fiscal policies, and nominal income are fixed, the lower saving implies a worsening in the country’s
current account position. Thus, the Harberger-Laursen-Metzler (hereafter H-L-M) effect states that a deterioration in the terms of trade will cause a reduction in the current account balance.
More recent contributions have viewed the current account in an explicit intertemporal setting in which the spending and saving decisions of forward-looking individuals are derived from the
maximization of an intertemporal utility function subject to lifetime budget constraints. Examples include Sachs (1981, 1982), Obstfeld (1982), Svensson and Razin (1983), Greenwood (1984), Persson
and Svensson (1985), Bean (1986), Edwards (1987), and Frenkel and Razin (1987).
A central finding in this literature is that a permanent deterioration in the terms of trade will not have a large impact on the current account (assuming that the initial equilibrium is stationary
and that trade is initially balanced).^1 The reason is that, with a permanent increase in the relative price of importables, both real income and real spending are likely to fall by similar amounts
(assuming homothetic preferences), and the current account, being the difference between these two magnitudes (assuming no historical debt commitment, so that initial current and trade account
balances are identical), will therefore remain unaffected.
In contrast, a temporary deterioration in the terms of trade has an ambiguous impact on the current account. On the one hand, the “consumption-smoothing” motive dictates that agents will maintain
spending in the face of a temporary decline in real income. This force favors a worsening in the current account position if the temporary deterioration in the terms of trade occurs in the current
period, but it favors an improvement in today’s current account if the shock to the terms of trade is expected to occur in the future.
On the other hand, a temporary current (expected future) deterioration in the terms of trade raises (lowers) the cost of current consumption in terms of future consumption. This rise (fall) in the
consumption-based real interest rate associated with a temporary current (expected future) deterioration in the terms of trade encourages (lowers) saving and hence favors an improvement in the
current account in the case of a current shock and a worsening in the current account in the case of a future shock. This is the intertemporal substitution effect.
Thus, the net effect on the current account resulting from a temporary change in the terms of trade depends on which of these two influences—the consumption-smoothing motive or the intertemporal
substitution effect—is stronger.
One aspect that has been largely ignored in the literature to date concerns the role of nontradable goods.^2 In general, a change in the terms of trade alters both the level and composition of
aggregate real spending, part of which falls on nontradable goods. The result, from an initial position of equilibrium in the home-goods sector, will be either excess demand or supply of nontradable
goods. To ensure market clearing, a new relative price structure—a new path of the equilibrium real exchange rate—is required. The new value of the real exchange rate in turn feeds back to affect the
real trade balance.
The purpose of this paper is therefore to consider the effect of terms of trade disturbances on the trade balance in a real, intertemporal, optimizing, general equilibrium model of a small open
economy in which some commodities are assumed not to be tradable internationally. This allows one to address two related questions. First, how do changes in the terms of trade affect the path of the
equilibrium real exchange rate? Second, how do movements in the real exchange rate induced by the change in the terms of trade affect the relationship between the terms of trade and the trade
balance? Put differently, how, in models with non-tradable goods, do changes in the terms of trade affect the balance of trade? Distinctions are drawn among temporary current, (expected) future, and
permanent changes in the terms of trade. The results are presented both diagrammatically and analytically.
The motivation for extending the two-tradable-good (importable and exportable) model of a small country to allow for a home-goods sector is by now clear. First, at a theoretical level, the terms of
trade and the real exchange rate are interesting macroeconomic relative price variables, and identifying whether one expects them to be positively or negatively correlated (as well as the magnitude
of the correlation) is a useful task. Second, any expenditure-switching policy that alters the internal terms of trade faced by domestic agents will in general have a nonzero effect on the real
exchange rate that policymakers may wish to take into account. Third, the results may shed some light on empirical regularities in the comovement between the terms of trade and the real exchange
rate. Finally, as mentioned previously, the real exchange rate is potentially an important variable through which terms of trade shocks are transmitted to the current account.
In this paper, the response of the trade balance to a change in the terms of trade is decomposed into two parts. First, there is a direct effect—as discussed in Svensson and Razin (1983) and Frenkel
and Razin (1987)—that in the context of this model may be viewed as the effect of a terms of trade disturbance holding constant the path of the real exchange rate. There is also an indirect effect
operating through the response of the real exchange rate to a change in the terms of trade and the feedback of the real exchange rate to the trade balance.
It will be shown that the indirect effect is in general nonzero, so that, quantitatively, the response of the trade balance to a change in the terms of trade will differ according to whether the
model incorporates non-tradable goods. Further, it is shown that, because the direct and indirect effects depend on different parameters of the model, they may be either of the same or opposite sign.
Moreover, for certain values of the parameters, the indirect effect may be opposite in sign to the direct effect and may even dominate the latter; thus, qualitatively, the two types of model may
differ in their predictions.^3
The remainder of the paper is organized as follows. Section I sets out the analytical framework, which is similar to the one employed in Svensson and Razin (1983) and Frenkel and Razin (1987) except
that it allows for one of the goods to be nontradable. Section II considers the effects of terms of trade shocks on the path of the equilibrium real exchange rate, and Section III uses these results
to determine the total effect of a change in the terms of trade on the balance of trade. Section IV reviews the main results of the paper and presents possible extensions. The paper concludes with a
brief technical Appendix.
I. Analytical Framework
Consider a two-period, real model of a small open economy in which there are three goods: an importable, an exportable, and a nontradable good. In this economy, there is a representative agent who
maximizes utility subject to the following budget constraints:
$\begin{array}{cc}& {c}_{x0}+{p}_{m0}{c}_{m0}+{p}_{n0}{c}_{n0}+\left(1+{r}_{x,-1}\right){B}_{-1}\\ =& {\stackrel{-}{Y}}_{x0}+{p}_{m0}{\stackrel{-}{Y}}_{m0}+{p}_{n0}{\stackrel{-}{Y}}_{n0}+{B}_{0,\
where c[xt], c[mt], and c[nt] denote consumption levels of exportables, importables, and nontradables, respectively; Ӯ[xt], Ӯ[mt], and Ӯ[nt] denote the endowments of exportables, importables, and
nontradables, respectively; p[mt] and p[nt] denote the relative price of importables and nontradables;B[-1] is the level of initial debt (which may be positive or negative): B[0] represents borrowing
between periods 0 and 1 (which, if negative, represents net lending) and r[xt] represents the rate of interest (for borrowing and lending) between periods t—1 and t, t = 0,1. The numeraire is chosen
to be the exportable commodity, so that all assets and liabilities (as well as interest rates) are measured in units of the exportable. There is no loss of generality in this choice of numeraire so
long as relative price changes are fully anticipated, since in this case an interest parity relationship will prevail between alternative debt instruments denominated in terms of different
commodities. See Frenkel and Razin (1987, pp. 170 and 182) for the consequences of the choice of numeraire in the case of unanticipated relative price movements.
The real exchange rate will, for the purposes of this paper, be defined as the inverse of the relative price of nontradable goods in terms of exportables; that is, 1/p[nt]. A rise in p[nt] denotes a
real appreciation, and a fall in p[nt] denotes a real depreciation.
In a model with three goods, there are in general two real exchange rates: one in terms of exportables (l//p[nt]), and the other in terms of importables (p[mt]/p[nt]). So long as the price of
importables relative to exportables, p[mt,] is constant, the choice of either definition of the real exchange rate is completely innocuous. This is not the case when the shock being considered is a
change in the commodity terms of trade; in this case, a change in p[mt] will obviously have different effects on l/p[nt] and P[mt]/P[nt]· In subsequent analysis of the real exchange rate effects of
various shocks, only the first definition is considered—that is, the effect on the relative price of exportables in terms of the home good, 1/p[nt]. Simple algebra is then required to determine the
effects of these shocks on the alternative definition of the real exchange rate, p[mt/]p[nt], or on any weighted average of the two definitions, such as the consumption-based measure of the real
exchange rate.
Preferences are defined over the six goods c[xo], c[mo], c[no], c[x1], c[m1], and c[n1], and it will be assumed that the intertemporal utility function is weakly separable through time (see, for
example, Goldman and Uzawa (1964) or Deaton and Muellbauer (1980) on separability). Thus, lifetime utility U(c[x0], c[m0], c[n0], c[x1], c[m1], c[n1]) may be written as
where C[0](.) and C[1](.) are the subutilities, which are functions of the consumption levels of the three commodities in periods 0 and 1, respectively. In addition, it will be assumed that the
subutility functions are themselves homothetic, so that, without any further loss of generality, they may be taken to be linearly homogeneous functions of the consumption vector in each of the two
With these assumptions, the consumer may be viewed as solving a two-stage optimization problem. In the first stage, the consumer chooses levels of c[xt], c[mt], and c[nt] to minimize the cost of
attaining subutility level C[t]. In other words, he solves
subject to
${C}_{t}\left({c}_{xt},{c}_{mt},{c}_{nt}\right)\ge {\stackrel{-}{C}}_{t}$
for t =0,1. The solution to this problem yields demands for the three goods, which are functions of the temporal relative prices, p[mt] and p[nt], and of total spending in that period, P[t]C[t,]
where P[t] is the price or marginal cost of a unit of subutility (or real spending), C[t]. The consumption-based price index, P[t], is a function of the relative prices, p[mt] and p[nt].
In the second stage, the consumer chooses real spending levels C[0], C[1] to maximize lifetime utility subject to an intertemporal wealth constraint. In other words, he solves
subject to
${P}_{0}{C}_{0}+{\alpha }_{x1}{P}_{1}{C}_{1}\le {W}_{0},$
where P[0], P[1] are the consumption-based price indices solved for in the first stage and W[0] represents wealth (in terms of exportables), defined as the present value of the economy’s current and
future endowment net of historical debt commitment. From equations (1) and (2), it is easy to verify that
${W}_{0}=\text{(}{\stackrel{-}{Y}}_{x0}+{p}_{m0}{\stackrel{-}{Y}}_{m0}+{p}_{n0}{\stackrel{-}{Y}}_{n0}\text{)+}{\alpha }_{x1}\text{(}{\stackrel{-}{Y}}_{x1}+{p}_{m1}{\stackrel{-}{Y}}_{m1}+{p}_{n1}{\
where α[x1] is the world discount factor, which is equal to 1/(1 + r[x0]). It is noteworthy that, in the absence of a historical debt commitment (B[-1]=0), the intertemporal budget constraint (with
equality) of the representative agent is identical to the condition that, over the lifetime of this economy (that is, during periods 0 and 1), the present value of the sum of the trade account
balances equals zero. This condition reflects, therefore, the assumption of perfect capital mobility subject only to the economy’s intertemporal solvency constraint.
Normalizing the intertemporal budget constraint by dividing by P[0] yields the constraint relevant for the second stage of the consumer’s problem:
${C}_{0}+{\alpha }_{c1}{C}_{1}\le {W}_{c0},\phantom{\rule[-0.0ex]{12em}{0.0ex}}\left(4\right)$
where α[c1] = (P[1]/P[0])α[x1], and W[c0] = W[0]/P[0]. In equation (4), all variables are measured in real terms; that is, in terms of units of period-0 subutility, C[0]. The solution to the
second-stage problem yields demands for C[0] and C[1] as functions of the intertemporal relative price, α[c1], and real lifetime wealth, W[c0].^4
II. Terms of Trade Shocks and the Real Exchange Rate
There are three main channels through which a change in the terms of trade alters the equilibrium real exchange rate. First, a change in the commodity terms of trade, whether brought about by a
change in the world relative price of importables or by policy actions (such as a tariff) affecting the domestic price, leads to substitution among goods within the period. Thus, for example, a
deterioration in the terms of trade in period 0 leads to increased consumption of nontradable goods in period 0 if the two goods are net (Hicksian) substitutes or to decreased consumption if they are
net complements, all other things being held constant (including the level of utility, or welfare). This is the intratemporal or simply temporal substitution effect.
Second, if the rise in the relative price of importables is confined to period 0, the real (consumption-based) rate of interest also rises. This is so because a temporary current deterioration in the
terms of trade (a rise in P[m0]) raises the consumption-based price index, P[0], whereas tomorrow’s price index, P[l], is constant because p[m1] is assumed to be constant. Since the ratio P[0]/P[1]
rises, the cost of current consumption relative to future consumption has risen. This induces substitution of aggregate spending from period 0 to period 1. This rise in tomorrow’s consumption and
fall in today’s consumption, brought about by the change in the intertemporal relative price while other factors are held constant, is the intertemporal substitution effect.
Third, in addition to these intratemporal and intertemporal effects, a rise in p[m0] reduces welfare. The magnitude of this welfare effect depends on the volume of imports at the initial terms of
To gain some insight into the quantitative impact of a change in the terms of trade on the real exchange rate, consider the market-clearing conditions in the market for nontradable goods in each of
the two periods:
${c}_{n0}\text{(}{p}_{n0},{p}_{m0},{P}_{0}{C}_{0}\left({\alpha }_{c1},{W}_{c0}\right)\right)={\stackrel{-}{Y}}_{n0}\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(5\right)$
${c}_{n1}\text{(}{p}_{n1},{p}_{m1},{P}_{1}{C}_{1}\left({\alpha }_{c1},{W}_{c0}\right)\right)={\stackrel{-}{Y}}_{n1}.\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(6\right)$
Equations (5) and (6) state that domestic demand must equal the exogenous endowment of nontradables, in each period. Of course, there are no corresponding conditions for tradable goods because trade
account imbalances allow discrepancies between demand and supply of tradables, period by period.
Total differentiation of equations (5) and (6) yields solutions for the endogenous variables, p[n0] and p[n1], as functions of the exogenous variables, the terms of trade, the world rate of interest,
and the endowments.
Because the focus here is on the first of these three variables, it is assumed in what follows that there are no supply shocks (endowments are fixed) and that there are no shocks to the world rate of
interest. A useful diagrammatic apparatus for the interpretation of the effect of terms of trade shocks on the equilibrium real exchange rate is provided in Figure 1.^5 The N[0]N[0] and N[1]N[1]
schedules represent the loci of combinations of p[no] and p[n1] that clear the period-0 and period-1 markets for nontradable goods, respectively. These schedules are drawn for given values of the
exogenous variables, p[m0] and p[ml]. For convenience, it is assumed that initially p[n0] = p[n1]. As shown in the Appendix, the two schedules have the following slopes:
$\frac{d\mathrm{log}{p}_{n1}}{d\mathrm{log}{p}_{n0}}{|}_{{N}_{0}{N}_{0}}=\frac{{\beta }_{m0}{\sigma }_{nm}+{\beta }_{x0}{\sigma }_{nx}+{\beta }_{n0}\gamma \sigma }{\gamma {\beta }_{n1}\sigma }\
$\frac{d\mathrm{log}{p}_{n1}}{d\mathrm{log}{p}_{n0}}{|}_{{N}_{1}{N}_{1}}=\frac{\left(1-\gamma \right){\beta }_{n0}\sigma }{{\beta }_{m1}{\sigma }_{nm}+{\beta }_{x1}{\sigma }_{nx}+{\beta }_{n1}\left
(1-\gamma \right)\sigma },\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(8\right)$
where β[mt], β[xt], and β[nt] are the period-t expenditure shares of importables, exportables, and nontradables. respectively: σ[ij] is the Allen elasticity of substitution between goods i and j, and
σ is the average saving propensity (defined as the ratio of future spending to lifetime wealth, in present value). It is easily verified that both schedules are positively sloped and that the N[0]N
[0] schedule is necessarily steeper than the N[1]N[1] schedule.
The intuition of this result is straightforward. Consider a rise in p[no] from A to B in Figure 1. This creates excess supply for c[n0] (through both a temporal and intertemporal substitution effect)
and excess demand for c[n1] (through an intertemporal substitution effect). A rise in p[nl] eliminates the excess supply for c[n0] (through an intertemporal substitution effect) and the excess demand
for c[n1] (through both a temporal and intertemporal substitution effect). The rise in p[n1] required to clear the period-0 market (to point D in Figure 1), however, is larger than the rise required
to clear the period-1 market (to point C in Figure 1). The reason is essentially that a change in the relative price of nontradable goods in period t always has a larger effect on excess demand in
period t than in the other period.^6
Temporary Current Changes in the Terms of Trade
Consider a temporary current deterioration in the terms of trade—that is, ${\stackrel{̂}{p}}_{m0}>0$ and ${\stackrel{̂}{p}}_{m1}>0$ where a circumflex (ˆ) above a variable denotes a proportional
change. The rise in the relative price of imports in period-0 affects both the N[0]N[0] and the N[1]N[1] schedules. In the Appendix, it is shown that the horizontal shifts of these loci are,
$\frac{d\mathrm{log}{p}_{n0}}{d\mathrm{log}{p}_{m0}}{|}_{{N}_{0}{N}_{0}}=\frac{{\beta }_{m0}\left\{{\sigma }_{nm}-\left[\left(1-\gamma \right)\left(1-{\mu }_{m0}\right)+\gamma \sigma \right]\right\}}
{\left[{\beta }_{m0}{\sigma }_{nm}+{\beta }_{x0}{\sigma }_{nx}+{\beta }_{n0}\gamma \sigma \right]},\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(9\right)$
$\frac{d\mathrm{log}{p}_{n0}}{d\mathrm{log}{p}_{m0}}{|}_{{N}_{1}{N}_{1}}=\frac{-{\beta }_{m0}\left(1-\gamma \right)\left[\sigma -\left(1-{\mu }_{m0}\right)\right]}{{\beta }_{n0}\left(1-\gamma \right)
\sigma },\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(10\right)$
where µ[nt] is the ratio of endowment to consumption of importables in period t, t = 0,1.
In equation (9) it can be seen that whether the N[0]N[0] schedule shifts to the right or to the left depends only on whether ${{\sigma }_{nm}}_{<}^{>}\text{[}\left(1-\gamma \right)\left(1-{\mu }_{m0}
\right)+\gamma \sigma \right]$. The intuition of this result can be seen by focusing on equation (5). There, a rise in p[m0] affects the demand for through three separate channels: (1) a temporal
substitution effect, since enters directly as an argument in the demand for c[n0] function; (2) the price index effect, since a rise in p[mo] raises the consumption-based price index, P[0], and hence
raises the value of spending P[0] C[0]; and (3) the real spending effect, since a rise in p[m0] alters both the real rate of interest and the real value of wealth and hence affects the period-0
demand for real spending, C[0]. Each of these three effects will be considered in turn.
The magnitude of the (gross) temporal substitution effect depends on the elasticity of C[no] with respect to p[mo] which, by the Slutsky decomposition, is equal to β[mo](σ[nm] - 1). Obviously, the
gross substitutability or complementarity of the two goods is determined by whether ${{\sigma }_{nm}}_{<}^{>}1$.
The price index effect is always positive. A percentage rise in p[m0] raises P[0], the consumption-based price index, by β[m0], the expenditure share of importables in period-0 spending. The rise in
P[0] raises total spending, P[0] C[0](.), and, by the homotheticity assumption, creates an excess demand for C[n0] equal to β[m0] times
The real spending effect is always negative; that is, a temporary current deterioration in the terms of trade always reduces current-period real spending. There are two channels at work here: a real
wealth effect and an intertemporal substitution effect. First, from the budget constraint, a rise in p[m0], lowers the real value of wealth, W[c0] (and hence the demand for c[n0]), by the amount β
[m0][(1—γ)µ[m0]—1]. Second, the rise in p[m0] lowers the real discount factor, and this reduces the demand for C[0] by an amount equal to the product of the change in α[c1], which equals -β[m0], and
the elasticity of C[0] with respect to α[c1] which equals γ(σ- 1). Summing these two effects yields the real spending effect, ^— β[m0][(1 - γ)(1—µ[m0])+ γσ], which is unambiguously negative. This
result accords with intuition because, in the case of a temporary current deterioration in the terms of trade, real wealth falls and the real rate of interest rises. These two effects are mutually
reinforcing as they affect real spending, C[0].
Figure 2.
Effect of a Temporary Current Deterioration in Terms of Trade on Path of Real Exchange Rate
Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A001
Figure 2.
Effect of a Temporary Current Deterioration in Terms of Trade on Path of Real Exchange Rate
Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A001
Figure 2.
Effect of a Temporary Current Deterioration in Terms of Trade on Path of Real Exchange Rate
Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A001
Finally, as is easily verified, the sum of the temporal substitution, price index, and real spending effects yields precisely the numerator of the expression in equation (9). The N[0]N[0] schedule
will shift to the right if σ[nm] > [(1 - γ) |(1-µ[m0])+γσ]; that is, if the temporal substitution effect dominates the real spending effect, and conversely. Note further that the welfare effect is
zero if σ[m0]= 1· This is the case in which, at the initial terms of trade, the small country is in the autarky equilibrium. In this case, only the relative magnitudes of the temporal and
intertemporal (compensated) elasticities (that is, whether ${{\sigma }_{nm}}_{<}^{>}\gamma \sigma$) determine the direction of the shift in the N[0]N[0] schedule.
Consider now the market-clearing condition in period-1, equation (6). A rise in p[m0] affects the demand for home goods in the future period only through the real spending effect. In contrast to the
period-0 equilibrium condition, there is neither a temporal substitution effect nor a price index effect.
Real spending in period 1, C[1], is influenced by way of two separate channels: the rise in p[m0] raises the consumption-based rate of interest, which in turn raises the demand for c[n1] (the
intertemporal substitution effect), while the negative real wealth effect lowers the demand for c[n1]. Specifically, the intertemporal substitution effect is equal to the product of the change in the
consumption-based discount factor,—β[m0,] and the elasticity of period-1 real spending with respect to the real discount factor, -[(1 - γ)σ + γ]. The sum of this intertemporal substitution effect and
the real wealth effect, given previously β[m0][(1 -γ)µ[m0]- 1], yields precisely the expression in the numerator of equation (10). As can be seen, the real spending effect on the demand for c[n1] is
ambiguous, reflecting a conflict between the negative wealth and the positive intertemporal substitution effects. If µ[m0] = 1 so that, at the initial terms of trade, the economy is in the autarky
equilibrium, the N[1]N[1] schedule necessarily shifts to the left. This is so because, in the neighborhood of the autarky equilibrium, the temporary terms of trade deterioration raises the demand for
c[n1] through the intertemporal substitution channel alone; there is no mitigating welfare effect.
In Figure 2, two possible equilibria are depicted. In both panels, it is assumed that the terms of trade shock occurs around the autarky equilibrium; that is, µ[m0] = 1. In panel A, the temporal
elasticity is assumed to exceed the intertemporal elasticity. In this case, the rise in p[m0] shifts the N[0]N[0] schedule to the right. In panel B, the N[0]N[0] schedule shifts to the left,
reflecting the assumption that σ[nm] < γσ. In both panels, the N[1]N[1] schedule shifts to the left. As can be seen in panel A, the equilibrium moves from point A to point B, and a temporary
deterioration in the terms of trade necessarily leads to a real appreciation in both periods.^8 In panel B, the equilibrium moves from point A to point B’, and there is a fall in p[n0] and a rise in
p[nl]. This result is not general, however, as experimenting with the figure will reveal. Finally, note that, although the shock is confined to period 0, part of the adjustment in the real exchange
rate occurs in period 1 when there is no change in any “fundamental.”
In the Appendix, the following general result is derived for the equilibrium response of p[no] (the inverse of today’s real exchange rate) to a temporary current deterioration in the terms of trade
(the result for p[nI] is also given in the Appendix):
$\begin{array}{cc}\frac{d\mathrm{log}{p}_{n0}}{d\mathrm{log}{p}_{m0}}& =\text{{}{\sigma }_{nm}-\text{[}\left(1-\gamma \right)\left(1-{\mu }_{m0}\right)+\gamma \sigma \right]\right\}{\mathrm{\Delta }}
_{1}\\ & +\left[{\sigma }_{nm}-\left(1-{\mu }_{m0}\right)\right]{\mathrm{\Delta }}_{2},\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(11\right)\end{array}$
$\begin{array}{cc}{\mathrm{\Delta }}_{1}& =\frac{{\beta }_{m0}\left({\beta }_{m1}{\sigma }_{nm}+{\beta }_{x1}{\sigma }_{nx}\right)}{\left({\beta }_{m0}{\sigma }_{nm}+{\beta }_{x0}{\sigma }_{nx}\
right)\left[{\beta }_{m1}{\sigma }_{nm}+{\beta }_{x1}{\sigma }_{nx}+{\beta }_{n1}\left(1-\gamma \right)\sigma \right]+{\beta }_{n0}\gamma \sigma \left({\beta }_{m1}{\sigma }_{nm}+{\beta }_{x1}{\sigma
}_{nx}\right)}\\ {\mathrm{\Delta }}_{2}& =\frac{{\beta }_{m0}{\beta }_{n1}\left(1-\gamma \right)\sigma }{\left({\beta }_{m0}{\sigma }_{nm}+{\beta }_{x0}{\sigma }_{nx}\right)\left[{\beta }_{m1}{\sigma
}_{nm}+{\beta }_{x1}{\sigma }_{nx}+{\beta }_{n1}\left(1-\gamma \right)\sigma \right]+{\beta }_{n0}\gamma \sigma \left({\beta }_{m1}{\sigma }_{nm}+{\beta }_{x1}{\sigma }_{nx}\right)},\end{array}$
and Δ[1] > 0, Δ[2] > 0. Thus, the following proposition may be stated.
Proposition 1. The nature of the response of the real exchange rate to a temporary current disturbance in the terms of trade depends on the relative magnitudes of the temporal, intertemporal, and
welfare effects. If nontradables and importables are Hicksian substitutes, the temporal substitution effect favors a contemporaneous real appreciation, whereas the (net) intertemporal and welfare
effects favor a real depreciation.^9
An (Expected) Future Change in the Terms of Trade
Consider now the effect that an (expected) future deterioration in the terms of trade would have on the path of the real exchange rate; that is.
${\stackrel{̂}{p}}_{m0}=0$ and ${\stackrel{̂}{p}}_{m1}>0$. In this case, the horizontal shifts in the N[0]N[0] and N[1]N[1] schedules are given by
$\frac{d\mathrm{log}{p}_{n0}}{d\mathrm{log}{p}_{m1}}{{|}_{{N}_{0}}}_{{N}_{0}}=\frac{{\beta }_{m1}\gamma \left[\sigma -\left(1-{\mu }_{m1}\right)\right]}{{\beta }_{m0}{\sigma }_{nm}+{\beta }_{x0}{\
sigma }_{nx}+{\beta }_{n0}\gamma \sigma }\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(12\right)$
$\frac{d\mathrm{log}{p}_{n0}}{d\mathrm{log}{p}_{m1}}{{|}_{{N}_{1}}}_{{N}_{1}}=\frac{-{\beta }_{m1}\left\{{\sigma }_{nm}-\left[\left(1-\gamma \right)\sigma +\gamma \left(1-{\mu }_{m1}\right)\right]\
right\}}{{\beta }_{n0}\left(1-\gamma \right)\sigma }.\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(13\right)$
Consider first equation (12). A rise in p[ml] affects the period-0 equilibrium condition through two channels. By lowering the real interest rate, the future deterioration in the terms of trade
causes substitution of aggregate spending (part of which falls on nontradable goods) from period 1 to period 0. The magnitude of this effect is governed by the intertemporal elasticity of
substitution, σ, and is positive in terms of its impact on today’s relative price of home goods, p[n0]. In contrast, the future deterioration in the terms of trade lowers welfare and hence reduces
the demand for home goods today. The magnitude of this effect, which is negative in terms of its impact on today’s relative price of home goods, is proportional to the ratio of imports to consumption
of importables in period 1, (1—µ[m1]). Overall, the N[0]N[0] locus shifts to the right if the intertemporal substitution effect outweighs the welfare effect—that is, if σ > (1—µ[m1])—and conversely.
Consider now the N[1]N[1] schedule. From equation (13), we see that an (expected) future deterioration in the terms of trade affects the period-1 market for nontradable goods in a way that is
analogous to the effect of a current terms of trade shock on the period-0 market for home goods. The only real difference arises from changes in the values of the various elasticities over time,
which are in turn functions of the underlying parameters: the expenditure shares, the temporal and intertemporal elasticities of substitution, the average propensity to save, and the ratio of imports
to consumption of importables. Accordingly, if importables and nontradables are Hicksian substitutes, the future deterioration in the terms of trade raises the demand for c[nl] through the temporal
substitution effect. In addition to the temporal substitution effect, there is a negative effect on aggregate real spending, C[1]. that tends to reduce the demand for c[n1], This effect operates
through two channels: a negative real wealth effect, which equals—β[m1] γ(1—µ[m1]), and a negative intertemporal substitution effect, which equals—β[m1] (1—γ)σ. The latter effect reflects the fall in
the real rate of interest caused by the increase in the price of importables that is expected in the future. Thus, the overall shift in the N[1]N[1] schedule reflects the sum of the positive temporal
substitution effect (if the goods are Hicksian substitutes) and the negative real spending effect.
Figure 3.
Effect of an (Expected) Future Deterioration in Terms of Trade on Path of Real Exchange Rate
Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A001
Figure 3.
Effect of an (Expected) Future Deterioration in Terms of Trade on Path of Real Exchange Rate
Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A001
Figure 3.
Effect of an (Expected) Future Deterioration in Terms of Trade on Path of Real Exchange Rate
Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A001
In Figure 3, a benchmark case is depicted in which the economy operates in the autarky equilibrium in period 1, so that ε[m1] is equal to unity. In this case, the rise in p[m1] necessarily causes the
N[0]N[0] schedule to shift to the right. The figure shows two possibilities for the N[1]N[1] schedule. In panel A, the temporal elasticity of substitution is assumed to exceed the (absolute value of
the) compensated elasticity of period-1 real spending with respect to the real discount factor. In that case, the rise in p[m1] causes excess demand for c[nl], and the N[1] N[1] schedule shifts to
the left. In panel B, the opposite case is considered, in which σ[nm] < (1 - γ)σ.
In panel A, the equilibrium moves from point A to point B, and the real exchange rate necessarily appreciates today as well as in the future. In general, however, whether p[n0] rises more or less
than p[nl] (both cases are possible) depends on the relative magnitudes of σ and σ[nm], the temporal and intertemporal elasticities of substitution. Thus, as in the case of a temporary current shock
to the terms of trade, the real exchange rate may either over- or undershoot its new long-run value.^10 Note also that, even though no “fundamental” has changed in period 0, part of the adjustment of
the real exchange rate occurs in that period.
In panel B, the new equilibrium is at point B’, at which p[n0] rises and p[n1] falls. This result is not general, however, as experimenting with the figure will reveal.
Recall now the general case: µ[m1] ≠ 1. In the Appendix it is shown that the equilibrium response of today’s relative price of nontradable goods to a future deterioration in the terms of trade (the
response of p[n1] is also given in the Appendix) can be given by
$\frac{d\mathrm{log}{p}_{n0}}{d\mathrm{log}{p}_{m1}}={\mathrm{\Delta }}_{3}\text{[}\sigma -\left(1-{\mu }_{m1}\right)\right]+{\mathrm{\Delta }}_{4}\left[{\sigma }_{nm}-\left(1-{\mu }_{m1}\right)\
$\begin{array}{cc}{\mathrm{\Delta }}_{3}& =\frac{\gamma {\beta }_{m1}\left({\beta }_{m1}{\sigma }_{nm}+{\beta }_{x1}{\sigma }_{nx}\right)}{\left({\beta }_{m0}{\sigma }_{nm}+{\beta }_{x0}{\sigma }_
{nx}\right)\left[{\beta }_{m1}{\sigma }_{nm}+{\beta }_{x1}{\sigma }_{nx}+{\beta }_{n1}\left(1-\gamma \right)\sigma \right]+{\beta }_{n0}\gamma \sigma \left({\beta }_{m1}{\sigma }_{nm}+{\beta }_{x1}{\
sigma }_{nx}\right)}\\ {\mathrm{\Delta }}_{4}& =\frac{\gamma {\beta }_{m1}{\beta }_{n1}\sigma }{\left({\beta }_{m0}{\sigma }_{nm}+{\beta }_{x0}{\sigma }_{nx}\right)\left[{\beta }_{m1}{\sigma }_{nm}+
{\beta }_{x1}{\sigma }_{nx}+{\beta }_{n1}\left(1-\gamma \right)\sigma \right]+{\beta }_{n0}\gamma \sigma \left({\beta }_{m1}{\sigma }_{nm}+{\beta }_{x1}{\sigma }_{nx}\right)},\end{array}$
and where Δ[3] > 0 and Δ[4] > 0. The analysis of anticipated future shocks to the terms of trade yields Proposition 2.
Proposition 2. An (expected) future change in the terms of trade will in general alter the real exchange rate in the present; that is, in periods before any “fundamental” has changed. A future
deterioration in the terms of trade will cause a real appreciation today if the temporal elasticity of substitution between importables and nontradables, σ[nm], and the intertemporal elasticity of
substitution, σ, both exceed the critical value, (1 - μ[m1]). which equals the ratio of imports to consumption of importables in period 1. If both elasticities fall short of the critical value,
however, a future deterioration in the terms of trade causes a real depreciation in the present.
Figure 4.
Effect of a Permanent Deterioration in Terms of Trade on Path of Real Exchange Rate
Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A001
Figure 4.
Effect of a Permanent Deterioration in Terms of Trade on Path of Real Exchange Rate
Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A001
Figure 4.
Effect of a Permanent Deterioration in Terms of Trade on Path of Real Exchange Rate
Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A001
Permanent Changes in the Terms of Trade
Consider now the effect of a permanent deterioration in the terms of trade; that is, ${\stackrel{̂}{p}}_{m0}={\stackrel{̂}{p}}_{m1}={\stackrel{̂}{p}}_{m}>0$. If it is assumed that the expenditure share
and production-consumption ratio of importables do not vary over time, the horizontal shifts in the N[0]N[0], and N[1]N[1] schedules are
$\frac{d\mathrm{log}{p}_{n0}}{d\mathrm{log}{p}_{m}}{|}_{{N}_{0}{N}_{0}}=\frac{{\beta }_{m}\left[{\sigma }_{nm}-\left(1-{\mu }_{m}\right)\right]}{{\beta }_{m}{\sigma }_{nm}+{\beta }_{x0}{\sigma }_{nx}
+{\beta }_{n0}\gamma \sigma }\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(15\right)$
$\frac{d\mathrm{log}{p}_{n0}}{d\mathrm{log}{p}_{m}}{|}_{{N}_{1}{N}_{1}}=\frac{-{\beta }_{m}\left[{\sigma }_{nm}-\left(1-{\mu }_{m}\right)\right]}{{\beta }_{n0}\left(1-\gamma \right)\sigma }.\phantom
Thus, a permanent deterioration in the terms of trade has two effects. First, by permanently raising the relative price of importables, the change in the terms of trade permanently alters the
temporal composition of spending. The magnitude of the temporal substitution effect is governed by σ[nm] the elasticity of substitution between importables and nontradables. Second, the permanent
deterioration in the terms of trade lowers welfare. The magnitude of this welfare effect, which is always negative in terms of its impact on the demand for c[n0] or c[n1] is governed by (1 - µ[m]),
the (assumed to be constant) ratio of imports to consumption of importables. The N[0]N[0] schedule clearly must shift to the right, and the N[1]N[1] schedule to the left, if σ[nm] > (1 - µ[m]), and
These two cases are illustrated in Figure 4. In panel A, it is assumed that σ[nm] > (l—µ[m]). In that case, a permanent deterioration in the terms of trade leads to a real appreciation in both
periods. In panel B, it is assumed that σ[nm] < (1 - µ[m]), so that the permanent deterioration in the terms of trade leads to a real depreciation in both periods. The intuition of this result is
clear: if σ[nm] >(1—µ[m]), the positive substitution effect outweighs the negative welfare effect so that the rise in p[m] raises (permanently) the demand for nontradables. Given the supply, a rise
in both p[n0], and p[n1] is necessary to clear the home-goods sector. Conversely, if σ[nm] >(1—µ[m]), the negative welfare effect dominates, and a permanent deterioration in the terms of trade lowers
permanently the demand for home goods, resulting in a real depreciation in both periods.
In the Appendix, it is shown that the equilibrium response of today’s relative price of nontradable goods to a permanent deterioration in the terms of trade (the response of p[n1] is also given in
the Appendix) is
$\frac{d\mathrm{log}{p}_{n0}}{d\mathrm{log}{p}_{m}}={\mathrm{\Delta }}_{5}\text{[}{\sigma }_{nm}-\left(1-{\mu }_{m}\right)\right],\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(17\right)$
${\mathrm{\Delta }}_{5}=\frac{{\beta }_{m}\left({\beta }_{m}{\sigma }_{nm}+{\beta }_{x1}{\sigma }_{nm}+{\beta }_{x1}{\sigma }_{nx}+{\beta }_{n1}\sigma \right)}{\left[{\beta }_{m}{\sigma }_{nm}+{\beta
}_{x1}{\sigma }_{nx}+{\beta }_{n1}\left(1-\gamma \right)\sigma \right]\left({\beta }_{m}{\sigma }_{nm}+{\beta }_{x0}{\sigma }_{nx}\right)+{\beta }_{n0}\gamma \sigma \left({\beta }_{m}{\sigma }_{nm}+
{\beta }_{x1}{\sigma }_{nx}\right)}>0.$
Note that, from equation (17), the effect of a permanent change in the terms of trade on the real exchange rate is very similar to the effect derived in the context of static (one-period) models (for
example, Dornbusch (1974) or Neary (1988)). The analysis of permanent terms of trade changes leads to the following proposition.
Proposition 3. The effect of a permanent terms of trade disturbance on the real exchange rate depends on the relative magnitudes of the temporal elasticity of substitution between importables and
nontradables, σ[nm]. and the ratio of imports to consumption of importables, (1—µ[m]). If the value of σ[nm] exceeds this critical ratio, a permanent terms of trade deterioration causes a real
appreciation in both periods, and conversely. That intertemporal considerations are absent from the analysis is a consequence of the assumption of constant expenditure shares. Under this assumption,
a permanent change in the terms of trade does not alter the (consumption-based) real rate of interest.
III. The Harberger-Laursen-Metzler Effect in the Presence of Nontradable Goods
The effect that various shocks to the terms of trade have on the path of the real exchange rate is an important ingredient in the analysis of the H-L-M effect. In particular, the total effect of a
terms of trade change on the trade balance can be decomposed into a direct effect (that is, the effect with the real exchange rate held constant) and an indirect effect (operating through changes in
the real exchange rates caused by the shock to the terms of trade, which in turn feed back to alter the trade balance). Accordingly, one may write
$\frac{d\left(T{A}_{c}{\right)}_{0}}{d\mathrm{log}{p}_{m0}}=\frac{\partial \left(T{A}_{c}{\right)}_{0}}{\partial \mathrm{log}{p}_{m0}}+\underset{t=0}{\overset{1}{\mathrm{\Sigma }}}\frac{\partial \
left(T{A}_{c}{\right)}_{0}}{\partial \mathrm{log}{p}_{nt}}\frac{d\mathrm{log}{p}_{nt}}{d\mathrm{log}{p}_{m0}}\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(18\right)$
$\frac{d\left(T{A}_{c}{\right)}_{0}}{d\mathrm{log}{p}_{m1}}=\frac{\partial \left(T{A}_{c}{\right)}_{0}}{\partial \mathrm{log}{p}_{m1}}+\underset{t=0}{\overset{1}{\mathrm{\Sigma }}}\frac{\partial \
left(T{A}_{c}{\right)}_{0}}{\partial \mathrm{log}{p}_{nt}}\frac{d\mathrm{log}{p}_{nt}}{d\mathrm{log}{p}_{m1}}\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(19\right)$
$\frac{d\left(T{A}_{c}{\right)}_{0}}{d\mathrm{log}{p}_{m}}=\frac{\partial \left(T{A}_{c}{\right)}_{0}}{\partial \mathrm{log}{p}_{m}}+\underset{t=0}{\overset{1}{\mathrm{\Sigma }}}\frac{\partial \left
(T{A}_{c}{\right)}_{0}}{\partial \mathrm{log}{p}_{nt}}\frac{d\mathrm{log}{p}_{nt}}{d\mathrm{log}{p}_{m}}\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(20\right)$
where (TA[c])[0] is the consumption-based trade balance in period 0, which is defined as
$\left(T{A}_{c}{\right)}_{0}=\left(GD{P}_{c}{\right)}_{0}-{C}_{0}\left({\alpha }_{c1,}{W}_{c0}\right),\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(21\right)$
The first term on the right-hand side in each of equations (18)-(20) corresponds to the direct effect, and the terms inside the summation signs represent the indirect effect of a terms of trade
change on the balance of trade.
Differentiating equations (21) and (22) yields
$\frac{\partial \left(T{A}_{c}{\right)}_{0}}{\partial \mathrm{log}{p}_{n0}}={\beta }_{n0}\text{[}\left(1-{\mu }_{c0}\right)+\gamma \sigma \right]{C}_{0}\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(23\
$\frac{\partial \left(T{A}_{c}{\right)}_{0}}{\partial \mathrm{log}{p}_{n1}}={\beta }_{n1}\left(\gamma \sigma \right){C}_{0},\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(24\right)$
where µ[c0] is the ratio of real gross domestic product (GDP) to real spending in period 0, a positive number that is greater or less than unity as the period-0 trade balance is in surplus or
As can be seen from equation (23), a temporary current rise in the relative price of home goods, p[n0], affects the consumption-based trade balance through two separate channels: a real income
effect, β[n0](1 - µ[c0]), and a real spending effect, β[n0]γσ. The real income effect is positive if µ[c0] < 1 (that is, if at the initial terms of trade the country has a trade deficit) and is
negative if µ[c0] > 1 (that is, if initially the country has a trade surplus). The reason is clear: if µ[c0] < 1, there is excess demand for tradables in period 0, so that a fall in their relative
price (a rise in p[n0]) raises real income. Conversely, if µ[c0] > 1, there is excess supply of tradables in period 0, so that a fall in their relative price lowers real GDP.
The only mechanism through which a change in p[n0] affects real spending, C[0], is intertemporal substitution. This is so because of the assumption that the home-goods market clears in each period.
Because non-tradable goods are neither in excess demand nor excess supply, there is no aggregate welfare effect from a change in their relative price. However, a rise in p[n0] raises the real rate of
interest relevant for consumption decisions, and this increase reduces current spending, which corresponds to an improvement in the current-period trade balance.
Consider now equation (24). A rise in p[nl] affects the period-0 trade balance only by altering real spending, C[0], because there is no effect of a change in p[n1] on current real income, (GDP[c])
[0]. Again, and for the same reason as above, real spending is affected only by altering the intertemporal terms of trade. The rise in p[n1] raises the discount factor relevant for consumption
decisions, which in turn raises current-period real spending. The increase in real spending corresponds to a worsening of the period-0 trade balance.
Permanent Changes in the Terms of Trade
Substituting the relevant expressions into equation (20) and assuming that the expenditure shares and importables production-consumption ratio do not vary over time allow the total effect of a terms
of trade change on the period-0 real trade balance to be written as
$\frac{\partial \left(T{A}_{c}{\right)}_{0}}{\partial \mathrm{log}{p}_{m}}={\beta }_{m}\left(1-{\mu }_{c0}\right){C}_{0}+{\beta }_{n}\left(1-{\mu }_{c0}\right){C}_{0}\left[{\sigma }_{nm}-\left(1-{\mu
}_{m}\right)\right]{\mathrm{\Delta }}_{6},\phantom{\rule[-0.0ex]{5em}{0.0ex}}\left(25\right)$
${\mathrm{\Delta }}_{6}={\beta }_{m}/\left({\beta }_{m}{\sigma }_{nm}+{\beta }_{x}{\sigma }_{nx}\right)>0.$
Consider the first term in equation (25), which represents the direct effect. As can be seen, the sign of the direct effect is determined solely by the initial position of the country’s trade
balance: the sign is positive if, at the initial relative price structure, the economy runs a trade deficit (µ[c0] < 1); it is negative if the country has a trade surplus (µ[c0] > 1), If initially
the trade account is balanced, then the permanent rise in the relative price of imports lowers real income and spending by the same amount, and the trade balance is unchanged.
The second term on the right-hand side of equation (25) is the indirect effect, and its sign depends on two factors: first, whether ${{\sigma }_{nm}}_{<}^{>}1-{\mu }_{m}$; second, whether ${{\mu }_
{c0}}_{<}^{>}1$. The sign of the expression [σ[nm] - (1 - µ[m])] determines the sign of the change in the relative price of home goods as a result of the deterioration in the terms of trade. If σ[nm]
> 1 - µ[m], the permanent rise in p[m] leads to a permanently lower real exchange rate, and conversely. The term [1 - µ[c0]] translates the change in the real exchange rate into a change in the
balance of trade.^11
The analysis suggests that if (1 - µ[m]) > σ[nm], so that a terms of trade deterioration causes a real depreciation, the direct and indirect effects of the terms of trade change will be opposite in
sign. It can be verified that, for certain values of the parameters, the indirect effect may even outweigh the direct effect.
Thus, although the assumption of no nontradable goods (a value of β[n] close to zero) implies that, from an initial position of trade deficit, a permanent terms of trade deterioration always improves
the trade balance, the analysis here suggests that a comovement consisting of (1) deteriorating (improving) terms of trade, (2) a negative and worsening (positive and improving) trade balance, and
(3) real depreciation (real appreciation) is a theoretical possibility.^12 These types of comovement would be difficult to explain in the context of models without nontradable goods. This analysis
suggests a fourth proposition.
Proposition 4. The response of the current account to a permanent disturbance in the terms of trade will be qualitatively similar in models with and without nontradable goods if the elasticity of
substitution between home goods and importables, σ[nm], exceeds the ratio of imports to consumption of importables, 1—µ[m], or if trade is initially balanced. For values of σ[nm] falling short of
this critical ratio, the behavior of the trade balance may differ qualitatively in the two types of model. For example, a deterioration in the terms of trade may lead to a worsened trade balance
(from an initial position of deficit), and an improvement in the terms of trade may lead to an increase in the trade surplus. In the first case, the worsened trade balance will be accompanied by a
real depreciation, whereas in the second case the larger surplus will be accompanied by a real appreciation.
Temporary Changes in the Terms of Trade
To sharpen the analysis, a benchmark case is considered in which, in the initial equilibrium, the trade account is balanced.^13 Substituting the relevant expressions into equations (18) and (19)
$\frac{d\left(T{A}_{c}{\right)}_{0}}{d\mathrm{log}{p}_{m0}}={\beta }_{m}\gamma \text{[}\sigma -\left(1-{\mu }_{m}\right)\right]{C}_{0}+{\beta }_{n}\gamma \sigma \left[{\sigma }_{nm}-\sigma \right]{C}
_{0}{\mathrm{\Delta }}_{7}\phantom{\rule[-0.0ex]{6em}{0.0ex}}\left(26\right)$
where Δ[7]= β[m][β[m] σ[nm]+ β[x] σ[nx] + β[n] σ]^-1 > 0.
Consider first the direct effect (the first term on the right-hand side) in equation (26). A rise in p[m0] has three effects on the period-0 trade balance. First, real GDP falls by the amount β[m]
(1—µ[m]); that is, in proportion to the (assumed to be constant) volume of imports. This is the negative real income effect associated with a deterioration in the terms of trade. Second, the rise in
p[m0] raises the consumption-based rate of interest, which lowers real spending, C[0]. The magnitude of this effect is governed by the product of β[m], which equals the fall in the real discount
factor, and γσ, the compensated elasticity of C[0] with respect to the discount factor. Third, the rise in p[m0] lowers lifetime real wealth. The proportional fall in lifetime wealth is a fraction,
(1 - γ), of the fall in current period real GDP, β[m](1—µ[m]). The fall in real wealth lowers the volume of spending, C[0], by the amount β[m](1—γ)(1—µ[m]), given the assumption of homotheticity.
Summing these three effects yields precisely the first expression in equation (26).
The direct effect is therefore positive or negative according to whether ${\sigma }_{<}^{>}\left(1-{\mu }_{m}\right)$. This result shows that, in response to a temporary deterioration in the terms of
trade, the trade balance need not behave as a “shock absorber” and move into deficit.^14 The reason is of course that a terms of trade shock is a particular kind of real income shock. Like a negative
supply shock, a deterioration in the terms of trade lowers real income. Unlike a negative supply shock, however, a deterioration in the terms of trade raises the real rate of interest (for this small
open economy). The rise in the rate of interest reduces real spending through the intertemporal substitution channel. If this force is sufficiently powerful—that is, if σ >(l—µ[m])—then a temporary
adverse movement in the terms of trade will actually cause the trade balance to move into surplus.
Equation (26) states that the sign of the indirect effect (second term on the right-hand side) of a temporary current deterioration in the terms of trade depends on the relative magnitudes of the
temporal and intertemporal elasticities of substitution. The intuition of this result is as follows. A rise in p[m0] induces substitution among goods within period 0. The magnitude of this
substitution is governed by σ[nm], the temporal elasticity of substitution between nontradables and importables. If the two goods are Hicksian substitutes, the temporal substitution effect raises the
real exchange rate ratio, p[n0]/p[n1].
In contrast, a rise in p[m0] raises the consumption-based rate of interest and induces substitution of aggregate spending (part of which falls on home goods) from period 0 to period 1. This
intertemporal substitution effect is negative in its impact on the price ratio, p[n0]/p[n1]. Its magnitude is governed by a, the intertemporal elasticity of substitution.
Thus, if σ[nm] > σ, so that the temporal substitution effect dominates the intertemporal substitution effect, the rise in p[m0] raises the price ratio, p[n0]/p[n1], and the sign of the indirect
effect is necessarily positive. This is the case of equilibrium overshooting, according to which the (absolute value of the) short-run (period-0) change in the real exchange rate exceeds the long-run
(period-1) change. Conversely, if a temporary current deterioration lowers the home-goods price ratio, which will occur if σ[nm] < σ, then the sign of the indirect effect is negative. In this case of
equilibrium undershooting, there is a real appreciation in the long run (period 1) relative to the short run (period 0).^15
The analysis above suggests that the direct and indirect effects of temporary terms of trade disturbances depend on different parameters of the model. Specifically, the sign of the direct effect
depends on the relative magnitudes of the intertemporal elasticity of substitution, σ, and the share of imports in the consumption of importables, 1 - µ[m]. On the other hand, the sign of the
indirect effect depends on the relative magnitudes of the temporal and intertemporal elasticities of substitution, σ and σ[nm]. This observation shows that the direct and indirect effects may
actually be of opposite sign. Furthermore, for certain parameter configurations, equation (26) reveals that the indirect effect may dominate the direct effect (see the Appendix). Thus, as in the case
of permanent shocks, the real exchange rate may be an important variable through which terms of trade shocks are transmitted to the current account.
The preceding analysis has concentrated on temporary current, rather than (expected) future, shocks to the terms of trade. The underlying symmetry between equations (26) and (27) suggests that a
separate analysis of future shocks is not required.^16 One can note briefly, however, that in the case of an (expected) future deterioration in the terms of trade, the direct effect on today’s trade
balance will be positive if σ < (1 - µ[m]). This is the case in which the current account acts as a shock absorber, moving into surplus in anticipation of a real income loss in period 1. If σ > (1—µ
[m]), however, then the direct effect will worsen the trade balance as a result of an (expected) future deterioration in the terms of trade. The intuition here is that the expected rise in p[ml]
lowers the real rate of interest, which raises real spending today, Co; this effect dominates the consumption-smoothing motive (or welfare effect), which by itself lowers real spending and hence
improves the trade balance. The direct effect is therefore negative if the welfare (or consumption-smoothing) effect is weak relative to the intertemporal substitution effect—that is, if σ > (1—µ[m])
—and conversely.
The sign of the indirect effect of a rise in p[m1] depends on the response of the time path of the real exchange rate, p[n0]/p[n1]. The parameters that influence this time path are the temporal and
intertemporal elasticities of substitution. If σ[nm] > σ, then a rise in p[ml] lowers the price ratio, p[n0]/p[n1], which in turn lowers the consumption-based rate of interest. Real spending, C[0],
therefore rises, and the indirect effect contributes to a worsening of the trade balance. In contrast, if σ[nm] < σ, a rise in p[m1] raises the price ratio. p[n0]/p[n1], which in turn contributes to
a higher real rate of interest. The indirect effect contributes to a fall in C[0], hence to an improvement in the period-0 trade balance. A further proposition is suggested.
Proposition 5, Temporary terms of trade disturbances will in general have different effects in models with or without nontradable goods. The fundamental reason underlying these differences is that
the time path of the real exchange rate (which is a key determinant of the real rate of interest and, hence, of real spending and the trade balance) is an endogenous variable that responds to
disturbances in the terms of trade. For certain parameter configurations, a temporary terms of trade shock will lead to a deterioration in the trade balance when the response of the real exchange
rate is taken into account even if, with the real exchange rate held constant, the trade balance improves, and conversely.
Figure 5 illustrates such a result. Real income and spending in period 0 are plotted on the horizontal axis, and the corresponding period-1 variables are shown on the vertical axis. The initial real
income and spending points are denoted I[0] and S[0], respectively. It is assumed, in Figure 5, that the initial trade balance is zero, so that I[0], S[0] occur at the same point. The solid budget
line in Figure 5 represents the lifetime solvency constraint, and its slope is (minus) unity plus the real rate of interest.
Consider now a temporary current deterioration in the terms of trade. This causes the real income point to shift from I[0] to I[1]. At a constant real discount factor, the real spending point shifts
from S[0] to S[1]. The period-0 trade balance unambiguously turns negative, with the magnitude of the deficit being equal to the horizontal distance between I[1] and S[1].
Figure 5.
Harberger-Laursen-Metzler Effect with Nontradable Goods: Effect of a Temporary Current Deterioration in Terms of Trade on the Trade Balance
Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A001
Figure 5.
Harberger-Laursen-Metzler Effect with Nontradable Goods: Effect of a Temporary Current Deterioration in Terms of Trade on the Trade Balance
Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A001
Figure 5.
Harberger-Laursen-Metzler Effect with Nontradable Goods: Effect of a Temporary Current Deterioration in Terms of Trade on the Trade Balance
Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A001
However, in general the real rate of interest will not remain unchanged. First, the current deterioration in the terms of trade tends to raise the real rate of interest, with the real exchange rate
held constant. This change is captured by a clockwise pivot of the budget line through point I[1], where the new slope equals (minus) ${R}_{c0}^{\prime }>{R}_{c0}$. Real spending moves to a point
such as S[2]. along a new Engel curve (not shown) corresponding to the higher real interest rate. The movement from the pair (I[0],S[0]) to the pair (I[1],S[2]) corresponds to the direct effect in
equation (26). Accordingly, it is assumed that the direct effect on the trade balance is negative.
Consider now the indirect effect, and suppose that the temporary terms of trade deterioration results in a steeper time profile of the real exchange rate (that is. p[no]/p[n1] rises). This effect
raises the real rate of interest and causes a further clockwise pivot of the budget line through point I[1]. The slope of the new budget line is equal to (minus) R”[c0] > R’[c0], and spending moves
to a point such as S[3], at the intersection of a new Engel curve (not shown) and the budget line. At S[3], there is a trade surplus corresponding to the horizontal distance between I[1] and S[3].
Thus, although the direct effect, which corresponds to the movement from (I[0], S[0]) to (I[1], S[2]) lowers the trade balance, the total effect, which corresponds to the movement from (I[0], S[0])
to (I[1], S[3]), yields a trade surplus. The fundamental reason for these different results is the endogenous movement in the time profile of the real exchange rate.
IV. Conclusions and Extensions
This paper has used an intertemporal, real model of a small country, in which optimizing agents consume three goods in each period, to determine to what extent the introduction of a nontradables
sector alters the relationship between changes in the terms of trade and the balance of trade. An answer to this question required an understanding of how the two temporal relative prices—the terms
of trade and the real exchange rate—are linked. Therefore, an analysis of the effects of various terms of trade shocks on the real exchange rate was undertaken. These changes in the real exchange
rate represent a separate and distinct channel through which changes in the terms of trade affect a country’s trade balance.
Specifically (and schematically), the effects of temporary shocks to the terms of trade were found to depend critically on two factors: first, the relative magnitudes of temporal and intertemporal
elasticities of substitution; second, the relative magnitudes of the intertemporal elasticity of substitution and the ratio of imports to consumption of importables. The first factor determines the
effect of the terms of trade change on the time profile of the real exchange rate (which is a key determinant of the real rate of interest and, hence, of real spending and the trade balance), whereas
the second determines the effect of the terms of trade change on the trade balance, with the real exchange rate held constant. It was shown that, for certain parameter configurations, the predictions
of models that incorporate nontradable goods may differ from those of models that do not.^17 The real exchange rate is potentially an important transmission mechanism of terms of trade shocks to the
current account.
The analysis of permanent shocks revealed that the initial trade balance position and the relative magnitudes of the temporal elasticity of substitution and the ratio of imports to consumption of
importables are important factors that determine the behavior of the trade balance. This finding contrasts with previous analysis (not incorporating nontradable goods), which showed that the initial
borrowing position of the country was the main factor determining the behavior of the current account as a result of a permanent terms of trade shock.
One possible extension of the model developed here would involve analyzing the role that capital mobility plays in determining the comovement of the terms of trade and the real exchange rate. In this
paper, agents were assumed to have perfect access to the world capital market; an assumption of restricted access to world financial markets would be worth examining.
A second extension might involve relaxing the small-country assumption in order to analyze, within the context of a two-country general equilibrium model of the world economy, how various shocks
(such as changes in fiscal and commercial policies, or supply shocks) affect the comovements of world real interest rates, real exchange rates at home and abroad, and the terms of trade.^18
The Basic Model
This Appendix provides some guide posts to the main derivations for equations contained in the body of the paper.
On numerous occasions in the text, use is made of the relationship between gross elasticities and Hicks-Allen elasticities of substitution. The Allen elasticity of substitution between goods i and j,
σ[ij] = σ[ij], equals ${\stackrel{-}{\eta }}_{ij}/{\beta }_{j}$, where
is the compensated elasticity of demand of good i with respect to a change in price p[j]. Using this definition, the Slutsky decomposition of a total elasticity into its corresponding substitution
and income effect components, and the homogeneity property of demand functions, one obtains a relationship between gross elasticities and an expenditure-share-weighted average of the elasticities of
substitution and total spending (or wealth) elasticities. If, in addition to homothetic subutility functions, one assumes that the intertemporal utility function is itself homothetic, the
elasticities of demand with respect to spending and of spending with respect to lifetime wealth will both be equal to unity. The following relationships used in the paper are now readily derivable:
${\eta }_{c0\alpha }=\gamma \left(\sigma -1\right)\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(28\right)$
${\eta }_{c1\alpha }=-\left[\left(1-\gamma \right)\sigma +\gamma \right]\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(29\right)$
${\eta }_{ntpnt}=-{\beta }_{mt}{\sigma }_{nm}-{\beta }_{x1}{\sigma }_{nx}-\beta {}_{nt}\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(30\right)$
${\eta }_{ntpnt}={\beta }_{mt}\left({\sigma }_{nm}-1\right),\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(31\right)$
where σ, the intertemporal elasticity of substitution, is defined as
$\sigma =\frac{\partial \mathrm{log}\left({C}_{1}/{C}_{0}\right)}{\partial \mathrm{log}\left[\left(\partial U/\partial {C}_{0}\right)/\left(\partial U/\partial {C}_{1}\right)\right]}.\phantom{\rule
Note that, since σ>0 and 0<γ<1, η[c1α] is negative. However, whether ${{\eta }_{c0\alpha }}_{<}^{>}0$ depends on the intertemporal elasticity of substitution. If σ > 1 (σ < 1), then a rise in the
“price” of C[1], α[c1], raises (lowers) demand for C[0], and η[c0α,] is positive (negative). Note also that, since -[β[mt] σ[nm] + β[xt]σ[nx]] is a compensated effect, it is nonpositive because of
the negative semidefiniteness of the Slutsky substitution matrix.
Many of the expressions in the paper follow from a solution to the system of market-clearing conditions in the nontradable-goods sector:
${c}_{no}\text{[}{p}_{n0},{p}_{m0},{P}_{0}{C}_{0}\text{(}{\alpha }_{c1},{W}_{c0}\text{)}\text{]}={\stackrel{-}{Y}}_{n0}\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(33\right)$
${c}_{n1}\text{[}{p}_{n1},{p}_{m1},{P}_{1}{C}_{1}\text{(}{\alpha }_{c1},{W}_{c0}\text{)}\text{]}={\stackrel{-}{Y}}_{n1}\phantom{\rule[-0.0ex]{8em}{0.0ex}}\left(34\right)$
Totally differentiated, these become
$\left({\eta }_{n0pn0}+{\beta }_{n0}\right){\stackrel{̂}{p}}_{n0}+\left({\eta }_{n0pm0}+{\beta }_{m0}\right){\stackrel{̂}{p}}_{m0}+{\eta }_{c0\alpha }{\stackrel{̂}{\alpha }}_{c1}+{\stackrel{̂}{W}}_{c0}=0
$\left({\eta }_{n1pn1}+{\beta }_{n1}\right){\stackrel{̂}{p}}_{n1}+\left({\eta }_{n1pm1}+{\beta }_{m1}\right){\stackrel{̂}{p}}_{m1}+{\eta }_{c1\alpha }{\stackrel{̂}{\alpha }}_{c1}+{\stackrel{̂}{W}}_{c0}=
where use has been made of the fact that the elasticity of the price index P, with respect to a change in one of the temporal relative prices (p[nt] or p[mt]) is simply the corresponding expenditure
share (β[NT] or β[MT]). It is assumed throughout that there are no supply shocks (endowments are constant) and that the world discount factor, α[x1], is given. In this case, the discount factor
relevant for domestic consumption, α[c1], evolves according to
${\stackrel{̂}{\alpha }}_{c1}=\beta {}_{n1}{\stackrel{̂}{p}}_{n1}+{\beta }_{m1}{\stackrel{̂}{p}}_{m1}-{\beta }_{n0}{\stackrel{̂}{p}}_{n0}-{\beta }_{m0}{\stackrel{̂}{p}}_{m0}.\phantom{\rule[-0.0ex]{8em}
Recalling that real wealth is given by
$\begin{array}{cc}{W}_{c0}=& \left\{\left[{\stackrel{̂}{Y}}_{x0}+{p}_{m0}{\stackrel{̂}{Y}}_{m0}+{p}_{n0}{\stackrel{̂}{Y}}_{n0}\right]+{\alpha }_{x1}\left[{\stackrel{̂}{Y}}_{x1}+{p}_{m1}{\stackrel{̂}{Y}}_
{m1}+{p}_{n1}{\stackrel{̂}{Y}}_{n1}\right]\\ & -\left(1+{r}_{x.-1}\right){B}_{-1}\right\}/{P}_{0},\phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(38\right)\end{array}$
one can totally differentiate equation (38) to obtain
${\stackrel{̂}{W}}_{c0}=\text{[}\left(1-\gamma \right){\mu }_{m0}-1\right]{\beta }_{m0}{\stackrel{̂}{p}}_{m0}-\gamma {\beta }_{n0}{\stackrel{̂}{p}}_{n0}+\gamma {\mu }_{m1}{\beta }_{m1}{\stackrel{̂}{p}}_
{m1}+\gamma \beta n1{\stackrel{̂}{p}}_{n1}.\phantom{\rule[-0.0ex]{6em}{0.0ex}}\left(39\right)$
Substituting equations (28)-(31), (37). and (39) into equations (34) and (35) yields the system
$\begin{array}{cc}& \left[\begin{array}{cc}-{\beta }_{m0}{\sigma }_{nm}-{\beta }_{x0}{\sigma }_{nx}-{\beta }_{n0}\gamma \sigma & \gamma {\beta }_{n1}\sigma \\ {\beta }_{n0}\left(1-\gamma \right)\
sigma & -{\beta }_{m1}{\sigma }_{nm}-{\beta }_{x1}{\sigma }_{nx}-{\beta }_{n1}\left(1-\gamma \right)\sigma \end{array}\right]\left[\begin{array}{c}{\stackrel{̂}{p}}_{n0}\\ {\stackrel{̂}{p}}_{n1}\end
{array}\right]\\ =& \left[\begin{array}{cc}-{\beta }_{m0}\left\{{\sigma }_{nm}-\left[\left(1-\gamma \right)\left(1-{\mu }_{m0}\right)+\gamma \sigma \right]\right\}& -{\beta }_{m1}\gamma \left[\sigma
-\left(1-{\mu }_{m1}\right)\right]\\ -{\beta }_{m0}\left(1-\gamma \right)\left[\sigma -\left(1-{\mu }_{m0}\right)\right]& -{\beta }_{m1}\left\{{\sigma }_{nm}-\left[\gamma \left(1-{\mu }_{m1}\right)+\
left(1-\gamma \right)\sigma \right]\right\}\end{array}\right]\left[\begin{array}{c}{\stackrel{̂}{p}}_{m0}\\ {\stackrel{̂}{p}}_{m1}\end{array}\right].\phantom{\rule[-0.0ex]{2em}{0.0ex}}\left(40\right)\
This system underlies the derivation of the slopes of the N[0] N[0] and N[1] N[1] schedules shown in Figures 1–4 as well as their shifts in response to various terms of trade changes.
Using equation (40) allows one to solve for
and in terms of and :
$\begin{array}{cc}{\stackrel{̂}{p}}_{n0}& ={\mathrm{\Delta }}^{-1}\text{[}{\beta }_{m0}\text{{}{\sigma }_{nm}-\text{[}\left(1-\gamma \right)\left(1-{\mu }_{m0}\right)+\gamma \sigma \right]\right\}\
left[{\beta }_{m1}{\sigma }_{nm}+{\beta }_{x1}{\sigma }_{nx}+{\beta }_{n1}\left(1-\gamma \right)\sigma \right]\\ & +{\beta }_{m0}\left(1-\gamma \right)\left[\sigma -\left(1-{\mu }_{m0}\right)\right]\
gamma {\beta }_{n1}\sigma \right]{\stackrel{̂}{p}}_{m0}\\ & {+\mathrm{\Delta }}^{-1}\left[\gamma {\beta }_{m1}\left[\sigma -1\left(1-{\mu }_{m1}\right)\right]\left[{\beta }_{m1}{\sigma }_{nm}+{\beta }
_{x1}{\sigma }_{nm}+{\beta }_{n1}-\gamma \right)\sigma \right]\\ & +{\beta }_{m1}\left\{{\sigma }_{nm}-\left[\gamma \left(1-{\mu }_{m1}\right)+\left(1-\gamma \right)\sigma \right]\right\}\gamma {\
beta }_{n1}\sigma \right]{\stackrel{̂}{p}}_{m1}\phantom{\rule[-0.0ex]{4em}{0.0ex}}\left(41\right)\end{array}$
$\begin{array}{cc}{\stackrel{̂}{p}}_{n1}& ={\mathrm{\Delta }}^{-1}\left[{\beta }_{m0}\left(1-\gamma \right)\left[\sigma -\left(1-{\mu }_{m0}\right)\right]\left[{\beta }_{m0}{\sigma }_{nm}+{\beta }_
{x0}{\sigma }_{nx}+{\beta }_{n0}\gamma \sigma \right]\\ & +{\beta }_{m0}\left(1-\gamma \right){\beta }_{n0}\sigma \left\{{\sigma }_{nm}-\left[\left(1-\gamma \right)\left(1-{\mu }_{m0}\right)+\gamma \
sigma \right]\right\}\right]{\stackrel{̂}{p}}_{m0}\\ & +{\mathrm{\Delta }}^{-1}\left[{\beta }_{m1}\left\{{\sigma }_{nm}-\left[\gamma \left(1-{\mu }_{m1}\right)+\left(1-\gamma \right)\sigma \right]\
right\}\left[{\beta }_{m0}{\sigma }_{nm}+{\beta }_{x0}{\sigma }_{nx}+{\beta }_{n0}\gamma \sigma \right]\\ & +{\beta }_{m1}\left(1-\gamma \right){\beta }_{n0}\gamma \sigma \left[\sigma -\left(1-{\mu }
$\mathrm{\Delta }=\left[{\beta }_{m0}{\sigma }_{nm}+{\beta }_{x0}{\sigma }_{nx}\right]\left[{\beta }_{m1}{\sigma }_{nm}+{\beta }_{x1}{\sigma }_{nx}+{\beta }_{n1}\left(1-\gamma \right)\sigma \right]+
{\beta }_{n0}\gamma \sigma \left[{\beta }_{m1}{\sigma }_{nm}+{\beta }_{x1}{\sigma }_{nx}\right]>0.$
Equations (11) and (14) of the text correspond to the coefficients of
and in equation (41). To determine the effects of permanent shocks, set ${\stackrel{̂}{p}}_{m0}={\stackrel{̂}{p}}_{m1}={\stackrel{̂}{p}}_{m}$ in equation (41). equation (42) is used to determine the
effect of terms of trade changes on the relative price of home goods in period 1.
Substitution of the relevant terms in equations (41) and (42) into equations (18)-(20) of the text underlies the main derivations in Section III of the paper.
Finally, some numerical examples are provided to illustrate the claim that the qualitative response of the trade account to a change in the terms of trade may depend on whether the model incorporates
nontradable goods. Consider first a permanent change in the terms of trade. Rewriting equation (25) of the text yields
$\frac{d\left(TAc{\right)}_{0}}{d\mathrm{log}{p}_{m}}=\frac{{\beta }_{m}\left(1-{\mu }_{c0}\right){C}_{0}}{\left[{\beta }_{m}{\sigma }_{nm}+{\beta }_{x}{\sigma }_{nx}\right]}\text{[}\left(1-{\beta }_
{x}\right){\sigma }_{nm}+{\beta }_{x}{\sigma }_{nx}-{\beta }_{n}\left(1-{\mu }_{m}\right)\right].\phantom{\rule[-0.0ex]{5em}{0.0ex}}\left(43\right)$
Consider the case in which the economy receives no endowment of importables (complete specialization), so that ³[m] = 0. Then let σ[nm] - σ[nx] = a, where 0 < a < 1 (all goods are net substitutes).
If the share of nontradables. β[n], is larger than a, the indirect effect will dominate the direct effect.
Consider now a temporary (current) change in the terms of trade. Rewriting equation (26) yields
$\frac{d\left(T{A}_{c}{\right)}_{0}}{d\mathrm{log}{p}_{m0}}=\frac{{\beta }_{m}\gamma {C}_{0}}{\left[{\beta }_{m}{\sigma }_{nm}+{\beta }_{x}{\sigma }_{nx}+{\beta }_{n}\sigma \right]}.\text{{[}\sigma -
\left(1-{\mu }_{m}\right)\right]\left[{\beta }_{m}{\sigma }_{nm}+{\beta }_{x}{\sigma }_{nx}+{\beta }_{n}\sigma \right]+{\beta }_{n}\sigma \left({\sigma }_{nm}-\sigma \right)\right\}.\phantom{\rule
Suppose again that µ„ = 0 (complete specialization).
Suppose now that the intertemporal utility function is logarithmic. Then the direct effect in equations (26) and (27) of the text is zero. In other words, with the real exchange rate held constant, a
temporary (current or future) terms of trade shock has no effect on the trade balance. When home goods are present in the model, however, it is clear that a temporary current deterioration in the
terms of trade results in a trade surplus if σ[nm] > 1 and in a trade deficit if σ[nm] < 1.
Consider another example. Maintaining the assumption that µ[m]=0, suppose that σ>1, so that the direct effect is positive in equation (26). Consider the following parameter values: σ[mn] = σ[nx] = a,
0 < a < 1, σ = 1 + a, and β[m] = β^x, = β^n = 1/3. If a< VT/3, then the direct effect is positive, but the total effect is negative.
Finally, consider the following parameter configuration: µ[m]=0, σ, 0 < a < 1, σ[nm] = σ[nx] = 1+a, β[m] = β[n] = β[x] = 1/3. If $a>\sqrt{2/3}$, then the direct effect is negative, but the total
effect is positive.
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Mr. Ostry is an economist in the Financial Studies Division of the Research Department. He holds a doctorate from the University of Chicago, as well as degrees from the London School of Economics and
Political Science. Oxford University, and Queen’s University (Canada).
This paper draws on two previous working papers (Ostry (1987, 1988a)) and the author’s Ph.D. dissertation (Ostry (1988b, Chapter 2)). The author thanks Jacob A. Frenkel, John Huizinga, and Assaf
Razin, who served on his thesis committee, for their encouragement and helpful comments. He also thanks his colleagues at Chicago, especially Carlos Végh for detailed comments.
This is the case in real models that abstract from physical capital and endogenous time preference. For a discussion of this issue in a monetary model with currency substitution, see Végh (1988; in
this issue of Staff Papers).
Marion (1984) discussed the effect of changes in the price of oil (as opposed to changes in the price of a final consumption good) in the context of a two-sector model with nontradable goods.
Greenwood (1984) examined the effect of terms of trade shocks on the real exchange rate and trade balance, but he considered the case of two goods on the production side and two goods on the
consumption side rather than the more general three-good model developed here. This accounts for some differences in the conclusions reached in the two papers. The papers by Edwards (1987) and Ostry
(1987) were developed independently and deal, in part, with similar issues.
The Appendix provides several numerical examples illustrating the reversals that may obtain when nontradable goods are introduced into the model.
The consumption-based discount factor, α[c1], is related to the (consumption-based) real rate of interest in the usual manner, namely α[c1] = (1 + r[c0])-1 where r[c0] is the (consumption-based) real
rate of interest. The relationship between the real rate of interest and the exogenous world rate of interest is given by r[c0] = (1 + r[x0])(P[0]/P[1]) -1. Therefore, a rise in P[0] or a fall in P
[1] raises the real rate of interest. (See, for example, Dornbusch (1983) or Frenkel and Razin (1987) on the consumption rate of interest.)
This verbal justification holds precisely for a flat spending profile over the life cycle; that is, γ = 1 - γ. However, it is easy to verify that, algebraically, the slope in equation (7) is always
larger than the slope in equation (8), even if γ ≠ 1 - γ.
Note that the sum of the gross temporal substitution effect and the price index effect equals β[m0] σ[nm], which is the net or compensated temporal substitution effect. The sign of the net temporal
substitution effect depends of course on whether ${{\sigma }_{nm}}_{<}^{>}0$; that is, on whether importables and nontradables are Hicksian substitutes or complements.
Panel A illustrates the case of equilibrium overshooting, according to which the short-run change in the real exchange rate exceeds the long-run change in absolute value (see Edwards (1987) for
further discussion of this issue). As shown below, whether there is equilibrium over- or undershooting is determined by the relative magnitudes of the temporal and intertemporal elasticities of
If σ[nm] > max {[(l - γ)(l - µ[m0]) + γσ], (1 - µ[m0])}, a temporary current deterioration in the terms of trade causes a real appreciation in the current period. If σ[nm] > min {[(l - γ)(l - µ[m0])
+ γσ], (1 - µ[m0])}, a temporary current deterioration in the terms of trade causes a real depreciation in the current period.
Panel A, for example, illustrates the case of equilibrium undershooting, according to which the long-run change in the real exchange rate exceeds the short-run change (in absolute value).
If trade is balanced initially, both the direct and indirect effects are zero in the case of a permanent change in the terms of trade.
Note that this is an equilibrium phenomenon and has nothing to do with lags in the adjustment of the current account to relative price changes, which underly the “J-curve.”
The role of initial trade account imbalances was discussed in the previous subsection. The ingredients necessary for consideration of the general case in which µ[c0] ≠ 1 are given in the Appendix.
As noted earlier, overshooting (undershooting) of the real exchange rate is defined as the short-run change being greater (less) than the long-run change (in absolute value), rather than as the
short-run level being higher (lower) than the long-run level.
This symmetry stems from the assumption of constant expenditure shares, constant production-consumption ratio of importables, and initially balanced trade.
Further differences arise if trade is initially not balanced, since in this case there are real GDP revaluation effects associated with real exchange rate changes. The direction of these effects was
discussed in the subsection on permanent terms of trade shocks; because the revaluation effects operate in essentially the same manner here, they were excluded in the subsection of the paper dealing
with temporary disturbances.
Ostry (1988b). Chapter 3, examines the effect of government purchases in such a model; Chapter 4 considers commercial policies. | {"url":"https://www.elibrary.imf.org/view/journals/024/1988/004/article-A001-en.xml","timestamp":"2024-11-14T17:05:50Z","content_type":"text/html","content_length":"861670","record_id":"<urn:uuid:00cf15ef-0d88-420c-abc5-239b00e97340>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00600.warc.gz"} |
Coupled Heat and Mass Transfer
I ultimately want to solve a coupled heat and mass transfer problem of two regions. One region is a liquid desiccant solution and the other region is humid air. The liquid desiccant takes up water
content from the air stream. Meanwhile sensible heat is transferred. Furthermore, latent heat is "generated" at the boundary, as the water changes phase from the air (vapor) to the solution (liquid).
I am using the official OpenFoam v7 on an Ubuntu machine.
• The humidity/water content in both air and solution is modelled by a passive scalar, which is called cw (mass fraction of water to total mass).
• Both solutions are incompressible with constant thermophysical properties (besides the vapor pressure, which depends on temperature and water content).
• Laminar steady state flow in both regions.
• Ideal gases.
• Indeformable, flat interface.
• On the interface between air and solution the temperatures (for heat transfer) and water vapor pressure (for mass transfer) are equal.
In order to solve this problem I started with the chtMultiRegionFoam solver, as it includes multiple regions and already an implementation of the sensible heat transfer. I first solve for the flow
field (U,p) and afterwards freeze the field and solve for the coupled heat and mass Transfer (T, cw). For both solvers, the air region is solved first and afterwards the solution region.
The water vapor pressure (ps) in the air region can be obtained directly from the water content and the system pressure ps_air=ps_air(cw_air,p_air). In the solution, the water vapor pressure is a
function of the water content and the solution temperature ps_sol = ps_sol(cw_sol,T_sol). This relation is given by a fit of experimental data (fit is conducted as an Antoine equation, with
coefficients A,B,C as a function of the water content cw_sol).
The idea behind the region coupling I got from Brent A. Craven and Robert L. Campbell on a presentation given at the 6th OpenFOAM Workshop (13–16 June 2011).
For the temperature interface I created a boundary condition which satisfies a thermodynamic equilibrium (equal temperatures at interface in both regions) and the energy balance. For this reason, the
temperature at the interface on the air side is set equal to the temperature at the interface on the solution side (optionally with a relaxation factor). For the solution region I set the gradient in
order to fulfill the energy balance. Therefore, the heat transferred to the solution region is equal to the sensible heat flow of the air region plus the latent heat transferred due to the absorbed
mass flux.
For the water content interface I created a boundary condition analog to the temperature boundary condition. In the air region, the water vapor pressure at the interface is set equal to the water
vapor pressure of the solution at the interface (again optionally with a relaxation factor). For the solution region the gradient of the water content is set to fulfill the mass balance. The gradient
is calculated according to the Eckert Schneider relation for equimolar diffusion dotM/A = (rho D snGrad(cw))/(1-cw), where dotM is the absorbed mass flux, A the surface area of the cell, rho the
density of the solution/air, D the diffusion coefficient of water in the solution/air and cw the water content.
So far, the simulation is working. However, there are two things I am concerned about:
1. The residuals of the water content stay somewhat high (1e-2 for the air region, 1e-4 for the solution region)
2. A step in the gradient for both temperature and water content inside the air region.
Especially the second thing concerns me. Doesn’t this imply, that the energy and mass balance are not fulfilled inside the air region?
This is the geometry which is simulated:
The air region is colored green and the solution region is colored red. Both are very long in z- and x- direction, but fairly thin in y-direction. The height (z) is 0.7m, the length (x) is 0.1m and
the width (y) is 6e.4m, 2e-3m (solution/air). The air flows in x direction (with a no slip condition at the interface) and the solution flows in z-direction (with a no slip condition at the wall, y=
These are the residuals of the flow problem in the air region (U,p):
As there are no velocities in z-direction the somewhat high residuals for Uz should not be a problem.
These are the residuals of the flow problem in the solution region (U,p):
As there are no velocities in the x-direction the somewhat high residuals for Ux should not to be a problem.
These are the residuals of the heat and mass transfer problem in the air region (T,cw):
These are the residuals of the heat and mass transfer problem in the solution region (T,cw):
This is the temperature distribution in the air region for different heights (z) in respect to the width (y).
A non-steady gradient is visible close to the interface (y=0).
This is the water content distribution in the air region for different heights (z) in respect to the width (y).
A non-steady gradient is visible close to the interface (y=0).
However, the total absorbed mass flux, which the solution takes up from the air stream, does not change anymore over successive iterations:
So my questions are:
• Do you know what causes the high residuals of the solution of the water content in the heat and mass transfer problem in both regions?
• What causes the unsteady gradient in the air region for both temperature and water content? Doesn't this imply, that the energy and mass balance are not fulfilled inside the air region?
To me it looks like the fixed gradient boundary condition gives problems to the solver and is actually not fulfilled.
If you need any further information about the solver (solver settings, discretization schemes, mesh, etc.) just leave a reply and I will update the post. The post already got way longer than I
wanted, therefore I didn't want to fill it with more content.
I already thank you a lot in advance! | {"url":"https://www.cfd-online.com/Forums/openfoam-solving/227458-coupled-heat-mass-transfer.html","timestamp":"2024-11-03T16:00:59Z","content_type":"application/xhtml+xml","content_length":"78599","record_id":"<urn:uuid:639b7eee-7b94-43cc-90f7-5a755f22e741>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00762.warc.gz"} |
1218 - Acman and his GirlFriend
1218 - Acman and his GirlFriend
ACman has been studying in HUST for nearly four years, and he thinks Wuhan is a very beautiful city even though he doesn’t like the weird weather. One day, he invited his girl friend Brandy to go
shopping, they found that there were many shopping malls in the city center. Brandy had no idea which mall should they enter, because they had only limited time for shopping. She asked ACman for
a help. ACman loved Maths as much as Brandy, he considered the beauty in Maths as the most beauty in the world. There were N(1 <= N <= 100000) shopping malls lying on a line, and the distance
between the adjacent malls was one meter. Every malls has his own height, and different malls might be the same height. You could select n continuous malls on condition that n must be M(1<=M<=N)
at least for some reason, and the beautiful value was defined as the ratio of the sum of the n malls’ heights and the number of the continuous selected malls, namely, n. What was the most
beautiful value?
As a good programmer, Could you help ACman solve the problem? If you can solve it correctly, ACman will give you a beautiful balloon.
The first line of input is an integer giving number of cases to follow. For each case:
The first line contains two integers N, M separated by a single space.
The second line contains N integers which represent the heights of the N shopping malls on a line from left to right. The heights of shopping malls are positive integers and less than 100000.
Print a single integer which is the most beautiful value rounded to three digits after the decimal point.
sample input
sample output
ACman, HUST Programming Contest 2007 | {"url":"http://hustoj.org/problem/1218","timestamp":"2024-11-13T16:24:17Z","content_type":"text/html","content_length":"8871","record_id":"<urn:uuid:1ca5abd6-57c9-40e1-a3a3-ff4e610a0c61>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00483.warc.gz"} |
How do you differentiate f(x)=(x^2+4x+6)^5? | HIX Tutor
How do you differentiate #f(x)=(x^2+4x+6)^5#?
Answer 1
$f ' \left(x\right) = 10 \left(x + 2\right) {\left({x}^{2} + 4 x + 6\right)}^{4}$
differentiate using the #color(blue)"chain rule"#
#"Given " f(x)=g(h(x))" then"#
#• f'(x)=g'(h(x))xxh'(x)larrcolor(red)" chain rule"#
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Answer 2
To differentiate ( f(x) = (x^2 + 4x + 6)^5 ), you can use the chain rule and the power rule. The chain rule states that if you have a function inside another function, you differentiate the outer
function first, then multiply by the derivative of the inner function. Applying the chain rule to this problem, the derivative of ( f(x) ) is ( 5(x^2 + 4x + 6)^4 \cdot (2x + 4) ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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A music data processing device includes at least one processor, configured to perform the following: performing calculations of Fast Fourier Transform on input data generated from music data inputted
for respective processing units; and for each of bin numbers corresponding to respective calculation points of the Fast Fourier Transform, calculating and outputting a shift amount, as a phase error,
that is obtained by subtracting, from a phase in a current processing unit obtained from the Fast Fourier Transform calculations, a sum of a phase in a previous processing unit obtained from the Fast
Fourier Transform calculations and a normalized phase displacement, wherein the normalized phase displacement is a change in phase that is supposed to occur when the processing unit advances one unit
with a bin number frequency corresponding to the bin number.
BACKGROUND OF THE INVENTION Technical
The present invention relates to a process for determining a tuning value from music data and a process for determining a chord.
Background Art
The tuning value (for example, the frequency of an A4 tone) of an acoustic signal can be determined from the music data by using an autocorrelation function if it is a single tone.
In addition to the method using an autocorrelation function, Fourier transform processing is a method often used to analyze acoustic signals. In particular, a method using FFT (Fast Fourier
Transform) allows high-speed processing on a computer and is used in many signal analyses.
Yousei Matsuoka, Mizuki Watabe, “Music chord recognition technology and its applications,” NTT DOCOMO Technical Journal Vol. 25 No. 2, Jul. 2017, uses a combination of FFT and chroma vector
technology to identify chords.
One of the advantages of the present disclosure is that it is possible to obtain accurate tuning values that are necessary when playing an instrument along with a song or when determining the chord
progression of a song.
Additional or separate features and advantages of the invention will be set forth in the descriptions that follow and in part will be apparent from the description, or may be learned by practice of
the invention. The objectives and other advantages of the invention will be realized and attained by the structure particularly pointed out in the written description and claims thereof as well as
the appended drawings.
To achieve these and other advantages and in accordance with the purpose of the present invention, as embodied and broadly described, in one aspect, the present disclosure provides a music data
processing device, comprising at least one processor, configured to perform the following: performing calculations of Fast Fourier Transform on input data generated from music data inputted for
respective processing units; and for each of bin numbers corresponding to respective calculation points of the Fast Fourier Transform, calculating and outputting a shift amount, as a phase error,
that is obtained by subtracting, from a phase in a current processing unit obtained from the Fast Fourier Transform calculations, a sum of a phase in a previous processing unit obtained from the Fast
Fourier Transform calculations and a normalized phase displacement, wherein the normalized phase displacement is a change in phase that is supposed to occur when the processing unit advances one unit
with a bin number frequency corresponding to the bin number.
In the music data processing device above, the at least one processor may be configured to perform the following: for each of bin numbers, calculating a current frequency that is a frequency obtained
by multiplying the bin number frequency by a ratio of a sum of the phase error and the normalized phase displacement to the normalized phase displacement; calculating a tentative scale note based on
a ratio of the current frequency to a frequency of a reference note; calculating a scale note shift amount based on a decimal part of the tentative scale note; and calculating a tuning value for the
music data based on the scale note shift amount.
In another aspect, the present disclosure provides a music data processing device, comprising at least one processor, configured to perform the following: performing calculations of Fast Fourier
Transform on input data generated from music data inputted for respective processing units; for each of bin numbers corresponding to respective calculation points of the Fast Fourier Transform,
calculating and outputting a shift amount, as a phase error, that is obtained by subtracting, from a phase in a current processing unit obtained from the Fast Fourier Transform calculations, a sum of
a phase in a previous processing unit obtained from the Fast Fourier Transform calculations and a normalized phase displacement, wherein the normalized phase displacement is a change in phase that is
supposed to occur when the processing unit advances one unit with a bin number frequency corresponding to the bin number; determining a current frequency for each of the bin numbers based on the
phase error; and determining a chord in the music data based on the determined current frequency for each of the bin numbers.
In the music data processing device above, said at least one processor may perform the following in determining the chord: for each of the bin numbers corresponding to the respective calculation
points of the Fast Fourier Transform, calculating a true scale note for each bin number based on the tuning value for the music data and the current frequency calculated for each of the bin numbers;
generating a chroma vector, which is a vector whose feature quantity is an amplitude intensity of a frequency for each tone number scale note, by distributing and synthesizing values of amplitudes
that are obtained for respective bin numbers from the Fast Fourier Transform calculations into a prescribed scale note range of tone number scale notes based on an integer part and a decimal part of
the true scale note calculated for each bin number and on the amplitude for each bin number; and determining the chord in the music data based on the chroma vector.
In other aspects, the present disclosure provides a method to be executed by at least one processor in a music data processing device, comprising the above-described processes, and a
computer-readable non-transitory storage medium storing a program executable by at least one processor in a music data processing device, comprising the above-described processes.
It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory, and are intended to provide further explanation of the
invention as claimed.
FIG. 1 is a block diagram showing an example of a hardware configuration of a music data processing device.
FIG. 2 is a block diagram of a chord determination processing.
FIG. 3 is a flowchart illustrating an example of processing by a tuning value determining process.
FIG. 4A is a flowchart illustrating a processing example of the chroma vector generation process.
FIG. 4B is a flowchart illustrating a processing example of the chord determination process.
FIG. 5A is a diagram showing an example of a chord constituent note table for a major chord.
FIG. 5B is a diagram showing an example of a chord constituent note table for a minor chord.
Hereinafter, embodiments for carrying out the present invention will be described in detail with reference to the drawings. FIG. 1 is a block diagram showing an example of a hardware configuration of
a music data processing device 100 that can determine a tuning value of a music data and perform chord determination based on the determined tuning value.
The music data processing device 100 is a user terminal that is, for example, a smartphone terminal, a tablet terminal, or a personal computer such as a so-called laptop computer operated by a user.
The music data processing device 100 includes a CPU (Central Processing Unit) 101 as at least one processor, a ROM (Read Only Memory) 102, a RAM (Random Access Memory) 103, an input unit 104
configured by, for example, a touch panel display; a display/output unit 105, and a communication unit 106 connected to, for example, the Internet or a local area network in order to communicate with
a server device or other user terminals, all of which are interconnected by a system bus 107. Other blocks that are commonly included in user terminals and are not directly related to the operation
of this embodiment (for example, microphones, speakers, call functions, cameras, etc.) are omitted, but needless to say, these units may be included.
The CPU 101 executes the control operation of the music data processing device 100 of FIG. 1 by executing the control program stored in the ROM 102 while using the RAM 103 as a work memory. Further,
the ROM 102 stores, in addition to the above-mentioned control program and various fixed data, for example, data of a “chord constituent note table” shown in FIGS. 5A and 5B, which will be described
later. The control program and the like may not be stored in advance in the ROM 102, but may be downloaded and installed as appropriate via a network such as the Internet via the communication unit
Further, although the control program is stored in the ROM 102 in this embodiment, the control program is not limited to this, and may be stored in a removable storage medium such as a USB memory,
CD, DVD, etc., or may be stored in a storage medium of a server. The music data processing device 100 may acquire a control program from such a storage medium and execute it.
An example of the processing of this embodiment executed by the computer in FIG. 1 will be described in detail below. References such as “CPU 101,” “ROM 102,” or “RAM 103” are intended to refer to
CPU 101, ROM 102, or RAM 103 in FIG. 1.
First, a processing unit PU, which is a processing unit that corresponds to one batch, is defined by the following equation (1).
$PU = SR / PR ( sample / times ) ( 1 )$
Here, SR is the sampling rate (samples/second), and PR is the FFT processing rate (times/second). Note that the FFT window size (sample) is WL.
FIG. 2 is a block diagram of a chord determination processing process 200 executed by the CPU 101 and the like of the music data processing device 100 of FIG. 1. The chord determination process 200
is executed by the processing described later in the music data processing device 100, and the corresponding hardware includes at least one of the CPU 101, the ROM 102, and the RAM 103 as described
above. The program executed in the chord determination process 200 is stored in the ROM 102 when the user terminal, which is the music data processing device 100 in FIG. 1 is shipped from a factory
to users, and may be loaded from the ROM 102 to the RAM 103. Alternatively, the program executed in the chord determination process 200 may be such that the user provides the user terminal of the
music data processing device 100 with a so-called application (app) having the functions of the chord determination process 200 by downloading and installing it via the communication unit 106 in FIG.
1 into the RAM 103 from a vendor company's website or the like via a network such as the Internet.
The chord determination process 200 is generally composed of a tuning value determination process 201, a chroma vector generation process 202, and a chord determination process 203.
The tuning value determining process 201 executes a waveform data reading process S210, an amplitude/phase calculation process S211, a phase error calculation process S212, a frequency determining
process S213, and a tuning value determining process S214.
The chroma vector generation process 202 executes 88-tone chroma vector generation processing S220 and 12-tone chroma vector generation processing S221.
The chord determination process 203 executes beat tracking processing S230 and chord determination processing S231.
FIG. 3 is a flowchart showing a more detailed processing example of the tuning value determining process 201 of FIG. 2. The CPU 101 treats information, such as [FFT window data], [complex data]
([amplitude] [phase]), [phase error], [current frequency], and [tentative scale note], [tentative scale note integer part] and [tentative scale note decimal part], [tentative scale note decimal part
center of gravity], and [tuning value], which are described as [main information], in the block of the tuning value determining process 201 of FIG. 2, as variables stored in the RAM 103
In FIG. 3, the CPU 101 reads the waveform data of music data to be subjected to chord determination, which is read from an external network (such as the Internet) via the ROM 102 or the communication
unit 106, into the RAM 103 sequentially with the processing unit PU that was defined by the above-mentioned equation (1): [Samples/times]=SR (sampling rate [samples/seconds])/PR (FFT processing rate
[times/seconds]). Here, first, it is determined whether the final data of the waveform data has been read (step S300 in FIG. 3).
If the final data has not been read and the determination in step S300 is NO, the CPU 101 executes the waveform data reading process S210 described in FIG. 2. Specifically, the CPU 101 loads new
waveform data of the PU sample into the RAM 103, and also sets FFT window data having an FFT window size WL [samples] from the ROM 102 into a FIFO (First In, First Out) buffer, which is a register or
the like, in the RAM 103 or a memory built in the CPU 101.
Next, the CPU 101 executes the amplitude/phase calculation process S211 described in FIG. 2.
Specifically, the CPU 101 first multiplies a sample of the music data on the RAM 103 with a sample of the FFT window data in the FIFO buffer for each corresponding sample so that the center sample of
the latest processing unit PU (sample) of music data that has been read into the RAM 103 for each processing unit PU [sample] and the center sample of the FFT window data set in the FIFO buffer are
Next, the CPU 101 performs an FFT operation on the multiplication result data for the FFT window size WL [samples].
Furthermore, the CPU 101 obtains complex data that is the result of the FFT calculation for each FFT bin number “bin” (hereinafter referred to as the bin number “bin”) obtained as a result of the FFT
calculation, and calculates the amplitude and phase from the complex data. Here, the calculation point of the FFT calculation is equal to the FFT window size WL [sample], but the calculation results
from 0 to (WL/2)-1 and the calculation results from WL/2 to WL-1 have a mirror image relationship. Therefore, the bin number “bin” corresponds to the FFT calculation point and takes a value of 0≤bin
<(FFT window size WL)/2, which is half the number of calculation points.
Now, if the real part of the complex data at the bin number “bin” is re (bin) and the imaginary part is im (bin), the amplitude Amp (bin) and the phase Phs (bin) at the bin number “bin” are
calculated by the following formulae (2) and (3).
$Amp ( bin ) = Sqrt ( re ( bin ) × re ( bin ) + im ( bin ) × im ( bin ) ) ( 2 )$
Here, “Sqrt(n)” is a calculation function that calculates the square root of n.
$Phs ( bin ) = A tan ( im ( bin ) , re ( bin ) ) ( 3 )$
Here, “Atan (y, x)” is a calculation function that calculates the arctangent of y with respect to x.
Returning to the explanation of FIG. 3, after the above amplitude/phase calculation process S211, the CPU 101 executes the phase error calculation process S212 described in FIG. 2.
Specifically, the CPU 101 first calculates, for each bin number bin (0≤bin<(FFT window size WL)/2) corresponding to the FFT calculation point, the FFT bin frequency BFQ(bin) (bin number frequency)
according to the calculation shown in the formula (4) below.
$BFQ ( bin ) = SR × bin / WL ( 4 )$
As shown in equation (4) above, the FFT bin frequency BFQ (bin) (Hz=1/sec) is calculated by multiplying the ratio of the bin number “bin” to the FFT window size WL (samples) by the sample rate SR
(samples/second) of the music data. That is, the FFT bin frequency BFQ (bin) is a frequency determined depending on the FFT calculation point indicated by the bin number “bin”.
Next, for each bin number “bin”, the CPU 101 calculates a normalized phase displacement NPD(bin), which is the phase amount to be displaced when the processing unit PU is advanced in one unit (=SR/PR
[sample]) with the FFT bin frequency BFQ(bin) calculated in the formula (4) above according to the calculation shown by the following equation (5).
$NPD ( bin ) = 2 π × BFQ ( bin ) × PU / SR ( 5 )$
Here, π is the circumference ratio, pi.
Next, for each bin number “bin”, the CPU 101 performs the calculation shown in the following equation (6) to derive a phase error ePhs (bin), which is a shift amount that is obtained by subtracting,
from the Phs1 (bin) in the current processing unit calculated from the complex data that is the FFT calculation result by the calculation shown in equation (3), the result of adding the normalized
phase displacement NPD (bin) calculated by the calculation shown in equation (5) to the phase Phs0 (bin) in the previous processing unit calculated the same way.
$ePhs ( bin ) = Phs 1 ( bin ) - ( Phs 0 ( bin ) + NPD ( bin ) ) % ( 2 π ) ( 6 )$
Note that the phase error ePhs (bin) does not exceed the range of ±1 by adjusting the sampling rate SR [number of samples/second], FFT processing rate PR [times/second], and FFT window size WL
[sample] defined by equation (1). Here, “%” is a remainder calculation expression, and the right side of equation (6) means the remainder obtained by dividing (Phs0 (bin)+NPD (bin)) by 2π is
subtracted from Phs1 (bin).
The sum of the phase Phs0 (bin) in the previous processing unit and the normalized phase displacement NPD (bin) calculated by the calculation shown in equation (5) should match the phase Phs1 (bin)
in the current processing unit. However, in reality, in the case where the frequency of the music data is different from the FFT bin frequency BFQ (bin) calculated by the calculation shown in
equation (4), the above two do not match, so the above difference between Phs1 (bin) and (Phs0 (bin)+NPD (bin)) %(2π) is calculated using formulae (4), (5) and (6), as the phase error ePhs (bin), in
terms of the reminder after dividing by 2π.
Returning to the explanation of FIG. 3, after the above phase error calculation process S212, the CPU 101 executes the frequency determining process S213 described in FIG. 2.
Specifically, the CPU 101 first calculates the current frequency cFq (bin) and tentative scale note vNt (bin) from the phase error ePhs (bin) calculated in the phase error calculation processing S212
for each bin number “bin” (0≤bin<(FFT window size WL)/2).
The result of adding the phase error ePhs (bin) calculated by the calculation shown by equation (6) to the normalized phase displacement NPD (bin) calculated by the calculation shown by equation (5)
is divided by the normalized phase displacement NPD (bin) to calculate the ratio of the actual phase to the normalized phase displacement NPD (bin).
Then, the CPU 101 calculates the current frequency cFq (bin) for each bin number “bin” by multiplying the FFT bin frequency BFQ (bin) calculated by the calculation of equation (4) by said ratio in
accordance with the equation (7).
$cFq ( bin ) = BFQ ( bin ) × ( NPD ( bin ) + ePhs ( bin ) / NPD ( bin ) ( 7 )$
Further, the CPU 101 uses the current frequency cFq (bin) calculated by the calculation shown by the equation (7) for each bin number bin to calculate the tentative scale note vNt (bin) by the
calculation shown by the following equation (8). calculate.
$vNt ( bin ) = Log ( cFq ( bin ) / 440. , 2. ) × 12 + 69 ( 8 )$
Here, “69” is the scale note number of A4 note. Further, Log (x, 2.0) is an arithmetic function that calculates the base 2 logarithm of x.
As shown in the above equation (8), the tentative scale note vNt (bin) at the bin number “bin” is calculated by calculating the base-2 logarithm of the result of dividing the current frequency cFq
(bin) at the bin number “bin” by the frequency of 440 Hz of the A4 reference tone which is the primary tone frequency of a prescribed pitch, and by multiplying the result by 12, and adding the scale
note number of A4 note=69.
Subsequently, when the total value of the amplitude Amp (bin) of the complex data, which is calculated by the equation (2), for all of the bin numbers “bin” (0≤bin<(FFT window size WL)/2), is greater
than a prescribed value, the CPU 101 calculates the tentative scale note integer part ivNt (bin)) as shown in the following equation (9), by rounding off the decimal part of the tentative scale note
vNt (bin) calculated by equation (8) for each bin number “bin”.
$ivNt ( bin ) = the result of rounding the decimal part of vNt ( bin ) ( 9 )$
Furthermore, as shown in the following equation (10), the CPU 101 calculates, for each bin number “bin”, the tentative scale note decimal part fvNt (bin) by subtracting the calculated tentative scale
note integer part ivNt (bin) calculated by equation (9) from the tentative scale note vNt (bin) calculated by equation (8).
$fvNt ( bin ) = vNt ( bin ) - ivNt ( bin ) ( 10 )$
Since the calculation shown in equation (9) is a rounding calculation, the tentative scale note decimal part fvNt (bin) calculated by the calculation shown in equation (10) fits in the range of −0.5
or more and less than 0.5. The tentative scale note decimal part fvNt (bin) calculated by the calculation shown in equation (10) can be considered as the scale note shift amount for each bin number
The CPU 101 further calculates the tentative scale note decimal part gravity center Flt, which is the center of gravity of the tentative scale note decimal part fvNt (bin) with the amplitude Amp
(bin) (see formula (2)) over the range of the bin numbers “bin's” in which the tentative scale note integer part ivNt (bin) calculated by the calculation shown in equation (9) is within a
predetermined note range (for example, 36 (C2) to 95 (B6)), using the following formula (11).
$Flt = ∑ bin = min bin max bin Amp ( bin ) · fvNt ( bin ) ∑ bin = min bin max bin Amp ( bin ) ( 11 )$
Here, bin is the bin number, mini[n ]is the minimum bin number of the predetermined range, max[bin ]is the maximum bin number of the predetermined note range, and Amp (bin) is the amplitude at the
bin number “bin” calculated by the calculation of equation (2), and fvNt (bin) is the tentative scale note decimal part of the bin number “bin” calculated by the calculation of equation (10). Note
that in order to satisfy the above equation, the amplitudes must be equal to or greater than a predetermined threshold.
In other words, the CPU 101 calculates the tentative scale note decimal part fvNt (bin) by the calculation shown in equation (10) for each bin number “bin” within the predetermined range within one
processing unit PU, and by calculating, for example, the center of gravity of the tentative scale note decimal part fvNt (bin), the tentative scale note decimal part center of gravity Flt, which is
the scale note shift amount for each processing unit corresponding to one processing unit, is calculated.
Returning to the explanation of FIG. 3, after the above frequency determining process S213, the CPU 101 returns to the determination process of step S300. In this way, the CPU 101 repeatedly performs
the waveform data reading process S210, the amplitude/phase calculation process S211, the phase error calculation process S212, and the frequency determining process S213 for the respective
processing units PU's [sample] until it is determined that the processes from the waveform data reading process S210 to the frequency determining process S213 are completed with the last data.
Through this iterative process, the CPU 101 can calculate the tentative scale note decimal part center of gravity Flt, which is the scale note shift amount for the corresponding processing unit, for
each of the processing units PU's [sample] from the beginning to the end of the music data.
When the music data for chord determination is read to the final data for each processing unit PU [sample] and the processing from the waveform data reading processing S210 to the frequency
determining process S213 is completed and the determination in step S300 in FIG. 3 becomes YES, the CPU 101 executes the tuning value determining process S214 explained in FIG. 2.
Specifically, the CPU 101 first calculates the tentative scale note decimal part center gravity average value aFlt, which is the average value of the tentative scale note decimal part gravity centers
Flt obtained for the respective processing units PU's [sample] from the beginning to the end of the music data.
It can be said that this tentative scale note decimal part gravity center average value aFlt corresponds to the scale note shift amount of the entire music data.
Then, the CPU 101 determines the tuning value sTun by using the above-described calculated tentative scale note decimal part center of gravity average value aFlt by the calculation shown in the
formula (12) below.
$sTun = 440. × Pow ( 2. , aFlt / 12 ) ( 12 )$
Here, Pow(x, y) is a calculation function that calculates x to the power of y.
In the calculation shown by the above formula (12), the CPU 101 calculates 2 to the power of [the result of dividing the tentative scale note decimal part center of gravity average value aFlt
corresponding to the scale note shift amount of the entire music data by 12]. This way, the scale note shift rate per note is calculated, and the scale note shift rate is multiplied by the primary
tone frequency of a prescribed scale note, for example, the frequency 440.0 (Hz) of the A4 reference tone so as to calculate the tuning value sTun for the music data.
As described above, the tuning value sTun for the entire music data can be calculated by the tuning value determining process 201 of FIG. 2 shown in the flowchart of FIG. 3.
FIG. 4A is a flowchart showing a more detailed processing example of the chroma vector generation process 202 in FIG. 2. The CPU 101 treats information, such as [true scale notes], [88-tone chroma
vector], and [12-tone chroma vector], which are described as “main information” in the block of the chroma vector generation process 202 in FIG. 2, as variables stored in the RAM 103.
First, the CPU 101 executes the 88-tone chroma vector generation processing S220 described in FIG. 2.
Specifically, the CPU 101 calculates, by the calculation shown in the formula (13), the true scale note sNt (bin) for each bin number “bin” based on the tuning value sTun over the entire music data
calculated by the calculation shown by equation (12) in the tuning value determining process 201 of FIGS. 2 and 3 and on the current frequency cFq (bin) for each bin number bin (0≤bin<(FFT window
size WL)/2) calculated by the calculation shown by equation (7) in the frequency determining process S213 in FIG. 2.
$sNt ( bin ) = Log ( cFq ( bin ) / sTun , 2. ) × 12 + 69 ( 13 )$
Here, “69” is the scale note number of the A4 note, similar to when the tentative scale note vNt (bin) was calculated by the calculation shown in equation (8). As shown in the above equation (13),
the true scale note sNt (bin) at the bin number “bin” is calculated by calculating the base-2 logarithm of the division result of dividing the current frequency cFq (bin) at the bin number “bin” by
the tuning value sTun calculated by the calculation shown in the equation (12), multiplying the result by 12, and by adding the result to A4 scale note number=69.
Next, as shown in equation (14) below, the CPU 101 calculates the true scale note integer part iNt (bin) by cutting off the decimal part of the true scale note sNt (bin) calculated by the calculation
shown in equation (13) for each bin number “bin”.
$iNt ( bin ) = Result of cutting off the decimal part of sNt ( bin ) ( 14 )$
Furthermore, as shown in the following equation (15), the CPU 101 calculates, for each bin number “bin”, the true scale note decimal part fNt (bin) by subtracting the calculated true scale note
integer part iNt (bin) obtained by the calculation shown in equation (14) from the true scale note sNt (bin) calculated by the calculation shown in equation (13).
$fNt ( bin ) = sNt ( bin ) - iNt ( bin ) ( 15 )$
Since the equation (14) cut off the decimal part, the true scale note decimal part fNt (bin) calculated by the calculation shown in the equation (15) falls within the range of 0.0 or more and less
than 1.0.
Next, the CPU 101 converts the amplitude Amp (bin) calculated by the calculation shown by equation (2) for each bin number “bin” into tone number scale notes distributed and synthesized in a
predetermined scale note range based on the true scale note integer part iNt (bin) calculated for each bin number “bin” by the calculation shown by equation (14) and on the true scale note decimal
part fNt (bin) calculated for each bin number “bin” by the calculation shown in equation (15), so as to generate, a chroma vector CRV [n], which is a vector whose feature quantity is the amplitude
intensity of the frequency for each tone number scale note.
More specifically, if the 88-tone chroma vector is expressed as CRV88 [n] (n: 0-87) in the entire musical range of the music data, for example, in the 88-tone scale from A0 (21) to C8 (108), the CPU
101 generates an 88-tone chroma vector CRV88 [n] by the respective distribution (synthesis) operations shown in the following equations (16) and (17).
$CRV 88 [ iNt ( bin ) - 21 ] += Amp ( bin ) × ( 1. - fNt ( bin ) ) ( 16 )$ $CRV 88 [ iNt ( bin ) + 1 - 21 ] += Amp ( bin ) × fNt ( bin ) ( 17 )$
Here, “+=” is a compound assignment operator, which means adding the value on the left side and the value on the right side of the expression and putting the result into the variable on the left
Returning to the explanation of FIG. 4A, the CPU 101 executes the 12-tone chroma vector generation processing S221 described in FIG. 2 after the 88-tone chroma vector generation processing S220.
Specifically, the CPU101 performs a resynthesis operation to round to a 12-tone scale after noise is removed based on the minimum value of the three adjacent scale notes (n−1, n, n+1) of the 88-tone
chroma vector CRV88 [n] (n: 0 to 87) calculated by the respective calculations shown in equations (16) and (17).
Here, if the 12-tone chroma vector is expressed as CRV12 [m] (m: 0 to 11), it is calculated by the resynthesis operation shown by the following equation (18).
$CRV 12 [ ( n + 21 ) %12 ] += CRV 88 [ n ] ( 18 )$
Here, n: 0 to 87, and %12 is a remainder operation divided by 12.
As described above, the chroma vector generation process 202 of FIG. 2 shown in the flowchart of FIG. 4A is performed to generate the 88-tone chroma vector CRV88 [n] (n: 0 to 87) and the 12-tone
chroma vector CRV12 [m] (m: 0 to 11).
FIG. 4B is a flowchart showing a detailed processing example of the chord determination process 203 in FIG. 2. The CPU 101 treats information, such as [beat tracking information] (tempo value, bar
position, beat position), [beat length 12-tone chroma vector], [chord constituent note table], and [chord determination result], which are described as [main information] in the block of the chord
determination process 203 of FIG. 2, as variables stored in the RAM 103.
First, the CPU 101 executes the beat tracking process S230 described in FIG. 2.
Specifically, the CPU 101 detects [Beat tracking information] (tempo value, bar position, beat position) based on the changes in volume of music data and in the constituent sounds, for example, based
on the change in the 12-tone chroma vector CRV12 [m] calculated by equation (18) in the 12-tone chroma vector generation process S221 in FIG. 4A in the chroma vector generation process 202 in FIG. 2.
Next, the CPU 101 executes the chord determination processing S231 described in FIG. 2.
Specifically, the CPU 101 determines the length of time for chord determination based on [beat tracking information] calculated in the beat tracking process S230 of FIG. 4B, and creates [beat length
12-tone chroma vector] whose element value is the sum of the element values of the 12-tone chroma vector CRV12 [m] obtained by the calculation shown in equation (18) for the corresponding time
Furthermore, the CPU 101 multiplies the above-mentioned [beat length 12-note chroma vector] by the values of [chord constituent note table] weighted by the constituent notes/non-constituent notes of
a possible chord to find a chord at which the largest/maximum value is achieved and stores such a chord to [chord determination result] by the calculation shown in the following equation (19).
$Total maximum value of [ Beat length 12 - tone chroma vector ] × [ Chord constituent note table ] → [ Chord determination result ] ( 19 )$
Note that the CPU 101 shifts the [chord constituent note table] 12 times to take into account the difference in root notes.
FIGS. 5A and 5B are diagram illustrating table examples of the weighting structure of the [chord constituent note table] with the C (do) note as the reference (root note). The table of FIG. 5A shows
a case of a major chord, and the table of FIG. 5B shows a case of a minor chord.
In FIGS. 5A and 5B, the weight of the constituent notes takes a positive value, and is set to 20 for each triad (60 in total). Non-constituent notes take negative values, and in particular major/
minor thirds, which are the difference between major/minor chords, have large absolute values.
In this disclosure, the term “at least” means, unless otherwise specified, that, for example, “at least one of A, B, and C” means “(A), (B), (C), (A and B), (A and C), (B and C), or (A, B and C),
including combinations of the plurality or numbers greater than or equal to the indicated number. For example, if C is plural, the term “at least one of A, B, and C” means “(A), (B), (at least one or
more of C), (A and B), (A and at least one or more C), (B and at least one or more C), or (A, B, and at least one or more C). If there is more than one A or more than one B, it will be interpreted in
the same way as above.
Conventionally, frequency information in analysis results obtained by FFT is a composite of discrete values for respective FFT bin numbers, and is not suitable for detecting frequencies that take
continuous values such as tuning values of the entire music. In response to this issue, according to embodiments of the present invention, it is now possible to obtain accurate tuning values needed
when playing an instrument to match the music or determining the chord progression of the music, making it easier to perform tuning operations. Further, based on this tuning value, it becomes
possible to obtain more accurate chord determination results. Note that the embodiments described above are presented as examples, and are not intended to limit the scope of the invention. The
embodiment can be implemented in various other forms, and various omissions, substitutions, and changes can be made without departing from the spirit of the invention. This embodiment and its
modifications are included within the scope and gist of the invention, as well as within the scope of the invention described in the claims and its equivalents. Therefore, it will be apparent to
those skilled in the art that various modifications and variations can be made in the present invention without departing from the spirit or scope of the invention. Thus, it is intended that the
present invention cover modifications and variations that come within the scope of the appended claims and their equivalents. In particular, it is explicitly contemplated that any part or whole of
any two or more of the embodiments and their modifications described above can be combined and regarded within the scope of the present invention.
1. A music data processing device, comprising at least one processor, configured to perform the following:
performing calculations of Fast Fourier Transform on input data generated from music data inputted for respective processing units; and
for each of bin numbers corresponding to respective calculation points of the Fast Fourier Transform, calculating and outputting a shift amount, as a phase error, that is obtained by subtracting,
from a phase in a current processing unit obtained from the Fast Fourier Transform calculations, a sum of a phase in a previous processing unit obtained from the Fast Fourier Transform
calculations and a normalized phase displacement, wherein the normalized phase displacement is a change in phase that is supposed to occur when the processing unit advances one unit with a bin
number frequency corresponding to the bin number.
2. The music data processing device according to claim 1, wherein the at least one processor is configured to perform the following:
for each of bin numbers, calculating a current frequency that is a frequency obtained by multiplying the bin number frequency by a ratio of a sum of the phase error and the normalized phase
displacement to the normalized phase displacement;
calculating a tentative scale note based on a ratio of the current frequency to a frequency of a reference note;
calculating a scale note shift amount based on a decimal part of the tentative scale note; and
calculating a tuning value for the music data based on the scale note shift amount.
3. The music data processing device according to claim 1, wherein the at least one processor calculates the bin number frequency corresponding to the bin number by multiplying a sampling rate of the
music data by a ratio of the bin number to a window size of window data that is multiplied onto the music data for each sampling prior to the Fast Fourier Transform.
4. The music data processing device according to claim 2, wherein the at least one processor executes the following:
(a) calculating the decimal part of the tentative scale note as a scale note shift amount for each bin number;
(b) calculating a scale note shift amount for each processing unit by performing the process (a) for all of the bin numbers within a prescribed note range within the processing unit; and
(c) calculating a scale note shift amount for an entirety of the music data by performing the process (b) for all of the processing units that span over the entirety of the music data.
5. The music data processing device according to claim 2, wherein the at least one processor calculates the tuning value for the music data by calculating a scale note shift rate per note from the
scale note shift amount and multiplying the scale note shift rate by a primary tone frequency of a prescribed scale note.
6. The music data processing device according to claim 1, further comprising:
determining a current frequency for each of the bin numbers based on the phase error; and
determining a chord in the music data based on the determined current frequency for each of the bin numbers.
7. The music data processing device according to claim 6, wherein said at least one processor performs the following in determining the chord:
for each of the bin numbers corresponding to the respective calculation points of the Fast Fourier Transform, calculating a true scale note for each bin number based on the tuning value for the
music data and the current frequency calculated for each of the bin numbers;
generating a chroma vector, which is a vector whose feature quantity is an amplitude intensity of a frequency for each tone number scale note, by distributing and synthesizing values of
amplitudes that are obtained for respective bin numbers from the Fast Fourier Transform calculations into a prescribed scale note range of tone number scale notes based on an integer part and a
decimal part of the true scale note calculated for each bin number and on the amplitude for each bin number; and
determining the chord in the music data based on the chroma vector.
8. The music data processing device according to claim 7, wherein the at least one processor performs the following:
generating, as said chroma vector, an n-note chroma vector corresponding to an n-note scale of an entire musical range having a number of notes n (n>12), and a 12-tone chroma vector that is
converted from the n-note chroma vector by rounding to a 12-tone scale;
detecting a tempo value, a bar position and a beat position, as beat tracking information, based on changes in the 12-tone chroma vector;
determining a time length for chord determination based on the beat tracking information;
generating a beat length 12-tone chroma vector whose element value is a sum of the element values of the 12-tone chroma vector for the time length; and
outputting, as a chord determination result, a chord that attains the largest value in a multiplication result of the beat length 12-tone chroma vector with values of chord constituent note
tables having weights in accordance with constituent notes and non-constituent notes of the chord.
9. The music data processing device according to claim 6, wherein the at least one processor calculates the bin number frequency corresponding to the bin number by multiplying a sampling rate of the
music data by a ratio of the bin number to a window size of window data that is multiplied onto the music data for each sampling prior to the Fast Fourier Transform.
10. A method to be executed by at least one processor in a music data processing device, comprising:
performing calculations of Fast Fourier Transform on input data generated from music data inputted for respective processing units; and
for each of bin numbers corresponding to respective calculation points of the Fast Fourier Transform, calculating and outputting a shift amount, as a phase error, that is obtained by subtracting,
from a phase in a current processing unit obtained from the Fast Fourier Transform calculations, a sum of a phase in a previous processing unit obtained from the Fast Fourier Transform
calculations and a normalized phase displacement, wherein the normalized phase displacement is a change in phase that is supposed to occur when the processing unit advances one unit with a bin
number frequency corresponding to the bin number.
11. The method according to claim 10, wherein the method includes the following:
for each of bin numbers, calculating a current frequency that is a frequency obtained by multiplying the bin number frequency by a ratio of a sum of the phase error and the normalized phase
displacement to the normalized phase displacement;
calculating a tentative scale note based on a ratio of the current frequency to a frequency of a reference note;
calculating a scale note shift amount based on a decimal part of the tentative scale note; and
calculating a tuning value for the music data based on the scale note shift amount.
12. The method according to claim 10, further comprising:
determining a current frequency for each of the bin numbers based on the phase error; and
determining a chord in the music data based on the determined current frequency for each of the bin numbers.
13. The method according to claim 12, wherein the method includes the following in determining the chord:
for each of the bin numbers corresponding to the respective calculation points of the Fast Fourier Transform, calculating a true scale note for each bin number based on the tuning value for the
music data and the current frequency calculated for each of the bin numbers;
generating a chroma vector, which is a vector whose feature quantity is an amplitude intensity of a frequency for each tone number scale note, by distributing and synthesizing values of
amplitudes that are obtained for respective bin numbers from the Fast Fourier Transform calculations into a prescribed scale note range of tone number scale notes based on an integer part and a
decimal part of the true scale note calculated for each bin number and on the amplitude for each bin number; and
determining the chord in the music data based on the chroma vector.
14. A computer-readable non-transitory storage medium storing a program executable by at least one processor in a music data processing device, the program causing the at least one processor to
perform the following:
performing calculations of Fast Fourier Transform on input data generated from music data inputted for respective processing units; and
for each of bin numbers corresponding to respective calculation points of the Fast Fourier Transform, calculating and outputting a shift amount, as a phase error, that is obtained by subtracting,
from a phase in a current processing unit obtained from the Fast Fourier Transform calculations, a sum of a phase in a previous processing unit obtained from the Fast Fourier Transform
calculations and a normalized phase displacement, wherein the normalized phase displacement is a change in phase that is supposed to occur when the processing unit advances one unit with a bin
number frequency corresponding to the bin number.
15. The computer-readable non-transitory storage medium according to claim 14, wherein the program causes the at least one processor to perform the following:
for each of bin numbers, calculating a current frequency that is a frequency obtained by multiplying the bin number frequency by a ratio of a sum of the phase error and the normalized phase
displacement to the normalized phase displacement;
calculating a tentative scale note based on a ratio of the current frequency to a frequency of a reference note;
calculating a scale note shift amount based on a decimal part of the tentative scale note; and
calculating a tuning value for the music data based on the scale note shift amount.
16. The computer-readable non-transitory storage medium according to claim 14, wherein the program causes the at least one processor to further perform the following:
determining a current frequency for each of the bin numbers based on the phase error; and
determining a chord in the music data based on the determined current frequency for each of the bin numbers.
17. The computer-readable non-transitory storage medium according to claim 16, wherein the program causes the at least one processor to perform the following in determining the chord:
for each of the bin numbers corresponding to the respective calculation points of the Fast Fourier Transform, calculating a true scale note for each bin number based on the tuning value for the
music data and the current frequency calculated for each of the bin numbers;
generating a chroma vector, which is a vector whose feature quantity is an amplitude intensity of a frequency for each tone number scale note, by distributing and synthesizing values of
amplitudes that are obtained for respective bin numbers from the Fast Fourier Transform calculations into a prescribed scale note range of tone number scale notes based on an integer part and a
decimal part of the true scale note calculated for each bin number and on the amplitude for each bin number; and
determining the chord in the music data based on the chroma vector.
Patent History
Publication number
: 20240339095
: Apr 4, 2024
Publication Date
: Oct 10, 2024
CASIO COMPUTER CO., LTD.
Yuji TABATA
Application Number
: 18/626,661
International Classification: G10H 1/00 (20060101); | {"url":"https://patents.justia.com/patent/20240339095","timestamp":"2024-11-14T13:56:01Z","content_type":"text/html","content_length":"124548","record_id":"<urn:uuid:1341077c-0ae9-40c6-adb8-e868a378e146>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00401.warc.gz"} |
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
By proving these particular identities, prove the existence of general cases.
Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1. | {"url":"https://nrich.maths.org/tags/identities","timestamp":"2024-11-14T01:36:27Z","content_type":"text/html","content_length":"47723","record_id":"<urn:uuid:2bdf016f-bf95-4d95-ba68-6a79ba4238aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00664.warc.gz"} |
All You Need to Know About STEP Cambridge: 3 Tips to Prepare Well
The STEP Cambridge is a significant mathematical competition that the University of Cambridge conducts. The history of STEP relates to multiple subjects such as biology, chemistry, English
literature, and numeric domains. However, it was only in 2003 that the authorities decided to shift toward a unified subject, Mathematics, to ensure better assessment.
The same is why STEP today stands tall as one of the most renowned and efficient tests to determine and evaluate the skills of some clever minds. The content below will illuminate some fantastic
perks that make it worth attempting. So, without any further discussion, let’s jump in.
What is STEP Cambridge?
STEP is an abbreviation of Sixth Term Examination Papers. It is an admission university undergraduate test which tests the mathematical capabilities of a student. From 2024 onwards, the assessment
will be administered by the OCR.
The official target audience of STEP has remained the University of Cambridge, Imperial College, London, and the University of Warwick.
Candidates are offered grade 1 in STEP 2 and Step 3, at least, depending on various situations and requirements.
STEP math paper pattern
The paper pattern of the STEP is one crucial thing to understand when preparing. It helps to know what kind of questions can come in the future and prepare yourself for all the pressure and time
The whole paper is based on the A-level mathematical concept and AS-level concepts with some modifications. The paper has about 12 questions, of which eight are pure, two relate to mechanics, and 2
to statistics.
Structure of the cambridge assessment is divided into 3 basic categories: STEP 1, STEP 2 and STEP 3. Following content sheds light upon each paper’s structure.
STEP I:
STEP I primarily focuses on assessing candidates’ mathematical problem-solving skills and their understanding of fundamental mathematical concepts. The same is why, content aligns with topics covered
in the first year of A Level Mathematics. Thus, ensuring a solid foundation in core mathematical principles. The specification is based on A-level math. However, some modifications exist, such as 11
questions, out of which eight are pure, and 3 are further questions on mechanics and probabilities.
Question Types:
Questions may involve applications of calculus, algebra, and basic geometry. The main focus remains on the application of methods learned at the A Level stage
While challenging, STEP I serves as an introduction to the more advanced problem-solving skills required in the subsequent papers.
STEP II:
The main aim of STEP II is to introduce students to more advanced mathematical concepts and assess problem-solving skills in a broader mathematical context. In other words, STEP II is a step ahead of
STEP I.
It is based on the A-level mathematical concept and AS-level concepts with some modifications. The paper has about 12 questions, of which eight are pure, two relate to mechanics, and 2 to statistics.
The content expands to include more complex calculus, algebraic techniques, and additional topics. It includes the differential equations and coordinate geometry.
Question Types:
Questions become more complex at this stage. Thus, requiring a deeper understanding of mathematical principles. Candidates needs to demonstrate creativity in problem-solving.
Integration of Concepts:
STEP II often integrates multiple mathematical concepts within a single problem. Hence, encouraging candidates to approach problem-solving in a holistic manner.
STEP III:
Lastly, STEP 3 comprises A Level mathematics and A-level further concepts with some changes and modifications. The paper again has 12 questions, of which eight are pure, two relate to mechanics, and
2 to statistics. The students at this level are given more complex problems than usual to determine their capabilities and efficiency in mathematics.
The paper is the representation of the pinnacle of the Cambridge STEP examination. It features the most challenging questions. It extends the content covered in both STEP I and STEP II.
The content of STEP III goes beyond standard A Level Mathematics. That means, it delves into more abstract and advanced mathematical reasoning. Topics may include group theory, complex analysis, and
further applications of calculus.
Question Types:
Questions are highly abstract and may involve proofs. Thus, requiring candidates to demonstrate a deep understanding of mathematical theory and the ability to reason at an advanced level.
Overall, STEP III challenges candidates to synthesize knowledge from various areas of mathematics and apply advanced reasoning skills to solve complex problems.
STEP Real Questions
To enable you to prepare well and acquire important information, here we discuss the real-time questions of STEP 1, 2 and 3 respectively.
STEP I:
• Solve the differential equation
Dx dy +2y=3x 2
STEP II:
• 1. Given the function
• f(x)=ex sin x
find the values of x for which
• f′′(x)=0
• 2. Prove that the equation
• x3+y3=z3 has no solutions in positive integers.
STEP III:
• 1. Let G be a group of order 56 Prove that G is not simple.
• 2. Determine the radius of convergence of the power series ∑=1∞nnn!x n
How to Prepare for Step, Cambridge
Below are some points to pass the assessment with flying colors.
1. Enroll in a mathematics course
Enrolling in math mathematics can stimulate a similar environment you deal with within the examination. WuKongSch has a challenging math course to help you in this regard. Specially designed for the
students, the course aims to develop problem-solving skills and teach you to solve problems efficiently and quickly. In addition, you can get help from professional teachers who not only explain but
also ensure to keep you engaged throughout the learning session.
2: Speed up your preparation
STEP Assessment is all about mathematical calculations. Different problems require different solving spans, depending upon the capability of the participant. However, during the assessment, you have
limited time.
The same is why it is crucial to speed up your attempting limit to get done with all the problems within the defined time limit. For this, practice daily and set a limit, such as 5 minutes or 10
minutes, to solve the problem. Try to finish it within the span and keep track of the progress.
3. Work clever, not hard
The key to success in the STEP is working smart instead of hard. That means, instead of working hours and hours, covering all the topics, getting your hands on the past paper and observing patterns.
Look at what kinds of questions are continuously repeated and pay more attention to those.
You can also consider cheat sheets to access formulas and solutions quickly.
Are there any resources provided by Cambridge for STEP preparation?
Yes, Cambridge provides official STEP preparation materials for helping students. It includes past papers, solutions, and guidance notes. You can access them from their official website.
Can international students take the STEP Cambridge assessment?
Yes. International students are allowed to take the STEP Cambridge assessment. In addition, some Cambridge colleges may require it as part of the admission process.
Is there a passing score for the STEP Cambridge assessment?
The STEP assessment does not have any specific passing requirement. Instead, the participants are ranked based on their performance in the test. The difficulty of the questions is taken into account
during the marking process
In a nutshell, STEP Cambridge is an assessment for Sixth Term Examination Papers. The main aim is to enhance the human capabilities related to mathematics.
The content above is your ultimate guide about STEP, which discusses the top perks you can avail yourself of by attempting the paper. In addition, it also sheds light upon some points to score
Graduated from the University of New South Wales. He has over 8 years of experience teaching elementary and high school mathematics and science. As a rigorous and steady mathematics teacher, Nathan
has always been well received by students 1-12 grades. | {"url":"https://www.wukongsch.com/blog/step-cambridge-post-27883/","timestamp":"2024-11-08T01:11:13Z","content_type":"text/html","content_length":"131071","record_id":"<urn:uuid:30d5c54c-dd07-4c3b-b8dc-ae44e8f8e2b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00543.warc.gz"} |
Chapter 4: Quadratic Relations
CHAPTER 4
Homework Solutions (Click the link on the left, password is class #)
☑ Day 1 (Nov 11) Intro to Quadratic Relations
We are learning to…
• To graph a curve of best fit with and without technology
• That a parabola is the graph of a quadratic relation
• To identify the key features of a parabola (AOS, vertex, y-int, zeros, max/min value)
• Graph a curve of best fit with and without technology
• Recognize that a parabola is the graph of a quadratic relation
• Determine the second difference to see if a relation is quadratic
• Identify the key features of a parabola (AOS, vertex, y-int, zeros, max/min value)
: p.172 #1-5, 9, and 10
☑ Day 2 (Nov 12) Transformations of Quadratic Relations 1
We are learning to…
• To use proper terminology to explain the transformations of a, h, and k in the equation
• To sketch the graph of the parabola by applying transformations to
I can
• Identify the effect that a, h and k on the graph of y = x^2
• Use proper terminology to explain the transformations of a, h, and k in the equation y = a(x-h)^2 + k
: p.178 #2, 3, 4cdfg, 5cd, 6, and 7
☑ Day 3 (Nov 13) Transformations of Quadratic Relations 2
We are learning to…
• To use proper terminology to explain the transformations of a, h, and k in the equation y = a(x-h)^2 + k
• To sketch the graph of the parabola by applying transformations to y = x^2
I can
• Identify the effect that a, h and k on the graph of y = x^2
• Use proper terminology to explain the transformations of a, h, and k in the equation y = a(x-h)^2 + k
: Pg 178 #1, 4abeh, 5ab, 9-14
☑ Day 4 (Nov 16) Mapping Notation
We are learning to…
• To use proper terminology to explain the transformations of a, h, and k in the equation y = a(x-h)^2 + k
• To sketch the graph of the parabola by applying transformations to y = x^2
I can…
• Identify the effect that a, h and k on the graph of y = x^2
• Use proper terminology to explain the transformations of a, h and k in the equation y = a(x-h)^2 + k
: p.185 #1 and 2
4 - Graphing Transformations & Mapping Notation SOLUTIONS
File Size: 1084 kb
File Type: pdf
Download File
☑ Day 5 (Nov 17) Writing an Equation for a Parabola
We are learning to…
• To use proper terminology to explain the transformations of a, h, and k in the equation y = a(x-h)^2 + k
• To create an equation using the graph of a parabola
I can…
• Create an equation using the graph of a parabola
: p.185 #3-10
☑ Day 6 (Nov 18) Quadratic Relations Applications
We are learning to solve real life problems using a graph or an equation of a quadratic relation
I can solve real life problems using a graph or an equation of a quadratic relation
EXTRA PRACTICE: p.186 #11-14, 17-19, and 20
☑ Day 7 (Nov 19) Quadratic Relations Applications II
We are learning to solve real life problems using a graph or an equation of a quadratic relation
I can solve real life problems using a graph or an equation of a quadratic relation
EXTRA PRACTICE: Handout
☑ Day 8 (Nov 20) Factored Form
We are learning to identify the x-intercepts from an equation in factored form
SUCCESS CRITERIA: I can determine the x-intercepts from a quadratic equation in factored form.
EXTRA PRACTICE: p.192 #1-3, 4ace, 5-13
☑ Day 9 (Nov 24) Unit Review
We are learning to refine all skills learned this unit.
I can complete all tasks required in this unit
Pg 202 #1-10
Pg 204 #1-11, 13
Day 10 (Nov 25) Unit Test | {"url":"http://300math.weebly.com/chapter-4-quadratic-relations.html","timestamp":"2024-11-13T08:28:24Z","content_type":"text/html","content_length":"64646","record_id":"<urn:uuid:3a364418-fba6-400e-a02c-3026ae65f2e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00344.warc.gz"} |
Package ecdh - The Go Programming Language
Package ecdh
Overview ▸
Overview ▾
Package ecdh implements Elliptic Curve Diffie-Hellman over NIST curves and Curve25519.
type Curve interface {
// GenerateKey generates a random PrivateKey.
// Most applications should use [crypto/rand.Reader] as rand. Note that the
// returned key does not depend deterministically on the bytes read from rand,
// and may change between calls and/or between versions.
GenerateKey(rand io.Reader) (*PrivateKey, error)
// NewPrivateKey checks that key is valid and returns a PrivateKey.
// For NIST curves, this follows SEC 1, Version 2.0, Section 2.3.6, which
// amounts to decoding the bytes as a fixed length big endian integer and
// checking that the result is lower than the order of the curve. The zero
// private key is also rejected, as the encoding of the corresponding public
// key would be irregular.
// For X25519, this only checks the scalar length.
NewPrivateKey(key []byte) (*PrivateKey, error)
// NewPublicKey checks that key is valid and returns a PublicKey.
// For NIST curves, this decodes an uncompressed point according to SEC 1,
// Version 2.0, Section 2.3.4. Compressed encodings and the point at
// infinity are rejected.
// For X25519, this only checks the u-coordinate length. Adversarially
// selected public keys can cause ECDH to return an error.
NewPublicKey(key []byte) (*PublicKey, error)
// contains filtered or unexported methods
func P256 ¶ 1.20
func P256() Curve
P256 returns a Curve which implements NIST P-256 (FIPS 186-3, section D.2.3), also known as secp256r1 or prime256v1.
Multiple invocations of this function will return the same value, which can be used for equality checks and switch statements.
func P384 ¶ 1.20
func P384() Curve
P384 returns a Curve which implements NIST P-384 (FIPS 186-3, section D.2.4), also known as secp384r1.
Multiple invocations of this function will return the same value, which can be used for equality checks and switch statements.
func P521 ¶ 1.20
func P521() Curve
P521 returns a Curve which implements NIST P-521 (FIPS 186-3, section D.2.5), also known as secp521r1.
Multiple invocations of this function will return the same value, which can be used for equality checks and switch statements.
func X25519() Curve
X25519 returns a Curve which implements the X25519 function over Curve25519 (RFC 7748, Section 5).
Multiple invocations of this function will return the same value, so it can be used for equality checks and switch statements.
PrivateKey is an ECDH private key, usually kept secret.
These keys can be parsed with crypto/x509.ParsePKCS8PrivateKey and encoded with crypto/x509.MarshalPKCS8PrivateKey. For NIST curves, they then need to be converted with crypto/ecdsa.PrivateKey.ECDH
after parsing.
type PrivateKey struct {
// contains filtered or unexported fields
func (*PrivateKey) Bytes ¶ 1.20
func (k *PrivateKey) Bytes() []byte
Bytes returns a copy of the encoding of the private key.
func (*PrivateKey) Curve ¶ 1.20
func (k *PrivateKey) Curve() Curve
func (*PrivateKey) ECDH ¶ 1.20
func (k *PrivateKey) ECDH(remote *PublicKey) ([]byte, error)
ECDH performs an ECDH exchange and returns the shared secret. The PrivateKey and PublicKey must use the same curve.
For NIST curves, this performs ECDH as specified in SEC 1, Version 2.0, Section 3.3.1, and returns the x-coordinate encoded according to SEC 1, Version 2.0, Section 2.3.5. The result is never the
point at infinity.
For X25519, this performs ECDH as specified in RFC 7748, Section 6.1. If the result is the all-zero value, ECDH returns an error.
func (*PrivateKey) Equal ¶ 1.20
func (k *PrivateKey) Equal(x crypto.PrivateKey) bool
Equal returns whether x represents the same private key as k.
Note that there can be equivalent private keys with different encodings which would return false from this check but behave the same way as inputs to [ECDH].
This check is performed in constant time as long as the key types and their curve match.
func (*PrivateKey) Public ¶ 1.20
func (k *PrivateKey) Public() crypto.PublicKey
Public implements the implicit interface of all standard library private keys. See the docs of crypto.PrivateKey.
func (*PrivateKey) PublicKey ¶ 1.20
func (k *PrivateKey) PublicKey() *PublicKey
PublicKey is an ECDH public key, usually a peer's ECDH share sent over the wire.
These keys can be parsed with crypto/x509.ParsePKIXPublicKey and encoded with crypto/x509.MarshalPKIXPublicKey. For NIST curves, they then need to be converted with crypto/ecdsa.PublicKey.ECDH after
type PublicKey struct {
// contains filtered or unexported fields
func (*PublicKey) Bytes ¶ 1.20
func (k *PublicKey) Bytes() []byte
Bytes returns a copy of the encoding of the public key.
func (*PublicKey) Curve ¶ 1.20
func (k *PublicKey) Curve() Curve
func (*PublicKey) Equal ¶ 1.20
func (k *PublicKey) Equal(x crypto.PublicKey) bool
Equal returns whether x represents the same public key as k.
Note that there can be equivalent public keys with different encodings which would return false from this check but behave the same way as inputs to ECDH.
This check is performed in constant time as long as the key types and their curve match. | {"url":"https://golang.google.cn/pkg/crypto/ecdh/","timestamp":"2024-11-13T18:10:43Z","content_type":"text/html","content_length":"40590","record_id":"<urn:uuid:161ebffd-50ca-490e-9960-c05c33e5bc43>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00209.warc.gz"} |
International Online Medical Council (IOMC)
A physical theory is a model of physical events. It is judged by the extent to which its predictions agree with empirical observations. The quality of a physical theory is also judged on its ability
to make new predictions which can be verified by new observations. A physical theory differs from a mathematical theorem in that while both are based on some form of axioms, judgment of mathematical
applicability is not based on agreement with any experimental results.A physical theory similarly differs from a mathematical theory, in the sense that the word "theory" has a different meaning in
mathematical terms.[b]
A physical theory involves one or more relationships between various measurable quantities. Archimedes realized that a ship floats by displacing its mass of water, Pythagoras understood the relation
between the length of a vibrating string and the musical tone it produces.[4][5] Other examples include entropy as a measure of the uncertainty regarding the positions and motions of unseen particles
and the quantum mechanical idea that (action and) energy are not continuously variable.Theoretical physics consists of several different approaches. In this regard, theoretical particle physics forms
a good example. For instance: "phenomenologists" might employ (semi-) empirical formulas and heuristics to agree with experimental results, often without deep physical understanding.[c] "Modelers"
(also called "model-builders") often appear much like phenomenologists, but try to model speculative theories that have certain desirable features (rather than on experimental data), or apply the
techniques of mathematical modeling to physics problems.[d] Some attempt to create approximate theories, called effective theories, because fully developed theories may be regarded as unsolvable or
too complicated. Other theorists may try to unify, formalise, reinterpret or generalise extant theories, or create completely new ones altogether.[e] Sometimes the vision provided by pure
mathematical systems can provide clues to how a physical system might be modeled;[f] e.g., the notion, due to Riemann and others, that space itself might be curved. Theoretical problems that need
computational investigation are often the concern of computational physics.Theoretical advances may consist in setting aside old, incorrect paradigms (e.g., aether theory of light propagation,
caloric theory of heat, burning consisting of evolving phlogiston, or astronomical bodies revolving around the Earth) or may be an alternative model that provides answers that are more accurate or
that can be more widely applied. In the latter case, a correspondence principle will be required to recover the previously known result.[6][7] Sometimes though, advances may proceed along different
paths. For example, an essentially correct theory may need some conceptual or factual revisions; atomic theory, first postulated millennia ago (by several thinkers in Greece and India) and the
two-fluid theory of electricity[8] are two cases in this point. However, an exception to all the above is the wave–particle duality, a theory combining aspects of different, opposing models via the
Bohr complementarity principle.
Relevant Topics in General Science | {"url":"https://www.iomcworld.org/medical-journals/theoretical-science-uses-50829.html","timestamp":"2024-11-14T20:45:26Z","content_type":"text/html","content_length":"36178","record_id":"<urn:uuid:339bce22-18f5-4f82-9fd5-00cf224d1de7>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00195.warc.gz"} |
Math Colloquia - Connes's Embedding Conjecture and its equivalent
I will talk on Cannes's Embedding Conjecture, which is considered as one of the most important open problems in the field of operator algebras. It asserts that every finite von Neumann algebra is
approximable by matrix algebras in suitable sense. It turns out, most notably by Kirchberg, that Cannes's Embedding Conjecture is equivalent to surprisingly
many other important conjectures which touches almost all the subfields of operator algebras and also to other branches of mathematics such as quantum information theory and noncommutative real
algebraic geometry. | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=colloquia&sort_index=room&order_type=asc&page=2&document_srl=105723&l=en","timestamp":"2024-11-05T09:30:18Z","content_type":"text/html","content_length":"43843","record_id":"<urn:uuid:bca78634-db0d-4abc-8a00-72aded8c222b>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00218.warc.gz"} |
Mastering the Q-Stick Indicator: Coding and Application Insights
Written on
Chapter 1: Understanding the Q-Stick Indicator
As we continue to innovate, develop, and back-test various technical indicators, our primary aim remains the same: to discover a balanced approach that enables us to achieve steady and reliable
returns over time. Navigating through intricate market conditions can be challenging, but with commitment and discipline, success is attainable. In this article, we will explore the Q-Stick
indicator, a contrarian oscillator designed to forecast market reversals.
Recently, I published a new book following the success of my earlier work, "Trend Following Strategies in Python." This new release includes a variety of advanced contrarian indicators and
strategies, along with a GitHub page for ongoing code updates. If you're interested, you can purchase the PDF version for 9.99 EUR via PayPal. Please ensure you include your email address in the
payment note to receive the file at the correct location. Once acquired, remember to download it via Google Drive.
Section 1.1: The Role of Exponential Moving Averages
Moving averages play a critical role in confirming trends and capitalizing on them. They are among the most recognized technical indicators, primarily due to their simplicity and consistent
effectiveness in enhancing analytical insights. These averages help identify support and resistance levels, set targets, and gain a clearer understanding of market trends. Their adaptability makes
them a vital component of any trader's toolkit.
Here’s a look at the EURUSD hourly data with a 200-hour simple moving average.
Mathematically, a moving average is calculated by dividing the total sum of observations by the number of observations. In practice, it acts as a dynamic support and resistance tool, guiding where to
place trades when the market approaches these levels. Below is an example code for implementing a moving average:
# The function to add a number of columns inside an array
def adder(Data, times):
for i in range(1, times + 1):
new_col = np.zeros((len(Data), 1), dtype=float)
Data = np.append(Data, new_col, axis=1)
return Data
# The function to delete a number of columns starting from an index
def deleter(Data, index, times):
for i in range(1, times + 1):
Data = np.delete(Data, index, axis=1)
return Data
# The function to delete a number of rows from the beginning
def jump(Data, jump):
Data = Data[jump:, ]
return Data
# Example of adding 3 empty columns to an array
my_ohlc_array = adder(my_ohlc_array, 3)
# Example of deleting the 2 columns after the column indexed at 3
my_ohlc_array = deleter(my_ohlc_array, 3, 2)
# Example of deleting the first 20 rows
my_ohlc_array = jump(my_ohlc_array, 20)
The OHLC acronym stands for Open, High, Low, and Close, which is a standard format for historical data.
The following code applies the moving average function on the 'my_data' array for a lookback period of 200, focusing on the column containing closing prices. The calculated moving average will be
stored in an additional column.
my_data = ma(my_data, 200, 3, 4)
Unlike the simple moving average, which gives equal weight to all observations, the exponential moving average (EMA) assigns greater weight to more recent data points. This makes it more responsive
to current market conditions. The formula is as follows:
The smoothing factor is commonly set to 2. Increasing the smoothing factor (alpha) further emphasizes recent observations. Below is the Python function for calculating the EMA:
def ema(Data, alpha, lookback, what, where):
alpha = alpha / (lookback + 1.0)
beta = 1 - alpha
# First value is a simple SMA
Data = ma(Data, lookback, what, where)
# Calculating first EMA
Data[lookback + 1, where] = (Data[lookback + 1, what] * alpha) + (Data[lookback, where] * beta)
# Calculating the rest of EMA
for i in range(lookback + 2, len(Data)):
Data[i, where] = (Data[i, what] * alpha) + (Data[i - 1, where] * beta)
except IndexError:
return Data
We will utilize the EMA to compute the Q-Stick Indicator, emphasizing the concepts rather than the specific coding details. Most of my strategies can be found within my published works. The key
takeaway is to grasp the techniques and strategies effectively.
Recently, I joined forces with Lumiwealth to offer a variety of courses on algorithmic trading, blockchain, and machine learning. I highly recommend checking out their detailed, hands-on courses.
Section 1.2: The Q-Stick Indicator Explained
The Q-Stick Indicator, created by Tushar Chande, assesses momentum by evaluating the difference between closing and opening prices, followed by a smoothing process. It resembles the rate of change
indicator but incorporates a smoothing factor to reduce the impact of outliers. The formula can be expressed as follows:
The default lookback period for the Q-Stick is typically set to 14, meaning we will compute the difference between closing and opening prices for each period and then apply a 14-period exponential
moving average.
Here's how the Q-Stick Indicator can be coded in Python:
def q_stick(Data, ema_lookback, opening, close, where):
for i in range(len(Data)):
Data[i, where] = Data[i, close] - Data[i, opening]
Data = ema(Data, 2, ema_lookback, where, where + 1)
return Data
The 'Data' variable refers to the OHLC array you are working with, while 'ema_lookback' indicates the chosen lookback period. The 'opening' and 'closing' variables refer to the respective columns in
the OHLC array, and 'where' determines the placement of the Q-Stick values.
While the Q-Stick Indicator is effective, it is unbounded, necessitating a strategy for trading it, such as employing moving average crosses or neutrality surpass methods.
To gain insight into the current market landscape, consider my weekly market sentiment report. This report evaluates the positioning and predicts future trends across major markets using both
straightforward and complex models.
Chapter 2: Conclusion and Best Practices
In summary, my objective is to contribute positively to the realm of objective technical analysis by promoting transparent methodologies that should be back-tested prior to implementation. This
approach aims to dispel the notion of technical analysis being subjective or lacking scientific foundation.
Medium offers a wealth of engaging articles. After reading numerous pieces, I felt inspired to begin writing my own. If you're interested, consider joining Medium using my referral link at no extra
cost to you.
Whenever you encounter a trading technique or strategy, I recommend following these steps:
1. Maintain a critical mindset and eliminate emotional biases.
2. Conduct back-testing using realistic simulation and conditions.
3. If you identify potential, optimize the strategy and run a forward test.
4. Always factor in transaction costs and simulate slippage in your tests.
5. Incorporate risk management and position sizing in your evaluations.
Finally, even after taking these precautions, remain vigilant and monitor your strategy closely, as market dynamics can shift, potentially rendering your approach unprofitable.
In this video, viewers will learn about technical indicators using Python, focusing on practical coding examples to enhance their trading strategies.
This video demonstrates how to utilize the new "Qgrid" indicator on standard candlestick charts, providing insights into its application and effectiveness. | {"url":"https://zhaopinboai.com/mastering-the-q-stick-indicator.html","timestamp":"2024-11-12T22:53:09Z","content_type":"text/html","content_length":"17280","record_id":"<urn:uuid:bf2b2e28-9dd6-4d03-b3a0-e9b404197ff2>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00654.warc.gz"} |
Electrical Formulas - Explanation, Solved Examples and FAQs
The branch of physics that deals with electricity, electronics, and electromagnetic concepts, is known as electrical. Electrical formulae are very helpful in determining the value of a parameter in
any electrical circuit. Voltage, current, power, resistance, and other electrical formulae are the most often used.
Understanding how the various units of electricity can work together can certainly help from a system of water pipes. Voltage represents water pressure, the current is represented flow rate, and
resistance represents pipe size in this analogy. Ohm's Law, volts, amps, ohms, and watts are all significant fundamental components of electricity. According to Ohm's Law, the voltage is equal to the
current flowing in a circuit multiplied by the resistance of that circuit. The base unit for measuring voltage is known as volts. The ampere, abbreviated as "amp" or "A," is the fundamental unit of
electric current in the International System of Units.
Some commonly used electrical formulae are included below, and they may be useful to you.
Electric Field Formula
An electric field is a region created by an electric charge around it, the influence of which can be observed when another charge is introduced into the field's region.
The Electric Field formula is given by,
E = F/q
F = Force
q = Charge
[Image will be uploaded soon]
Potential Difference Formula
The potential between two points (E) in an electrical circuit is defined as the amount of work (W) done by an external agent in moving a unit charge (Q) from one point to another. The potential
difference formula is expressed as,
E = W/Q
E =The electrical potential difference between two points.
W = Work done in moving a charge from one point to another.
Q = Quantity of charge in coulombs.
[Image will be uploaded soon]
Electric Power Formula
Electric power may be defined as the rate at which work is completed. The watt is the SI unit for power and is written as P. The time, voltage, and charge are all connected by the power formula.
Ohm's law can be used to change the formula. The formula of electric power is as follows:
P = VI
The formula of electric power in term of Ohm’s law is as follows,
P = I^2R
P = V^2/R
Q= Electric charge
V= Voltage
t= Time
R= Resistance
Electric Potential Formula
The charge possessed by an object and the relative position of an object with respect to other electrically charged objects is the two elements that give an object its electric potential energy.
When an item moves against an electric field, it gains energy that is known as electric potential energy. The electric potential is calculated by dividing the potential energy by the quantity of
charge for any charge.
The electric potential energy formula at any point around a point charge is given by:
V = Electric potential energy
q = Point charge
r = Distance between any point around the charge to the point charge
k = Coulomb constant; k = 9.0 × 10^9 N
[Image will be uploaded soon]
Electric Flux Formula
The electric flux is the total number of electric field lines passing through a given area in a given period of time.
The electric flux formula is expressed as,
𝜑[p] = EA
When the same plane is tilted at an angle ϴ, the projected area is Acosϴ, and the total flux through the surface is:
𝜑 = EA cosƟ
E = Magnitude of the electric field.
A = Area of the surface.
Ɵ = Angle made by the plane.
[Image will be uploaded soon]
Electric Current Formula
An electric current is the constant flow of electrons in an electric circuit. When a potential difference is applied across a wire or terminal, electrons move. The rate of change of electric charge
via a circuit is known as electric current. This current is equal to the circuit's voltage and resistance. The symbol for it is I, and the SI unit is Amperes. The electric charge and the time are
related to the electric current.
The electric current formula, according to Ohm's law, will be,
V = Voltage
R = Resistance
I = Current
[Image will be uploaded soon]
Electric Charge Formula
Electric Charge is the property of subatomic particles that causes to experience a force when placed in an electromagnetic field. The S.I unit of electric charge is coulomb and the symbol is Q. The
electric charge formula is given by,
Q = I x t
Q = Electric Charge
I = Electric current
t = Time
[Image will be uploaded soon]
Solved Examples
Ex.1. A Wire Carrying a Voltage of 21 Volts is Having a Resistance of 7⍵. Calculate the Electric Current.
Given: Voltage = 21 V,
Resistance R = 7 ω
The electric current formula is,
I = 3 Amperes
Hence electric current is 3 Amp.
Ex.2. A force of 13 N is Acting on the Charge at 9 μ C at any Point. Determine the Electric Field Intensity at that Point.
Force F = 13 N
Charge q = 9 μ C
Electric field formula is given by
E = 1.45 10^-6 N/C
Ex.3. If The Current And Voltage of An Electric Circuit Are Given As 3.5A And 16V Respectively. Calculate The Electrical Power?
Given measures are,
I = 3.5A and V = 16V
The formula of electric power is,
P = VI
P = 16 × 3.5 = 25
P = 56 watts.
FAQs on Electrical Formulas
1. What is the Work Formula?
The work is equal to the force times the distance is given by,
W = fd
If the force is being exerted at an angle to the displacement, the work done is,
W= fd cosӨ
2. What are Types of Current?
Answer: Direct current (DC) and alternating current (AC) are the two types of current electricity.
Electrons travel in one direction with direct current. Direct current is generated batteries. Electrons flow in both directions in alternating current.
3. What are Voltage and Current?
Answer: The difference in charge between two points is known as voltage. The rate at which charge travels is known as current. | {"url":"https://www.vedantu.com/formula/electrical-formulas","timestamp":"2024-11-10T10:53:08Z","content_type":"text/html","content_length":"268807","record_id":"<urn:uuid:2dce5c95-5d3b-4f9a-86f9-6ed5de084dbe>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00313.warc.gz"} |
Chen, Nan
• Office: Room 709A, William M.W. Mong Engineering Building
• Phone: (852) 3943-8237 Fax: (852) 2603-5505
• Email: @se.cuhk.edu.hk
Research Interests
• Quantitative Methods in Finance and Risk Management
• Monte Carlo Simulation
• Applied Probability
My Google Scholar Citations Citations: 1222, h-index: 17, i10-index: 17. (as of 4/10/2024)
Working Papers
• N. Chen and M. Liu. (2023). Adversarial Reinforcement Learning: A Duality-Based Approach to Solving Overlearning Issue in Deep Learning Approximation for Optimal Control.
• N. Chen, Y. Xu, R. Zhang, and M. Zhong. (2023). A Two Timescale Evolutionary Game Approach to Multi-Agent Reinforcement Learning.
• N. Chen, M. Dai, Q. Ding, and C. Yang. (2023). Patience is a Virtue: Optimal Investment in the Presence of Market Resilience. Major Revision with Management Science. [full pdf] [abstract]
• N. Chen, X. Wan, and N. Yang. (2024). Explicit Pathwise Expansion for Multivariate Diffusions and Its Application. Major Revision under Journal of Applied Probability. [full pdf] [abstract]
• N. Chen and H. Xu. (2024). The Complementary Role of Digital Engagement in Social Activeness of Elderly Life: Evidence from Urban Region of China.
• N. Chen, X. Giroud, L. Qin, and N. Wang. (2024). Corporate Investment and Savings Demand. [abstract] | {"url":"https://www1.se.cuhk.edu.hk/~nchenweb/Publications.html","timestamp":"2024-11-05T02:58:05Z","content_type":"application/xhtml+xml","content_length":"72894","record_id":"<urn:uuid:e4d8da60-da29-4d02-b736-042ef5a1b51e>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00434.warc.gz"} |
Problem Set II Solution - Programming Help
For this problem set you have to use the data in the ascii file yogurt 2018.txt. The data consists of observations on 429 households making 2567 yogurt purchases. They purchase each time one of three
brands. The five variables in the data set are, (i) the household id, running from 1 to 430, (ii) the choice made by the household, running from 1 to 3, (iii) the price for that household when they
made their decision, in cents, of yogurt brand 1, (iv) the price, in cents, of yogurt brand 2, (v) the price, in cents, of yogurt brand 3.
Let j index the choice, running from 1 to 3, t index the purchase, running from 1 to T[i], and i the household, running from 1 to 429. The number of purchases made by each household differs. For
example, the first two purchases come from household 1, the next two from household 2, and the next eight from the third household.
We focus on a discrete choice model where the utility for individual i associated with choice j, in purchase t is
Uijt = αj + β · Pijt + ijt,
where P[ijt] is the price of brand j for household i at purchase time t. We assume the ε[ijt] are independent across time, choice and household, with an extreme value distribution. Normalize α[1] =
0, so that there are three free parameters, α[2], α[3], and β .
1. First, calculate the mean price for each brand, by the choice made. That is, calculate the average price of brand 1, 2 and 3 for households choosing brand 1, calculate the average price of brand
1, 2 and 3 for households choosing brand 2, and calculate the average price of brand 1, 2 and 3 for households choosing brand 3. Do the patterns
Imbens, Problem Set II, MGTECON640/ECON292 Fall ’18 2
make sense? That is, are the prices for brand j lowest among the households choosing brand j?
2. Next we want to estimate the conditional logit model. Because of the independence assumption on the ε[ijt], we can ignore the fact that some purchases come from the same household, and the
likelihood function is
L(α[2], α[3], β) =
N ^Ti
Y Y 1Y[ijt] =1 · exp(β Pi1t) + 1Y[ijt] =2 · exp(β Pi2t + α2) + 1Y[ijt] =2 · exp(β Pi3t + α3)
i=1 t=1 exp(β P[i1t]) + exp(β P[i2t] + α[2]) + exp(β P[i3t] + α[3])
First show that at
β −0.0400
α[2] = 0.5000
α[3] −1.0000
the log likelihood function is equal to -2660.1.
3. Calculate the analytic first derivatives of the log likelihood function at these values for the parameters. Hint: the first derivative with respect to β is -6948.8.
3. Calculate the analytic second derivatives with respect to the parameters. Hint: the second derivative with respect to β is approximately -505130.
3. Next, use the Newton-Rahpson algorithm to find the maximum likelihood estimators for β , α[2], and α[3].
3. Next we explore a random coefficients version of the conditional logit model. Rather than use continuous mixtures, as is more common in the literature, we use a compu-tationally simpler version
with a binary mixture. We will let the coefficient on price vary by household. Thus, we model the latent utility as
Uijt = αj + βi · Pijt + ijt,
Imbens, Problem Set II, MGTECON640/ECON292 Fall ’18 3
β[i] ∈ {β[L], β[H] }, with pr(β[i] = β[L]) = π = 0.4.
(More generally we may want to estimate the mixture probability π, but we wont do that here.)
Show that the likelihood function for this mixture model is
L(α[2], α[3], β[L], β[H] , π) = ^Y
T[i] 1Y[ijt] =1 · exp(βLPi1t) + 1Y[ijt] =2 · exp(βLPi2t + α2) + 1Y[ijt] =2 · exp(βLPi3t + α3)
π · Y
t=1 exp(β[L]P[i1t]) + exp(β[L]P[i2t] + α[2]) + exp(β[L]P[i3t] + α[3])
T[i] 1[Y][ijt] [=1] · exp(β[H] P[i1t]) + 1[Y][ijt] [=2] · exp(β[H] P[i2t] + α[2]) + 1[Y][ijt] [=2] · exp(β[H] P[i3t] + α[3])
+(1−π)·^Y .
exp(β[H] P[i1t]) + exp(β[H] P[i2t] + α[2]) + exp(β[H] P[i3t] + α[3])
7. Plot the log likelihood function, for π = 0.4, at
α[2] αˆ[2]
= ,
α[3] ˆ αˆ[3]
β[L] β − c
β[H] ˆ + c
as a function of c, from c = 0 to c = 0.2.
Compare the value of the log likelihood function at the value of c that maximimes it with the value at c = 0. Does it appear that allowing for the heterogeneity in price sensitivity is important?
8. Estimate the random effects model using the EM algorithm. Report parameter esti-mates for α[2], α[3], β[L], and β[H] . | {"url":"https://www.edulissy.org/product/problem-set-ii-solution/","timestamp":"2024-11-09T11:11:40Z","content_type":"text/html","content_length":"201484","record_id":"<urn:uuid:203a351f-b632-485d-a5bc-87bc3b71a8aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00262.warc.gz"} |
This post is the fourth in a series of tutorials on IIR Butterworth filter design. So far we covered lowpass [1], bandpass [2], and band-reject [3] filters; now we’ll design highpass filters. The
general approach, as before, has six steps:
Find the poles of a lowpass analog prototype filter with Ωc = 1 rad/s. Given the -3 dB frequency of the digital highpass filter, find the corresponding frequency of the analog highpass filter
(pre-warping). Transform the...
In this post, I show how to design IIR Butterworth band-reject filters, and provide two Matlab functions for band-reject filter synthesis. Earlier posts covered IIR Butterworth lowpass [1] and
bandpass [2] filters. Here, the function br_synth1.m designs band-reject filters based on null frequency and upper -3 dB frequency, while br_synth2.m designs them based on lower and upper -3 dB
frequencies. I’ll discuss the differences between the two approaches later in this...
In this post, I present a method to design Butterworth IIR bandpass filters. My previous post [1] covered lowpass IIR filter design, and provided a Matlab function to design them. Here, we’ll do
the same thing for IIR bandpass filters, with a Matlab function bp_synth.m. Here is an example function call for a bandpass filter based on a 3rd order lowpass prototype:
N= 3; % order of prototype LPF fcenter= 22.5; % Hz center frequency, Hz bw= 5; ...
Cedron Dawg ●
January 6, 2018
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas to calculate the phase and amplitude of a pure complex tone from a DFT
bin value and knowing the frequency. This is a much simpler problem to solve than the corresponding case for a pure real tone which I covered in an earlier blog article[1]. In the noiseless single
tone case, these equations will be exact. In the presence of noise or other tones...
Cedron Dawg ●
December 17, 2017
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving alternative exact formulas for the bin values of a real tone in a DFT. The derivation
of the source equations can be found in my earlier blog article titled "DFT Bin Value Formulas for Pure Real Tones"[1]. The new form is slighty more complicated and calculation intensive, but it is
more computationally accurate in the vicinity of near integer frequencies. This...
While there are plenty of canned functions to design Butterworth IIR filters [1], it’s instructive and not that complicated to design them from scratch. You can do it in 12 lines of Matlab code. In
this article, we’ll create a Matlab function butter_synth.m to design lowpass Butterworth filters of any order. Here is an example function call for a 5th order filter:
N= 5 % Filter order fc= 10; % Hz cutoff freq fs= 100; % Hz sample freq [b,a]=...
Half-band filters are lowpass FIR filters with cut-off frequency of one-quarter of sampling frequency fs and odd symmetry about fs/4 [1]*. And it so happens that almost half of the coefficients are
zero. The passband and stopband bandwiths are equal, making these filters useful for decimation-by-2 and interpolation-by-2. Since the zero coefficients make them computationally efficient, these
filters are ubiquitous in DSP systems.
Here we will compute half-band...
Cedron Dawg ●
November 6, 2017
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by extending the exact two bin formulas for the frequency of a real tone in a DFT to the three bin
case. This article is a direct extension of my prior article "Two Bin Exact Frequency Formulas for a Pure Real Tone in a DFT"[1]. The formulas derived in the previous article are also presented in
this article in the computational order, rather than the indirect order they were...
With the growth in the Internet of Things (IoT) products, the number of applications requiring an estimate of range between two wireless nodes in indoor channels is growing very quickly as well.
Therefore, localization is becoming a red hot market today and will remain so in the coming years.
One question that is perplexing is that many companies now a days are offering cm level accurate solutions using RF signals. The conventional wireless nodes usually implement synchronization...
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas for the frequency of a real tone in a DFT. This time it is a two bin
version. The approach taken is a vector based one similar to the approach used in "Three Bin Exact Frequency Formulas for a Pure Complex Tone in a DFT"[1]. The real valued formula presented in this
article actually preceded, and was the basis for the complex three bin...
One of the basic DSP principles states that a sampled time signal has a periodic spectrum with period equal to the sample rate. The derivation of can be found in textbooks [1,2]. You can also
demonstrate this principle numerically using the Discrete Fourier Transform (DFT).
The DFT of the sampled signal x(n) is defined as:
$$X(k)=\sum_{n=0}^{N-1}x(n)e^{-j2\pi kn/N} \qquad (1)$$
X(k) = discrete frequency spectrum of time sequence x(n)
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas for the phase and amplitude of a non-integer frequency real tone in a
DFT. The linearity of the Fourier Transform is exploited to reframe the problem as the equivalent of finding a set of coordinates in a specific vector space. The found coordinates are then used to
calculate the phase and amplitude of the pure real tone in the DFT. This article...
This blog discusses a little-known filter characteristic that enables real- and complex-coefficient tapped-delay line FIR filters to exhibit linear phase behavior. That is, this blog answers the
What is the constraint on real- and complex-valued FIR filters that guarantee linear phase behavior in the frequency domain?
I'll declare two things to convince you to continue reading.
Declaration# 1: "That the coefficients must be symmetrical" is not a correct
Some might argue that measurement is a blend of skepticism and faith. While time constraints might make you lean toward faith, some healthy engineering skepticism should bring you back to statistics.
This article reviews some practical statistics that can help you satisfy one common question posed by skeptical engineers: “How precise is my measurement?” As we’ll see, by understanding how to
answer it, you gain a degree of control over your measurement time.
An accurate, precise...
Neil Robertson ●
September 26, 2021
Digitizing a signal using an Analog to Digital Converter (ADC) usually requires an anti-alias filter, as shown in Figure 1a. In this post, we’ll develop models of lowpass Butterworth and Chebyshev
anti-alias filters, and compute the time domain and frequency domain output of the ADC for an example input signal. We’ll also model aliasing of Gaussian noise. I hope the examples make the
textbook explanations of aliasing seem a little more real. Of course, modeling of...
Analog to digital converters (ADC’s) have several imperfections that affect communications signals, including thermal noise, differential nonlinearity, and sample clock jitter [1, 2]. As shown in
Figure 1, the ADC has a sample/hold function that is clocked by a sample clock. Jitter on the sample clock causes the sampling instants to vary from the ideal sample time. This transfers the jitter
from the sample clock to the input signal.
In this article, I present a Matlab...
This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by framing it in a graphical interpretation. The bin calculation formula is shown to be the
equivalent of finding the center of mass, or centroid, of a set of points. Various examples are graphed to illustrate the well known properties of DFT bin values. This treatment will only consider
real valued signals. Complex valued signals can be analyzed in a similar manner with...
Vincent Herrmann ●
October 11, 2016
In the previous blog post I described the workings of the Fast Wavelet Transform (FWT) and how wavelets and filters are related. As promised, in this article we will see how to construct useful
filters. Concretely, we will find a way to calculate the Daubechies filters, named after Ingrid Daubechies, who invented them and also laid much of the mathematical foundations for wavelet analysis.
Besides the content of the last post, you should be familiar with basic complex algebra, the...
In Part 1, I presented a Matlab function to model an ADC with jitter on the sample clock, and applied it to examples with deterministic jitter. Now we’ll investigate an ADC with random clock jitter,
by using a filtered or unfiltered Gaussian sequence as the jitter source. What we are calling jitter can also be called time jitter, phase jitter, or phase noise. It’s all the same phenomenon.
Typically, we call it jitter when we have a time-domain representation,...
Cedron Dawg ●
November 11, 2022
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by explaining how a tweak to a well known frequency approximation formula makes it better, and
another tweak makes it exact. The first tweak is shown to be the first of a pattern and a novel approximation formula is made from the second. It only requires a few extra calculations beyond the
original approximation to come up with an approximation suitable for most...
Cedron Dawg ●
April 29, 2017
This is an article that is a digression from trying to give a better understanding to the Discrete Fourier Transform (DFT).
A method for building a table of Base 10 Logarithms, also known as Common Logarithms, is featured using math that can be done with paper and pencil. The reader is assumed to have some familiarity
with logarithm functions. This material has no dependency on the material in my previous blog articles.
If you were ever curious about how...
The Discrete Fourier Transform (DFT) is used to find the frequency spectrum of a discrete-time signal. A computationally efficient version called the Fast Fourier Transform (FFT) is normally used to
calculate the DFT. But, as many have found to their dismay, the FFT, when used alone, usually does not provide an accurate spectrum. The reason is a phenomenon called spectral leakage.
Spectral leakage can be reduced drastically by using a window function in conjunction...
In Part 1, I presented a Matlab function to model an ADC with jitter on the sample clock, and applied it to examples with deterministic jitter. Now we’ll investigate an ADC with random clock jitter,
by using a filtered or unfiltered Gaussian sequence as the jitter source. What we are calling jitter can also be called time jitter, phase jitter, or phase noise. It’s all the same phenomenon.
Typically, we call it jitter when we have a time-domain representation,...
This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving an analytical formula for the DFT of pure real tones. The formula is used to explain
the well known properties of the DFT. A sample program is included, with its output, to numerically demonstrate the veracity of the formula. This article builds on the ideas developed in my previous
two blog articles:
Some might argue that measurement is a blend of skepticism and faith. While time constraints might make you lean toward faith, some healthy engineering skepticism should bring you back to statistics.
This article reviews some practical statistics that can help you satisfy one common question posed by skeptical engineers: “How precise is my measurement?” As we’ll see, by understanding how to
answer it, you gain a degree of control over your measurement time.
An accurate, precise...
FFMPEG is a set of libraries and a command line tool for encoding and decoding audio and video in many different formats. It is a free software project for manipulating/processing multimedia data.
Many open source media players are based on FFMPEG libraries.
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving an exact formula for the frequency of a real tone in a DFT. According to current
teaching, this is not possible, so this article should be considered a major theoretical advance in the discipline. The formula is presented in a few different formats. Some sample calculations are
provided to give a numerical demonstration of the formula in use. This article is...
Rick Lyons ●
October 31, 2013
●1 comment
In digital signal processing (DSP) we're all familiar with the processes of bandpass sampling an analog bandpass signal and downsampling a digital bandpass signal. The overall spectral behavior of
those operations are well-documented. However, mathematical expressions for computing the translated frequency of individual spectral components, after bandpass sampling or downsampling, are not
available in the standard DSP textbooks. The following three sections explain how to compute the...
Many of us are familiar with modeling a continuous-time system in the frequency domain using its transfer function H(s) or H(jω). However, finding the time response can be challenging, and
traditionally involves finding the inverse Laplace transform of H(s). An alternative way to get both time and frequency responses is to transform H(s) to a discrete-time system H(z) using the
impulse-invariant transform [1,2]. This method provides an exact match to the continuous-time...
The discrete frequency response H(k) of a Finite Impulse Response (FIR) filter is the Discrete Fourier Transform (DFT) of its impulse response h(n) [1]. So, if we can find H(k) by whatever method,
it should be identical to the DFT of h(n). In this article, we’ll find H(k) by using complex exponentials, and we’ll see that it is indeed identical to the DFT of h(n).
Consider the four-tap FIR filter in Figure 1, where each block labeled Ts represents a delay of one... | {"url":"https://www.dsprelated.com/blogs-5/mp/all/Tutorials.php","timestamp":"2024-11-03T21:41:28Z","content_type":"text/html","content_length":"70719","record_id":"<urn:uuid:90dd7489-9cae-4cd2-ac83-75120e6080fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00230.warc.gz"} |
Quantum Computers: Understanding Their Basic Mechanics and Limitless Potential - FutureEnTech
Quantum Computers: Understanding Their Basic Mechanics and Limitless Potential
Since the advent of computers, humans have relentlessly pursued advancements and innovations to enhance their capabilities consistently.
One of the most critical and visionary mission humans have ever undertaken was the Apollo 11 mission to land the first human on the moon’s surface. It had the most sophisticated computing and
electronic systems, called Apollo Guidance Computer.
Can you guess the RAM and ROM Apollo Guidance Computer had?
The Apollo Guidance Computer had RAM of 4KB and 32KB of ROM.
Yes, it is astonishing. It is like we have created our own YAKA in the last 50 years regarding computing abilities and digital advancement.
Our era is defined by rigorous technological progress. Our relentless pursuit of advancement and sophistication has led us to another milestone in the realm of computers, the beginning of quantum
Quantum computers have emerged as a captivating and revolutionary step forward.
In this simple exploration, we will embark on a journey through the intricate workings of quantum computers and try to shed light on their operation and the boundless possibilities they present.
The Quantum Foundation: A Paradigm Shift
I would like to start with something not so obvious (not that anything is obvious when it comes to Quantum mechanics): it might come as a surprise to you when I say that the working mechanics of
computers that we usually taught in our school or college physics are grossly oversimplified. Yes, we have a solid understanding of how computers work. However, there is continuous debate and
discussion between physicists and computer scientists about the fundamental workings of classical computers and the underlying principles of computing.
If anyone tells you that they understand how Quantum Computers work, do not believe them!
Unlike classical computers, Quantum computers operate on the principles of quantum mechanics. Quantum mechanics is a field notorious for challenging the very fabric of conventional reality.
Classical computers work on binary code. They use bits (0 and 1) as the fundamental data unit; we also refer to 0 as the ground state and 1 as the excited state.
Unlike classical computers, which employ bits as the fundamental unit of data, quantum computers utilize quantum bits or qubits. These qubits are based on a well-known phenomenon called
superposition. Superposition is a Quantum phenomenon where a quantum state can exist in multiple states simultaneously. Remember the cat that is both dead and alive? Yes, it is that, Schrödinger’s
Superposition allows the qubits to exist in multiple states simultaneously, like Schrödinger’s cat. This unique property gives quantum computers an innate advantage over their classical counterparts,
allowing them to process extensive volumes of data with exponential speed and efficiency. Quantum computers can process information in ways that were once deemed unfathomable.
Classical computers use bits where probabilistic states do not play any role. It is either an excited state (1) or a ground state (0). However, qubits are fundamentally different. There is no
absolute probability of them being either a 1 or a 0. They can be either both simultaneously or anything in between!
You can think of qubits as sinusoidal (or sine) waves. A qubit is neither the maxima nor the minima. But it’s likely to be either. When a number of these qubits interact, they follow the interference
patterns like waves. They interact constructively or destructively. This is the basics of working a Quantum Computer; when we manipulate how these constructive and destructive interference patterns
occur, we are essentially harnessing the quantum phenomenon for our intended use.
Entanglement: The Key to Unprecedented Processing Power
Quantum Entanglement is another famous quantum phenomenon that is involved in the working of Quantum Computers. Even Einstein dismissed this phenomenon as a “spooky action at a distance.”
Quantum entanglement is a strange and intriguing phenomenon where the state of one entangled particle is intricately transmitted and intertwined with the state of the other, even if they are on two
opposite ends of the universe.
In the case of quantum computers, entanglement equips quantum computers to undertake intricate calculations with astonishing precision. As the number of entangled qubits increases, the processing
capability of quantum computers grows exponentially, unveiling solutions to challenges that would be insurmountable for classical computing systems.
Quantum Gates: Manipulating and controlling the Building Blocks
Yes, these complex and counter-intuitive quantum phenomena exist, but how do they make computers work?
The answer is Quantum Gates. Quantum gates govern the manipulation of qubits and the execution of computation. These gates allow us to control the transformations and interactions of qubits, alter
their state and enable intricate calculations.
The classic counterparts of quantum gates are logic gates. However, it’s important to note that quantum gates play a pivotal role in forming quantum circuits, enabling information processing that
defies the constraints of classical logic. The capacity to alter qubits at their most fundamental level is the foundation of quantum computers’ unparalleled processing power. Some examples of quantum
gates (to feed your curiosity) are the Hadamard gate, Pauli X gate, Pauli Y gate, and Identity gate.
Shor’s Algorithm: A Glimpse into Quantum Computing Potential & Supremacy
Developed in 1994 by a prominent American mathematician Peter Shor, Shor’s algorithm has the capability to factor large numbers at an exponentially accelerated rate compared to any classical
algorithm. It is one of the most astonishing showcases of quantum computing’s potential.
So how does Shor’s algorithm show the supremacy of quantum computers?
One of the most complex problems in mathematics is finding an integer’s prime factors. The security of our online transaction hinges on the idea that it is practically impossible to factor integers
with a thousand or more digits.
It is a profound subject that deserves its own book.
Simply put, a quantum computer could factor integers with a thousand or more digits using Shor’s algorithm. Yes, classical computers, too, can run Shor’s algorithm, but they are painfully shown. A
quantum computer containing only a few hundred qubits has the potential to break down numbers that would be practically impossible (taking hundreds of years of computer processing) for a classical
computer to solve.
What a supercomputer or a server farm would take years to do; can be done on a quantum computer within minutes or even seconds.
The implications are profound: quantum computers could potentially render many contemporary encryption methods obsolete, presenting both remarkable opportunities and intricate challenges in the realm
of cybersecurity and data protection.
Quantum Computers in Action: Realizing Real-World Impact
The capabilities of quantum computers transcend far beyond the field of cryptography and mathematics. Quantum computers have tremendous potential to transform diverse areas such as optimization,
material science, drug discovery, and artificial intelligence. Their ability to swiftly analyze extensive and elaborate datasets paves the way for a revolution across industries. From handling
intricate optimization quandaries to accelerating the development of novel materials and drugs, quantum computers have the power to unlock realms of science and technology that were once deemed
Looking At the Future: The Limitless Horizons of Quantum Computing
From quantum error correction to qubit stability and hardware development, the field of quantum computing is rapidly evolving with the persistent efforts of computer scientists, engineers, and
Our ruthless pursuit of quantum supremacy—the threshold at which quantum computers surpass the most advanced classical computing systems—propels the field forward at a breathtaking pace. The power,
influence, and transformative potential of a quantum computer are becoming more and more evident with each milestone.
We are approaching a future where harnessing the quantum realm’s remarkable properties will unravel mysteries that have long eluded our grasp. This article is researched and prepared by Ubaid
Shareef, Digital Marketer at BrainerHub Solutions – Custom software development company in India
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Viewed From The Motivation Of The Students Of Sultan Agung Junior High School In Purworejo
Mujiyem, Sapti and Suparwati, Suparwati (2011) An Experiment Of Mathematics Teaching Using SAVI Approach And Conventional Approach Viewed From The Motivation Of The Students Of Sultan Agung Junior
High School In Purworejo. PROCEEDINGS International Seminar and the Fourth National Conference on Mathematics Education. ISSN 978-979-16353-7-0
P - 36.pdf
Download (157kB) | Preview
The objective of this research is to investigate whether Mathematics teaching using SAVI approach can make better achievement in learning Mathematics than conventional approach viewed from the
student’s motivation of Sultan Agung Junior High School in Purworejo on the circle material. This research is a quasi experimental research with 2×3 factorial design. The subject of the research is
the 8th-grade students of Sultan Agung Junior High School in Purworejo in the academic year 2010/2011. The sample of this research are 60 students which consist of experimental group and control
group. The data were collected by using test of learning achievement in Mathematics and a questionnaire of student’s motivation. The test instruments were validated by expert. In the pre-requisite
test, analysis variance precondition test using Liliefors test for normality and Bartlett test for homogeneity test. With ∝ =0,05, samples come from normal distributed population and homogeneous. The
hypothesis testing using two-way ANAVA with different cell with α = 0,05. It shows: (1) Fc = 4.378 > Ft = 4.024, it means Mathematics teaching using SAVI approach gives a better achievement in
learning Mathematics than using conventional approach; (2) Fc =20.822 > Ft= 3.174, it means the achievement in learning Mathematics of the students who have higher motivation is better than those who
have lower motivation; and (3) Fc = 1.617 < Ft = 3.174, it means the difference characteristic between the Mathematics teaching using SAVI approach and conventional approach for every students’
motivation in learning Mathematics is the same. Key Words: Mathematics Teaching, SAVI, Motivation
Actions (login required) | {"url":"http://eprints.uny.ac.id/1346/","timestamp":"2024-11-10T08:39:13Z","content_type":"application/xhtml+xml","content_length":"25568","record_id":"<urn:uuid:7ccfa07f-6627-4a8a-8851-0add29ee3144>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00253.warc.gz"} |
How to Check Data Inside Column In Pandas?
To check data inside a column in pandas, you can use the unique() method to see all unique values in that column. You can also use the value_counts() method to get a frequency count of each unique
value in the column. Additionally, you can use boolean indexing to filter the dataframe based on specific conditions in the column.
How to drop rows with missing values in a specific column in pandas?
You can drop rows with missing values in a specific column in pandas by using the dropna() method along with the subset parameter. Here's an example:
1 import pandas as pd
3 # Create a sample dataframe
4 data = {'A': [1, 2, None, 4],
5 'B': [4, None, 6, 7]}
7 df = pd.DataFrame(data)
9 # Drop rows with missing values in column 'A'
10 df = df.dropna(subset=['A'])
12 print(df)
This will drop rows with missing values in column 'A' and output the updated dataframe without those rows.
How to extract specific rows based on values in a column in pandas?
You can use the pandas library in Python to extract specific rows based on values in a column. Here is an example code snippet that demonstrates how to do this:
1 import pandas as pd
3 # Create a sample DataFrame
4 data = {
5 'Name': ['Alice', 'Bob', 'Charlie', 'David'],
6 'Age': [25, 30, 35, 40],
7 'Gender': ['F', 'M', 'M', 'M']
8 }
9 df = pd.DataFrame(data)
11 # Extract rows where the value in the 'Gender' column is 'M'
12 filtered_rows = df[df['Gender'] == 'M']
14 print(filtered_rows)
In this example, we create a DataFrame with three columns: 'Name', 'Age', and 'Gender'. We then use the df[df['Gender'] == 'M'] syntax to extract rows where the value in the 'Gender' column is 'M'.
This will return a new DataFrame containing only the rows where the condition is met.
You can modify the condition inside the square brackets to extract rows based on different values or conditions in the specified column.
How to check for outliers in a column in pandas?
One common way to check for outliers in a column in pandas is by using the interquartile range (IQR) method.
Here's a step-by-step guide on how to do this:
1. Calculate the first quartile (25th percentile) and third quartile (75th percentile) of the column using the quantile() method in pandas.
1 Q1 = df['column_name'].quantile(0.25)
2 Q3 = df['column_name'].quantile(0.75)
1. Calculate the interquartile range (IQR) by subtracting the first quartile from the third quartile.
1. Define the lower and upper bounds for outliers by multiplying 1.5 with the IQR and adding/subtracting it to the first and third quartile, respectively.
1 lower_bound = Q1 - 1.5 * IQR
2 upper_bound = Q3 + 1.5 * IQR
1. Identify the outliers by filtering the values in the column that fall outside the lower and upper bounds.
1 outliers = df[(df['column_name'] < lower_bound) | (df['column_name'] > upper_bound)]
Now you have a DataFrame containing the outliers in the specified column. You can further investigate or handle these outliers as needed.
What is the best way to handle missing values in a column in pandas?
The best way to handle missing values in a column in pandas is to either drop the rows with missing values, fill in the missing values with a specific value, or use more advanced techniques like
interpolation or machine learning algorithms to impute the missing values.
Here are some common methods for handling missing values in pandas:
1. Drop rows with missing values:
1 df.dropna(subset=['column_name'], inplace=True)
1. Fill in missing values with a specific value:
1 df['column_name'].fillna(value, inplace=True)
1. Fill in missing values with the mean, median, or mode of the column:
1 mean = df['column_name'].mean()
2 df['column_name'].fillna(mean, inplace=True)
1. Interpolate missing values using the interpolate() function:
1 df['column_name'].interpolate(method='linear', inplace=True)
1. Use machine learning algorithms like KNN or Random Forest to impute missing values:
1 from sklearn.impute import KNNImputer
2 imputer = KNNImputer(n_neighbors=2)
3 df['column_name'] = imputer.fit_transform(df['column_name'].values.reshape(-1, 1))
Each method has its own advantages and disadvantages, so it is important to consider the nature of the missing data and the characteristics of the dataset before choosing the appropriate method. | {"url":"https://freelanceshack.com/blog/how-to-check-data-inside-column-in-pandas","timestamp":"2024-11-05T10:45:47Z","content_type":"text/html","content_length":"417635","record_id":"<urn:uuid:315d07f6-6735-4af0-8741-dea6a09b0c21>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00132.warc.gz"} |
icrometers to Arpent
Micrometers to Arpent Converter
⇅ Switch toArpent to Micrometers Converter
How to use this Micrometers to Arpent Converter 🤔
Follow these steps to convert given length from the units of Micrometers to the units of Arpent.
1. Enter the input Micrometers value in the text field.
2. The calculator converts the given Micrometers into Arpent in realtime ⌚ using the conversion formula, and displays under the Arpent label. You do not need to click any button. If the input
changes, Arpent value is re-calculated, just like that.
3. You may copy the resulting Arpent value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Micrometers to Arpent?
The formula to convert given length from Micrometers to Arpent is:
Length[(Arpent)] = Length[(Micrometers)] / 58521599.953856885
Substitute the given value of length in micrometers, i.e., Length[(Micrometers)] in the above formula and simplify the right-hand side value. The resulting value is the length in arpent, i.e., Length
Calculation will be done after you enter a valid input.
Consider that a high-precision engineering tool has a tolerance of 10 micrometers.
Convert this tolerance from micrometers to Arpent.
The length in micrometers is:
Length[(Micrometers)] = 10
The formula to convert length from micrometers to arpent is:
Length[(Arpent)] = Length[(Micrometers)] / 58521599.953856885
Substitute given weight Length[(Micrometers)] = 10 in the above formula.
Length[(Arpent)] = 10 / 58521599.953856885
Length[(Arpent)] = 1.70877078e-7
Final Answer:
Therefore, 10 µm is equal to 1.70877078e-7 arpent.
The length is 1.70877078e-7 arpent, in arpent.
Consider that a state-of-the-art microchip measures 200 micrometers in thickness.
Convert this thickness from micrometers to Arpent.
The length in micrometers is:
Length[(Micrometers)] = 200
The formula to convert length from micrometers to arpent is:
Length[(Arpent)] = Length[(Micrometers)] / 58521599.953856885
Substitute given weight Length[(Micrometers)] = 200 in the above formula.
Length[(Arpent)] = 200 / 58521599.953856885
Length[(Arpent)] = 0.00000341754156
Final Answer:
Therefore, 200 µm is equal to 0.00000341754156 arpent.
The length is 0.00000341754156 arpent, in arpent.
Micrometers to Arpent Conversion Table
The following table gives some of the most used conversions from Micrometers to Arpent.
Micrometers (µm) Arpent (arpent)
0 µm 0 arpent
1 µm 1.709e-8 arpent
2 µm 3.418e-8 arpent
3 µm 5.126e-8 arpent
4 µm 6.835e-8 arpent
5 µm 8.544e-8 arpent
6 µm 1.0253e-7 arpent
7 µm 1.1961e-7 arpent
8 µm 1.367e-7 arpent
9 µm 1.5379e-7 arpent
10 µm 1.7088e-7 arpent
20 µm 3.4175e-7 arpent
50 µm 8.5439e-7 arpent
100 µm 0.00000170877 arpent
1000 µm 0.00001708771 arpent
10000 µm 0.00017087708 arpent
100000 µm 0.00170877078 arpent
A micrometer (µm) is a unit of length in the International System of Units (SI). One micrometer is equivalent to 0.000001 meters or approximately 0.00003937 inches.
The micrometer is defined as one-millionth of a meter, making it an extremely precise measurement for very small distances.
Micrometers are used worldwide to measure length and distance in various fields, including science, engineering, and manufacturing. They are especially important in fields that require precise
measurements, such as semiconductor fabrication and microscopy.
An arpent is a historical unit of length used primarily in French-speaking regions and in land measurement. One arpent is approximately equivalent to 192.75 feet or 58.66 meters.
The arpent was used in various regions, including France and the former French colonies, to measure land and property. Its length could vary slightly depending on the specific region and historical
Arpents were used in land surveying and agriculture, particularly in historical and regional contexts. Although less common today, the unit provides historical insight into land measurement practices
and regional variations in measurement standards.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Micrometers to Arpent in Length?
The formula to convert Micrometers to Arpent in Length is:
Micrometers / 58521599.953856885
2. Is this tool free or paid?
This Length conversion tool, which converts Micrometers to Arpent, is completely free to use.
3. How do I convert Length from Micrometers to Arpent?
To convert Length from Micrometers to Arpent, you can use the following formula:
Micrometers / 58521599.953856885
For example, if you have a value in Micrometers, you substitute that value in place of Micrometers in the above formula, and solve the mathematical expression to get the equivalent value in Arpent. | {"url":"https://convertonline.org/unit/?convert=micrometers-arpents","timestamp":"2024-11-02T01:54:32Z","content_type":"text/html","content_length":"91530","record_id":"<urn:uuid:1cd8030e-ed52-4a32-8d62-c8d9fd2735ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00419.warc.gz"} |
Math Vindication! - Diana Lesire Brandmeyer
6 thoughts on “Math Vindication!”
1. That is funny! I liked math well enough, I suppose, (until calculus). Word problems were always my downfall.
2. Great to visit with Ma & Pa Kettle. Funny.
3. LOL!! I LOVE this!! Makes me think of the old Abott & Costello skit where 13 x 7 = 28! I”m the opposite of Algebra — had to work REALLY hard for a C in high school and barely passed College
Algebra … but geometry I aced without trying.
1. What 13×7 doesn't equal 28?
4. I actually love math. Though, I don't really do much of it anymore. I've always been pretty good with algebra and statistics, though I hated geometry!
1. April I can't imagine loving math. Oddly I didn't mind geometry.
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://dianabrandmeyer.com/math-vindication/","timestamp":"2024-11-07T03:33:05Z","content_type":"text/html","content_length":"151484","record_id":"<urn:uuid:03d9a431-e27a-4b5d-b5e9-f1072773af72>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00868.warc.gz"} |
dspgv: computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x - Linux Manuals (l)
dspgv (l) - Linux Manuals
dspgv: computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x
DSPGV - computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x
ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, INFO )
CHARACTER JOBZ, UPLO
INTEGER INFO, ITYPE, LDZ, N
DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ), Z( LDZ, * )
DSPGV computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A
and B are assumed to be symmetric, stored in packed format, and B is also positive definite.
ITYPE (input) INTEGER
Specifies the problem type to be solved:
= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x
JOBZ (input) CHARACTER*1
= aqNaq: Compute eigenvalues only;
= aqVaq: Compute eigenvalues and eigenvectors.
UPLO (input) CHARACTER*1
= aqUaq: Upper triangles of A and B are stored;
= aqLaq: Lower triangles of A and B are stored.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
AP (input/output) DOUBLE PRECISION array, dimension
(N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = aqUaq, AP(i
+ (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = aqLaq, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, the contents of AP are destroyed.
BP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows: if UPLO = aqUaq, BP(i + (j-1)*j/2)
= B(i,j) for 1<=i<=j; if UPLO = aqLaq, BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. On exit, the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T, in the same
storage format as B.
W (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) DOUBLE PRECISION array, dimension (LDZ, N)
If JOBZ = aqVaq, then if INFO = 0, Z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(B)*Z = I. If JOBZ =
aqNaq, then Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ = aqVaq, LDZ >= max(1,N).
WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: DPPTRF or DSPEV returned an error code:
<= N: if INFO = i, DSPEV failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero. > N: if INFO = n + i, for 1 <= i <= n, then the leading minor of
order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. | {"url":"https://www.systutorials.com/docs/linux/man/docs/linux/man/docs/linux/man/l-dspgv/","timestamp":"2024-11-06T14:21:18Z","content_type":"text/html","content_length":"11417","record_id":"<urn:uuid:915d1978-ad99-4135-8b8f-42d4969210b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00107.warc.gz"} |
Problem Design for Multiple Choice Questions
I gave my students a problem from the 2002 AMC 10-A:
Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5}, and Sergio randomly selects a number from the set {1, 2, …, 10}. The probability that Sergio’s number is larger than the
sum of the two numbers chosen by Tina is: (A) 2/5, (B) 9/20, (C) 1/2, (D) 11/20, (E) 24/25.
Here is a solution that some of my students suggested:
On average Tina gets 6. The probability that Sergio gets more than 6 is 2/5.
This is a flawed solution with the right answer. Time and again I meet a problem at a competition where incorrect reasoning produces the right answer and is much faster, putting students who
understand the problem at a disadvantage. This is a design flaw. The designers of multiple-choice problems should anticipate mistaken solutions such as the one above. A good designer would create a
problem such that a mistaken solution leads to a wrong answer — one which has been included in the list of choices. Thus, a wrong solution would be punished rather than rewarded.
Readers: here are three challenges. First, to ponder what is the right solution. Second, to change parameters slightly so that the solution above doesn’t work. And lastly, the most interesting
challenge is to explain why the solution above yielded the correct result.
6 Comments
1. bons:
well the right way to do it is to multiply the probability that Tina’s two numbers sum to x, where x is between 3 and 9, by the probability that Sergio beats Tina, and add those probabilities
together. I suppose the solution above yields the correct result because 6 is the median number as well.
23 December 2010, 5:44 pm
2. Jonathan:
Solution to problem:
1/2 wins or ties 3/10
1/3 wins or ties 4/10
1/4 wins or ties 5/10
2/3 wins or ties 5/10
1/5 wins or ties 6/10
2/4 wins or ties 6/10
2/5 wins or ties 7/10
3/4 wins or ties 7/10
3/5 wins or ties 8/10
4/5 wins or ties 9/10
As each pair occurs with probability 1/10, we get (3+4+5+5+6+6+7+7+8+9)/10/10 = 60/100
Same problem, slightly altered, average still 6, but answer is not 6/10
If we change the first set to {0,1,3,4,7} the average holds, but the result falls.
0/1 wins or ties 1/10
0/3 wins or ties 3/10
0/4 wins or ties 4/10
1/3 wins or ties 4/10
1/4 wins or ties 5/10
0/7 wins or ties 7/10
3/4 wins or ties 7/10
1/7 wins or ties 8/10
3/7 wins or ties 10/10
4/7 wins or ties 10/10
As each pair occurs with probability 1/10, we get (1+3+4+4+5+7+7+8+10+10)/10/10 = 59/100
In the first example, the probability that a sum is greater than or equal to a random number from 1 to 10 is directly proportional to the sum (all sums are between 3 and 10). In the second
example, one sum is greater than 10, breaking the symmetry that previously kept the probability proportional to the sum.
23 December 2010, 9:07 pm
3. Jonathan:
Hmm, first pairs of numbers should be the sum from the first set, so not
1/2 wins or ties 3/10,
but rather
1+2 wins or ties 3/10 of the time.
24 December 2010, 9:04 am
4. JBL:
Nice counter-example, Jonathan. Of course, we can write down an argument that sounds very much like the student’s argument but is actually valid, and in fact this argument is in some respects
more elegant than a brute-force solution — we build off Jonathan’s discussion and don’t have to do any arithmetic.
27 December 2010, 10:22 am
5. JBL:
Here are two follow-up questions:
1) Let S be a finite multiset of integers (i.e., I can have repetitions) with mean 6. Tina randomly selects a number from S, and Sergio randomly selects a number between 1 and 10. Let p be the
probability that Sergio picks a larger number than Tina. As S varies, what is the range of possible values of p?
2) Let S be a finite set of integers with mean 3. Tina randomly selects two distinct numbers from S, and Sergio randomly selects a number between 1 and 10. Let p be the probability that the
number Sergio picks is larger than the sum of the numbers Tina picks. As S varies, what is the range of possible values of p?
27 December 2010, 12:32 pm
6. Tanya Khovanova:
Here is a comment from a_shen in my LJ discussion on the same subject:
Once I heard another example: how many faces will have the union of a pyramid based on a square whose faces are equilateral triangles and a regular tetrahedron that is glued to one of these
faces: it was said that the right answer to this problem had a negative correlation with the result of the entire test…
27 December 2010, 10:11 pm | {"url":"https://blog.tanyakhovanova.com/2010/12/problem-design-for-multiple-choice-questions/","timestamp":"2024-11-10T16:27:19Z","content_type":"text/html","content_length":"64402","record_id":"<urn:uuid:ea1c0656-a6ca-400a-9659-e5adf7eaadd6>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00834.warc.gz"} |
[Solved] Pytorch: loss.backward (retain_graph = true) of back propagation error
The backpropagation method in RNN and LSTM models, the problem at loss.backward()
The problem tends to occur after updating the pytorch version.
Problem 1:Error with loss.backward()
Trying to backward through the graph a second time (or directly access saved tensors after they have already been freed). Saved intermediate values of the graph are freed when you call .backward() or
autograd.grad(). Specify retain_graph=True if you need to backward through the graph a second time or if you need to access saved tensors after calling backward.
(torchenv) star@lab407-1:~/POIRec/STPRec/Flashback_code-master$ python train.py
Prolem 2: Use loss.backward(retain_graph=True)
one of the variables needed for gradient computation has been modified by an inplace operation: [torch.FloatTensor [10, 10]], which is output 0 of AsStridedBackward0, is at version 2; expected
version 1 instead. Hint: enable anomaly detection to find the operation that failed to compute its gradient, with torch.autograd.set_detect_anomaly(True).
Some pitfalls about loss.backward() and its argumenttain_graph
First of all, the function loss.backward() is very simple, is to calculate the gradient of the current tensor associated with the leaf nodes in the graph
To use it, you can of course use it directly as follows
optimizer.zero_grad() clearing the past gradients.
loss.backward() reverse propagation, calculating the current gradient.
optimizer.step() updates the network parameters according to the gradient
or this case
for i in range(num):
optimizer.zero_grad() clears the past gradients.
loss.backward() back-propagate and compute the current gradient.
optimizer.step() update the network parameters according to the gradient
However, sometimes, such errors occur: runtimeerror: trying to backward through the graph a second time, but the buffers have already been free
This error means that the mechanism of pytoch is that every time. Backward() is called, all buffers will be free. There may be multiple backward() in the model, and the gradient stored in the buffer
in the previous backward() will be free because of the subsequent call to backward(). Therefore, here is retain_Graph = true, using this parameter, you can save the gradient of the previous backward
() in the buffer until the update is completed. Note that if you write this:
optimizer.zero_grad() clearing the past gradients.
loss1.backward(retain_graph=True) backward propagation, calculating the current gradient.
loss2.backward(retain_graph=True) backward propagation, calculating the current gradient.
optimizer.step() updates the network parameters according to the gradient
Then you may have memory overflow, and each iteration will be slower than the previous one, and slower and slower later (because your gradients are saved and there is no free)
the solution is, of course:
optimizer.zero_grad() clearing the past gradients.
loss1.backward(retain_graph=True) backward propagation, calculating the current gradient.
loss2.backward() backpropagate and compute the current gradient.
optimizer.step() updates the network parameters according to the gradient
That is: do not add retain to the last backward()_Graph parameter, so that the occupied memory will be released after each update, so that it will not become slower and slower.
Someone here will ask, I don’t have so much loss, how can such a mistake happen? There may be a problem with the model you use. Such problems occur in both LSTM and Gru. The problem exists with the
hidden unit, which also participates in backpropagation, resulting in multiple backward(),
in fact, I don’t understand why there are multiple backward()? Is it true that my LSTM network is n to N, that is, input n and output n, then calculate loss with n labels, and then send it back?
Here, you can think about BPTT, that is, if it is n to 1, then gradient update requires all inputs of the time series and hidden variables to calculate the gradient, and then pass it forward from the
last one, so there is only one backward(), In both N to N and N to m, multiple losses need to be backwarded(). If they continue to propagate in two directions (one from output to input and the other
along time), there will be overlapping parts. Therefore, the solution is very clear. Use the detach() function to cut off the overlapping backpropagation, (here is only my personal understanding. If
there is any error, please comment and point it out and discuss it together.) there are three ways to cut off, as follows:
hidden = hidden.detach()
hidden = Variable(hidden.data, requires_grad=True)
Similar Posts: | {"url":"https://debugah.com/solved-pytorch-loss-backward-retain_graph-true-of-back-propagation-error-21181/","timestamp":"2024-11-15T04:39:27Z","content_type":"text/html","content_length":"36554","record_id":"<urn:uuid:47932a68-f7ba-4eea-ab25-9e30e89d9980>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00604.warc.gz"} |
Kilograms to Tons Conversion (kg to t) - Inch Calculator
Kilograms to Tons Converter
Enter the weight in kilograms below to convert it to tons.
Do you want to convert tons to kilograms?
How to Convert Kilograms to Tons
To convert a measurement in kilograms to a measurement in tons, multiply the weight by the following conversion ratio: 0.001102 tons/kilogram.
Since one kilogram is equal to 0.001102 tons, you can use this simple formula to convert:
tons = kilograms × 0.001102
The weight in tons is equal to the weight in kilograms multiplied by 0.001102.
For example,
here's how to convert 500 kilograms to tons using the formula above.
tons = (500 kg × 0.001102) = 0.551156 t
Kilograms and tons are both units used to measure weight. Keep reading to learn more about each unit of measure.
What Is a Kilogram?
One kilogram is equal to 1,000 grams, 2.204623 pounds, or 1/1,000 of a metric ton.
The formal definition of the kilogram changed in 2019. One kilogram was previously equal to the mass of the platinum-iridium bar, known as the International Prototype of the Kilogram, which was
stored in Sèvres, France.
The 2019 SI brochure now defines the kilogram using the Planck constant, and it is defined using the meter and second.^[1] It is equal to the mass of 1,000 cubic centimeters, or milliliters, of
The kilogram, or kilogramme, is the SI base unit for weight and is also a multiple of the gram. In the metric system, "kilo" is the prefix for thousands, or 10^3. Kilograms can be abbreviated as kg;
for example, 1 kilogram can be written as 1 kg.
Learn more about kilograms.
What Is a Ton?
One ton, also known as a short ton or a US ton, is equal to 2,000 pounds and is mostly used in the US as a unit of weight.^[2] One ton is equal to 907.18474 kilograms.
The ton is a US customary unit of weight. A ton is sometimes also referred to as a short ton. Tons can be abbreviated as t; for example, 1 ton can be written as 1 t.
The short ton should not be confused with the long ton, which is used mostly in the United Kingdom, or the metric ton which is used in most other countries.
Learn more about tons.
Kilogram to Ton Conversion Table
Table showing various
kilogram measurements
converted to tons.
Kilograms Tons
1 kg 0.001102 t
2 kg 0.002205 t
3 kg 0.003307 t
4 kg 0.004409 t
5 kg 0.005512 t
6 kg 0.006614 t
7 kg 0.007716 t
8 kg 0.008818 t
9 kg 0.009921 t
10 kg 0.011023 t
20 kg 0.022046 t
30 kg 0.033069 t
40 kg 0.044092 t
50 kg 0.055116 t
60 kg 0.066139 t
70 kg 0.077162 t
80 kg 0.088185 t
90 kg 0.099208 t
100 kg 0.110231 t
200 kg 0.220462 t
300 kg 0.330693 t
400 kg 0.440925 t
500 kg 0.551156 t
600 kg 0.661387 t
700 kg 0.771618 t
800 kg 0.881849 t
900 kg 0.99208 t
1,000 kg 1.1023 t
1. International Bureau of Weights and Measures, The International System of Units, 9th Edition, 2019, https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf
2. United States Department of Agriculture, Weights, Measures, and Conversion Factors for Agricultural Commodities and Their Products, https://www.ers.usda.gov/webdocs/publications/41880/
More Kilogram & Ton Conversions | {"url":"https://www.inchcalculator.com/convert/kilogram-to-ton/","timestamp":"2024-11-07T13:38:34Z","content_type":"text/html","content_length":"67710","record_id":"<urn:uuid:438ad4b3-f776-4a1a-a051-f8ef7b881ee0>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00302.warc.gz"} |
Crash Course
The name of the fipp package is an acronym for “functionals for implicit prior partitions” and the package mainly provides users with tools that can be used to inspect implicit assumptions on the
prior distribution of partitions of a mixture model. This implies that the package is mostly tailored towards applied Bayesian statisticians working on clustering, mixture modeling and related areas
(if you are using packages such as BayesMix and PReMiuM, you are the right audience). Specifically, it covers models that assign a prior distribution on the number of components in the mixture model
such as Dirichlet Process Mixtures and models broadly classified as Mixture of Finite Mixtures (MFMs, as coined by Miller and Harrison (2018)).
Nonetheless, we believe some of the functionality may also be of interest to researchers in the broad area of mixture models, such as those who prefer to use an EM-based approach, most notably
implemented in the mclust package.
This vignette is designed as a stand-alone and self-contained tour of the fipp package, although we suggest the interested reader to have a look at our paper Greve et al. (2020) for further
Models considered in this package
So far, the package includes three important Bayesian mixture models that assign a prior on the number of components commonly denoted as \(K\) as well as on the concentration parameter or the
parameter for the Dirichlet distribution either denoted as \(\alpha\) or \(\gamma\). The Dirichlet parameter induces a sequence \(\gamma_K\) for each \(K\).
The three models considered are: Dirichlet Process Mixtures (DPMs), Static Mixtures of Finite Mixtures (Static MFMs) and Dynamic Mixtures of Finite Mixtures (Dynamic MFMs).
The important parameters for these models, \(K\) and \(\gamma_K\), are specified as follows:
\(K\) \(K = \infty\) \(K-1\sim P(K)\) \(K-1\sim P(K)\)
\(K=1,2,\ldots\) \(K=1,2,\ldots\)
\(\gamma_K\) \(\alpha\) \(\gamma\) \(\alpha/K\)
The concentration or Dirichlet parameter, regardless of the model, is usually given a prior, such as a gamma distribution which has support on the positive real line.
For the Static and Dynamic MFMs, \(P(K)\) must be a distribution with support on the nonnegative integers. Following the suggestion in Frühwirth-Schnatter, Malsiner-Walli, and Grün (2020), the
package implements the translated prior which specifies the prior on \(K-1\) with support on the nonnegative integers.
An introduction to the theory behind the model and parameters
All three models considered in this package deal with mixture models which may be written as
\[ p(y_i) = \sum^{K}_{k=1}\eta_kf_{\tau}(y_i|\theta_k) \]
for observations \(y_i,i = 1,2,\ldots N\).
Parameters \(\eta_k\)’s are weights that sum up to one, while \(f_{\tau}(\cdot|\theta)\) refers to the component density which follows a distribution \(\tau(\theta)\).
Furthermore, \(\eta_k\)’s are drawn from the symmetric Dirichlet distribution as follows:
\[ \pmb{\eta}_K|K,\gamma_K \sim Dir_K(\gamma_K,\gamma_K,\ldots,\gamma_K) \]
where \(\pmb{\eta}_K = (\eta_1,\eta_2,\ldots,\eta_K)\).
An important quantity hidden in the equations above is the number of data clusters \(K_+\) defined as
\[ K_+ = K - \#\{\text{components $k$ not associated to any $y_i$}\}. \]
In other words, \(K_+\) is the number of groups in the partition that separates \(y_i,\ i= 1,2,\ldots,N\). Crucially, when we talk about a “k-cluster solution” for a data set, we refer not to \(K\)
but rather to \(K_+\) being equal to \(k\). Thus, we believe that \(K_+\) and characteristics of the partitions captured through some functionals are more practically relevant quantities when
specifying suitable priors for a Bayesian cluster analysis than \(K\), and thus prior specifications regarding those should be done appropriately in some way.
To address this, the central aim of the fipp package is to translate the prior specifications with respect to \(K\) and the concentration or Dirichlet parameter \(\gamma_K\) into the number of
clusters \(K_+\) and various functionals computed over the partitions.
One can use results obtained from this package to conduct a sanity check on prior specifications w.r.t. \(P(K)\) and \(\gamma_K\) in terms of the implied distribution on \(K_+\) and moments of
functionals computed over the induced partitions.
Demonstration 1: induced prior on the number of clusters
First, we demonstrate how the fipp package can be used to obtain a prior pmf on the number of clusters \(K_+\) from specifically selected \(P(K)\) and \(\gamma_K\). First, let’s load the package.
The function (to be precise, it is a closure as it returns a function) in question is called nClusters().
Let’s start with the implied number of clusters \(K_+\) for a DPM with the concentration parameter \(\alpha = 1/3\) (based on the recommendation in Escobar and West (1995)) and a sample size of \(N =
100\). We evaluate the pmf between \(K_+ = 1\) to \(K_+ = 30\). Since \(K\) is fixed to \(\infty\) in DPMs, there is no prior on the number of components \(P(K)\) to compare the resulting
distribution on \(K_+\) to.
As we can see, this specification implicitly implies that the probability of \(K_+\) being above 10 is very small. Also, the induced prior on \(K_+\) generally prefers a sparse solution of \(K_+\)
with values between 1 and 3 for \(N = 100\) observations and the probability of homogeneity (that is the probability of \(K_+ = 1\)) is about 1/5.
Now, let’s see what kind of inference we can draw for the Static MFM. Here, we replicate the specification used in Richardson and Green (1997) with \(U(1,30)\) for \(K\) (which translates to \(K-1 \
sim U(0,29)\)) and \(\gamma = 1\) for the concentration parameter.
This prior appears uninformative as we assume that \(K\) is uniformly distributed on \([1, 30]\) and \(\gamma = 1\) also corresponds to the uniform distribution on the simplex. However, it turns out
to be still informative for \(K_+\) as we can clearly see from the result below.
While the distribution on \(K\) is uniform and thus uninformative, that of \(K_+\) induced by the uniform distribution on \(K\) and \(\gamma = 1\) is not uninformative. It has a clear peak around 19.
This highlights the well known problem of using an uniform distribution for \(P(K)\). When the sample size is relatively small, it gives a false sense of uninformativeness despite specifying a prior
with a clear peak (for higher \(N\) it does get more and more uniformly distributed).
Click here for a brief explanation of why this happens. There are several factors that are in play to produce the above result. Crucially, the higher the concentration parameter \(\gamma\), the more
likely it is that all \(K\) components are filled, with many components being filled even for the relatively small sample size \(N\). Therefore, despite the small \(N\), \(K_+ \approx K\) and thus
the induced prior on \(K_+\) also approximates the specified \(P(K)\) (one can check this with the code above by increasing \(\gamma\) to values such as 100). The reverse happens when \(\gamma\) is
small and the result obtained for the DPM more resembles results obtained for a Static MFM with a small \(\gamma\) value.
The above example provides insights into the well known fact that the seemingly innocent choice of specifying a uniform prior on \(K\) and the component weights \(\pmb{\eta}_K\) results in a rather
problematic behavior in terms of the prior on \(K_+\) and could also skew the resulting posterior (see Nobile (2004) for details). In recent work on MFMs different alternative distributions with
support on \(\mathbb{Z}_{>0}\) have been suggested.
Again, we can use the function returned by the closure nClusters() to inspect several choices of such \(P(K)\)’s. The resulting function allows computations shared across all \(P(K)\)’s to only be
run once regardless of the number of \(P(K)\)’s considered. Here we examine two choices: \(Pois(3)\) and \(Geom(0.3)\).
Similarly, we can do the same comparison w.r.t. the Dynamic MFM with \(\alpha = 1\) (note that this is not related in any way to the Static MFM with \(\gamma = 1\)). Again, computations shared
between these \(P(K)\)’s will not be re-run, thus allowing users to experiment with multitudes of \(P(K)\)’s in practical applications.
Here, the distributions on \(K_+\) and \(K\) do not coincide, especially for the Poisson case. We leave the details to the paper Frühwirth-Schnatter, Malsiner-Walli, and Grün (2020). To put it
simply, as the concentration parameter is now \(\alpha/K\), greater values of \(K\) induce smaller values \(\alpha/K\) and \(K_+\) tends to be less than \(K\), thus skewing the distribution of \(K_+
\) more towards the right than that of \(K\).
For further details concerning the difference between the Static and Dynamic MFM w.r.t. the induced prior on \(K_+\), we refer to Frühwirth-Schnatter, Malsiner-Walli, and Grün (2020) and Greve et al.
Demonstration 2: relative entropy of the induced prior partitions
Another important function (again it is a closure to be precise) available in the package is fipp(). It allows the user to compute moments (currently only up to the 2nd moment) of any additive
symmetric functional over the induced prior partitions specified through picking \(P(K)\), \(\gamma_K\) and \(K_+\).
Here, we pick the relative entropy as functional which is given by
\[ -\frac{1}{\log(K_+)}\sum_{i=1}^{K_+}\frac{N_i}{N}\log\Bigg(\frac{N_i}{N}\Bigg) \]
with \(K_+\) the number of clusters and \(N_i\) the number of observations in the \(i\)-th cluster. This functional takes values between \((0,1)\) and the closer this functional is to 1, the more
evenly distributed the \(N_i\)’s are and vise versa.
Let’s start with the DPM with \(\alpha = 1/3\) as before and with the specific case where \(K_+ = 4\). Function fipp() determines the functional based on the sum of the results of a vectorized
function of the induced unordered cluster sizes \((N_i)_{i=1,\ldots,K_+}\). This implies that the functional must be additive and symmetric. In addition, the vectorized function must be supplied in
its log form for computational reasons.
Therefore, for the relative entropy, the vectorized function supplied to fipp() determines \(\log(N_i)-\log(N)+\log(\log(N)-\log(N_i))\). The resulting prior mean and standard deviation of the
functional (default specification returns mean/variance, which can be changed to 1st/2nd moments) will be divided by \(\log(K_+)\) and \(\log(K_+)^2\), respectively.
## Statistics computed over the prior partitions: Relative entropy
## Model: DPM (alpha = 1/3)
## conditional on: K+ = 4
## mean = 0.5654722
## sd = 0.1987301
It turns out that the value of the concentration parameter \(\gamma_K \equiv \alpha\) does not play any role in the relative entropy of the prior partitions for DPMs (try adjusting alpha = and see
for your self) nor does \(P(K)\) as \(K\) is fixed to \(\infty\). Thus, DPM induces a fixed structure a-priori on the partitions conditional on \(K_+\).
Now let’s see what the prior mean and standard deviation of the relative entropy of the Static MFM with \(\gamma = 1\) and two priors on \(K-1\) (\(Pois(3)\) and \(Geom(0.3)\)) conditional on \(K_+ =
4\) is.
## Statistics computed over the prior partitions: Relative entropy
## Model: static MFM (gamma = 1)
## conditional on: K+ = 4
## case 1 with K-1 ~ dpois(3): mean = 0.7920192 sd = 0.1310269
## case 2 with K-1 ~ dgeom(.3): mean = 0.7920192 sd = 0.1310269
Here, we observe that the choice of \(P(K)\) does not affect the relative entropy of the induced prior partitions for the Static MFM. Nonetheless, the choice of \(\gamma_K\equiv \gamma\) will still
affect the relative entropy unlike the DPM.
Now, let’s see what the Dynamic MFM with \(\alpha = 1\) and two priors on \(K-1\) (\(Pois(3)\) and \(Geom(0.3)\)) conditional on \(K_+ = 4\) gives.
## Statistics computed over the prior partitions: Relative entropy
## Model: dynamic MFM (alpha = 1)
## conditional on: K+ = 4
## case 1 with K-1 ~ dpois(3): mean = 0.6337794 sd = 0.1860941
## case 2 with K-1 ~ dgeom(.3): mean = 0.6269217 sd = 0.1879739
As we can see, the relative entropy of the induced prior partitions for the Dynamic MFM depends on both the concentration parameter \(\gamma_K\equiv \alpha\) as well as the prior on \(K\). Therefore,
it is the most flexible of the three models considered when it comes to embedding prior information regarding the characteristics of the partitions captured through relative entropy.
Demonstration 3: expected number of clusters which contain less than \(10\%\) of the observation each
Now let’s consider other functionals. For example, we may be interested in the a-priori expected number (and corresponding standard deviations) of clusters which contain less than \(10\%\) of the
The vectorized function for this functional required by fipp() can simply be written as \(\log(\mathbb{I}_{N_i<0.1*N})\) where \(\mathbb{I}_{.}\) stands for the indicator function. Let’s reuse the
specifications in Demo 2 by only replacing the functional. We do not show the code for the sake of conciseness.
## Statistics computed over the prior partitions:
## Number of clusters with less than 10% of the obs
## Model: DPM (alpha = 1/3)
## conditional on: K+ = 4
## mean = 1.880596
## sd = 0.780884
## Statistics computed over the prior partitions:
## Number of clusters with less than 10% of the obs
## Model: static MFM (gamma = 1)
## conditional on: K+ = 4
## case 1 with K-1 ~ dpois(3): mean = 1.003997 sd = 0.7399482
## case 2 with K-1 ~ dgeom(.3): mean = 1.003997 sd = 0.7399482
## Statistics computed over the prior partitions:
## Number of clusters with less than 10% of the obs
## Model: dynamic MFM (alpha = 1)
## conditional on: K+ = 4
## case 1 with K-1 ~ dpois(3): mean = 1.646882 sd = 0.793471
## case 2 with K-1 ~ dgeom(.3): mean = 1.670937 sd = 0.7938899
We see that specifications used for DPMs and Dynamic MFMs result in expectation in about 2 clusters out of 4 having less than \(10\%\) of the observations. This indicates rather unevenly sized
partitions which is in line with the relative entropy results derived earlier.
The Static MFM with \(\gamma=1\) seems to produce partitions that are more evenly distributed with the expected number of clusters containing less than \(10\%\) observations being close to 1. Again,
this aligns with the relative entropy results obtained for the Static MFM earlier.
Other symmetric additive functionals that might be of interest, such as the expected number of singleton clusters (\(\mathbb{I}_{N_i = 1}\)), might also easily be used together with the fipp() | {"url":"https://pbil.univ-lyon1.fr/CRAN/web/packages/fipp/vignettes/fippCrashCourse.html","timestamp":"2024-11-12T16:13:36Z","content_type":"application/xhtml+xml","content_length":"123460","record_id":"<urn:uuid:b29cf744-99a2-4038-8f8b-889134fb44e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00560.warc.gz"} |
Quality Factor of Passive Filter Calculator | Calculate Quality Factor of Passive Filter
What are the benefits of a high Quality Factor?
A high Quality Factor results in a sharper filter with better selectivity. This is desirable for many applications, such as audio and radio receivers, telecommunication systems, and medical
How to Calculate Quality Factor of Passive Filter?
Quality Factor of Passive Filter calculator uses Quality Factor = (Angular Resonant Frequency*Inductance)/Resistance to calculate the Quality Factor, The Quality Factor of Passive Filter formula is
defined as the ratio of the reactive energy stored in the filter to the energy lost in one cycle of oscillation. Quality Factor is denoted by Q symbol.
How to calculate Quality Factor of Passive Filter using this online calculator? To use this online calculator for Quality Factor of Passive Filter, enter Angular Resonant Frequency (ω[n]), Inductance
(L) & Resistance (R) and hit the calculate button. Here is how the Quality Factor of Passive Filter calculation can be explained with given input values -> 8.335557 = (24.98*50)/149.9. | {"url":"https://www.calculatoratoz.com/en/quality-factor-of-passive-filter-calculator/Calc-42485","timestamp":"2024-11-12T15:01:29Z","content_type":"application/xhtml+xml","content_length":"114505","record_id":"<urn:uuid:6bd6bdee-2f4d-4d87-a04d-9dfa80b253a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00680.warc.gz"} |
How to know if the math tuition your child is attending is good
Understandably, parents will always be wondering if the math tuition their child is attending is good. They will be thinking if their child’s results will improve after attending tuition. How to know
if the math tuition is good? Here are 4 ways that you can gauge whether the math tuition is good for your child.
1) Is your child able to understand the math lessons?
After a few lessons (preferably around 4 lessons), ask your child if he or she is able to understand the math lessons well. If your child is able to, that’s great! Ask more questions like why are
they able to understand better as compared to not being able to understand in school. Knowing all these allow you to diagnose the issue of your child’s lack of performance in math subject. At the
same time, you can at least feel better that your child is in good hands in the math tuition he or she is attending.
If your child is not able to understand the math lessons well, probe further. Is it because your child cannot understand the explanation provided by the math tutor? Is it because your child cannot
keep up with the pace of the class hence your child requires dedicated attention? It is key to communicate with your child from time to time to find the root of the issue and address it early on. It
is very important for math tutors to ensure that their students understand the math lessons well. No students should be left behind because the lack of understanding will continue to pile up until
2) Is your child able to get his or her questions answered?
Ask your child if he or she is able to ask math questions and whether or not it gets answered. Sometimes, we have to always put ourselves in students’ shoes. We have to see things from their
perspective. Think about it this way. You need help with math from your tutor and your tutor is very patient and willing to help you until you understand. You will feel thankful to have such a tutor
and you would be more willing to ask questions without feeling afraid.
On the contrary, if your tutor is someone who expects you to know after teaching you a few times yet at times you still do not understand the math content, it is quite likely that you would be afraid
to ask questions anymore. I have students telling me that some of their past math tutors did not want to answer their questions. These tutors simply tell the students that they have explained a lot
of times and the students should figure it out themselves. To me, that’s a sign of frustration that these past math tutors are showing. Not only have they avoided answering their students’ questions,
at the same time they have created a detrimental effect on the students by not being supportive and likely causing the students to be afraid to ask questions in the future.
3) Is your child comfortable with the lesson configuration?
There was once I noticed a student who wasn’t able to cope well with group tuition setting and I immediately pulled her out of the group tuition, informed the parents and changed from group tuition
to 1 to 1 tuition. These are things that I can notice first hand when conducting lessons but parents won’t usually see it. Parents have to ask their child attending the math tuition whether they are
comfortable with the lesson configuration (group tuition). Some students require dedicated attention (1 to 1 lesson) perhaps because they are slow or they need more attention from the math tutor.
Understanding this early is key to also helping your child in his or her academics.
4) What’s the chemistry like between your child and the math tutor?
It is important to know whether your child has a good chemistry with your math tutor. As a math tutor myself, I see myself as a mentor to my students in addition to teaching them mathematics.
Building good rapport with students is very important. Why is it so? They know that you care for them. Being able to understand them and speak to them as friends rather than being someone of a higher
position is important to them.
When tutors are able to build rapport with their students, it serves as a platform where students will listen to their tutors’ advice. Also, students are able to ask questions freely without being
afraid of getting judged. I always tell my students that they should treat me as their mentor and friend, not as someone of a higher position than them.
If you are interested to know about the benefits of attending math tuition, you can check out 4 Key Benefits Of Math Tuition. | {"url":"https://odysseymathtuition.com/how-to-know-if-the-math-tuition-your-child-is-attending-is-good/","timestamp":"2024-11-06T18:03:21Z","content_type":"text/html","content_length":"120942","record_id":"<urn:uuid:601c5db1-e0de-4eec-b0d9-bc1730cf946f>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00611.warc.gz"} |
Online calculator
Secant method
The secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f.
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must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/3706/. Also, please do not modify any references to the original work (if any) contained
in this content.
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Different bar chart and histogram
Histograms are used to show distributions of variables while bar charts are used to compare variables. Histograms plot quantitative data with ranges of the data grouped into bins or intervals while
Difference Between a Histogram and a Bar Chart. Although they look the same, bar charts and histograms have one important difference: they plot different types of data. Plot discrete data on a bar
chart, and plot continuous data on a histogram What’s the difference between discrete and continuous data?). The blue rectangles here are called bars. Note that the bars have equal width and are
equally spaced, as mentioned above. This is a simple bar diagram. Histogram. A bar diagram easy to understand but what is a histogram? Unlike a bar graph that depicts discrete data, histograms depict
continuous data. The continuous data takes the form of class intervals.
1 Sep 2017 The fundamental difference between histogram and bar graph will help you to identify the two easily is that there are gaps between bars in a 4 Jan 2012 why many other graphics that look
similar have very different grammars.” The figures below show an example of a histogram and a bar chart. What is the difference between a bar graph and a histogram? Hi,. There are two differences,
one is in the type of data that is presented and the other in the way 21 Mar 2016 Here is the main difference between bar charts and histograms. With bar charts, each column represents a group
defined by a categorical variable; and with
The following section provides further information about histograms and bar charts. a noticeable difference, a 20 mm range in height seems quite reasonable.
Bar charts are used to display relative sizes of the counts of variables. Histograms are used to display the shape of distribution of the data. Drawing : Bar graph is drawn in such a way that there
is proper spacing between bars in a bar graph that indicates discontinuity (because it is not continuous). The most common objects are: - Point: `geom_point()` - Bar: `geom_bar()` - Line: `geom_line
()` - Histogram: `geom_histogram()` In this tutorial, you are interested in the geometric object geom_bar() that create the bar chart. Bar chart: count. Your first graph shows the frequency of
cylinder with geom_bar(). Here is the main difference between bar charts and histograms. With bar charts, each column represents a group defined by a categorical variable; and with histograms, each
column represents a group defined by a quantitative variable. Histograms are used to show distributions of variables while bar charts are used to compare variables. Histograms plot quantitative data
with ranges of the data grouped into bins or intervals while Difference Between a Histogram and a Bar Chart. Although they look the same, bar charts and histograms have one important difference: they
plot different types of data. Plot discrete data on a bar chart, and plot continuous data on a histogram What’s the difference between discrete and continuous data?). The blue rectangles here are
called bars. Note that the bars have equal width and are equally spaced, as mentioned above. This is a simple bar diagram. Histogram. A bar diagram easy to understand but what is a histogram? Unlike
a bar graph that depicts discrete data, histograms depict continuous data. The continuous data takes the form of class intervals.
21 Mar 2016 Here is the main difference between bar charts and histograms. With bar charts, each column represents a group defined by a categorical variable; and with
4 Jan 2012 why many other graphics that look similar have very different grammars.” The figures below show an example of a histogram and a bar chart. What is the difference between a bar graph and a
histogram? Hi,. There are two differences, one is in the type of data that is presented and the other in the way 21 Mar 2016 Here is the main difference between bar charts and histograms. With bar
charts, each column represents a group defined by a categorical variable; and with
Histogram charts. Use a histogram when you want to show the distribution of a data set across different buckets or ranges. The height of each bar represents the
A frequency distribution shows how often each different value in a set of data occurs. It looks very much like a bar chart, but there are important differences I also have a nominal array called
RACE , which is 1250x1, and has 5 different races chosen. I would like to somehow make a histogram that looks like this, which I Sensitivity histograms can be used to compare the results of
different sensitivity simulations. Cumulative, if checked, causes a cumulative histogram to be
The differences between histogram and bar graph can be drawn clearly on the following grounds: Histogram refers to a graphical representation; that displays data by way A histogram represents the
frequency distribution of continuous variables. Histogram presents numerical data whereas bar
A frequency distribution shows how often each different value in a set of data occurs. It looks very much like a bar chart, but there are important differences I also have a nominal array called
RACE , which is 1250x1, and has 5 different races chosen. I would like to somehow make a histogram that looks like this, which I Sensitivity histograms can be used to compare the results of
different sensitivity simulations. Cumulative, if checked, causes a cumulative histogram to be The following section provides further information about histograms and bar charts. a noticeable
difference, a 20 mm range in height seems quite reasonable. Bar graphs and histograms are used to compare the sizes of different group/ categories. Bar Graph (Chart) bargraph. General
Characteristics: • Column label How to make Histograms in Python with Plotly. More generally, in plotly a histogram is an aggregated bar chart, with several possible aggregation however traces with
barmode = "overlay" and on different axes (of the same axis type) can One of the most common statistical charts, histograms histogram chart is fundamentally different from the bar
6 Oct 2015 The bar chart is one of the most diverse chart types out there. Each of them serves a different purpose. Vertical v.s horizontal However, if the values (not a trend) is what you are after,
feel free to use a histogram. Grouped. Histograms and pareto charts are better than plain bar charts. you call them, Bar graphs and Column charts are used to compare different categories of data.
Difference Between Bar Graph and Histogram • First and foremost, a histogram is a development from the bar graph, • Bar graphs are used to plot categorical or qualitative data while histograms are
used • Bar graphs are used to compare variables while histograms are used to show The differences between histogram and bar graph can be drawn clearly on the following grounds: Histogram refers to a
graphical representation; that displays data by way A histogram represents the frequency distribution of continuous variables. Histogram presents numerical data whereas bar Firstly, a bar chart
displays and compares categorical data, while a histogram accurately shows the distribution of quantitative data. Unlike bar charts that present distinct variables, the elements in a histogram are
grouped together and are considered ranges. Bar charts and histograms can both be used to compare the sizes of different groups. A Bar chart is made up of bars plotted on a graph. Histogram is a
chart representing a frequency distribution; heights of the bars represent observed frequencies. In other words a histogram is a graphical display of data using bars of different heights. Bar charts
are used to display relative sizes of the counts of variables. Histograms are used to display the shape of distribution of the data. Drawing : Bar graph is drawn in such a way that there is proper
spacing between bars in a bar graph that indicates discontinuity (because it is not continuous). | {"url":"https://topbinhqwtne.netlify.app/werkmeister539ke/different-bar-chart-and-histogram-383","timestamp":"2024-11-03T18:24:20Z","content_type":"text/html","content_length":"36910","record_id":"<urn:uuid:33ad7b21-6375-4493-add9-20bbdcd3f077>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00174.warc.gz"} |
PHY 765: Graduate Advanced Physics,
PHY 765: Graduate Advanced Physics, Fall 2016
Instructor: Prof. Thomas Barthel
Lectures: Mondays and Fridays 1:25PM-2:40PM in Physics 299
Office hours: Mondays and Fridays 2:40PM-3:45PM in Physics 287
Teaching assistant: Gu Zhang (office 294)
Tutorials: Wednesdays 4:40PM-5:55PM in Physics 299
Grading: Problem Sets (60%), Final exam (40%)
This is a graduate course on the wonderful world of quantum many-body physics. It covers several essential phenomena that (almost) every physicist should have heard of and prepares for more
specialized studies of quantum optics, quantum information, condensed matter, and quantum field theory.
After warming up with generalizations of the postulates of quantum physics (density matrices, quantum channels, POVM), we will start looking at systems of identical particles and derive the
approximative Hartree-Fock equations. On the basis of these, we will discuss the electronic structure of atoms and molecules. Quantum many-body systems are efficiently described using the formalism
of 2nd quantization. With this tool at hand, we will study the electron gas, Landau's Fermi liquid theory (describing the normal state of metals), the BCS theory and Ginzburg-Landau theory of
superconductivity, and the quantum Hall effect. Switching from fermions to bosons, we will discuss the Bose gas, Bose-Einstein condensation, and the effects of interactions on the basis of the
Bogoliubov theory and Gross-Pitaevskii equations (depletion, phonons, healing length, superfluidity). If time permits, we will also discuss the Bose-Hubbard model with its superfluid-Mott transition
and aspects of quantum magnetism. We will end our tour with topics from quantum information theory and quantum computation such as no-cloning, teleportation, Bell inequalities, quantum algorithms,
error correction, and entanglement.
Knowledge of single-particle quantum mechanics on the level of courses PHY 464 or PHY 764 is expected.
Lecture Notes
[Are provided on the Sakai site PHYSICS.765.01.F16.]
You are encouraged to discuss homework assignments with fellow students. But the written part of the homework must be done individually and cannot be a copy of another student's solution. (See the
Duke Community Standard.) However, you are allowed to work in groups of two. In that case, both partners still need to hand in a handwritten copy of their solution and should additionally always
specify the name of their partner.
Homework due dates are strict (for the good of all), i.e., late submissions are not accepted. If there are grave reasons, you can ask for an extension early enough before the due date.
[Are provided on the Sakai site PHYSICS.765.01.F16.]
Useful literature
Quantum mechanics (single-particle).
• Sakurai "Modern Quantum Mechanics", Addison Wesley (1993)
• Shankar "Principles of Quantum Mechanics" 2nd Edition, Plenum Press (1994)
• Le Bellac "Quantum Physics", Cambridge University Press (2006)
• Schwabl "Quantum Mechanics", 4th Edition, Springer (2007)
• Baym "Lectures on Quantum Mechanics", Westview Press (1974)
Quantum many-body physics.
• Coleman "Introduction to Many-Body Physics", Cambridge University Press (2015)
• Nazarov, Danon "Advanced Quantum Mechanics", Cambridge University Press (2013)
• Stoof, Gubbels, Dickerscheid "Ultracold Quantum Fields", Springer (2009)
• Pethick, Smith "Bose-Einstein Condensation in Dilute Gases", Cambridge University Press (2002)
• Altland, Simons "Condensed Matter Field Theory" 2nd Edition, Cambridge University Press (2010)
• Negele, Orland "Quantum Many-Particle Systems", Westview Press (1988, 1998)
• Bruus, Flensberg "Many-Body Quantum Theory in Condensed Matter Physics", Oxford University Press (2004)
• Ashcroft, Mermin "Solid State Physics", Harcourt (1976)
• Ibach, Lüth "Solid state physics" 4th Edition, Springer (2009)
Quantum information and computation.
• Nielsen, Chuang "Quantum Computation and Quantum Information", Cambridge University Press (2000)
• Preskill "Quantum Computation", Lecture Notes (2015)
• Wilde "Quantum Information Theory", 2nd Edition, arXiv:1106.1445 (2016)
• Bruss, Leuchs "Lectures on Quantum Information", Wiley (2007) | {"url":"http://webhome.phy.duke.edu/~barthel/L2016-08_GAP_phy765/","timestamp":"2024-11-03T21:57:16Z","content_type":"text/html","content_length":"7635","record_id":"<urn:uuid:6d1a40de-10f8-4485-b0c7-a9aebb180ba6>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00504.warc.gz"} |
Why your A/B testing results may be false?
In the digital age, and with the support of technology. Testing have become very popular in the business environment. Business setup up different groups to target a goal and try to find a group that
will produce the best result. This type of setup is called A/B testing. The goal that you are trying to reach should have a meaningful purpose. I found that many businesses setup goals that lack a
meaningful purpose. For instance, finding out that we have more male customers than female customers. Our customers are mainly located in the Taipei city. Ok, great and then? How is this information
going to help the business? How to setup a meaningful goal is out of the scope of this article, but in general you should try to tie your goals to business profits.
Ok, going back to our topic on A/B testing result. A/B testing are used very commonly in businesses today. In the sales and marketing field, businesses produce different banners on the website and
observe which one has the best click through rate. Businesses produce different content in their email campaign to observe which type of content will lead to the most conversions. In the customer
service field, businesses provide different customer service level and try to conclude if the different customer service level will affect the retention rate of the customer…etc. The application
could be found in many areas.
Lets say, at the end of the day, we get to a conclusion that A is better than B for achieving the target goal. However, have you considered the error in your statistical analysis? What is the chance
that A is better than B because of error? In other words, A is better than B in your test due to statistical error. In reality A is not better than B. For instance, you come to the conclusion that A
is better than B. However, there is a 50% chance that this result is caused by error. In this case you cannot jump to the conclusion that A is better than B and will need to increase your sample
size. How to pick your sample is also another big topic but we will not discuss it in this article. In general, we always try to achieve “random sampling” within a controlled group. I found that most
company that performs A/B testing never get to the stage of validating their result. Never validating the chance that the result is caused by an error and does not reflect the truth.
Ok, let’s go into more details about how you calculate your “chance of error”. In general, you can assume a normal distribution. You make a claim, which we call the Null Hypothesis, denoted as H0.
The null hypothesis is the formal basis for testing statistical significance. To state a proposition that there is no association. For instance, we can’t claim that campaign A is better than campaign
B. Then there is the alternative hypothesis, denoted as H1. The alternative hypothesis proposes that there is an association. The alternative hypothesis cannot be tested directly. It is accepted by
exclusion if the test of statistical significance rejects the null hypothesis. Under this scenario, can you get two types of error: type1 and type2. Type 1 error is when you reject the null
hypothesis when the null hypothesis is true. Type 2 error is when you fail to reject the null hypothesis when the null hypothesis is false. Type 1 error is also called the p- value. For instance a p
value of 5% indicate the there is a 5% chance that you are wrong when you reject the null hypothesis.
Therefore always calculate the value of the Type 1 error, to help you draw a conclusion on whether your result is likely to reflect the truth or is whether it is caused by error. | {"url":"https://www.ideasexecution.com/post/why-your-a-b-testing-results-may-be-false","timestamp":"2024-11-09T03:50:49Z","content_type":"text/html","content_length":"1050486","record_id":"<urn:uuid:7f2a53a2-58e3-4ca7-a467-e9bd99e70ddb>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00263.warc.gz"} |
Graphical decrypt
Graphical decrypt --- Introduction ---
Warning. This exercise is probably very hard even prohibitive for those who don't know primitive polynomials over finite fields! In this case please prefer Decrypt which is mathematically much more
Graphical decrypt is an exercise on the algebraic cryptology based on pseudo-random sequences generated by primitive polynomials over a finite field [q]. You will be presented a picture composed of n
×n pixels, crypted by such a sequence. This picture has q colors, each color representing an element of [q].
And your goal is to decrypt this crypted picture, by finding back the primitive polynomial as well as the starting terms which determine the pseudo-random sequence.
Now with q = and difficulty level = .
One should remark however that this is just an exercise for teaching purposes. Even with the highest difficulty level, it is still incomparably easier than the real algebraic crypting in the real
life... The most recent version
This page is not in its usual appearance because WIMS is unable to recognize your web browser.
Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.
Description: decrypt a picture crypted by a psudo-random sequence. interactive exercises, online calculators and plotters, mathematical recreation and games
Keywords: interactive mathematics, interactive math, server side interactivity, algebra,coding, finite field, cryptology, primitive polynomial, cyclic code | {"url":"http://www.designmaths.net/wims/wims.cgi?lang=en&+module=U4%2Fcoding%2Fgrdecrypt.en","timestamp":"2024-11-13T06:24:37Z","content_type":"text/html","content_length":"5770","record_id":"<urn:uuid:5014147c-a54d-4dff-adc8-bbd4c96e4647>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00694.warc.gz"} |
Welcome New Members...
Top Posters In This Topic
hey just wanna see sum pic so i can work out what ones i wanna get so i can work where i wanna put them and do u do banners
Hey all my name's Tom. I just bought a ke55 on monday for $300 with no plates so i've gotta see what i need to do to get a pink slip or whatever, it runs pretty well though and i don't think it'll
need too much done to it. I'm not all that great with cars but i'm learning so please forgive my stupidity at times =P I'll get some pics up of the car when it's light out.
P.S. I'm from Tarragindi
Edited by TommyGun
hey ... thanks for the pic i will work out my money nd get back to u on what sticker i wanna get .. how do i pay if i wanna get a sticker
cheerz cammo
hey all I'm jess, yes thats right a girl, is it that usual to see chick on here, i dunno lol, I'm 17 and my first car is my rolla, i live in bundy, home of the rum, but never seem to see anybody with
done up rollas, so hopefully i will be the first.
anyways i have and 84 ke70, like a conary yellow, i hate it, i love green so am getting it painted new Holden green, and am getting my engine rebuilt and worked on at the moment, I'm excited, but
sucks i couldn't put something else in it because of all the new laws.......
anyways thanks for ur time, cheers
P.S. was wondering if i could get a rollaclub sticker for my car
Edited by Rolla_Baby
Hey, welcome to the club. Your not the only girl but we don't have many. Your car sounds like it coming along nicely, post us up a pic if you like :P
As for stickers just send me a PM with which ones you want and how many and we'll take it from there.
Ok so i got around to taking some pictures of my ke55 and a few of her problems, still working on a name for her.
Dent in the bonnet.
Dent in the rear end.
I put my mates cars plates on it so i could take it for a little drive to try and work out what the noise it's making is.
Edited by TommyGun
WoW mate, sweet ride you have there. Looks great inside. Hey if ya need anything give me a holler. I think I might have a rear bar ay.
Thanks man. The seats have a few little tears in them and the drivers seat is cloth, wouldn't mind finding one the same as the passenger seat to replace it with. I'm gonna take it into a mechanic
tomorrow and see what i'll need to do to get it road worthy. I've got kind of a bad feeling that this noise in the clutch/gearbox is gonna end up costing me a bit... =(
P.S. It even came with a free rolla club sticker =D
Edited by TommyGun
Is it manual?? 4 Speed?? I'll be selling off the majority of my part collection soon. Let us know if you'd be keen to upgrade to a 5 speed.
Have pics of the seats??
I'll get some pics of the seats up tomorrow. Yeah it's a 4 spd manual. I'd be really really keen on upgrading to a 5 spd seeing as i seem to have a problem with my gearbox/clutch.
Here's a few photo's of the rips in the seats as well as the previous owners ghetto speaker installation.
Front passenger seat.
Rear seat.
Ghetto speaker setup. | {"url":"https://www.rollaclub.com/board/topic/29822-welcome-new-members/page/2/","timestamp":"2024-11-13T13:08:16Z","content_type":"text/html","content_length":"290521","record_id":"<urn:uuid:f2344b2e-8033-4adf-a55d-d0c77a7d5ab8>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00661.warc.gz"} |
Principal Component Analysis (PCA)
The Curse of Dimensionality:
Often, data Scientists get datasets which have thousand of features.
These create two kind of problems:
• Increase in computation time:
Majority of the machine learning algorithms they rely on the calculation of distance for model building and as the number of dimensions increases it becomes more and more computation-intensive to
create a model out of it.
One more point to consider is that as the number of dimension increases, points are going far away from each other.
• Hard (or almost impossible) to visualise the relationship between features:
Humans are bound by their perception of a maximum of three dimensions.
We can’t comprehend shapes/graphs beyond three dimensions.
So, if we have an n-dimensional dataset, the only solution left to us is to create either a 2-D or 3-D graph out of it.
Disadvantages of having more dimensions:
• Training time increases
• Data Visualization becomes difficult
• Computational resources requirement increases
• Chances of overfitting is high
• Difficult to explore the data.
Two ways to remove curse of dimensionality
1. Feature Selection- Drop less important feature
2. Dimensionality Reduction- Derive new feature from set of feature which is called feature extraction
Among many algorithm, we will discuss PCA here.
Principal Component Analysis (PCA)
• The principal component analysis is an unsupervised machine learning algorithm used for feature selection using dimensionality reduction techniques.
• As the name suggests, it finds out the principal components from the data.
• PCA transforms and fits the data from a higher-dimensional space to a new, lower-dimensional subspace
• This results into an entirely new coordinate system of the points where the first axis corresponds to the first principal component that explains the most variance in the data.
• The PCA algorithm is based on some mathematical concepts such as:
□ Variance and Covariance
□ Eigenvalues and Eigen factors
What are the principal components?
• Principal components are the derived features which explain the maximum variance in the data.
• The first principal component explains the most variance, the 2nd a bit less and so on.
• Each of the new dimensions found using PCA is a linear combination of the original features.
Some common terms used in PCA algorithm:
• Dimensionality: It is the number of features or variables present in the given dataset. More easily, it is the number of columns present in the dataset.
• Correlation: It signifies that how strongly two variables are related to each other. Such as if one changes, the other variable also gets changed. The correlation value ranges from -1 to +1.
Here, -1 occurs if variables are inversely proportional to each other, and +1 indicates that variables are directly proportional to each other.
• Orthogonal: It defines that variables are not correlated to each other, and hence the correlation between the pair of variables is zero.
• Eigenvectors: If there is a square matrix M, and a non-zero vector v is given. Then v will be eigenvector if Av is the scalar multiple of v.
• Covariance Matrix: A matrix containing the covariance between the pair of variables is called the Covariance Matrix.
Explained Variance Ratio
• It represents the amount of variance each principal component is able to explain.
• The total variance is the sum of variances of all individual principal components.
• The fraction of variance explained by a principal component is the ratio between the variance of that principal component and the total variance.
For example,
• Variance of PC1 is 50 and
• Variance of PC2 is 5.
So the total variance is 55.
EVR of PC1= Variance of PC1 / (Totalvariance)=50/55=0.91
EVR of PC1= Varianceof PC2 / (Totalvariance)=5/55=0.09
Thus PC1 explains 91% of the variance of data. Whereas, PC2 only explains 9% of the variance. Hence we can use only PC1 as the input for our model as it explains the majority of the variance.
In a real-life scenario, this problem is solved using the Scree Plots.
Steps involved in PCA:
1. Scaling the data: PCA tries to get the features with the maximum variance and the variance is high for high magnitude features. So we need to scale the data.
2. Calculate the covariance: to understand the variables that are highly correlated.
3. Calculate eigen vectors and eigen values (they are computed from covariance).
□ Eigen vectors determine the direction of new feature space.
□ Eigen values determine their magnitude ie., the scalar of the respective eigen vectors.
□ For example:
If you have 2 dimensional dataset, there will be 2 eigen vectors and their respective eigen values.
Reason for having the eigen vectors is to use the covariance matrix to understand where in the data, there is more amount of variance.
The covariance matrix generally gives the overall variance among all the variables in the data.
More the variance denotes more information about the data.
So eigen vector will tell where in the data, we have maximum variance.
4. Compute the Principal Components:
□ After identifying eigen vectors and eigen values, sort them in descending order. Highest eigen value is the most siginificant component.
□ PCs are the new features that are obtained and they posses most of the useful information that was scattered among the initial variables.
□ These PCs are orthogonal to each other ie., the correlation between 2 variables will be zero.
5. Reduce the dimensions of the data:
□ Eliminate the PCs that have least eigen value.
□ They are not important.
Scree Plots:
• Scree plots are the graphs that convey how much variance is explained by corresponding Principal components.
• The Scree Plot helps in deciding how many components to retain by identifying the “elbow” in the plot.
• If we see the above plot where the explained variance drops significantly from Component 1 to Component 2, and then the drop becomes smaller and more gradual from Component 3 onwards, the elbow
point would likely be at Component 2.
• This means that the first two principal components explain most of the variance, and adding more components has diminishing returns.
Python Implementation of PCA:
# import libraries
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline
# Load Dataset
data = pd.read_csv('/content/drive/MyDrive/Data Science/CDS-07-Machine Learning & Deep Learning/06. Machine Learning Model /10_Dimensionality-Reduction/PCA Class/glass.data')
index RI Na Mg Al Si K Ca Ba Fe Class
0 1 1.52101 13.64 4.49 1.10 71.78 0.06 8.75 0.0 0.0 1
1 2 1.51761 13.89 3.60 1.36 72.73 0.48 7.83 0.0 0.0 1
2 3 1.51618 13.53 3.55 1.54 72.99 0.39 7.78 0.0 0.0 1
3 4 1.51766 13.21 3.69 1.29 72.61 0.57 8.22 0.0 0.0 1
4 5 1.51742 13.27 3.62 1.24 73.08 0.55 8.07 0.0 0.0 1
Name: count, dtype: int64
EDA – Skipping
Data Preprocessing
index 0
RI 0
Na 0
Mg 0
Al 0
Si 0
K 0
Ca 0
Ba 0
Fe 0
Class 0
dtype: int64
# Creating x and y
x = data.drop(columns=['index','Class'],axis=1)
y = data.Class
# Splitting training & testing data
from sklearn.model_selection import train_test_split
x_train,x_test,y_train,y_test = train_test_split(x,y,test_size=0.2,random_state=73)
# Creating model
from sklearn.linear_model import LogisticRegression
lr = LogisticRegression()
y_predict = lr.predict(x_test)
from sklearn.metrics import accuracy_score
score1 = accuracy_score(y_test,y_predict)
Perform PCA
# Scaling down the data
from sklearn.preprocessing import StandardScaler
sc = StandardScaler()
scaled_data = sc.fit_transform(x)
array([[ 0.87286765, 0.28495326, 1.25463857, ..., -0.14576634,
-0.35287683, -0.5864509 ],
[-0.24933347, 0.59181718, 0.63616803, ..., -0.79373376,
-0.35287683, -0.5864509 ],
[-0.72131806, 0.14993314, 0.60142249, ..., -0.82894938,
-0.35287683, -0.5864509 ],
[ 0.75404635, 1.16872135, -1.86551055, ..., -0.36410319,
2.95320036, -0.5864509 ],
[-0.61239854, 1.19327046, -1.86551055, ..., -0.33593069,
2.81208731, -0.5864509 ],
[-0.41436305, 1.00915211, -1.86551055, ..., -0.23732695,
3.01367739, -0.5864509 ]])
# Creating new dataframe
new_data = pd.DataFrame(data=scaled_data,columns= x.columns)
RI Na Mg Al Si K Ca Ba Fe
0 0.872868 0.284953 1.254639 -0.692442 -1.127082 -0.671705 -0.145766 -0.352877 -0.586451
1 -0.249333 0.591817 0.636168 -0.170460 0.102319 -0.026213 -0.793734 -0.352877 -0.586451
2 -0.721318 0.149933 0.601422 0.190912 0.438787 -0.164533 -0.828949 -0.352877 -0.586451
3 -0.232831 -0.242853 0.698710 -0.310994 -0.052974 0.112107 -0.519052 -0.352877 -0.586451
4 -0.312045 -0.169205 0.650066 -0.411375 0.555256 0.081369 -0.624699 -0.352877 -0.586451
# Getting the optimal number of PCA
from sklearn.decomposition import PCA
pca = PCA()
array([2.79018192e-01, 2.27785798e-01, 1.56093777e-01, 1.28651383e-01,
1.01555805e-01, 5.86261325e-02, 4.09953826e-02, 7.09477197e-03,
# Scree plot
plt.xlabel('Number of Components')
plt.ylabel('Variance (%)') #for each component
plt.title('Explained Variance')
From the diagram, it can be seen 5 principal components explain almost 90% of the variance of the data .
So instead of giving all features as inputs, we’d only feed these 5 principal components of the data to the machine learning algorithm and we’d obtain a similar result
pca = PCA(n_components=5)
final_data = pca.fit_transform(new_data)
df = pd.DataFrame(data=final_data,
x1 = df
# Splitting training & testing data
from sklearn.model_selection import train_test_split
x1_train,x1_test,y_train,y_test = train_test_split(x1,y,test_size=0.2,random_state=73)
# Creating model
from sklearn.linear_model import LogisticRegression
lr = LogisticRegression()
y1_predict = lr.predict(x1_test)
from sklearn.metrics import accuracy_score
score2 = accuracy_score(y_test,y1_predict) | {"url":"https://data4fashion.com/principal-component-analysis-pca/","timestamp":"2024-11-07T07:24:51Z","content_type":"text/html","content_length":"161712","record_id":"<urn:uuid:626587ff-7263-41b6-9c81-125fb5328c85>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00673.warc.gz"} |
Calculus: Line integrals and Green's theorem
49 votes
Free Closed [?]
Line integral of scalar and vector-valued functions. Green's theorem and 2-D divergence theorem.
Introduction to the Line Integral. Line Integral Example 1. Line Integral Example 2 (part 1). Line Integral Example 2 (part 2). Position Vector Valued Functions. Derivative of a position Khan
vector valued function. Differential of a vector valued function. Vector valued function derivative example. Line Integrals and Vector Fields. Using a line integral to find the work done by Academy
a vector field example. Parametrization of a Reverse Path. Scalar Field Line Integral Independent of Path Direction. Vector Field Line Integrals Dependent on Path Direction. Path
Independence for Line Integrals. Closed Curve Line Integrals of Conservative Vector Fields. Example of Closed Line Integral of Conservative Field. Second Example of Line Integral of
Conservative Vector Field. Green's Theorem Proof Part 1. Green's Theorem Proof (part 2). Green's Theorem Example 1. Green's Theorem Example 2. Constructing a unit normal vector to a curve. 2
D Divergence Theorem. Conceptual clarification for 2-D Divergence Theorem. Introduction to the Line Integral. Line Integral Example 1. Line Integral Example 2 (part 1). Line Integral Example
2 (part 2). Position Vector Valued Functions. Derivative of a position vector valued function. Differential of a vector valued function. Vector valued function derivative example. Line
Integrals and Vector Fields. Using a line integral to find the work done by a vector field example. Parametrization of a Reverse Path. Scalar Field Line Integral Independent of Path
Direction. Vector Field Line Integrals Dependent on Path Direction. Path Independence for Line Integrals. Closed Curve Line Integrals of Conservative Vector Fields. Example of Closed Line
Integral of Conservative Field. Second Example of Line Integral of Conservative Vector Field. Green's Theorem Proof Part 1. Green's Theorem Proof (part 2). Green's Theorem Example 1. Green's
Theorem Example 2. Constructing a unit normal vector to a curve. 2 D Divergence Theorem. Conceptual clarification for 2-D Divergence Theorem.
Categories: Mathematics
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MATLAB Linear Systems Example
MATLAB Linear Systems Example
MATLAB Linear Systems Example
To enter matrix A in MATLAB, type:
A=[1 -2 -3; 1 2 –1; 2 4 –1]
This command generates a 3x3 matrix, which is displayed on your screen.
Then type
b=[1 2 3]’
to generate a column vector b (make sure you include the prime ’ at the end of the command).
In order to solve the system Ax=b using Gauss-Jordan elimination, you first need to generate the augmented matrix, consisting of the coefficient matrix A and the right hand side b:
Aaug=[A b]
You have now generated augmented matrix Aaug (you can call it a different name if you wish). You now need to use command “rref”, in order to reduce the augmented matrix to its reduced row echelon
form and solve your system:
C = rref(Aaug)
Can you identify the solution of the system after you calculated matrix C?
You can also solve the same system in MATLAB using command
x= A\b
The symbol between matrix A and vector b is a “backslash”. This command will generate a vector x, which is the solution of the linear system.
(Can we always use this method to solve linear systems in MATLAB? Experiment with different systems.)
Command "help" is a command you should use frequently. It shows you how MATLAB commands should be used. For example, type:
help rref
and you will get information on the usage of "rref". To find out more about command "help", type
help help
Command "help" is useful when you know the exact command you want to use and you want to find out details on its usage. Sometimes we do not know the exact command we should use for the problem we
need to solve. Then we can use command "lookfor". Type
lookfor echelon
and you will get as a result a number of MATLAB commands that have to do with row echelon forms.
You can also get help using command "doc". Type
help doc
for details.
To save your work, you can use command “diary”. Type
help diary
for more information on how to use the command.
A few more useful commands:
A’ is the transpose of A.
Command “inv” calculates the inverse of a matrix. Type:
Command “det” computes determinants (we will learn more about determinants shortly).
There are several MATLAB commands that generate special matrices.
Command “rand” generates matrices with random entries (rand(3,4) creates a 3x4 matrix with random entries). Command “eye” generates the identity matrix (try typing eye(3)).
Other such commands are “zeros” (for zero matrices) and “magic” (type help zeros and help magic for more information). | {"url":"https://math.njit.edu/students/undergraduate/matlab_linear_systems.php","timestamp":"2024-11-03T18:39:11Z","content_type":"text/html","content_length":"47427","record_id":"<urn:uuid:4e0baeb0-dfba-47f9-9755-e64b819604d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00557.warc.gz"} |
How Many Feet Are in a Roofing Square? - Ohio Roofing Supplies - Roofing Supplies in Ohio - Roofing Contractors
Whether you’re planning to install a new roof on your home or you’re remodeling an old one, knowing how many square feet are in a roofing square can make your job much easier. While roofing squares
come in various sizes, they must measure 100 square feet. One square measures 10′ x 10′, while another measures 20′ x 5′. To calculate your roof’s square footage, find out the height and width of
your roof. Roofing Company Powell
To figure the square footage of your roof, first measure the width and length of your roof. Then, divide the two numbers to find the square footage of each plane. Simple roofs have two planes, while
other styles have more. Next, add up all the square footage to determine the overall square footage. Then, you can hire a contractor to measure your roof. Make sure to note that you can’t ignore
headlap because it will only skew your measurements.
A roofing square is a great measuring tool to use for calculating the size of a roof. It helps roofing contractors calculate the size of a roof, as each square covers ten square feet. Then, they can
determine the materials and labor needed to complete the job. Knowing the square size also helps you calculate rafter connections and stairs. Once you know how many squares are in a roofing square,
it will be easier to estimate how much you need to purchase for a roofing job.
Getting a good estimate of your roof size is important to a roofing project. The roofing square is often referred to as a ‘roof square’, meaning it’s equivalent to 100 square feet. A roofing square
is important in roofing because it makes it easier to plan and estimate your material needs. If you need to replace the entire roof, you’ll need at least a thousand square feet of roofing material.
If you’re getting estimates for a new roof, you can use the roofing square to calculate the cost of the job. A roofing square represents the total area of the roof, so if your roof is a hexagon,
you’ll need more than one roofing material. For a new roof, asphalt shingles are an excellent choice. They’re affordable, easy to install, and come in an array of colors.
To calculate the square footage of a new roof, multiply the square footage of the two sides of the roof. Once you’ve calculated the total area, multiply it by two. Then, divide that amount by 100 to
get the square footage of your roofing square. Then, divide it by 100 to determine the number of squares you need. A square measuring a thousand square feet will have ten square feet.
Once you have the number of square feet you need, you can calculate the roofing square area. You can also use a two-foot level to guide your circular saw. To do this, take the length and width of
your roof and align the measurement to the peak. Once you’ve determined the size of your roof, you can multiply the length and width by the digits below the level. A free roofing square calculator is
available from Home Advisor. | {"url":"https://ohioroofingsupply.com/how-many-feet-are-in-a-roofing-square/","timestamp":"2024-11-03T07:23:53Z","content_type":"text/html","content_length":"283209","record_id":"<urn:uuid:73f3e5e7-86a8-4d97-81ae-22119779751d>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00816.warc.gz"} |
Video Riddles - Riddles.com
Last updated:
Watch these Video Riddles
Some of the best riddles are better if they are presented in a video riddle. Here is a collection of animated riddles to challenge and test your problem-solving skills.
In the early 1900's in England, there was two towns: town A and town B, and there was a huge river in between them. The only way across to each town was by a long bridge with a guard post in the
middle. It took 20 minutes to cross the bridge from one side to the other and 10 minutes each way from the guards tower. There were no hiding spots on the bridge so people couldn't sneak past, and
the guard comes out every 5 minutes or so to check to see if anyone is trying to cross. The guards tower was placed because of a law that stated that no one was aloud to leave their own town into the
other because of political reasons and anyone who was caught by the guard would be fined and told to turn back. Many people have tried to cross but have always been caught, even very fast runners
have only able to make it past the guards tower before being caught. But one night someone was able to make it across....How did they do it?
Answer: Did you get it? The answer is that they went up close to the guards tower, turned around and started walking back to the town they came from, the guard caught them and assumed that they came
the other way and sent them back! Hope you enjoyed this, please subscribe and check back soon for more riddles!
You and nine other individuals have been captured by super intelligent alien overlords. The aliens think humans look quite tasty, but their civilization forbids eating highly logical and cooperative
beings. Unfortunately, they're not sure whether you qualify, so they decide to give you all a test. Through its universal translator, the alien guarding you tells you the following: You will be
placed in a single-file line facing forward in size order so that each of you can see everyone lined up ahead of you. You will not be able to look behind you or step out of line. Each of you will
have either a black or a white hat on your head assigned randomly, and I won't tell you how many of each color there are. When I say to begin, each of you must guess the color of your hat starting
with the person in the back and moving up the line. And don't even try saying words other than black or white or signaling some other way, like intonation or volume; you'll all be eaten immediately.
If at least nine of you guess correctly, you'll all be spared. You have five minutes to discuss and come up with a plan, and then I'll line you up, assign your hats, and we'll begin. Can you think of
a strategy guaranteed to save everyone? Alex Gendler shows how.
Answer: Let's see how it would play out if the hats were distributed like this. The tallest captive sees three black hats in front of him, so he says "black," telling everyone else he sees an odd
number of black hats. He gets his own hat color wrong, but that's okay since you're collectively allowed to have one wrong answer. Prisoner two also sees an odd number of black hats, so she knows
hers is white, and answers correctly. Prisoner three sees an even number of black hats, so he knows that his must be one of the black hats the first two prisoners saw. Prisoner four hears that and
knows that she should be looking for an even number of black hats since one was behind her. But she only sees one, so she deduces that her hat is also black. Prisoners five through nine are each
looking for an odd number of black hats, which they see, so they figure out that their hats are white. Now it all comes down to you at the front of the line. If the ninth prisoner saw an odd number
of black hats, that can only mean one thing. You'll find that this strategy works for any possible arrangement of the hats. The first prisoner has a 50% chance of giving a wrong answer about his own
hat, but the parity information he conveys allows everyone else to guess theirs with absolute certainty. Each begins by expecting to see an odd or even number of hats of the specified color. If what
they count doesn't match, that means their own hat is that color. And every time this happens, the next person in line will switch the parity they expect to see. So that's it, you're free to go. It
looks like these aliens will have to go hungry, or find some less logical organisms to abduct.
One hundred green-eyed logicians have been imprisoned on an island by a mad dictator. Their only hope for freedom lies in the answer to one famously difficult logic puzzle. Can you solve it? Alex
Gendler walks us through this green-eyed
Answer: The answer to the riddle is in the video riddle above
Before he turned physics upside down, a young Albert Einstein supposedly showed off his genius by devising a complex riddle involving a stolen exotic fish and a long list of suspects. Can you resist
tackling a
brain teaser
written by one of the smartest people in history? Dan Van der Vieren shows how.
Answer: The key is that the person at the back of the line who can see everyone else's hats can use the words "black" or "white" to communicate some coded information. So what meaning can be assigned
to those words that will allow everyone else to deduce their hat colors? It can't be the total number of black or white hats. There are more than two possible values, but what does have two possible
values is that number's parity, that is whether it's odd or even. So the solution is to agree that whoever goes first will, for example, say "black" if he sees an odd number of black hats and "white"
if he sees an even number of black hats. Let's see how it would play out if the hats were distributed like this. The tallest captive sees three black hats in front of him, so he says "black," telling
everyone else he sees an odd number of black hats. He gets his own hat color wrong, but that's okay since you're collectively allowed to have one wrong answer. Prisoner two also sees an odd number of
black hats, so she knows hers is white, and answers correctly. Prisoner three sees an even number of black hats, so he knows that his must be one of the black hats the first two prisoners saw.
Prisoner four hears that and knows that she should be looking for an even number of black hats since one was behind her. But she only sees one, so she deduces that her hat is also black. Prisoners
five through nine are each looking for an odd number of black hats, which they see, so they figure out that their hats are white. Now it all comes down to you at the front of the line. If the ninth
prisoner saw an odd number of black hats, that can only mean one thing. You'll find that this strategy works for any possible arrangement of the hats. The first prisoner has a 50% chance of giving a
wrong answer about his own hat, but the parity information he conveys allows everyone else to guess theirs with absolute certainty. Each begins by expecting to see an odd or even number of hats of
the specified color. If what they count doesn't match, that means their own hat is that color. And every time this happens, the next person in line will switch the parity they expect to see.
Taking that internship in a remote mountain lab might not have been the best idea. Pulling that lever with the skull symbol just to see what it did probably wasn't so smart either. But now is not the
time for regrets because you need to get away from these mutant zombies...fast. Can you use math to get you and your friends over the bridge before the zombies arrive? Alex Gendler shows how.
Answer: At first it might seem like no matter what you do, you're just a minute or two short of time, but there is a way. The key is to minimize the time wasted by the two slowest people by having
them cross together. And because you'll need to make a couple of return trips with the lantern, you'll want to have the fastest people available to do so. So, you and the lab assistant quickly run
across with the lantern, though you have to slow down a bit to match her pace. After two minutes, both of you are across, and you, as the quickest, run back with the lantern. Only three minutes have
passed. So far, so good. Now comes the hard part. The professor and the janitor take the lantern and cross together. This takes them ten minutes since the janitor has to slow down for the old
professor who keeps muttering that he probably shouldn't have given the zombies night vision. By the time they're across, there are only four minutes left, and you're still stuck on the wrong side of
the bridge. But remember, the lab assistant has been waiting on the other side, and she's the second fastest of the group. So she grabs the lantern from the professor and runs back across to you. Now
with only two minutes left, the two of you make the final crossing. As you step on the far side of the gorge, you cut the ropes and collapse the bridge behind you, just in the nick of time.
You're stranded in a rainforest, and you've eaten a poisonous mushroom. To save your life, you need an antidote excreted by a certain species of frog. Unfortunately, only the female frog produces the
antidote. The male and female look identical, but the male frog has a distinctive croak. Derek Abbott shows how to use conditional probability to make sure you lick the right frog and get out alive.
How do you get out alive?
Answer: If you chose to go to the clearing, you're right, but the hard part is correctly calculating your odds. There are two common incorrect ways of solving this problem. Wrong answer number one:
Assuming there's a roughly equal number of males and females, the probability of any one frog being either sex is one in two, which is 0.5, or 50%. And since all frogs are independent of each other,
the chance of any one of them being female should still be 50% each time you choose. This logic actually is correct for the tree stump, but not for the clearing. Wrong answer two: First, you saw two
frogs in the clearing. Now you've learned that at least one of them is male, but what are the chances that both are? If the probability of each individual frog being male is 0.5, then multiplying the
two together will give you 0.25, which is one in four, or 25%. So, you have a 75% chance of getting at least one female and receiving the antidote. So here's the right answer. Going for the clearing
gives you a two in three chance of survival, or about 67%. If you're wondering how this could possibly be right, it's because of something called conditional probability. Let's see how it unfolds.
When we first see the two frogs, there are several possible combinations of male and female. If we write out the full list, we have what mathematicians call the sample space, and as we can see, out
of the four possible combinations, only one has two males. So why was the answer of 75% wrong? Because the croak gives us additional information. As soon as we know that one of the frogs is male,
that tells us there can't be a pair of females, which means we can eliminate that possibility from the sample space, leaving us with three possible combinations. Of them, one still has two males,
giving us our two in three, or 67% chance of getting a female. This is how conditional probability works. You start off with a large sample space that includes every possibility. But every additional
piece of information allows you to eliminate possibilities, shrinking the sample space and increasing the probability of getting a particular combination. The point is that information affects
probability. And conditional probability isn't just the stuff of abstract mathematical games. It pops up in the real world, as well. Computers and other devices use conditional probability to detect
likely errors in the strings of 1's and 0's that all our data consists of. And in many of our own life decisions, we use information gained from past experience and our surroundings to narrow down
our choices to the best options so that maybe next time, we can avoid eating that poisonous mushroom in the first place. | {"url":"https://www.riddles.com/video-riddles","timestamp":"2024-11-03T22:18:56Z","content_type":"text/html","content_length":"157257","record_id":"<urn:uuid:4734e314-e552-416c-a858-c0ab044d2a3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00657.warc.gz"} |
Prepare First, download and install CrypTool. If using a Windows® PC, install CrypTool 1. If using a MAC PC, install JCrypTool. Note: In order for the CrypTool program to - NursingEssays
First, download and install CrypTool.
• If using a Windows® PC, install CrypTool 1.
• If using a MAC PC, install JCrypTool.
Note: In order for the CrypTool program to function, you must first OPEN a text file. Only then will items on the toolbar become enabled.
Additional information for this assignment is covered in the Assignment 3 — CrypTool Lab document.
Part 1.1
1. Identify the digital signature schemes and MAC schemes that are supported by CrypTool. For each scheme, determine the key sizes supported by CrypTool and which key sizes are recommended by NIST.
2. Encrypt text using two digital signature schemes, measure the execution time for key generation (if applicable), signature generation, and signature verification. Measurements should be performed
multiple times, and then the analysis should use the average time for comparisons. In order to obtain measurable results, you will likely need to use long messages (i.e., 5 MB or larger).
3. Record results and draw conclusions. Identify the encryption algorithm used. Discuss the results of the measurements and any variations or trends observed when comparing the results from the
group members. Report the results and the analysis. Be sure to include details on the measured results and details on each member’s computer (i.e., processor type, speed, cache size, RAM size,
OS, etc.).
Part 1.2
1. Using CrypTool, generate an MD5 hash for a small plaintext document.
2. Record the hash result (cut and paste).
3. Change one character in the plaintext and regenerate a hash on the changed text.
4. Record the hash (cut and paste into your report).
5. Do you think you could find a hash collision (that is, can you find two different texts that produce the same hash)? Experiment a few times and draw a conclusion about whether this can be done
Part 1.3
1. Generate a large prime (at least 20 digits long) and a short prime (less than 10 digits in length) (HINT: Individual Procedures RSA Cryptosystem Generate prime numbers).
2. Determine which factorization algorithms are implemented within CrypTool (Indiv. Procedures -> RSA Demonstration -> Factorization of a Number).
3. Identify the characteristic features of these algorithms by reading the “Factorization algorithms” help within the tutorials section of CrypTool’s built-in help.
4. Try to factor the primes using some of these methods.
5. Report your results, identifying the methods you used. Be sure to include how long the factorization took (or how long it was estimated to take if it was not completed in a reasonable time).
6. Now, just enter numeric values to be factored (again, one long number and one short one).
7. What are the results? Were factors found? Are you good at guessing prime numbers?
Unit 3 Assignment.html
assignment deTails
Assignment 3
Outcomes addressed in this assignment:
Unit Outcomes:
• Evaluate asymmetric key cryptography, including the implementation of at least one asymmetrical algorithm.
• Explain the role asymmetric algorithms play in securing network protocols.
• Examine the similarities and differences between symmetric and asymmetric encryption.
Course Outcome:
IT543-2: Evaluate various cryptographic methods.
Assignment Instructions:
First, download and install CrypTool.
• If using a Windows® PC, install CrypTool 1.
• If using a MAC PC, install JCrypTool.
Note: In order for the CrypTool program to function, you must first OPEN a text file. Only then will items on the toolbar become enabled.
Additional information for this assignment is covered in the Assignment 3 — CrypTool Lab document.
Part 1.1
1. Identify the digital signature schemes and MAC schemes that are supported by CrypTool. For each scheme, determine the key sizes supported by CrypTool and which key sizes are recommended by NIST.
2. Encrypt text using two digital signature schemes, measure the execution time for key generation (if applicable), signature generation, and signature verification. Measurements should be performed
multiple times, and then the analysis should use the average time for comparisons. In order to obtain measurable results, you will likely need to use long messages (i.e., 5 MB or larger).
3. Record results and draw conclusions. Identify the encryption algorithm used. Discuss the results of the measurements and any variations or trends observed when comparing the results from the
group members. Report the results and the analysis. Be sure to include details on the measured results and details on each member’s computer (i.e., processor type, speed, cache size, RAM size,
OS, etc.).
Part 1.2
1. Using CrypTool, generate an MD5 hash for a small plaintext document.
2. Record the hash result (cut and paste).
3. Change one character in the plaintext and regenerate a hash on the changed text.
4. Record the hash (cut and paste into your report).
5. Do you think you could find a hash collision (that is, can you find two different texts that produce the same hash)? Experiment a few times and draw a conclusion about whether this can be done
Part 1.3
1. Generate a large prime (at least 20 digits long) and a short prime (less than 10 digits in length) (HINT: Individual Procedures RSA Cryptosystem Generate prime numbers).
2. Determine which factorization algorithms are implemented within CrypTool (Indiv. Procedures -> RSA Demonstration -> Factorization of a Number).
3. Identify the characteristic features of these algorithms by reading the “Factorization algorithms” help within the tutorials section of CrypTool’s built-in help.
4. Try to factor the primes using some of these methods.
5. Report your results, identifying the methods you used. Be sure to include how long the factorization took (or how long it was estimated to take if it was not completed in a reasonable time).
6. Now, just enter numeric values to be factored (again, one long number and one short one).
7. What are the results? Were factors found? Are you good at guessing prime numbers?
Written Assignment Requirements:
Written work should be free of spelling, grammar, and APA errors. Points deducted from the grade for writing, spelling, or grammar errors are at your instructor’s discretion.
For more information and examples of APA formatting, see the resources in Academic Tools.
Directions for Submitting Your Assignment:
Compose your assignment in a Word document, save it as IT543 YourName_Unit_3.doc, and submit it to the Dropbox for Unit 3. | {"url":"https://nursingessays.blog/solutions/prepare-first-download-and-install-cryptool-if-using-a-windows-pc-install-cryptool-1-if-using-a-mac-pc-install-jcryptool-note-in-order-for-the-cryptool-program-to/","timestamp":"2024-11-02T02:19:50Z","content_type":"text/html","content_length":"154753","record_id":"<urn:uuid:8542de1a-5045-41bb-85c9-55b34090e478>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00723.warc.gz"} |
Breadth-First Search Algorithm in Clojure
"All that is gold does not glitter, Not all those who wander are lost…" — J. R. R. Tolkien
Below is an implementation of a graph structure and breadth-first search (BFS) algorithm that allows each vertex or node to contain any payload you like – a map, a string, whatever.
Why a dang graph algorithm?
Graph traversal algorithms like Breadth-First Search and its close cousin, Depth-First Search, open up fast solutions to some interesting problems including:
• Graph coloring, used in drawing maps, creating compilers, and playing Sudoku
• Finding articulation vertices, which are likely points of failure in graphs such as telephone networks, internet networks, and more
• Making your way out of a maze by the shortest path possible
After starting a new job working with Python and JavaScript, I felt the itch to learn something in the functional tradition. Eventually I settled on Clojure, a dialect of the venerable Lisp. Working
with algorithms and data structures is one of the best ways to learn a new language, so I translated a breadth-first search algorithm into Clojure, and in the process found some interesting
differences between the Lispey code and the classic imperative style.
Much credit is due to the free book Problem Solving with Algorithms and Data Structures using Python, which provided the basis for the graph data structure implementation, and The Algorithm Design
Manual by Steven Skiena, which inspired the BFS algorithm used here. I recommend both to anyone who'd like to beef up their algorithmic chops.
The Graph and the Vertex
First off, let's define our data structures. Each Vertex has a key, a value, and a list of connected vertices. Each Graph is just a collection of vertices.
(defrecord Graph [vertices])
(defrecord Vertex
[key value connections parent])
The Graph type contains just one morsel of data:
• vertices: a Clojure map object, where each key is a Vertex object's key, and each value is the Vertex object itself.
The Vertex has a few more attributes:
• key: a unique identifier representing this vertex
• value: an arbitrary payload you'd like to associate with the vertex
• connections: a set of keys to vertices that are connected to this node (often called edges or links)
• parent: the key to this vertex's parent, to be used in the BFS implementation later
Fundamental operations
We'll use a few basic operations to experiment with our new structure:
• Add a new vertex to the graph
• Get an existing vertex from the graph
• Create an edge, or connection, between two existing vertices
• Check if two vertices are connected
We'll implement each of these in a purely functional way, meaning the function performing each operation has no side effects (it just returns a new output based on the input) and the function's
return value is always the same for the same inputs. This is a powerful way to program, drawing deep from the mighty roots of mathematics, and luckily Clojure makes it natural to work this way!
(defn add-vertex
"Add a new vertex with the given `key` and `value` to a `graph`."
[{:keys [vertices] :as graph} key value]
(->Graph (assoc
(->Vertex key value #{} nil))))
(defn get-vertex
[{:keys [vertices] :as graph} key]
(get vertices key))
(defn add-edge-directed
"Connects vertex `u` to `v` with a one-way / directed edge."
[u v]
(assoc u :connections (conj (:connections u) (:key v))))
(defn add-edge
"Creates an undirected edge between vertices `u` and `v`."
[{:keys [vertices] :as graph} u v]
;; Notice the thread "->" operation here, which inserts `vertices`
;; as the first argument of the two `assoc` calls
(-> vertices
(assoc (:key u) (add-edge-directed u v))
(assoc (:key v) (add-edge-directed v u)))))
(defn connected?
"Returns `true` if `u` has an edge to `v`; otherwise, returns `false`."
[u v]
(contains? (:connections u) (:key v)))
A couple of things to note here:
1. Each function that "modifies” the graph (add-vertex, add-edge-directed, and add-edge) returns a new Graph object, rather than editing the existing object. In terms of writing pure functions, this
is an example of "no side effects”.
2. Our add-edge function returns an undirected edge, meaning that if u is connected to v, then v is also connected to u. Depending on your problem, you might want to instead use add-edge-directed to
make a one-way connection.
In addition to the operations above, we'll add a couple of utility functions to print our graph:
;; These functions are, to use a word Nathaniel Hawthorne's grandfather would
;; use to describe our modern ways, "impure" because they have the side effect
;; of printing to the screen.
;; It turns out that all useful things are, indeed, impure.
(defn print-vertex [v]
(println (str
"\t" (:key v)
"\t- conn: " (:connections v)
"\t- parent: " (:parent v))))
(defn print-graph [{:keys [vertices] :as graph}]
(println "Graph:")
(doseq [v (map val vertices)]
(print-vertex v))
Now for the fun part: testing our new structure and seeing what it looks like!
Here we'll use the handy threading macro as-> to progressively transform our Graph object. Each call to add-vertex/edge returns a new Graph, which the as-> macro assigns to the name g for use in the
next line.
(as-> (->Graph {}) g
(add-vertex g "A" 1)
(add-vertex g "B" 2)
(add-edge g (get-vertex g "A") (get-vertex g "B"))
(print-graph g))
;; -- Output --
;; Graph:
;; A - conn: #{"B"} - parent:
;; B - conn: #{"A"} - parent:
(as-> (->Graph {}) g
(add-vertex g "Answer" 42)
(add-vertex g "Question" "Meaning?")
(add-vertex g "Thanks" "Fish")
(add-vertex g "Bob" {:occupation :painting-genius})
(add-edge g (get-vertex g "Answer") (get-vertex g "Question"))
(add-edge g (get-vertex g "Answer") (get-vertex g "Thanks"))
(print-graph g)
"Are Answer and Question connected? "
(connected? (get-vertex g "Answer") (get-vertex g "Question"))))
;; -- Output --
;; Graph:
;; Answer - conn: #{"Question" "Thanks"} - parent:
;; Question - conn: #{"Answer"} - parent:
;; Thanks - conn: #{"Answer"} - parent:
;; Bob - conn: #{} - parent:
;; Are Answer and Question connected? true
The Search they call Breadth First
Now for the actual Breadth-First Search algorithm! In pseudo-code, the algorithm can be described like this:
BFS(graph, start-node)
Mark all vertices in `graph` as "undiscovered"
Mark `start-node` as "discovered"
Create a FIFO (first-in first-out) queue `Q` initially containing `start-node`
While `Q` is not empty:
Pop the first element from `Q`, assign to `u`
Process vertex `u` as needed
For each vertex `v` adjacent to `u`:
Process edge `(u, v)` as needed
If `v` is "undiscovered":
Mark `v` as "discovered"
Set `v`'s parent to `u`
Add `v` to `Q`
Mark `u` as "completely explored"
This is in a somewhat imperative style, so we`ll make some adjustments to utilize the Power of the Lambda.
Our breadth first search function will accept a graph object and a starting vertex. It will return the graph with each vertex's :parent property updated to the appropriate parent's key for a BFS
;; For ClojureScript programs, replace both instances of
;; `clojure.lang.PersistentQueue/EMPTY` below with `#queue []`
(use '[clojure.string :only (join)])
(defn bfs
"Return `graph` with each vertex's `:parent` property updated."
[graph start]
(loop [discovered clojure.lang.PersistentQueue/EMPTY
discovered-map {(:key start) true}
u start
new-graph graph]
(if (nil? u)
;; Base case: all vertices have been explored
;; Queue any of the current node's neighbors that haven't been
;; discovered
(let [neighbor-keys
(filter #(not (contains? discovered-map %))
(:connections u))
(map #(assoc
(get-vertex graph %) :parent (:key u))
(as-> discovered ds
(into clojure.lang.PersistentQueue/EMPTY
(concat ds neighbors)))]
;; Further processing of `u` or the `(u, v)` edges can go here
(println (str "Exploring vertex " (:key u) ", neighbors: "
(join ", " neighbor-keys)))
;; Proceed to exploring the next vertice, adding `u`'s
;; neighbors to the discovered pile
(pop new-discovered)
(into discovered-map
(map #(hash-map % true) neighbor-keys))
(peek new-discovered)
;; Update each of `u`'s neighbors to show `u` is the parent
(reduce #(assoc-in %1 [:vertices %2 :parent] (:key u))
The discovered-map property here is just used to see which vertices have already been discovered, more efficiently than iterating over the discovered queue.
Appendix: Creating a Vertex Explorer with async.core
Notice the println in the above code? You can replace that with any processing you need to do on the graph, such as recording results in another data structure or emitting your next Sudoku move.
But what if your program has multiple uses for the BFS, or you don't want to embed application logic inside of the graph algorithm? In this case, we can harness Clojure's async.core system to create
a channel that emits each vertex as it gets explored.
This only requires a miniscule change to our function:
;; Note: for ClojureScript programs, replace both instances of
;; `clojure.lang.PersistentQueue/EMPTY` below with `#queue []`
(use '[clojure.string :only (join)]
:as async
:refer [>! <! go chan take]])
(defn bfs-channel
"Return a 2-tuple where the first element is `graph`, with each
vertex's `:parent` property updated, and the second element is an
async.core channel that emits each vertex as it is explored."
[graph start]
(let [out (chan)]
(loop [discovered clojure.lang.PersistentQueue/EMPTY
discovered-map {(:key start) true}
u start
new-graph graph]
(if (nil? u)
;; Base case: all vertices have been discovered
(take (count (:vertices new-graph)) out)]
;; Get all of the current node's neighbors that haven't been
;; discovered
(let [neighbor-keys
(filter #(not (contains? discovered-map %))
(:connections u))
(map #(assoc
(get-vertex graph %) :parent (:key u))
(as-> discovered ds
(into clojure.lang.PersistentQueue/EMPTY
(concat ds neighbors)))]
;; Emit the vertex being explored to `out`
(go (>! out u))
;; Proceed to exploring the next vertice, adding `u`'s
;; neighbors to the discovered pile
(pop new-discovered)
(into discovered-map
(map #(hash-map % true) neighbor-keys))
(peek new-discovered)
;; Update each of `u`'s neighbors to show `u` is the parent
(reduce #(assoc-in %1 [:vertices %2 :parent] (:key u))
Now we can read from the out channel, and do all processing outside of the bfs function.
Freakin' sweet!
I hope you've enjoyed reading this article half as much as I've enjoyed playing with these graphs.
If you think of ways to improve this (I'm sure there are some!) or want to work with me on a project, feel free to shoot me a message!
Good luck, and happy hacking. | {"url":"https://nathanclonts.com/breadth-first-search-algorithm-in-clojure/","timestamp":"2024-11-01T19:56:35Z","content_type":"text/html","content_length":"94035","record_id":"<urn:uuid:d414793d-85d4-4aec-93e1-e06699b88369>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00059.warc.gz"} |
Bihar Board 10th Maths Answer Key 1st & 2nd Sitting
After appearing in the Bihar Board 10th Math Exam 2022, Students are looking for BSEB 10th Math Answer Key.
Here you will get the answer key for the 10th class maths exam 2022. Bihar School Examination Committee has conducted the BSEB 10th Math Exam on 17th February 2022 in 01 Sitting and 2 Sitting.
In the Math Exam total of 100 Objective Asked, In which students have to answer any of 50 on the BSEB 10th Math OMR Sheet. Students can use the Answer Key of the 10th to predict Marks in the Exam.
students who have appeared in the Bihar Board 10th Math Exam can use the official bseb matric math answer key from the Below Link. | {"url":"https://bsebresult.in/web-stories/bihar-board-matric-maths-answer-key/","timestamp":"2024-11-13T06:49:51Z","content_type":"text/html","content_length":"65143","record_id":"<urn:uuid:99e9344a-a963-4f1b-be51-53672a6a14a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00506.warc.gz"} |
Wonseok Choi
Wonseok Choi
Building PRFs from TPRPs: Beyond the Block and the Tweak Length Bounds Abstract
A secure n-bit tweakable block cipher (TBC) using t-bit tweaks can be modeled as a tweakable uniform random permutation, where each tweak defines an independent random n-bit permutation. When an
input to this tweakable permutation is fixed, it can be viewed as a perfectly secure t-bit random function. On the other hand, when a tweak is fixed, it can be viewed as a perfectly secure n-bit
random permutation, and it is well known that the sum of two random permutations is pseudorandom up to 2n queries.A natural question is whether one can construct a pseudorandom function (PRF) beyond
the block and the tweak length bounds using a small number of calls to the underlying tweakable permutations. A straightforward way of constructing a PRF from tweakable permutations is to xor the
outputs from two tweakable permutations with c bits of the input to each permutation fixed. Using the multi-user security of the sum of two permutations, one can prove that the (t + n − c)-to-n bit
PRF is secure up to 2n+c queries.In this paper, we propose a family of PRF constructions based on tweakable permutations, dubbed XoTPc, achieving stronger security than the straightforward
construction. XoTPc is parameterized by c, giving a (t + n − c)-to-n bit PRF. When t < 3n and c = t/3 , XoTPt/3 becomes an (n + 2t/3 )-to-n bit pseudorandom function, which is secure up to 2n+2t/3
queries. It provides security beyond the block and the tweak length bounds, making two calls to the underlying tweakable permutations. In order to prove the security of XoTPc, we extend Mirror theory
to q ≫ 2n, where q is the number of equations. From a practical point of view, our construction can be used to construct TBC-based MAC finalization functions and CTR-type encryption modes with
stronger provable security compared to existing schemes.
The Committing Security of MACs with Applications to Generic Composition Abstract
Message Authentication Codes (MACs) are ubiquitous primitives deployed in multiple flavours through standards such as HMAC, CMAC, GMAC, LightMAC and many others. Its versatility makes it an essential
building block in applications necessitating message authentication and integrity check, in authentication protocols, authenticated encryption schemes, or as a pseudorandom or key derivation
function. Its usage in this variety of settings makes it susceptible to a broad range of attack scenarios. The latest attack trends leverage a lack of commitment or context-discovery security in AEAD
schemes and these attacks are mainly due to the weakness in the underlying MAC part. However, these new attack models have been scarcely analyzed for MACs themselves. This paper provides a thorough
treatment of MACs committing and context-discovery security. We reveal that commitment and context-discovery security of MACs have their own interest by highlighting real-world vulnerable scenarios.
We formalize the required security notions for MACs, and analyze the security of standardized MACs for these notions. Additionally, as a constructive application, we analyze generic AEAD composition
and provide simple and efficient ways to build committing and context-discovery secure AEADs.
Toward Full n-bit Security and Nonce Misuse Resistance of Block Cipher-based MACs Abstract
In this paper, we study the security of MAC constructions among those classified by Chen {\it et al.} in ASIACRYPT '21. Precisely, $F^{\text{EDM}}_{B_2}$~(or $\ewcdm$ as named by Cogliati and Seurin
in CRYPTO '16), $F^{\text{EDM}}_{B_3}$, $F^{\text{SoP}}_{B_2}$, $F^{\text{SoP}}_{B_3}$ (all as named by Chen {\it et al.}) are proved to be fully secure up to $2^n$ MAC queries in the
nonce-respecting setting, improving the previous bound of $\frac{3n}{4}$-bit security. In particular, $F^{\text{SoP}}_{B_2}$ and $F^{\text{SoP}}_{B_3}$ enjoy graceful degradation as the number of
queries with repeated nonces grows (when the underlying universal hash function satisfies a certain property called \emph{multi-xor-collision resistance}). To do this, we develop a new tool, namely
extended Mirror theory based on two independent permutations to a wide range of $\xi_{\max}$ including inequalities. We also present matching attacks on $F^{\text{EDM}}_{B_4}$ and $F^{\text{EDM}}_
{B_5}$ using $O(2^{3n/4})$ MAC queries and $O(1)$ verification query without using repeated nonces.
Improved Multi-User Security Using the Squared-Ratio Method Abstract
Proving security bounds in contexts with a large number of users is one of the central problems in symmetric-key cryptography today. This paper introduces a new method for information-theoretic
multi-user security proofs, called ``the Squared-Ratio method''. At its core, the method requires the expectation of the square of the ratio of observing the so-called good transcripts (from
Patarin's H-coefficient technique) in the real and the ideal world. Central to the method is the observation that for information-theoretic adversaries, the KL-divergence for the multi-user security
bound can be written as a summation of the KL-divergence of every single user. We showcase the Squared-Ratio method on three examples: the Xor of two Permutations by Bellare et al. (EUROCRYPT '98)
and Hall et al. (CRYPTO '98), the Encrypted Davies-Mayer by Cogliati and Seurin (CRYPTO '16), and the two permutation variant of the nEHtM MAC algorithm by Dutta et al. (EUROCRYPT '19). With this new
tool, we provide improved bounds for the multi-user security of these constructions. Our approach is modular in the sense that the multi-user security can be obtained directly from single-user
Multi-User Security of the Sum of Truncated Random Permutations 📺 Abstract
For several decades, constructing pseudorandom functions from pseudorandom permutations, so-called Luby-Rackoff backward construction, has been a popular cryptographic problem. Two methods are
well-known and comprehensively studied for this problem: summing two random permutations and truncating partial bits of the output from a random permutation. In this paper, by combining both
summation and truncation, we propose new Luby-Rackoff backward constructions, dubbed SaT1 and SaT2, respectively. SaT2 is obtained by partially truncating output bits from the sum of two independent
random permutations, and SaT1 is its single permutation-based variant using domain separation. The distinguishing advantage against SaT1 and SaT2 is upper bounded by O(\sqrt{\mu q_max}/2^{n-0.5m})
and O({\sqrt{\mu}q_max^1.5}/2^{2n-0.5m}), respectively, in the multi-user setting, where n is the size of the underlying permutation, m is the output size of the construction, \mu is the number of
users, and q_max is the maximum number of queries per user. We also prove the distinguishing advantage against a variant of XORP[3]~(studied by Bhattacharya and Nandi at Asiacrypt 2021) using
independent permutations, dubbed SoP3-2, is upper bounded by O(\sqrt{\mu} q_max^2}/2^{2.5n})$. In the multi-user setting with \mu = O(2^{n-m}), a truncated random permutation provides only the
birthday bound security, while SaT1 and SaT2 are fully secure, i.e., allowing O(2^n) queries for each user. It is the same security level as XORP[3] using three permutation calls, while SaT1 and SaT2
need only two permutation calls.
Toward a Fully Secure Authenticated Encryption Scheme From a Pseudorandom Permutation 📺 Abstract
In this paper, we propose a new block cipher-based authenticated encryption scheme, dubbed the Synthetic Counter with Masking (SCM) mode. SCM follows the NSIV paradigm proposed by Peyrin and Seurin
(CRYPTO 2016), where a keyed hash function accepts a nonce N with associated data and a message, yielding an authentication tag T, and then the message is encrypted by a counter-like mode using both
T and N. Here we move one step further by encrypting nonces; in the encryption part, the inputs to the block cipher are determined by T, counters, and an encrypted nonce, and all its outputs are also
masked by an (additional) encrypted nonce, yielding keystream blocks. As a result, we obtain, for the first time, a block cipher-based authenticated encryption scheme of rate 1/2 that provides n-bit
security with respect to the query complexity (ignoring the influence of message length) in the nonce-respecting setting, and at the same time guarantees graceful security degradation in the faulty
nonce model, when the underlying n-bit block cipher is modeled as a secure pseudorandom permutation. Seen as a slight variant of GCM-SIV, SCM is also parallelizable and inverse-free, and its
performance is still comparable to GCM-SIV.
Improved Security Analysis for Nonce-based Enhanced Hash-then-Mask MACs 📺 Abstract
In this paper, we prove that the nonce-based enhanced hash-then-mask MAC (nEHtM) is secure up to 2^{3n/4} MAC queries and 2^n verification queries (ignoring logarithmic factors) as long as the number
of faulty queries \mu is below 2^{3n/8}, significantly improving the previous bound by Dutta et al. Even when \mu goes beyond 2^{3n/8}, nEHtM enjoys graceful degradation of security. The second
result is to prove the security of PRF-based nEHtM; when nEHtM is based on an n-to-s bit random function for a fixed size s such that 1 <= s <= n, it is proved to be secure up to any number of MAC
queries and 2^s verification queries, if (1) s = n and \mu < 2^{n/2} or (2) n/2 < s < 2^{n-s} and \mu < max{2^{s/2}, 2^{n-s}}, or (3) s <= n/2 and \mu < 2^{n/2}. This result leads to the security
proof of truncated nEHtM that returns only s bits of the original tag since a truncated permutation can be seen as a pseudorandom function. In particular, when s <= 2n/3, the truncated nEHtM is
secure up to 2^{n - s/2} MAC queries and 2^s verification queries as long as \mu < min{2^{n/2}, 2^{n-s}}. For example, when s = n/2 (resp. s = n/4), the truncated nEHtM is secure up to 2^{3n/4}
(resp. 2^{7n/8}) MAC queries. So truncation might provide better provable security than the original nEHtM with respect to the number of MAC queries.
Highly Secure Nonce-based MACs from the Sum of Tweakable Block Ciphers 📺 Abstract
Tweakable block ciphers (TBCs) have proven highly useful to boost the security guarantees of authentication schemes. In 2017, Cogliati et al. proposed two MACs combining TBC and universal hash
functions: a nonce-based MAC called NaT and a deterministic MAC called HaT. While both constructions provide high security, their properties are complementary: NaT is almost fully secure when nonces
are respected (i.e., n-bit security, where n is the block size of the TBC, and no security degradation in terms of the number of MAC queries when nonces are unique), while its security degrades
gracefully to the birthday bound (n/2 bits) when nonces are misused. HaT has n-bit security and can be used naturally as a nonce-based MAC when a message contains a nonce. However, it does not have
full security even if nonces are unique.This work proposes two highly secure and efficient MACs to fill the gap: NaT2 and eHaT. Both provide (almost) full security if nonces are unique and more than
n/2-bit security when nonces can repeat. Based on NaT and HaT, we aim at achieving these properties in a modular approach. Our first proposal, Nonce-as-Tweak2 (NaT2), is the sum of two NaT instances.
Our second proposal, enhanced Hash-as-Tweak (eHaT), extends HaT by adding the output of an additional nonce-depending call to the TBC and prepending nonce to the message. Despite the conceptual
simplicity, the security proofs are involved. For NaT2 in particular, we rely on the recent proof framework for Double-block Hash-then-Sum by Kim et al. from Eurocrypt 2020.
Indifferentiability of Truncated Random Permutations Abstract
One of natural ways of constructing a pseudorandom function from a pseudorandom permutation is to simply truncate the output of the permutation. When n is the permutation size and m is the number of
truncated bits, the resulting construction is known to be indistinguishable from a random function up to $$2^{{n+m}\over 2}$$ queries, which is tight.In this paper, we study the indifferentiability
of a truncated random permutation where a fixed prefix is prepended to the inputs. We prove that this construction is (regularly) indifferentiable from a public random function up to $$\min \{2^
{{n+m}\over 3}, 2^{m}, 2^\ell \}$$ queries, while it is publicly indifferentiable up to $$\min \{ \max \{2^{{n+m}\over 3}, 2^{n \over 2}\}, 2^\ell \}$$ queries, where $$\ell $$ is the size of the
fixed prefix. Furthermore, the regular indifferentiability bound is proved to be tight when $$m+\ell \ll n$$.Our results significantly improve upon the previous bound of $$\min \{ 2^{m \over 2}, 2^\
ell \}$$ given by Dodis et al. (FSE 2009), allowing us to construct, for instance, an $$\frac{n}{2}$$-to-$$\frac{n}{2}$$ bit random function that makes a single call to an n-bit permutation,
achieving $$\frac{n}{2}$$-bit security.
Program Committees
Asiacrypt 2024 | {"url":"https://iacr.org/cryptodb/data/author.php?authorkey=11218","timestamp":"2024-11-02T13:58:12Z","content_type":"text/html","content_length":"43604","record_id":"<urn:uuid:7a9d7ed4-8e9d-4af9-b3c5-90988973d7cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00033.warc.gz"} |
Rimuru's Road
Standard Input (stdin)
Standard Output (stdout)
Memory limit:
64 megabytes
Time limit:
0.5 seconds
Rewind to the 17th century in the country of Tempest: the road from Tempest to the Sorcerer's Dynasty has just been completed. As a passionate fan of transport infrastructure projects, you start
journeying from one end of the road to the other, starting from Tempest. At Tempest, which is the starting point (0km), there are three types of facilities available: a bar, an outpost, and an inn.
From this starting point, there is a bar located every $A$ kilometres on the road, an outpost every $B$ kilometres, and an inn every $C$ kilometres.
However, the weather's quite hot today. At some point during your journey, you lose track of how far you are from a facility, and your phone's dead too. You only know how far along the road you
currently are, and the values $A$, $B$, and $C$ from above. How far from the closest facility are you from your current point?
The first line will contain three space-separated integers $A$, $B$, and $C$ ($1 \le A, B, C \le 1\,000\,000\,000$).
The second line will contain an integer $D$, the distance from Tempest to your current point on the road in kilometres ($0 \lt D \le 1\,000\,000\,000$).
The absolute distance to the closest facility in kilometres, regardless of whether it is a bar, outpost, or inn.
If there is more than one facility that is the closest and are of different types (e.g. an outpost and a bar), output this text on the line after the absolute distance: can't make up my mind
• Subtask 1 (60%): $1 \le A, B, C, D \le 500$
• Subtask 2 (40%): $1 \le A, B, C, D \le 1\,000\,000\,000$
Sample Explanation
In the first sample case, you are currently 29km away from Tempest. The closest bar is at the 30km mark (6 × 5km); the closest outpost is 28km (2 × 14km); the closest inn is 33km (3 × 11km). Both the
bar and the outpost (two different types of facilities) are 1km away from your current location, so on the next line is the message can't make up my mind
In the second sample case, you are 35km away from Tempest. The closest bar is 46km away from Tempest; the two closest outposts are 30km and 40km; the closest inn is 42km. There are two of the same
type of facility (two outposts) to choose from that are the closest, being 5km away from your location. | {"url":"https://train.nzoi.org.nz/problems/1113","timestamp":"2024-11-05T11:59:27Z","content_type":"text/html","content_length":"38792","record_id":"<urn:uuid:277457bb-b62a-40db-8fe5-43b4246dba92>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00002.warc.gz"} |
Returns the Student's left-tailed t-distribution. The t-distribution is used in the hypothesis testing of small sample data sets. Use this function in place of a table of critical values for the
Sample Usage
T.DIST(x,deg_freedom, cumulative)
The T.DIST function syntax has the following arguments:
• X Required. The numeric value at which to evaluate the distribution
• Deg_freedom Required. An integer indicating the number of degrees of freedom.
• Cumulative Required. A logical value that determines the form of the function. If cumulative is TRUE, T.DIST returns the cumulative distribution function; if FALSE, it returns the probability
density function.
• If any argument is nonnumeric, T.DIST returns the #VALUE! error value.
• If deg_freedom < 1, T.DIST returns an error value. Deg_freedom needs to be at least 1.
In order to use the T.DIST formula, start with your edited Excellentable:
Then type in the T.DIST formula in the area you would like to display the outcome:
By adding the values you would like to calculate the T.DIST formula for, Excellentable will generate the outcome: | {"url":"https://www.excellentable.com/help/t-dist","timestamp":"2024-11-12T07:07:29Z","content_type":"text/html","content_length":"37179","record_id":"<urn:uuid:90832e90-1e1c-4ccc-a153-52007e046045>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00626.warc.gz"} |
Barbara P. Lorde - Crispus Attucks Academy
M&M MATH
Barbara P. Lorde Crispus Attucks Academy
3813 S. Dearborn
CHICAGO IL 60609
(773) 535-1270
This lesson is designed to introduce math at the kindergarten level. All
students should be able to:
1. Learn basic colors
2. Sort according to colors
3. Count 1 through 10
Materials Needed:
For a class of thirty students you will need the following:
30 small bags of M&MS
30 coffee filters
30 papers with colors red, blue, orange, brown, yellow, and green written in
Have students guess how many M&M'S are in the bag.
Open bag of candy and put candy into coffee filter.
Sort all M&M's into sets by colors using the color sheet.
Name each color and count each set.
Once activity is finish have students put M&M's
back into their coffee filters.
Ask students to put two M&M in their mouth.
Have students count how many are left.
Eat two more. How many M&M'S do you have left?
Performance Assessment:
Each student will be orally assessed on colors, sorting, and counting 1-10.
The students should be able to name various colors.
Sort according to colors and count 1-10.
Primary Bears, Book 1- 1987 AIMS Education Foundation
Return to Mathematics Index | {"url":"https://smileprogram.info/phma9916.htm","timestamp":"2024-11-15T01:23:10Z","content_type":"text/html","content_length":"2244","record_id":"<urn:uuid:b4ad25b8-b91a-4685-8a61-28511f1584b4>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00371.warc.gz"} |
Parity game
Parity Games are (possibly) infinite games played on a graph by two players, 0 and 1. In such games, game positions are assigned priorities, i.e. natural numbers.
A play in a parity game is a maximal sequence of nodes following the transition relation. The winner of a finite play of a parity game is the player whose opponent is unable to move. The outcome
of an infinite play is determined by the priorities of the occurring positions. Typically, player 0 wins an infinite play if the highest infinitely often occurring priority is even. Player 1 wins
otherwise. Whence the name "parity".
Parity games lie in the third level of the borel hierarchy and are as such determined [D. A. Martin: Borel determinacy, The Annals of Mathematics, Vol 102 No. 2 pp. 363-371 (1975)] . Games
related to parity games were implicitly used in Rabin'sproof of decidability of second order theory of n successors, where determinacy of such games was proven [cite journal|last = Rabin|first =
MO|title= Decidability of second order theories and automata on infinite trees|journal = Trans. AMS|year=1969|pages=1–35|volume=141|doi= 10.2307/1995086] . Knaster-Tarski theorem leads to a
relatively simple proof of determinacy of parity gamesE. A. Emerson and C. S. Jutla: Tree Automata, Mu-Calculus and Determinacy, IEEE Proc. Foundations of Computer Science, pp 368-377 (1991),
ISBN 0-8186-2445-0] .
Moreover, parity games are history-free determined [A. Mostowski: Games with forbidden positions, University of Gdansk, Tech. Report 78 (1991)] , so that if a player has a winning strategy then
she has a winning strategy which depends only on the board position, and not on the history of the play.
Decision Problem
The "decision problem" of a parity game played on a finite graph consists of deciding the winner of a play from a given position. It has been shown that this problem is in NP and Co-NP, as well
as UP and co-UP. It remains an open question whether for any parity game the decision problem is solvable in PTime.
Given that parity games are history free determined, the decision problem can be seen as the following rather simple looking graph theory problem. Given a finite colored directed bipartite graph
with "n" vertices $V = V_0 cup V_1$, and "V" colored with colors from "1" to "m", is there a choice function selecting a single out-going edge from each vertex of $V_0$, such that the resulting
subgraph has the property that in each cycle the largest occurring color is even.
Related Games and Their Decision Problems
A slight modification of the above game, and the related graph-theoretic problem, makes the decision problem NP-hard. The modified game has the Rabin acceptance condition.Specifically, in the
above bipartite graph decision problem, the problem now is to determine if thereis a choice function selecting a single out-going edge from each vertex of $V_0$, such that the resulting subgraph
has the property that in each cycle (and hence each strongly connected component) it is the case that there exists an "i" and a node with color $2cdot i$, and no node with color $2cdot i-1$.
Note that as opposed to parity games, this game is no more symmetric with respect to players 0 and 1.
* E. Grädel, W. Thomas, T. Wilke (Eds.) : [http://www-mgi.informatik.rwth-aachen.de/Misc/LNCS2500/ Automata, Logics, and Infinite Games] , Springer LNCS 2500 (2003), ISBN 3540003886
* Complexity Zoo: [http://qwiki.caltech.edu/wiki/Complexity_Zoo#np NP]
External links
Parity Game Solvers:
* [http://www.tcs.ifi.lmu.de/~mlange/pgsolver/ PGSolver Collection]
Wikimedia Foundation. 2010. | {"url":"https://en-academic.com/dic.nsf/enwiki/3961023","timestamp":"2024-11-06T04:41:29Z","content_type":"text/html","content_length":"38145","record_id":"<urn:uuid:a57b11c8-236e-41a1-aa5f-ee7174ade4f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00293.warc.gz"} |
Theorems covered in MAS575 Combinatorics Spring 2017 - Sang-il Oum (엄상일) (엄상일)
Theorems covered in MAS575 Combinatorics Spring 2017
This post will give an incomplete list of theorems covered in MAS575 Combinatorics Fall 2017. This post will be continuously updated throughout this semester. (Last update: April 5, 2017.)
2017년 봄학기 MAS575 조합론 과목에서 다룬 정리들을 정리하였습니다. 빠진 것도 있습니다. 강의가 진행되면서 내용을 업데이트 하겠습니다.\(\newcommand\abs[1]{\lvert #1\rvert}\newcommand\F{\mathbb F}\
• Graham and Pollak (1971). Proof by Tverberg (1982)
□ The edge set $E(K_n)$ of the complete graph cannot be partitioned into less than $n-1$ copies of the edge sets of complete bipartite subgraphs.
• Lindström and Smet (1970).
□ Let $A_1,A_2,\ldots,A_{n+1}\subseteq [n]$. Then there exist subsets $I$ and $J$ of $[n+1]$ such that \[ I\cup J\neq \emptyset, I\cap J=\emptyset, \text{ and }\bigcup_{i\in I}A_i =\bigcup_{j\
in J} A_j.\]
• Lindström (1993)
□ Let $A_1,A_2,\ldots,A_{n+2}\subseteq [n]$. Then there exist subsets $I$ and $J$ of $[n+2]$ such that $I\cup J\neq \emptyset$, $I\cap J=\emptyset$, and \[\bigcup_{i\in I} A_i=\bigcup_{j\in J}
A_j\text{ and }\bigcap_{i\in I}A_i=\bigcap_{j\in J}A_j.\]
• Larman, Rogers, and Seidel (1977) [New in 2017]
□ Every two-distance set in $\mathbb R^n$ has at most $\binom{n+2}{2}+n+1$ points. (A set of points is a two-distance set if the set of distances between distinct points has at most two
• Blokhuis (1984) [New in 2017]
□ Every two-distance set in $\mathbb R^n$ has at most $\binom{n+2}{2}$ points.
• Erdős (1963)
□ Let $C_1,C_2,\ldots,C_m$ be the list of clubs and each club has at least $k$ members. If $m< 2^{k-1}$, then such an assignment of students into two lecture halls is always possible.
• Erdős, Ko, and Rado (1961). Proof by Katona (1972)
□ Let $k\le n/2$. Let $\mathcal F$ be a $k$-uniform intersecting family of subsets of an $n$-set. Then \[|\mathcal F|\le\binom{n-1}{k-1}.\]
• Fisher (1940), extended by Bose (1949). Related to de Brujin and Erdős (1948)
□ Let $k$ be a positive integer. Let $\mathcal F$ be a family on an $n$-set $S$ such that $|X\cap Y|=k$ whenever $X,Y\in \mathcal F$ and $X\neq Y$. Then $|\mathcal F|\le n$.
□ Corollary due to de Brujin and Erdős (1948): Suppose that $n$ points are given on the plane so that not all are on one line. Then there are at least $n$ distinct lines through at least two
of the $n$ points.
• Frankl and Wilson (1981). Proof by Babai (1988).
□ If a family $\mathcal F$ of subsets of $[n]$ is $L$-intersecting and $|L|=s$, then \[|\mathcal F|\le \sum_{i=0}^s \binom{n}{i}.\]
• Ray-Chaudhuri and Wilson (1975). Proof by Alon, Babai, and Suzuki (1991).
□ If a family $\mathcal F$ of subsets of $[n]$ is uniform $L$-intersecting and $|L|=s$, then \[|\mathcal F|\le \binom{n}{s}.\] (A family of sets is \emph{uniform} if all members have the same
• Deza, Frankl, and Singhi (1983)
□ Let $p$ be a prime. Let $L\subseteq\{0,1,2,\ldots,p-1\}$ and $|L|=s$.If
(i) $|A|\notin L+p\mathbb Z$ for all $A\in \mathcal F$,
(ii) $|A\cap B|\in L+p\mathbb Z$ for all $A,B\in \mathcal F$, $A\neq B$,
then \[|\mathcal F|\le \sum_{i=0}^s \binom{n}{i}.\]
• Alon, Babai, and Suzuki (1991)
□ Let $p$ be a prime. Let $k$ be an integer. Let $L\subseteq \{0,1,2,\ldots,p-1\}$ and $|L|=s$. Assume $s+k\le n$. If
(i) $|A|\equiv k \notin L+p\mathbb Z$ for all $A\in \mathcal F$,
(ii) $|A\cap B|\in L+p\mathbb Z$ for all $A,B\in \mathcal F$, $A\neq B$,
then \[|\mathcal F|\le \binom{n}{s}.\]
• Grolmusz and Sudakov (2002) [New in 2017]
□ Let $p$ be a prime. Let $L\subseteq\{0,1,\ldots,p-1\}$ with $|L|=s$ and $k\ge 2$. Let $\mathcal F$ be a family of subsets of $[n]$ such that
(i) $|A|\notin L+p\mathbb Z$ for all $A\in \mathcal F$ and
(ii) $|A_1\cap A_2\cap \cdots \cap A_k|\in L+p\mathbb Z$ for every collection of $k$ distinct members $A_1,A_2,\ldots,A_k$ of $\mathcal F$.
Then \[|\mathcal F|\le (k-1)\sum_{i=0}^s \binom{n}{i}.\]
• Grolmusz and Sudakov (2002) [New in 2017]
□ Let $\abs{L}=s$ and $k\ge 2$. Let $\mathcal F$ be a family of subsets of $[n]$ such that \[|A_1\cap A_2\cap \cdots \cap A_k|\in L\] for every collection of $k$ distinct members $A_1,A_2,\
ldots,A_k$ of $\mathcal F$. Then \[|\mathcal F|\le (k-1)\sum_{i=0}^s \binom{n}{i}.\]
• Sperner (1928)
□ If $\mathcal F$ is an antichain of subsets of an $n$-set, then \[ |\mathcal F|\le \binom{n}{\lfloor n/2\rfloor}.\]
• Lubell (1966), Yamamoto (1954), Meschalkin (1963)
□ If $\mathcal F$ is an antichain of subsets of an $n$-element set, then \[ \sum_{A\in \mathcal F} \frac1{\binom{n}{|A|}}\le 1.\]
• Bollobás (1965)
□ Let $A_1$, $A_2$, $\ldots$, $A_m$, $B_1$, $B_2$, $\ldots$, $B_m$ be subsets on an $n$-set such that
(a) $A_i\cap B_i=\emptyset$ for all $i\in[m]$,
(b) $A_i\cap B_j\neq\emptyset$ for all $i\neq j$.
Then\[\sum_{i=1}^m \frac1{\binom{|A_i|+|B_i|}{|A_i|}} \le 1.\]
• Bollobás (1965), extending Erdős, Hajnal, and Moon (1964)
□ If each family of at most $\binom{r+s}{r}$ edges of an $r$-uniform hypergraph can be covered by $s$ vertices, then all edges can also be covered by $s$ vertices.
• Lovász (1977)
□ Let $A_1$, $A_2$, $\ldots$, $A_m$, $B_1$, $B_2$, $\ldots$, $B_m$ be subsets such that $|A_i|=r$ and $|B_i|=s$ for all $i$ and
(a) $A_i\cap B_i=\emptyset$ for all $i\in[m]$,
(b) $A_i\cap B_j\neq\emptyset$ for all $i<j$.
Then $m\le \binom{r+s}{r}$.
□ Let $W$ be a vector space over a field $\F$. Let $U_1,U_2,\ldots,U_m,V_1,V_2,\ldots,V_m$ be subspaces of $W$ such that $\dim(U_i)=r$ and $\dim(V_i)=s$ for each $i=1,2,\ldots,m$ and
(a) $U_i\cap V_i=\{0\}$ for $i=1,2,\ldots,m$;
(b) $U_i\cap V_j\neq\{0\}$ for $1\le i<j\le m$.
Then $m\le \binom{r+s}{r}$.
• Füredi (1984)
□ Let $U_1,U_2,\ldots,U_m,V_1,V_2,\ldots,V_m$ be subspaces of a vector space $W$ over a field $\F$. If $\dim(U_i)=r$, $\dim(V_i)=s$ for all $i\in\{1,2,\ldots,m\}$ and for some $t\ge 0$,
(a) $\dim(U_i\cap V_i)\le t$ for all $i\in\{1,2,\ldots,m\}$,
(b) $\dim(U_i\cap V_j)>t$ for all $1\le i<j\le m$,
then $m\le \binom{r+s-2t}{r-t}$.
• Frankl and Wilson (1981)
□ The chromatic number of the unit distance graph of $\mathbb R^n$ is larger than $1.2^n$ for sufficiently large $n$.
• Kahn and Kalai (1993)
□ (Borsuk’s conjecture is false) Let $f(d)$ be the minimum number such that every set of diameter $1$ in $\mathbb R^d$ can be partitioned into $f(d)$ sets of smaller diameter. Then $f(d)> (1.2)
^{\sqrt d}$ for large $d$.
• Mehlhorn and Schmidt (1982) [New in 2017]
□ For a matrix C, $2^{\kappa(C)}\ge \operatorname{rank}(C)$. (Let $\kappa(C)$ be the minimum number of rounds in order to partition $C$ into almost homogeneous matrices, if in each round we can
split each of the current submatrices into two either vertically or horizontally. This is a parameter related to the communication complexity.)
• Lovász and Saks (1993)
• ?
□ There exists a randomized protocol to decide the equality of two strings of length $n$ using $O(\log n)$ bits.
The probablity of outputting an incorrect answer is less than $1/n$.
• Dvir (2009) [New in 2017]
□ Let $f\in \F[x_1,\ldots,x_n]$ be a polynomial of degree at most $q-1$ over a finite field $\F$ with $q=\abs{F}$ elements. Let $K$ be a Kakeya set. If $f(x)=0$ for all $x\in K$, then $f$ is a
zero polynomial.
□ If $K\subseteq\F^n$ is a Kakeya set, then \[\abs{K}\ge \binom{\abs{\F}+n-1}{n} \ge \frac{\abs{F}^n}{n!}.\]
• Ellenberg and Gijswijt (2017) [New in 2017]
□ If $A$ is a subset of $\F_3^n$ with no three-term arithmetic progression, then $\abs{A}\le 3N$ where \[N=\sum_{a,b,c\ge 0; a+b+c=n; b+2c\le 2n/3} \frac{n!}{a! b! c!}.\]Furthermore $3N\le o
• Tao (2016) [New in 2017]
□ Let $k\ge 2$ and let $A$ be a finite set and $\F$ be a field. Let $f:A^k\to \F$ be a function such that if $f(x_1,x_2,\ldots,x_k)\neq 0$, then $x_1=x_2=\cdots=x_k$. Then the slice rank of $f$
is equal to $\abs{\{x: f(x,x,\ldots,x)\neq0\}}$.
• Erdős and Rado (1960) [New in 2017]
□ There is a function $f(k,r)$ on positive integers $k$ and $r$ such that every family of $k$-sets with more than $f(k,r)$ members contains a sunflower of size $r$.
• Naslund and Sawin (2016) [New in 2017]
□ Let $\mathcal F$ be a family of subsets of $[n]$ having no sunflower of size $3$. Then \[\abs{\mathcal F}\le 3(n+1)\sum_{k\le n/3}\binom{n}{k}.\]
• Alon and Tarsi (1992)
□ Let $\F$ be a field and let $f\in \F[x_1,x_2,\ldots,x_n]$. Suppose that $\deg(f)=d=\sum_{i=1}^n d_i$ and the coefficient of $\prod_{i=1}^n x_i^{d_i}$ is nonzero. Let $L_1,L_2,\ldots,L_n$ be
subsets of $\F$ such that $|L_i|>d_i$. Then there exist $a_1\in L_1$, $a_2\in L_2$, $\ldots$, $a_n\in L_n$ such that \[ f(a_1,a_2,\ldots,a_n)\neq 0.\]
• Cauchy (1813), Davenport (1935)
□ Let $p$ be a prime and let $A,B$ be two nonempty subsets of $\mathbb{Z}_p$. Then \[|A+B|\ge \min(p,|A|+|B|-1).\]
• Dias da Silva and Hamidoune (1994). A proof by Alon, Nathanson, and Ruzsa (1995). Conjecture of Erdős and Heilbronn (1964).
□ Let $p$ be a prime and $A$ be a nonempty subset of $\mathbb{Z}_p$. Then \[ \lvert\{ a+a’: a,a’\in A, a\neq a’\}\rvert \ge \min(p,2|A|-3).\]
• Alon (2000) [New in 2017]
□ Let $p$ be an odd prime. For $k< p$ and every integers $a_1,a_2,\ldots,a_k,b_1,b_2,\ldots,b_k$, if $b_1,b_2,\ldots,b_k$ are distinct, then there exists a permutation $\pi$ of $\{1,2,\ldots,k
\}$ such that the sums $a_i+b_{\pi(i)}$ are distinct.
• Alon? [New in 2017]
□ If $A$ is an $n\times n$ matrix over a field $\F$, $\per(A)\neq 0$ and $b\in \F^n$, then for every family of sets $L_1$, $L_2$, $\ldots$, $L_n$ of size $2$ each, there is a vector $x\in L_1\
times L_2\times \cdots\times L_n$ such that \[(Ax)_i\neq b_i\] for all $i$.
• Alon? [New in 2017]
□ Let $G$ be a bipartite graph with the bipartition $A$, $B$ with $\abs{A}=\abs{B}=n$. Let $B=\{b_1,b_2,\ldots,b_n\}$. If $G$ has at least one perfect matching, then for every integer $d_1,d_2,
\ldots,d_n$, there exists a subset $X$ of $A$ such that for each $i$, the number of neighbors of $b_i$ in $X$ is not $d_i$.
• Erdős, Ginzburg, and Ziv (1961) [New in 2017]
□ Let $p$ be a prime. Every sequence $a_1,a_2,\ldots,a_{2p-1}$ of integers contains a subsequence $a_{i_1}$, $a_{i_2}$, $\ldots$, $a_{i_p}$ such that $a_{i_1}+a_{i_2}+\cdots a_{i_p}\equiv 0\
pmod p$.
• Alon, Friedland, and Kalai (1984) [New in 2017]
□ Every (multi)graph with average degree $>4$ and maximum degree $\le 5$ contains a $3$-regular subgraph.
• ?
□ Let $G$ be an undirected graph. Let $d(G)=\max_{H\subseteq G}\frac{|E(H)|}{|V(H)|}$. Then there is an orientation of $G$ such that the outdegree of each vertex is at most $\lceil d(G)\rceil$.
• Alon and Tarsi (1992)
□ A simple planar bipartite graph is $3$-choosable.
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Application of Interval Valued Fuzzy Linear Programming for Stock Portfolio Optimization
Application of Interval Valued Fuzzy Linear Programming for Stock Portfolio Optimization ()
1. Introduction
Portfolio investment is quoted securities investment, a narrow sense of investment. It refers to the behavior that an enterprise or individual buys negotiable securities such as stocks and bonds with
accumulated money to earn profits. For financial institutions, it refers to businesses that take securities as the business objects, and objectives of portfolio investment are mainly issuance and
purchase of government bonds, corporate bonds and stocks. Portfolio investment is mainly composed of three elements: income, risk and time. Risk refers to the uncertainty of the future income
situation, that is, the possibility of making a profit and a loss. It can be summed up as system risk and non-system risk. Income is value added after a period of holding a bond, equal to the ratio
between the actual income in the holding period and the bond buying price. It mainly includes income profit and capital gains. Portfolio investment has the following characteristics: 1) portfolio
investment is featured with a high degree of “market forces” [1] [2] [3] ; 2) portfolio investment is risk investment to quoted securities expected to lead income; 3) investment and speculation are
the two indispensable behaviors in portfolio investment; 4) portfolio investment in level-2 market will not increase total social capital, but redistribute between holders.
2. Algorithm for Sequencing of Fuzzy Numbers of Interval Value
2.1. Fuzzy Theory and Method
Professor L. A. Zadeh from California University of America published a famous paper in 1965 [4] . It is the first time to put forward the important concept of expressing fuzziness: membership
function, which broke through the classical set theory of Descartes in the late 19th century and laid the foundation of fuzzy theory. It mainly includes the content of fuzzy set theory, fuzzy logic,
fuzzy reasoning and fuzzy control.
Fuzzy theory is based on fuzzy set, with the basic spirit of accepting the existence of the phenomenon of fuzzy facts, and based on the research object of matters with fuzzy and uncertain concept,
and actively and rigorously quantifying it into information that the computer can handle, rather than resolving the model with complex mathematical analysis, i.e. model [5] [6] [7] .
2.2. Basic Concept of Fuzzy Sets of Interval Value
Fuzzy mathematics is a science to study and deal with fuzzy phenomenon. To resolve problems of system optimization, process control and the like in non- deterministic environment, it requires firstly
mastering the basic theory of fuzzy mathematics. There the basic concept of fuzzy sets, especially the concept of fuzzy number, operation method and basic theorem, is introduced basically.
Fuzzy set of interval value is a kind of generalized fuzzy set, the mark $\left[I\right]=\left\{\left[a,b\right]:a\le b,a,a\in \left[0,1\right]\right\}$ is introduced.
Definition 2.3.1: Set X as a nonempty set (that is, set in general), called $A:X\in \left[I\right]$ as fuzzy set of interval value on section of X.
Attention, because of $\forall a\in \left[0,1\right],a=\left[a,a\right]\in \left[I\right]$ , the fuzzy set of interval valued is generalization of fuzzy set. Generally speaking, if it is set fuzzily
in an interval value, the membership degree of an element is not a number, but a small section included in $\left[0,1\right]$ , so image of the fuzzy set is not a curve, but a belt area.
Definition 2.3.2: The closed interval $a=\left[{a}^{-},{a}^{+}\right]\text{\hspace{0.17em}}\left(0\le {a}^{-}\le {a}^{+}\le 1\right)$ in $\left[0,1\right]$ is called as a closed interval number on $\
left[0,1\right]$ , and all close interval numbers on $\left[0,1\right]$ is recorded as ${\stackrel{¯}{L}}_{1}$ .
$\stackrel{˜}{A}$ Is a subset of fuzzy number in an interval on X, $\stackrel{˜}{A}\left(•\right)$ is called as subordinating degree function of $\stackrel{˜}{A}$ , and $\stackrel{˜}{A}\left(x\right)
$ is called as degree of membership of x to fuzzy number set of interval value $\stackrel{˜}{A}$ . ${\stackrel{˜}{A}}^{+}\left(x\right)$ and ${\stackrel{˜}{A}}^{-}\left(x\right)$ are called as upper
and lower degree of membership (membership function) respectively fuzzy set of interval value, so it is determined that the fuzzy sets ${\stackrel{˜}{A}}^{+}$ and ${\stackrel{˜}{A}}^{-}$ are upper
and lower fuzzy sets of fuzzy set of interval value $\stackrel{˜}{A}$ .
Entire collection of fuzzy set of interval value on X is called as fuzzy power set of interval value. Exceptionally, when ${\stackrel{˜}{A}}^{+}$ and ${\stackrel{˜}{A}}^{-}$ degrade into common set,
the fuzzy set of interval value turns into common set of interval value.
Lemma $\stackrel{˜}{A}\subseteq \stackrel{˜}{B}⇔\forall x\in X,{\stackrel{˜}{A}}^{-}\left(x\right)\le {\stackrel{˜}{B}}^{-}\left(x\right),{\stackrel{˜}{A}}^{+}\left(x\right)\le {\stackrel{˜}{B}}^{+}\
left(x\right)$ .
2.3. Sequencing of Fuzzy Number of Interval Value
1) It is assumed that ${\mu }_{\stackrel{˜}{\alpha }}^{L}\left(x\right)$ expresses left membership function of fuzzy number $\stackrel{˜}{\alpha }=\left(\alpha ,\beta ,\gamma \right)$ , and it
continues on $\left[\alpha ,\beta \right]$ and decreases progressively strictly. It is assumed that ${g}_{\stackrel{˜}{\alpha }}^{L}\left(y\right)$ and ${g}_{\stackrel{˜}{\alpha }}^{R}\left(y\right)$
are inverse functions on ${\mu }_{\stackrel{˜}{\alpha }}^{L}\left(x\right)$ and ${\mu }_{\stackrel{˜}{\alpha }}^{R}\left(x\right)$ respectively, it is obvious that they continue on $\left[0,1\right]$
, and increases and decreases progressively strictly. Therefore, ${g}_{\stackrel{˜}{\alpha }}^{L}\left(y\right)$ and ${g}_{\stackrel{˜}{\alpha }}^{R}\left(y\right)$ are integrable respectively in
interval $\left[0,1\right]$ , recorded as follows:
${I}_{L}\left(\stackrel{˜}{\alpha }\right)={\int }_{0}^{1}{g}_{\stackrel{˜}{\alpha }}^{L}\left(y\right)\text{d}y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{R}\left(\stackrel{˜}{\alpha }\right)=
{\int }_{0}^{1}{g}_{\stackrel{˜}{\alpha }}^{R}\left(y\right)\text{d}y$
Definition 2.3.1: If $\stackrel{˜}{\alpha }$ is a fuzzy number, and the membership function definition is shown in (1), ${I}_{T}\left(\stackrel{˜}{\alpha }\right)$ is called as the entire expected
value of the fuzzy number. Including
${I}_{T}\left(\stackrel{˜}{\alpha }\right)=\frac{{I}_{L}\left(\stackrel{˜}{\alpha }\right)+{I}_{R}\left(\stackrel{˜}{\alpha }\right)}{2}$
According to above definition:
i) For $\stackrel{˜}{\alpha }=\left(\alpha ,\beta ,\gamma \right)$ , there is ${I}_{T}\left(\stackrel{˜}{\alpha }\right)=\frac{\alpha +\gamma +2\beta }{4}.$
In this way, if ${\stackrel{˜}{r}}_{i}=\left({r}_{i1},{r}_{i2},{r}_{i3}\right)$ , ${I}_{T}\left({\stackrel{˜}{r}}_{i}\right)=\frac{{r}_{i1}+{r}_{i3}+2{r}_{i2}}{4}.$
ii) As for the sum of n triangle fuzzy numbers ${\stackrel{˜}{\alpha }}_{1},{\stackrel{˜}{\alpha }}_{2},\cdots ,{\stackrel{˜}{\alpha }}_{n}$ , there is:
${I}_{T}\left(\underset{i=1}{\overset{n}{\sum }}{\stackrel{˜}{\alpha }}_{i}\right)=\underset{i=1}{\overset{n}{\sum }}{I}_{T}\left({\stackrel{˜}{\alpha }}_{i}\right).$
2) Definition 2.3.2: It is assumed that $A=\left[{A}^{-},{A}^{+}\right]$ is a fuzzy set of interval value on R and ${A}^{-}$ and ${A}^{+}$ are closed fuzzy numbers on R, and then A is a fuzzy number
of closed interval value. The collection of all fuzzy numbers of closed interval values are recorded as E.
Definition 2.3.3: It is assumed that $A\in E$ and $A=\left[{A}^{-},{A}^{+}\right]$ . If ${A}^{-}$ and ${A}^{+}$ are triangle fuzzy numbers, and then A is a fuzzy number in a triangle interval value.
The collection of all fuzzy numbers of triangle interval values is recorded as IFN.
Theorem: It is assumed that $A=\left[{A}^{-},{A}^{+}\right],B=\left[{B}^{-},{B}^{+}\right]$ are fuzzy numbers in two triangle interval values:
{0.17em}}{B}^{-}=\left({b}_{0}^{-},{b}_{1}^{-},{b}_{2}^{-}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}^{+}=\left({b}_{0}^{+},{b}_{1}^{+},{b}_{2}^{+}\right)$ ,
So $\forall k\in {R}^{+}$ , $A+B=\left[{A}^{-}+{B}^{-},{A}^{+}+{B}^{+}\right]$ ,
$A-B=\left[{A}^{-}-{B}^{-},{A}^{+}-{B}^{+}\right]$ ,
$kA=\left[k{A}^{-},k{A}^{+}\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(k\ge 0\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}kA=\left[k{A}^{+},k{A}^{-}\right]\text{\hspace{0.17em}}\
left(k\prec 0\right)$ .
Wherein ${A}^{-}±{B}^{-}=\left({a}_{0}^{-}±{b}_{0}^{-},{a}_{1}^{-}±{b}_{1}^{-},{a}_{2}^{-}±{b}_{2}^{-}\right)$
${A}^{+}±{B}^{+}=\left({a}_{0}^{+}±{b}_{0}^{+},{a}_{1}^{+}±{b}_{1}^{+},{a}_{2}^{+}±{b}_{2}^{+}\right)$ ,
$k{A}^{-}=\left(k{a}_{0}^{-},k{a}_{1}^{-},k{a}_{2}^{-}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}k{A}^{+}=\left(k{a}_{0}^{+},k{a}_{1}^{+},k{a}_{2}^{+}\right)$ .
As for triangle fuzzy numbers $a=\left({a}_{0},{a}_{1},{a}_{2}\right)$ and $b=\left({b}_{0},{b}_{1},{b}_{2}\right)$ , we apply the following two fuzzy order relations $a\le b⇔{a}_{2}\le {b}_{1}$ and
$a\le b⇔{a}_{0}+{a}_{1}+{a}_{2}\le {b}_{0}+{b}_{1}+{b}_{2}$ .
It is assumed $A=\left[{A}^{-},{A}^{+}\right],\text{\hspace{0.17em}}B=\left[{B}^{-},{B}^{+}\right]$ are fuzzy numbers of two triangle interval values and ${A}^{-}=\left({a}_{0}^{-},{a}_{1}^{-},{a}_
{+}=\left({b}_{0}^{+},{b}_{1}^{+},{b}_{2}^{+}\right)$ .
The sequencing method of fuzzy numbers of triangle interval values are shown as below:
1) $A\le B⇔{a}_{2}^{-}\le {b}_{1}^{-},{a}_{2}^{+}\le {b}_{1}^{+};$
2) $A\le B⇔{a}_{0}^{-}+{a}_{1}^{-}+{a}_{2}^{-}\le {b}_{0}^{-}+{b}_{1}^{-}+{b}_{2}^{-},\text{\hspace{0.17em}}{a}_{0}^{+}+{a}_{1}^{+}+{a}_{2}^{+}\le {b}_{0}^{+}+{b}_{1}^{+}+{b}_{2}^{+}.$
3. Stock Portfolio Investment Model Based on Fuzzy Linear Programming of Interval Value
3.1. Linear Programming
Linear programming is an important branch of operational research, which has been studied earlier, developed rapidly, used widely and matured. It is a mathematical method to assist people in
scientific management. In the management activities of enterprises, such as planning, production, transportation, technology and other issues, linear programming refers to the process of selecting
the most reasonable calculation method from various restrict combinations, establishing a linear programming model to obtain the best results [8] . The basic method to solve the linear programming
problem is simple method. The standard software of existing simplex method is available to linear programming problem with more than 10,000 of constraint conditions and decision variables on
computer. To improve the speed of solving problems, improved simplex method, original duality method, decomposition algorithm and various polynomial time algorithms are available.
1) Fuzzy linear programming
Fuzzy set theory developed in 1950s and fuzzy optimization method based on the theory provide effective methods and techniques for modeling and optimizing this kind of soft system. The study of fuzzy
optimization theory and method originated from the concept of fuzzy decision made by Bellman and Zadeh in 1970s and the decision model under the fuzzy environment. Many scholars have studied fuzzy
linear programming model, fuzzy multi-objective programming model, fuzzy integer programming model, fuzzy dynamic programming model, possible linear partitioning model and fuzzy nonlinear programming
model, and given methods knowing such models. At the same time, application of fuzzy sequencing, fuzzy set operation, sensitivity analysis, duality theory and fuzzy optimization in production
practice also become important research contents of fuzzy optimization theory and method [9] .
2) Interval number linear programming
The objective discussed by interval number linear programming problem, as a flexible mathematical programming, is that the value of the target coefficient is a closed interval, or the restraint
coefficient and objective function coefficients are interval numbers. Then according to the maximum inequality in equation and the minimum in equation, the possible interval of the objective function
solution is obtained, so as to translate interval number linear programming into classical multi-objective linear programming problem, and find an effective solution or the effective solution
interval. Similarly, when the objective function constraint conditions are all interval numbers, we can use the ordinal relation of interval numbers to measure the degree of variables to constraint
conditions, so as to gets any effective solution of interval number linear programming. With the method, the decision maker can get the expected effective solution according to estimated value of
optimization level of satisfaction level to objective function and constraint conditions.
3) Fuzzy linear programming of interval numbers
During decision-making, evaluation and other applications, because of the lack of understanding in nature of the development, people often use interval numbers to express the decision attributes, so
as to reduce the uncertainty of decision-making. Since the theory of fuzzy set was put forward, scholars began to carry out a series of researches in this direction, applying fuzzy set theory to
reasoning, signal transmission and fuzzy control, etc. [10] . In practical application, it is often not easy to determine the membership function of a fuzzy set, but it is relatively easy to
determine the membership degree of the interval value. Therefore, it provides a new way for sequencing of fuzzy numbers of interval value, which gets the common model of interval valued fuzzy linear
programming problem, and then solves corresponding problems.
3.2. Establishment of Interval Value Fuzzy Linear Programming Model for Stock Portfolio Investment
In reality, since the investment environment is quite complex, certain small changes will affect the choice of portfolio. In this case, in order to facilitate in discussing problems, the model
assumes that:
1) Investors evaluate the securities by the expected rate of return and the risk loss rate;
2) Securities are indefinite and can be divided;
3) There is no need to pay the transaction costs in the course of transaction;
4) Investors obey the assumption of non-satisfaction and assumption of avoiding risk;
5) Short selling operation is not allowed;
6) Interest rate of the bank is unchanged for the investors during the investment period.
It is supposed that investors (or asset managers) invest in n risk securities. ${r}_{i},A\left(i=1,2,\cdots ,n\right)$ is expected return rate and rate of risked return respectively, ${r}_{0}$ is
interest rate of the bank, ${x}_{i}\left(i=1,2,\cdots ,n\right)$ is the proportion of funds invested in the secondary securities, ${x}_{0}$ is the proportion of the total amount of investment in the
investment period. n + 1 vector quantity: $X=\left({x}_{0},{x}_{1},{x}_{2},\cdots ,{x}_{n}\right)$ is the expected return rate of an investment combination held by the investors, shown as follows:
$R={r}_{0}{x}_{0}+\underset{i=1}{\overset{n}{\sum }}{r}_{i}{x}_{i}$ .
If investors are rational, investors will pursue maximization of investment interest and expect the minimum risk of investment after buying some risk securities. If b is assumed as the risk
coefficient of portfolio investment, A is risk coefficient of i securities. Coefficient b reflects market risk of portfolio investment. When $\beta >1$ , risk of stock portfolio is greater than the
average market risk; when $b=1$ , risk of stock portfolio is equal to the average market risk; when $b<1$ , the risk of stock portfolio is less than the average market risk, in which case, the risk
of combination X is defined as the maximum value of various securities risks, namely
$V=\mathrm{max}\left({a}_{1}{x}_{1},{a}_{2}{x}_{2},\cdots ,{a}_{n}{x}_{n}\right)$
So we can find the effective combination according to the sum of the securities risk in the portfolio not more than a given level, and the maximum expected return rate of the combination. Its linear
programming model is as follows:
$\begin{array}{l}\mathrm{max}\text{\hspace{0.17em}}R={r}_{0}{x}_{0}+\underset{i=1}{\overset{n}{\sum }}{r}_{i}{x}_{i}\\ {P}_{1}:s.t.\left\{\begin{array}{l}Ax\le b\\ {x}_{0}+\underset{i=1}{\overset{n}
{\sum }}{x}_{i}=1\\ {x}_{x}\ge 0\text{\hspace{0.17em}}\left(i=0,1,2,\cdots ,n\right)\end{array}\end{array}$
Wherein, b is the portfolio risk amount that the investor is willing to bear, and it is the risk parameter given by the investor. When ${r}_{i},A\left(i=1,2,\cdots ,n\right)$ is a determined
parameter, the above model is a common linear programming problem. For the flexible consideration of income and risk, it is more reasonable to use the fuzzy number to describe ${r}_{i},A$ and b. In
this paper, ${r}_{i}$ is equipped with triangular fuzzy number and $A,b$ is equipped with triangular interval value fuzzy number, generally recorded as: ${\stackrel{˜}{r}}_{i}=\left({r}_{i1},{r}_
{i2},{r}_{i3}\right),A=\left({A}^{-},{A}^{+}\right),b=\left({b}^{-},{b}^{+}\right)$ , wherein, ${r}_{i1}\le {r}_{i2}\le {r}_{i3}$ , and its membership function is:
${\mu }_{{\gamma }_{i}}\left(x\right)=\left\{\begin{array}{l}0\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\prec {\gamma }_{i1}\\ \frac{x-{\gamma
}_{i1}}{{\gamma }_{i2}-{\gamma }_{i1}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\gamma }_{i1}\le x\prec {\gamma }_{i2}\\ \frac{{\gamma }_{i3}-x}{{\gamma }_{i3}-{\gamma }_{i2}}\text{ }\text
{\hspace{0.17em}}\text{\hspace{0.17em}}{\gamma }_{i2}\le x\prec {\gamma }_{i3}\\ 0\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\succ {\gamma }_
Wherein, x in ${\mu }_{{\gamma }_{i}}\left(x\right)$ is the variable of expected return rate, and the membership ${\mu }_{{\gamma }_{i}}\left(x\right)$ reflects the credibility of the expected return
rate of x as the i^th security. When $x={\gamma }_{i2}$ , membership ${\mu }_{{\gamma }_{i}}\left(x\right)=1$ , the credibility of expected return rate of ${\gamma }_{i2}$ as the i^th security
dominates; when value of x is near ${\gamma }_{i2}$ and expanded left and right, the membership ${\mu }_{{\gamma }_{i}}\left(x\right)$ decreased gradually, that is, corresponding credibility
gradually decreased; when $x\prec {\gamma }_{i1}$ or $x\succ {\gamma }_{i3}$ , the membership ${\mu }_{{\gamma }_{i}}\left(x\right)=0$ , that is, it is incredible when the expected return rate of i^
th security is less than ${\gamma }_{i1}$ or more than ${\gamma }_{i3}$ .
Finally, translate above linear programming model ${P}_{1}$ into ${P}_{2}$ :
$\begin{array}{l}\mathrm{max}\text{\hspace{0.17em}}\stackrel{˜}{R}={r}_{0}{x}_{0}+\underset{i=1}{\overset{n}{\sum }}{\stackrel{˜}{r}}_{i}{x}_{i}\\ {P}_{2}:s.t.\left\{\begin{array}{l}\stackrel{˜}{A}x\
le \stackrel{˜}{b}\\ {x}_{0}+\underset{i=1}{\overset{n}{\sum }}{x}_{i}=1\\ {x}_{x}\ge 0\text{\hspace{0.17em}}\left(i=0,1,2,\cdots ,n\right)\end{array}\end{array}$
3.3. Solution of Fuzzy Linear Programming Model of Portfolio Investment Interval Value
Definition 3.3.1: It is assumed that
Then (3.1)
Called as fuzzy linear programming of interval.
Algorithm of the solution (4.1) is shown as following:
Step 1: translate (4.1) into (4.2)
Step 2: It is assumed that
Step 3: two auxiliary models of (i) and (ii) can be gotten:
Step 4: resolve with formula (3.5) and form (3.6) with Lingo software or simplex method, then the best solution of the formula (3.5) and form (3.6) can be gotten. According to the above analysis, we
can think that the optimal solution is the optimal solution of formula 3.1.
4. Analysis of Examples
The petrochemical industry that is one of the pillar industries in our country plays a vital role in the development of the national economy. In recent years, frequent fluctuations are in oil prices,
a large number of funds are invested in the petroleum chemical industry, oil prices may be regulated upward within this year, the stock market reaction is fierce; in recent years, the housing price
keeps high, and even produces large changes in the price in one day. The real estate market is very active, occupying a certain proportion in the China GDP. Its importance is self-evident. Therefore,
it is of great significance to analyze the investment in China’s petrochemical and real estate securities markets.
In view of this, this paper selects the following 4 kinds of stocks:
S[1] CNPC (601857):the largest dominated oil and gas producer and distributor in China’s oil and gas industry, one of companies the largest sales incomes in China, and one of the largest oil
companies in the world, one of the largest state-owned enterprise groups in China.
S[2] CNPC (600028): one of the largest integrated energy and chemical companies in China, also China’s largest producer and supplier of petroleum products and main petrochemical products, and also
China’s second largest producer of crude oil.
S[3] Wanke A (000002): At present, the largest professional residential development enterprise in China, also the representative real estate blue chip stock in the stock market.
S[4] Poly Real Estate (600048): The large state-owned real estate enterprise, the operation platform of the real estate business of China Poly Group and the national first class real estate
development qualification enterprise.
To sum up, the above 4 kinds of stocks have great influence in their industry and are representative in the petroleum real estate stock. So we take this as a reference to demonstrate the validity and
reliability of the above models.
4.1. Data Preparation
The above 4 securities are selected and the following data are obtained on the basis of the annual financial reports of the companies in 2009. We consider the choice of an investor (an individual or
enterprise) in the portfolio of bank savings and the 4 stocks. The data of the expected return rate and the risk loss rate of the 4 stocks are listed as follows (Table 1, Table 2).
As For an investor (an individual or enterprise), if we assume the annual bank interest rate of current period isTable 3).
4.2. Numerical Example Solution
Now we consider how to determine the proportion of investment between savings and the 4 stocks, and make the correct portfolio investment, so that the expected return will reach the maximum value [5]
[7] when the existing risk loss rate and the risk coefficient are acceptable.
It is known from the analysis of the Chapter 4 that the following mathematical models are established firstly.
According to formula (3.5) gained based on fuzzy number sequencing of interval value, translate the formula (4.1) into the following auxiliary model:
It is known what is the optimal solution of the above linear programming with Lingo software:
The optimal value is
Then, according to formula (3.6), translate the formula (4.1) into the following auxiliary model:
It is known what is the optimal solution of above linear programming with Lingo software:
The optimal value is
4.3. In Summary
To sum up, the resolution is as follows (Table 4).
For the formula (4.2), when the bank interest rate is 0.07, the securities portfolio investment strategy chosen shall be like this: 83.15% of all capital is saved into the bank, 7.66% of all capital
is invested into security of S[1]CNPC, 9.19% is invested into security of S[3] Wanke A, so obtain the maximum expected return 8.07% on the premise of bearing risk coefficient
In the same way, for the formula (4.3), when the bank interest rate is 0.07, the securities portfolio investment strategy chosen shall be like this: 77.5% of all capital is saved into the bank, 22.5%
of all capital is invested into security of S[3][ ]
Wanke A, so obtain the maximum expected return 8.64% on the premise of no purchasing any risk security and bearing risk coefficient
It shows that when the risk coefficient that can be borne by investors’ change, the expected return is variable keeps changed on premise of fuzzy expected return rate and risk loss rate. That is to
say, investors can choose the risk coefficient that they can bear to reach the maximum value of the expected income.
Of course, such judgment is not blind. We can see that when the interval value of b is large, the combined income is also large, so the portfolio risk is larger. When the interval value of b is
small, the combined risk and portfolio risk are relatively small. That is, the greater the risk coefficient, the greater the income, the smaller the risk coefficient and the smaller the income.
Investors can determine their own portfolios according to their own conditions in order to meet their own interests [11] .
5. Conclusions
In this paper, we focus on fuzzy linear programming, interval number linear programming and interval valued fuzzy linear programming based on computational mathematics, such as model analysis,
algorithm analysis and numerical calculation. The main contents are as follows:
1) By using the definition and operation rules of fuzzy numbers of interval value, we find out the sequencing problem of fuzzy numbers of interval value, and apply it to the establishment of interval
valued fuzzy linear programming model.
2) As for
3) We collect specific data materials of expected return rate, etc. of 4 representative securities in petrochemical engineering and real estate industry, according to the established model, resolve
the portfolio investment conditions of investors between savings and the 4 kinds of securities with software programming, and verify reliability and validity of the model.
The result shows that due to uncertainty in various correlation coefficients in securities investment, it makes investment portfolio more flexible, more close to reality and more practice to describe
expected return rate, risk loss rate, affordable risk coefficient and other factors influencing investment effect, which plays a good guiding role for institutional organizations or individual | {"url":"https://scirp.org/journal/paperinformation?paperid=82566","timestamp":"2024-11-14T02:01:14Z","content_type":"application/xhtml+xml","content_length":"179455","record_id":"<urn:uuid:e1539ca0-5899-4d32-ba9c-c99309e0312e>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00804.warc.gz"} |
Differential equation
Equation XX-2 Differential Equation
It is not possible to integrate this equation but by looking at low and high concentrations we can get some idea of the plasma concentration versus time curve.
Low Cp approximation to first order
At low concentrations, where Km > Cp, Km + Cp is approximately equal to Km
where the Vm/Km is a constant term and the whole equation now looks like that for first order elimination, with Vm/Km a first order elimination rate constant.
Therefore at low plasma concentrations we would expect first order kinetics. Remember, this is the usual situation for most drugs. That is Km is usually larger than the plasma concentrations that are
High Cp approximation to zero order
For some drugs, higher concentrations are achieved, that is Cp > Km, then Km + Cp is approximately equal to Cp.
and we now have zero order elimination of drug. At high plasma concentrations we have zero order or concentration independent kinetics.
Figure XX-1 Linear Plot of Cp Versus Time Showing High Cp and Low Cp - Zero Order and First Order Elimination
From the plot.
High Cp, in the zero order part, the slope is fairly constant (straight line on linear graph paper) but steeper, that is, the rate of elimination is faster than at lower concentrations. [However, the
apparent rate constant is lower. This is easier to see on the semi-log graph.]
At higher concentrations the slope = -Vm. At lower concentrations we see an exponential decline in plasma concentration such as with first order elimination.
On semi-log graph paper we can see that in the zero order region the slope is more shallow, thus the rate constant is lower. The straight line at lower concentrations is indicative of first order
Figure XX-2 Semi-Log Plot of Cp Versus Time Showing High Cp and Low Cp
The presence of saturation kinetics can be quite important when high doses of certain drugs are given, or in case of over-dose. In the case of high dose administration the effective elimination rate
constant is reduced and the drug will accumulate excessively if saturation kinetics are not understood.
Figure XX-3 Linear Plot of
Phenytoin is an example of a drug which commonly has a Km value within or below the therapeutic range. The average Km value about 4 mg/L. The normally effective plasma concentrations for phenytoin
are between 10 and 20 mg/L. Therefore it is quite possible for patients to be overdosed due to drug accumulation. At low concentration the apparent half-life is about 12 hours, whereas at higher
concentration it may well be much greater than 24 hours. Dosing every 12 hours, the normal half-life, can rapidly lead to dangerous accumulation. At concentrations above 20 mg/L elimination maybe
very slow in some patients. Dropping for example from 25 to 23 mg/L in 24 hours, whereas normally you would expect it to drop from 25 -> 12.5 -> 6 mg/L in 24 hours. Typical Vm values are 300 to 700
mg/day. These are the maximum amounts of drug which can be eliminated by these patients per day. Giving doses approaching these values or higher would cause dangerous accumulation of drug. Figure 90
is a plot of
This page was last modified: 12 February 2001
Copyright 2001 David W.A. Bourne | {"url":"https://www.boomer.org/c/p1/Ch20/Ch2003.html","timestamp":"2024-11-10T22:01:12Z","content_type":"text/html","content_length":"7323","record_id":"<urn:uuid:4b57cd7a-b085-43f5-94c1-097754b1ad02>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00611.warc.gz"} |
A14011. Average Family Income (In 2017 Inflation Adjusted Dollars) [1] - Social Explorer Tables: ACS 2017 (5-Year Estimates) (SE) - ACS 2017 (5-Year Estimates) - Social Explorer
Mean income is the amount obtained by dividing the aggregate income of a particular statistical universe by the number of units in that universe. For example, mean household income is obtained by
dividing total household income by the total number of households. (The aggregate used to calculate mean income is rounded. For more information, see "Aggregate income.")
For the various types of income, the means are based on households having those types of income. For household income and family income, the mean is based on the distribution of the total number of
households and families including those with no income. The mean income for individuals is based on individuals 15 years old and over with income. Mean income is rounded to the nearest whole dollar.
Care should be exercised in using and interpreting mean income values for small subgroups of the population. Because the mean is influenced strongly by extreme values in the distribution, it is
especially susceptible to the effects of sampling variability, misreporting, and processing errors. The median, which is not affected by extreme values, is, therefore, a better measure than the mean
when the population base is small. The mean, nevertheless, is shown in some data products for most small subgroups because, when weighted according to the number of cases, the means can be computed
for areas and groups other than those shown in Census Bureau tabulations. (For more information on means, see "Derived Measures.") | {"url":"https://www.socialexplorer.com/data/ACS2017_5yr/metadata/?ds=SE&table=A14011","timestamp":"2024-11-07T04:14:22Z","content_type":"text/html","content_length":"40603","record_id":"<urn:uuid:2d3e44b6-b27a-4ac1-9b6f-70f114d0555e>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00499.warc.gz"} |
• Fluency with addition, subtraction, multiplication and division of whole numbers
and fractions.
• Ability to apply the any-order principle for multiplication and addition (commutative law and associative law) for whole numbers and fractions.
• Familiarity with the order of operation conventions for whole numbers.
Algebra is a fascinating and essential part of mathematics. It provides the written language in which mathematical ideas are described.
Many parts of mathematics are initiated by finding patterns and relating to different quantities. Before the introduction and development of algebra, these patterns and relationships had to be
expressed in words. As these patterns and relationships became more complicated, their verbal descriptions became harder and harder to understand. Our modern algebraic notation greatly simplifies
this task.
A well-known formula, due to Einstein, states that E = mc2. This remarkable formula gives the relationship between energy, represented by the letter E, and mass, represented by letter m. The letter c
represents the speed of light, a constant, which is about 300 000 000 metres per second. The simple algebraic statement E = mc2 states that some matter is converted into energy (such as happens in a
nuclear reaction), then the amount of energy produced is equal to the mass of the matter multiplied by the square of the speed of light. You can see how compact the formula is compared with the
verbal description.
We know from arithmetic that 3 × 6 + 2 × 6 = 5 × 6. We could replace the number 6 in this statement by any other number we like and so we could write down infinitely many such statements. All of
these can be captured by the algebraic statement 3x + 2x = 5x, for any number x. Thus algebra enables us to write down general statements clearly and concisely.
The development of mathematics was significantly restricted before the 17th century by the lack of efficient algebraic language and symbolism. How this notation evolved will be discussed in the
History section.
In algebra we are doing arithmetic with just one new feature − we use letters to represent numbers. Because the letters are simply stand-ins for numbers, arithmetic is carried out exactly as it is
with numbers. In particular the laws of arithmetic (commutative, associative and distributive) hold.
For example, the identities
2 + x = x + 2 2 × x = x × 2
(2 + x) + y = 2 + (x + y) (2 × x) × y = 2 × (x × y)
6(3x + 1) = 18x + 6
hold when x and y are any numbers at all.
In this module we will use the word pronumeral for the letters used in algebra. We choose to use this word in school mathematics because of confusion that can arise from the words such as ‘variable’.
For example, in the formula E = mc2, the pronumerals E and m are variables whereas c is a constant.
Pronumerals are used in many different ways. For example:
• Substitution: ‘Find the value of 2x + 3 if x = 4.’ In this case the pronumeral is given the value 4.
• Solving an equation: ‘Find x if 2x + 3 = 8.’ Here we are seeking the value of the pronumeral that makes the sentence true.
• Identity: ‘The statement of the commutative law: a + b = b + a.’ Here a and b can be any real numbers.
• Formula: ‘The area of a rectangle is A = lw.‘ Here the values of the pronumerals are connected by the formula.
• Equation of a line or curve: ‘The general equation of the straight line is y = mx + c.’
Here m and c are parameters. That is, for a particular straight line, m and c are fixed.
In some languages other than English, one distinguishes between ‘variables’ in functions and ‘unknown quantities’ in equations (‘incógnita’ in Portuguese/Spanish, ‘inconnue’ in French) but this does
not completely clarify the situation. The terms such as variable and parameter cannot be precisely defined at this stage and are best left to be introduced later in the development of algebra.
An algebraic expression is an expression involving numbers, parentheses, operation signs and pronumerals that becomes a number when numbers are substituted for the pronumerals. For example 2x + 5 is
an expression but +) × is not.
Examples of algebraic expressions are:
3x + 1 and 5(x2 + 3x)
As discussed later in this module the multiplication sign is omitted between letters and between a number and a letter. Thus substituting x = 2 gives
3x + 1 = 3 × 2 + 1 = 7 x2 + 3x) = 5(22 + 3 × 2) = 30.
In this module, the emphasis is on expressions, and on the connection to the arithmetic that students have already met with whole numbers and fractions. The values of the pronumerals will therefore
be restricted to the whole numbers and non-negative fractions.
Changing words to algebra
In algebra, pronumerals are used to stand for numbers. For example, if a box contains x stones and you put in five more stones, then there are x + 5 stones in the box. You may or may not know what
the value of x is (although in this example we do know that x is a whole number).
• Joe has a pencil case that contains an unknown number of pencils. He has three
other pencils. Let x be the number of pencils in the pencil case. Then Joe has x + 3 pencils altogether.
• Theresa has a box with least 5 pencils in it, and 5 are removed. We do not know how many pencils there are in the pencil case, so let z be the number of pencils in the box. Then there are z − 5
pencils left in the box.
• There are three boxes, each containing x marbles in each box, then the total number of marbles is 3 × x = 3x.
• If n oranges are to be divided amongst 5 people, then each person receives oranges. (Here we assume that n is a whole number. If each person is to get a whole number of oranges, then n must be a
multiple of 5.)
The following table gives us some meanings of some commonly occurring
algebraic expressions.
x + 3 • The sum of x and 3
• 3 added to x, or x is added to 3
• 3 more than x, or x more than 3
x − 3 • The difference of x and 3 (where x ≥ 3)
• 3 is subtracted from x
• 3 less than x
• x minus 3
3 × x • The product of 3 and x
• x is multiplied by 3, or 3 is multiplied by x
x ÷ 3 • x divided by 3
• the quotient when x is divided by 3
2 × x − 3 • x is first multiplied by 2, then 3 is subtracted
x ÷ 3 + 2 • x is first divided by 3, then 2 is added
Expressions with zeroes and ones
Zeroes and ones can often be eliminated entirely. For example:
+ 0 = x
1 × x = x
Algebraic notation
In algebra there are conventional ways of writing multiplication, division and indices.
Notation for multiplication
The × sign between two pronumerals or between a pronumeral and a number is usually omitted. For example, 3 × x is written as 3x and a × b is written as ab. We have been following this convention
earlier in this module.
It is conventional to write the number first. That is, the expression 3 × a is written as 3a and not as a3.
Notation for division
The division sign ÷ is rarely used in algebra. Instead the alternative fraction notation for division is used. We recall 24 ÷ 6 can be written as . Using this notation, x divided by 5
is written as , not as x ÷ 5.
Index notation
x × x is written as x2 and y × y × y is written as y3.
Write each of the following using concise algebraic notation.
A number x is multiplied by itself and then doubled.
A number x is squared and then multiplied by the square of a second number y.
A number x is multiplied by a number y and the result is squared.
ax × x × 2 = x2 × 2 = 2x2. b x2 × y2 = x2y2.
c (x × y)2 = (xy)2 which is equal to x2y2
• 2 × x is written as 2x • x1 = x (the first power of x is x)
• x × y is written as xy or yx • xo = 1
• x × y × z is written as xyz • 1x = x
• x × x is written as x2 • 0x = 0
• 4 × x × x × x = 4x3
• x ÷ 3 is written as
• x ÷ (yz) is written as
Assigning values to a pronumeral is called substitution.
If x = 4, what is the value of:
a x x + 3 cx − 1 d
a 5x = 5 × 4 = 20 x + 3 = 4 + 3 = 7c x − 1 = 4 − 1 = 3d = = 2
If x = 6, what is the value of:
a 3x + 4 b 2x + 3 c 2x − 5
d − 2 e + 2
a 3x + 4 = 3 × 6 + 4 = 22 b 2x + 3 = 2 × 6 + 3 = 15
c 2x − 5 = 2 × 6 − 5 = 7 d − 2 = − 2 = 1
e + 2 = + 2 = 4
Adding and subtracting like terms
Like terms
If you have 3 pencil case with the same number x of pencils in each, you have 3x pencils altogether.
If there are 2 more pencil cases with x pencils in each, then you have 3x + 2x = 5x pencils altogether. This can be done as the number of pencils in each case is the same. The terms 3x and 2x are
said to be like terms.
Consider another example. If Jane has x packets of chocolates each containing y chocolates, then she has x × y = xy chocolates.
If David has twice as many chocolates as Jane, he has 2 × xy = 2xy chocolates.
Together they have 2xy + xy = 3xy chocolates.
The terms 2xy and xy are like terms. Two terms are called like terms if they involve exactly the same pronumerals and each pronumeral has the same index.
Which of the following pairs consists of like terms:
a 3x, 5x 4x2, 8x 4x2y, 12x2y
5x and 3x are like terms.
4x2 and 8x are not like terms since the powers of x are different.
4x2y, 12x2y are like terms
Adding and subtracting like terms
The distributive law explains the addition and subtraction of like terms. For example:
xy + xy = 2 × xy + 1 × xy =(2 + 1)xy = 3xy
The terms 2x and 3y are not like terms because the pronumerals are different. The terms 3x and 3x2 are not like terms because the indices are different. For the sum 6x + 2y + 3x, the terms 6x and 3x
are like terms and can be added. There are no like terms for 2y, so by using the commutative law for addition the sum is
x + 2y + 3x = 6x + 3x + 2y = 9x + 2y.
The any-order principle for addition is used for the adding like terms.
Because of the commutative law and the associative law for multiplication (any-order principle for multiplication) the order of the factors in each term does not matter.
x × 3y = 15xy = 15yx
ab × 3b2a = 18a2b3 = 18b3a2
Like terms can be added and subtracted as shown in the example below.
Simplify each of the following by adding or subtracting like terms:
a 2x + 3x + 5x b 3xy + 2xy c 4x2 − 3x2
d 2x2 + 3x + 4x e 4x2y − 3x2y + 3xy2
a 2x + 3x + 5x = 10x b 3xy + 2xy = 5xy
c 4x2 − 3x2 = x2 d 2x2 + 3x + 4x = 2x2 + 7x
e 4x2y − 3x2y + 3xy2 = x2y + 3xy2
Brackets fulfill the same role in algebra as they do in arithmetic. Brackets are used in algebra in the following way.
Let x be the number. Then the expression is (x + 6) × 3. We now follow the convention that the number is written at the front of the expression (x + 6) without a multiplication sign. The preferred
form is thus 3(x + 6).
For a party, the host prepared 6 tins of chocolate balls, each containing n chocolate balls.
The host places two more chocolates in each tin. How many chocolates are there altogether in the tins now?
If n = 12, that is, if there were originally 12 chocolates in each box, how many chocolates are there altogether in the tins now?
The number of chocolates in each tin is now n + 2. There are 6 tins, and therefore there are 6 (n + 2) chocolates in total.
If n = 12, the total number of chocolates is 6 (n + 2) = 6 (12 + 2) = 6 × 14 = 84.
Each crate of bananas contains n bananas. Three bananas are removed from each crate.
How many bananas are now in each crate?
If there are 7 crates, how many bananas are there now in total?
Five extra seats are added to each row of seats in a theatre. There were s seats in each row and there are 20 rows of seats. How many seats are there now in total?
Use of brackets and powers
The following example shows the importance of following the conventions of order of operations when working with powers and brackets.
• 2x2 means 2 × x2
• (2x)2 = 2x × 2x = 4x2
For x = 3, evaluate each of the following:
a x2 b x)2
ax2 = 2 × x × x = 2 × 3 × 3 = 18 bx)2 = (2 × x)2 = (2 × 3)2 = 36
Evaluate each expression for x = 3.
a 10x3 (10x)3
Multiplying algebraic terms involves the any-order property of multiplication discussed for whole numbers.
The following shows how the any-order property can be used
3x × 2y × 2xy = 3 × x × 2 × y × 2 × x × y
= 3 × 2 × 2 × x × x × y × y
= 12x2y2
Simplify each of the following:
a 5 × 2a b 3a × 2a c 5xy × 2xy
5 × 2a = 10a
3a × 2a = 3 × a × 2 × a = 6a2
5xy × 2xy = 5 × x × y × 2 × x × y = 10x2y2
With practice no middle steps are necessary.
For example, 2a × 3a × 2a2 = 12a4
Arrays of dots have been used to represent products in the module, Multiplication of Whole Numbers.
For example 2 × 6 = 12 can be represented by 2 rows of 6 dots.
The diagram below represents two rows with some number of dots.
Let n be the number of dots in each row. Then there are 2 × n = 2n dots.
If an array is m dots by n dots then there are mn dots. If the array is m × n then by convention we have m rows and n columns. We can represent the product m × n = mn by such an array.
The pattern goes on forever.
How many dots are there in the nth diagram?
In the 1st diagram there are 1 × 1 = 12 dots.
In the 2nd diagram there are 2 × 2 = 22 dots.
In the 3rd diagram there are 3 × 3 = 32 dots.
In the nth diagram there will be n × n = n2 dots.
We can also represent a product like 3 × 4 by a rectangle.
cm by 4 cm rectangle is 12 cm2.
The area of a x cm by y cm rectangle is x × y = xy cm2.
(x and y can be any positive numbers.)
The area of a x cm by x cm square is x2 cm2.
(x can be any positive number.)
Find the total area of the two rectangles in terms of x and y.
The area of the rectangle to the left is xy cm2 and the area of the rectangle to the right is 2xy cm2. Hence the total area is xy + 2xy = 3xy cm2.
Some simple statements with numbers demonstrate the convenience of algebra.
The nth positive even number is 2n.
What is the square of the nth positive even number?
If the nth positive even number is doubled what is the result?
The square of the nth positive even number is (2n)2 = 4n2
The double of the nth positive even number is 2 × 2n = 4n.
a If b is even, what is the next even number?
b If a is a multiple of three, what are the next two multiples of 3?
c If n is odd and n ≥ 3, what is the previous odd number?
Think of a ‘number’. Let this number be x.
Write the following using algebra to see what you get.
• Multiply the number you thought of by 2 and subtract 5.
• Multiply the result by 3.
• Add 15.
• Subtract 5 times the number you first thought of.
Show that the sum of the first n odd numbers is n2.
Quotients of expressions involving pronumerals often occur. We call them algebraic fractions we will meet this again in the modules, Special Expansions and Algebraic Fractions.
Write each of the following in algebraic notation.
a A number is divided by 5, and 6 is added to the result.
b Five is added to a number, and the result is divided by 3.
a Let x be the number.
Dividing by 5 gives .
Adding 6 to this result gives + 6.
b Let the number be x.
Adding 5 gives x + 5.
Dividing this by 3 gives .
Notice that the vinculum acts as a bracket.
If x = 10, find the value:
a b + 3 c d
a = b + 3 = + 3
= 5 = 5
c = d =
= =
= 4 =
A vat contains n litres of oil. Forty litres of oil are then added to the vat.
a How many litres of oil are there now in the vat?
b The oil is divided into 50 containers. How much oil is there in each container?
a There is a total of n + 40 litres of oil in the vat.
b There are litres of oil in each container.
A shed contains n tonnes of coal. An extra 1000 tonnes are then added.
How many tonnes of coal are there in the shed now?
It is decided to ship the coal in 10 equal loads. How many tonnes of coal are there in each load?
Expanding brackets and collecting like terms
Expanding brackets
Numbers obey the distributive laws for multiplication over addition and subtraction. For example:
The distributive laws for division over addition and subtraction also hold as shown.
For example:
= −
As with adding like terms and multiplying terms, the laws that apply to arithmetic can be extended to algebra. This process of rewriting an expression to remove brackets is usually referred to as
expanding brackets.
Use the distributive law to rewrite these expressions without brackets.
a 5(x − 4) b 4(3x + 2) c 6(4 − 2x)
a 5(x − 4) = 5x − 20 b 4(3x + 2) = 12x + 8
c 6(4 − 2x) =24 − 12x
Use the distributive law to rewrite these expressions without brackets.
Collecting like terms
After brackets have been expanded like terms can be collected.
Expand the brackets and collect like terms:
a 2(x − 6) + 5x 3 + 3(x − 1) 3(2x + 4) + 6(x − 1)
a 2(x − 6) + 5x = 2x − 12 + 5x = 7x − 12 3 + 3(x − 1) = 3 + 3x − 3 = 3x
c 3(2x + 4) + 6(x − 1) = 6x + 12 + 6x − 6 = 12x + 6
Expand the brackets and collect like terms:
a 5(x + 2) + 2(x − 3) b 2(7 + 5x) + 4(x + 6)
c 3(2x + 7) + 2(x − 5)
A sound understanding of algebra is essential for virtually all areas of mathematics.
The introduction to algebra is continued in the modules, Negatives and the Index Laws in Algebra, Special Expansions and Algebraic Fractions and Fractions and the Index Laws in Algebra.
It was only in the 17th century that algebraic notation similar to that used today was introduced. For example, the notation used by Descartes (La Geometrie, 1637) and Wallis (1693) was very close to
modern notation. However, algebra has a very long history.
There are examples of the ancient Egyptians working with algebra. About 1650 BC, the Egyptian scribe Ahmes made a transcript of even more ancient mathematical scriptures dating to the reign of the
Pharaoh Amenemhat III. In 1858 the Scottish antiquarian, Henry Rhind, came into possession of Ahmes's papyrus. The papyrus is a scroll 33 cm wide and about 5.25m long filled with mathematical
problems. One of the problems is as follows:
100 measures of corn must be divided among 5 workers, so that the second worker gets as many measures more than the first worker, as the third gets more than the second, as the fourth gets more than
the third, and as the fifth gets more than the fourth. The first two workers shall get seven times fewer measures of corn than the three others. How many measures of corn shall each worker get? (The
answer involves fractional measures of corn. Answer: 1 , 10 , 20, 29 and 38 measures.)
Euclid (circa 300 BC) dealt with algebra in a geometric way and algebraic problems are solved without using algebraic notation of any form. Diophantus (circa 275 AD ) who is sometimes called the
father of algebra, produced a work largely devoted to the ideas of the subject.
Chinese and Indian authors wrote extensively on algebraic ideas and achieved a great deal in the solution of equations. The earliest Chinese text with algebraic ideas is The Nine Chapters on the
Mathematical Art, written around 250 BC. A later text is Shu-shu chiu-chang, or Mathematical Treatise in Nine Sections, which was written by the wealthy governor and minister Ch'in Chiu-shao (circa
1202 − circa 1261 AD).
In approximately 800 BC an Indian mathematician Baudhayana, in his Baudhayana Sulba Sutra, discovered Pythagorean triples algebraically, and found geometric solutions of linear equations and
quadratic equations of the forms, ax2 = c and of ax2 + bx = c.
In 680 AD the Indian mathematician Brahmagupta, in his treatise Brahma Sputa Siddhanta, worked with quadratic equations and determined rules for solving linear and quadratic equations. He discovered
that quadratic equations can have two roots, including both negative and irrational roots.
Indian mathematician Aryabhata, in his treatise Aryabhatiya, obtains whole-number solutions to linear equations by a method equivalent to the modern one.
Arab and Persian mathematicians had an interest in algebra and their ideas flowed into Europe. One algebraist of special prominence was al-Khwarizmi, whose al-jabr w’al muqabalah (circa 825 AD) gave
the discipline its name and gave the first systematic treatment of algebra.
The works of al-Khwarizmi were translated into European languages by several different translators during the twelfth century and so his and other Arab writers’ work were well known in Europe.
Fibonacci was the greatest European writer on algebra during the middle ages and his work Liber Quadratorum (circa 1225 AD) includes many different ingenious ways of solving equations.
Cardano, Tartaglia (16th century), Vieta (16th − 17th century) and others developed the ideas and notation of algebra. The Ars Magna (Latin: “The Great Art”) is an important book on Algebra written
by Gerolamo Cardano. It was first published in 1545 under the title Artis Magnæ, Sive de Regulis Algebraicis Liber Unus (Book number one about The Great Art, or The Rules of Algebra). There was a
second edition in Cardano’s lifetime, published in 1570. It is considered one of the three greatest scientific treatises of the Renaissance. The book included the solutions to the cubic and quartic
equations. The solution to one particular case of the cubic, x3 + ax = b (in modern notation), was communicated to him by Niccolò Fontana Tartaglia, and the quartic was solved by Cardano’s student
Lodovico Ferrari.
Development of algebraic notation
Here are some of the different notations used from the Middle Ages onwards together with their modern form. They reveal how useful modern algebraic notation is.
Trouame.1.n0.che gito al suo drat0 facia.12
x + x2 = 12 − Pacioli (1494)
4 Se. −51 Pri. −30 N. dit is ghelijc 45
4x4 − 51x − 30 = 45 − Vander Hoecke (1514)
cub p: 6 reb ælis 20.
x3 + 6x = 20 − Cardano (1545)
1 Q C −15 Q Q + 85 C −225 Q + 274 N æquator 120.
x6 − 15x4 + 85x3 − 225x2 + 274x = 120 − Vieta (c. 1590)
aaa − 3.bba ======= + 2.ccc
x3 − 3b2x = 2c3. − Harriot (1631)
A History of Mathematics: An Introduction, 3rd Edition, Victor J. Katz, Addison-Wesley, (2008)
Exercise 1
a n − 3 bananas in each crate 7(n − 3) bananas in total
Exercise 2
a 20(s + 5) seats in total
Exercise 3
a 270 27 000
Exercise 4
a b + 2 a + 3, a + 6 c n − 2
Exercise 5
x 2x − 5 3(2x − 5) = 6x − 15 6x 6x − 5x = x
Exercise 6
Sum of the first n odd numbers is: 1 + 3 + 5 + …. + (2n − 5) + (2n − 3) + (2n − 1)
Reverse the sum: (2n − 1) + (2n − 3) + (2n − 5) + … + 5 + 3 + 1
Add to the original sum pairing terms yields
(2n − 1 + 1) + (2n − 3 + 3) + … + (3 + 2n − 3) + (1 + 2n − 1) = 2n × n =2n2
The sum is n2
Exercise 7
a n + 1000 tonnes tonnes
Exercise 8
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Compound annual interest rate of 6 percent
18 Sep 2019 Compound interest is the numerical value that is calculated on the initial ( Where P = Principal, i = nominal annual interest rate in percentage terms, As an example, an investment that
has a 6% annual rate of return will 13 Nov 2019 Compound Annual Growth Rate (CAGR). Real-life interest=P[(1+i)n−1]where:P =Principlei=interest rate in percentage For example, an investment that has
a 6 % annual rate of return will double in 12 years (72 / 6%).
22 Aug 2019 The Annual Percentage Rate (APR) is a calculation of the overall cost quotes an interest rate of 4% per year compounded every 6 months the SA's Best Investment Rate at 13.33%* on Fixed
Deposit Investment. Guranteed Returns Receive your interest payouts monthly, every 6 or 12 months, or at maturity. *Based on Nominal Annual Compounding Annually (NACA) Interest Rate. frequencies of
compounding, the effective rate of interest and rate of discount, and to time t, for a principal of 1 unit, is r × t, where r is the constant of proportion Although the rate of interest is often
quoted in annual term, the interest accrued to 6. CHAPTER 1. Thus, the accumulated amount at time t + 1 m is a. ( t +. 1 m. ). 11 Jun 2018 Annual percentage rate, or APR, is one you should definitely
Most credit cards and revolving lines of credit use compound interest. APY stands for Annual Percentage Yield, which is a formula used to compare stated interest rates that have different compounding
periods. For example, if one Simple interest, often called the nominal annual percentage rate (APR), If a savings account paid a nominal interest rate of 6%, that was compounded
Compound Interest Calculator. Enter the Amount of Money To Start With: $ Enter The Interest Rate (ex. for 5%, enter 5): How Often Will It Be Compounded? How Many Years Will It Compound For? -- Your
result will display here -- What Would $1 Be Worth If What Would $1,000 Be Worth At An Annual 7% Interest Rate After 35 Years?--
Now we can choose different values, such as an interest rate of 6%: When interest is compounded within the year, the Effective Annual Rate is higher than the The annual percentage rate (APR) that
you are charged on a loan may not be the amount of interest you actually pay. The amount of interest In this video, we calculate the effective APR based on compounding the APR daily. 6 years ago. A
bank deposit paying simple interest at the rate of 6%/year grew to a sum of Eff(annual interest rate as a percentage, the number of compounding periods per You should check with your financial
institution to find out how often interest is being compounded on your particular investment. Yearly APY. Annual percentage
Compound interest is the concept of earning interest on your investment, then a long term savings account offering a rate of 4.2% effective annual interest rate ( eAPR). After 6 years, his deposits
total $4,320, and the interest paid only $869.
SA's Best Investment Rate at 13.33%* on Fixed Deposit Investment. Guranteed Returns Receive your interest payouts monthly, every 6 or 12 months, or at maturity. *Based on Nominal Annual Compounding
Annually (NACA) Interest Rate. frequencies of compounding, the effective rate of interest and rate of discount, and to time t, for a principal of 1 unit, is r × t, where r is the constant of
proportion Although the rate of interest is often quoted in annual term, the interest accrued to 6. CHAPTER 1. Thus, the accumulated amount at time t + 1 m is a. ( t +. 1 m. ).
13 Nov 2019 Compound Annual Growth Rate (CAGR). Real-life interest=P[(1+i)n−1]where:P =Principlei=interest rate in percentage For example, an investment that has a 6 % annual rate of return will
double in 12 years (72 / 6%).
where r is the annual interest rate and t is the number of years. Sometimes interest is compounded more often than annually, For example, if 6% interest is 12 Feb 2019 For example, if interest
compounds monthly, after the first month the For example, if a bank quotes you a 6 percent annual percentage rate, compound interest (CI) calculator - formulas & solved example problems to 1. to
calculate how much CI payable based on the yearly compounding frequency. compounding period or frequency and the interest rate R in percentage are the at 6% rate of interest for the total period of 5
years with quarterly compounding 22 Aug 2019 The Annual Percentage Rate (APR) is a calculation of the overall cost quotes an interest rate of 4% per year compounded every 6 months the SA's Best
Investment Rate at 13.33%* on Fixed Deposit Investment. Guranteed Returns Receive your interest payouts monthly, every 6 or 12 months, or at maturity. *Based on Nominal Annual Compounding Annually
(NACA) Interest Rate. frequencies of compounding, the effective rate of interest and rate of discount, and to time t, for a principal of 1 unit, is r × t, where r is the constant of proportion
Although the rate of interest is often quoted in annual term, the interest accrued to 6. CHAPTER 1. Thus, the accumulated amount at time t + 1 m is a. ( t +. 1 m. ). 11 Jun 2018 Annual percentage
rate, or APR, is one you should definitely Most credit cards and revolving lines of credit use compound interest.
11 Jun 2018 Annual percentage rate, or APR, is one you should definitely Most credit cards and revolving lines of credit use compound interest.
For example, if instead of a 6 percent annual percentage rate the bank quotes a 6 percent annual percentage yield, then first divide by 100 to get 0.06. Second, add 1 to 0.06 to get 1.06. Third,
raise 1.06 to the 1/12th power to get 1.004867551. Fourth, subtract 1 to find that the monthly interest rate as a decimal is 0.004867551. Compound interest formula. Compound interest, or 'interest on
interest', is calculated with the compound interest formula. Multiply the principal amount by one plus the annual interest rate to the power of the number of compound periods to get a combined figure
for principal and compound interest.
where r is the annual interest rate and t is the number of years. Sometimes interest is compounded more often than annually, For example, if 6% interest is 12 Feb 2019 For example, if interest
compounds monthly, after the first month the For example, if a bank quotes you a 6 percent annual percentage rate, | {"url":"https://brokerehcpsq.netlify.app/hanning73048luw/compound-annual-interest-rate-of-6-percent-ke","timestamp":"2024-11-09T09:28:06Z","content_type":"text/html","content_length":"34086","record_id":"<urn:uuid:202ed09c-3013-46d4-8e33-0379a2a8b52f>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00535.warc.gz"} |
(ii) The dipole is aligned parallel to the field. Find the work in rotating it through the angle of ${{180}^{0}}$.
Hint: The first part of the problem can be solved by using the formula for the force on a charge in an electric field. A dipole has two charges of equal magnitude and opposite signs. The second part
of the problem can also be solved by using the direct formula for the external work required to rotate a dipole in an electric field between two angles (that the dipole axis makes with the direction
of the electric field).
Formula used:
$W=pE\left( \cos {{\theta }_{1}}-\cos {{\theta }_{2}} \right)$
Complete step by step answer:
(i) A dipole consists of two charges of equal magnitude but opposite signs. When a dipole is placed in an electric field, the total force on it is the sum of the force on each charge due to the
electric field.
The force $\overrightarrow{F}$ on a charge $q$ placed in an electric field $\overrightarrow{E}$ is given by
$\overrightarrow{F}=q\overrightarrow{E}$ --(1)
We will consider a dipole with charges $+q$ and $-q$ placed in an electric field $\overrightarrow{E}$.
Now, using (1), we get the force $\overrightarrow{{{F}_{p}}}$ on charge $+q$ due to the electric field as
$\overrightarrow{{{F}_{p}}}=+q\overrightarrow{E}$ --(2)
Similarly, using (1), we get the force $\overrightarrow{{{F}_{n}}}$ on charge $-q$ due to the electric field as
$\overrightarrow{{{F}_{n}}}=-q\overrightarrow{E}$ --(3)
As explained above, the total force on the dipole due to the electric field will be the sum of the forces on the two charges. Hence, we get the total force $\overrightarrow{F}$ on the dipole placed
in the electric field as
Using (2) and (3), we get,
$\overrightarrow{F}=+q\overrightarrow{E}+\left( -q\overrightarrow{E} \right)=+q\overrightarrow{E}-q\overrightarrow{E}=0$
Hence, the net force on a dipole placed in an electric field is zero.
(ii) Now, when a dipole is placed in an electric field, external work is required to rotate it between two angles. The angles refer to the angle made by the dipole axis with the direction of the
electric field.
The external work $W$ required to rotate a dipole of dipole moment of magnitude $p$ in an electric field of magnitude $E$ is given by
$W=pE\left( \cos {{\theta }_{1}}-\cos {{\theta }_{2}} \right)$ --(4)
where ${{\theta }_{1}},{{\theta }_{2}}$ are the initial and final angles made by the dipole axis with the direction of the electric field.
Now, we will consider an electric dipole of dipole moment magnitude $p$ placed in an electric field of magnitude $E$.
Let ${{\theta }_{1}},{{\theta }_{2}}$ be the initial and final angles made by the dipole axis with the direction of the electric field.
It is given that the dipole is initially aligned along the electric field.
$\therefore {{\theta }_{1}}={{0}^{0}}$
Also, it is given that the dipole has to be rotated through ${{180}^{0}}$.
$\therefore {{\theta }_{2}}={{0}^{0}}+{{180}^{0}}={{180}^{0}}$
Using (4), the external work $W$ required to rotate the dipole will be
$W=pE\left( \cos \left( {{0}^{0}} \right)-\cos \left( {{180}^{0}} \right) \right)=pE\left( 1-\left( -1 \right) \right)=pE\left( 1+1 \right)=pE\left( 2 \right)=2pE$
$\left( \because \cos \left( {{0}^{0}} \right)=1,\cos \left( {{180}^{0}} \right)=-1 \right)$
Therefore, we have found the amount of external work required.
Note: The position where the dipole axis is parallel to the electric field is the position of stable equilibrium for the dipole. In this position, it has the lowest potential energy and hence, when a
slight deflection is given to the dipole, it will try to come back to this position. On the other hand, when the dipole axis is anti-parallel to the electric field direction, the potential energy of
the dipole is the highest and it is in a position of unstable equilibrium. If the dipole is given a slight deflection at this point it will keep on rotating to reach the point of stable equilibrium.
Students must remember this important concept. | {"url":"https://www.vedantu.com/question-answer/an-electric-dipole-is-held-in-a-uniform-electric-class-12-physics-cbse-5f45dc1155e8473a85de62f7","timestamp":"2024-11-02T02:56:12Z","content_type":"text/html","content_length":"190269","record_id":"<urn:uuid:43d53fd4-927c-41f4-a951-ce8e6c78e031>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00813.warc.gz"} |
A Level Maths Further Maths Guide - Keystone Tutors
Should I take Maths for A Level?
Mathematics has been the most popular A Level choice for some time now. This is due to Maths being a highly applicable subject in a range of professions, and is often a requirement for some
university degrees. While it is a popular choice, it is also a challenging subject that will require a lot of work to get top marks.
After completing GCSEs, students will have to make the important decision of choosing which subjects they will continue to study for the next two years. Most people do only 3 or 4 subjects for A
Level, so choosing subjects that you are eager to study in depth is vital, as you will spend a lot more time with each of these subjects. With this in mind, students should consider whether Maths is
the right choice for them from the perspective of ability and interest.
Many schools will only let students take A Level Maths if they achieved a 6 or higher in their GCSE Maths exam, though this may vary at different institutions. If you are on track for a 7,8, or 9 at
GCSE, then A Level Maths will likely be within your ability.
Finally, students that are aiming for a certain degree should confirm whether Maths is a required or suggested subject for their course. Potential Maths or Physics students should certainly be taking
Maths (and often Further Maths) at A Level, but many other STEM subjects also ask for a strong grade in Maths. For example, courses in Medicine, Psychology, and Computer Science may require students
to take Maths or a similar subject for A Level.
What is the difference between Maths and Further Maths?
Further Maths is a whole additional A Level, so completing Maths and Further Maths will account for two separate qualifications. Further Maths explores the topics from the Maths course in more
detail, as well as introducing more sophisticated ideas such as complex numbers or proof by induction.
Should I take Further Maths for A Level?
Further Maths is an excellent choice for students looking to study a STEM subject, particularly when applying for competitive courses. Many university courses recommend that applicants take Further
Maths, so make sure to check university websites for their entry requirements. As some schools do not offer Further Maths, it is often not a strict requirement for entry - that said, if it is a
recommended subject then it will likely be almost essential for easing the transition to higher education. For example, Mathematics degrees usually only list Further Maths as a recommended subject,
not a required course. However, plenty of material from Further Maths appears at degree level, so applicants are given a big head start if they took this course at A Level. Some other courses that
can recommend Further Maths are Computer Science, Engineering, or Biochemistry.
Further Maths is a popular course, with over 15,000 students studying the subject, but this is only a fraction of the students that take Mathematics. This is due to Further Maths having a less broad
appeal in university application, as well as the fact that Further Maths is much harder than the straight Maths course. Schools will ask that students achieve at least a 7 (but usually an 8 or 9) at
GCSE to take Further Maths at A Level, and since these courses are worth two A Level qualifications, students will find that about half (or more!) of their A Level work is centred on Mathematics.
Students should consider whether Further Maths will be useful in their future studies, as well as whether they enjoy Mathematics enough to complete two full A Levels in the subject. A prospective
Maths student will likely relish the opportunity to focus a large portion of their time on Mathematics, while a student hoping to study a course that doesn’t require Further Maths, like Psychology or
Law, may decide that the regular Maths qualification would be enough for them. There is no one-size-fits-all approach, so having a conversation with teachers or career advisors will be greatly
beneficial in finalising your decisions.
Does Further Maths help with A Level Maths?
Completing the Further Maths course will help students to consolidate their knowledge from A Level Maths, and as a result, students that take Further Maths will often achieve some excellent results
in the straight Maths course. That said, success in Further Maths is built on the foundation of the Maths course, so students that have had a lot of difficulties with Maths will likely have a lot
more trouble in the Further Maths course.
How to get an A* in A Level Maths
Independent Review - A Level Maths is a broad subject that many students find highly challenging, and a likely problem will be older topics getting forgotten as students move on to the next part of
the course, particularly when moving between Pure Maths, Mechanics, and Statistics. Hopefully, teachers will set their students regular tests on this past material to keep it fresh, but students can
take this into their own hands as well. Frequently looking back over the past month’s work will boost recollection and reduce the revision workload for the end of the year significantly.
Calculator Skills - While students will likely be very comfortable with the standard functions on a calculator by the time they take their A Levels, there are plenty of additional techniques that can
boost their score. For example, many exam-approved calculators have an integral calculator, and while this cannot be used to solve the question (showing your work is necessary!) it can definitely be
used to check your answer at the end.
Time Management - Maths exams are often quite time pressured, so being able to cope with these conditions will help students to achieve the highest mark possible. An excellent technique is to work
through the paper from start to finish, and skip any questions that you are unable to solve. This guarantees that you will complete every question that you are capable of finishing, and you won’t
waste time on a question that you won’t be able to figure out. After finishing the easier questions, you can go back to the difficult questions and hopefully scoop up a few extra marks before the end
of the test.
Past Paper Practice - Understanding the course material is essential for Maths, but putting that knowledge into practice is what will earn the best grades in the exam. Regular practice with past
paper questions will develop student’s problem solving techniques, and timed practice without notes will acclimate students to the style of examinations.
Common Pitfalls
We have gone through a range of examiner’s reports from the new A Level exams, and picked out a few common errors that arise across the years, as well as some personal tips from tuition experience.
This is by no means an exhaustive list, but focussing on some of these areas may help to boost your mark from an A to an A*.
Careless errors - Whether looking at students in Year 7 or Year 13, one of the most common places that students lose marks is in simple mistakes like misreading the question, failing to put the
correct units in their answer, or forgetting to check their negative signs in an expansion. Improving this skill comes with practice, but picking up some techniques for checking your work (such as
calculator functions) can help eliminate these mistakes.
Calculus - This is a notoriously challenging topic which is usually new for students in Year 12. Making sure to understand the process used in Differentiation and Integration fully will help secure
top marks, as harder A Level questions will demand deeper knowledge of these topics.
Trigonometric - Identities Using these identities is not that difficult in a vacuum, but it can be difficult for students to notice when they can be used in more open ended questions. Being confident
with these identities will help to avoid getting lost in these complex questions.
Modelling - A focus in the new specification is applying mathematical ideas to real life scenarios, and usually constructing or explaining a model for these situations. Generally, students have
struggled with questions that require an explanation, so becoming familiar with these questions is a must.
Large Data Set - This was a new addition to the Statistics paper, and has been generally poorly attempted in papers since. Taking some time to get familiar with the data set would help candidates aim
for the very top marks.
Resultant Forces - This is a very common Mechanics topic that is usually not too difficult for most students, but there are occasional questions that ask students to find more difficult forces, such
as resultant forces acting on a pulley, or questions involving scale pans.
How to get an A* in Further Maths
All of the A Level Maths advice above applies to Further Maths, and perhaps to a higher degree: independent review will help students stay on top of the large and varied workload, calculator aptitude
will help with checking answers and avoiding computational errors, and Further Maths exams tend to be even more time pressured than the Maths course so good exam technique and plenty of practice are
Don’t neglect the basics - Further Maths is an extension of the Maths A Level, and so strong fundamentals are necessary when tackling harder content. Taking some time to revisit the material from the
previous year’s work will help to build a full picture of the course, and ensuring mastery of core A Level skills will make it easy to transfer those skills to difficult Further Maths topics.
Ask for help - While this advice is true for any A level subject, Further Maths is an especially difficult course to grasp conceptually. Individual students will find different areas of the course
particularly challenging, and trying to keep up with the increased workload of A Levels can often lead to these difficult topics remaining a mystery throughout the year. When a topic doesn’t make
sense, talking to a friend, teacher, or tutor can help clear up any misconceptions. You might find Collisions from Further Mechanics challenging, while your neighbour may struggle with the Geometric
Distribution from Further Statistics, so having a chat about these areas together will likely help to address any concerns and strengthen your knowledge.
Common Pitfalls
As stated above, this list is based on information from examiners reports as well as personal tutoring experience. There are plenty of other tricky points in the Further Maths course, but these are a
few of the common areas that students find difficult.
Many of the usual errors made in the regular Maths course also apply to Further Maths, particularly careless errors as many Further Maths questions require a lot of work which raises the chance of
small arithmetic or typographic errors.
Induction - These questions are usually quite varied, and there is not a single approach that works for every type of question. Broad practice of this technique is key, especially as induction
questions can include algebra, series, matrices, and other topics.
Probability Generating Functions and Distributions - One of the most difficult Statistics topics, understanding the harder variants of PGF questions is likely to make a candidate stand out. Being
able to recognise when to use different distributions and knowing the conditions for when a distribution is applicable are also common areas for losing marks.
Significant Figures - In Mechanics papers, it is stated that the acceleration from gravity g is 9.8, while many candidates use the (more accurate) g value 9.81, which loses an accuracy mark.
Similarly, answers should be rounded to the number of significant figures given by data in the questions (which includes g = 9.8). These are very small errors that could make students narrowly miss
grade boundaries.
Open-ended Questions - A new focus of the curriculum is questions that have less structure, and the results so far have shown that students find these questions difficult to complete. Be prepared to
tackle questions that don’t guide you towards a given answer or, for example, ask you to calculate an integral without telling you which method to use. A lot of varied practice questions will be key
to mastering this idea.
What are the best universities for Mathematics?
The Complete University Guide ranks the best UK universities for Maths as follows:
1. Cambridge University
2. Oxford University
3. University of Warwick
4. Durham University
5. Imperial College London
6. University College London
7. London School of Economics and Political Science
8. University of Bristol
9. University of Bath
10. University of St Andrews
These rankings should not be taken as gospel however, as there are many other factors to consider when applying to a Maths course at university. Some universities will offer different joint courses
with Maths, such as the Mathematics and Philosophy course at Oxford, and the materials covered by the course will vary in different institutions, particularly in the later years of the course.
Overall ranking is a fairly good judge of a course’s quality, but taking a student’s individual interests and needs into account will be more beneficial than simply choosing the ‘best’ university!
Recommended resources for A Level Maths
A collection of practice papers and focussed exercises for A Level Maths and Further Maths.
A huge database of powerpoints and worksheets for all of the A Level papers.
Questions separated into each topics
Recommended Youtube Channels for A Level Maths and Further Maths
Tutors for A Level Maths and Further Maths
With tutors based in London and available online to families around the world, Keystone is one of the UK’s leading private tutoring organisations. We have a range of specialist A Level Maths and
Further Maths tutors who can assist students approaching A Levels and university admissions tests for Maths. | {"url":"https://www.keystonetutors.com/news/a-level-maths-further-maths-guide","timestamp":"2024-11-06T23:59:52Z","content_type":"text/html","content_length":"37895","record_id":"<urn:uuid:8d94eac6-89bb-4d31-8039-b6f8d59ff72b>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00708.warc.gz"} |
Numerical calculation of unsteady aerodynamic characteristics of cylinder models for supersonic laminar flow
Authors: Galaktionov A.Yu., Khlupnov A.I. Published: 06.10.2015
Published in issue: #5(104)/2015
DOI: 10.18698/0236-3941-2015-5-4-13
Category: Aviation and Rocket-Space Engineering | Chapter: Aerodynamics and Heat Transfer Processes in Aircrafts
Keywords: Navier-Stokes equations, dynamic of rotation, conjugate problem, numerical methods, aerodynamic damping, body of rotation, supersonic flow
The article describes computer calculations of unsteady aerodynamic characteristics of the axially symmetrical cylinder models with a fixed aspect ratio using the program created by the authors. A
numerical solution of the Navier -Stokes equation is obtained for the perfect gas model in laminar flow. The authors consider the solved problem as a conjugate one since it was necessary to obtain a
damping coefficient (an aerodynamic pitch damping moment), which required mathematic modeling of both the unsteady flows and free oscillations of the above mentioned bodies relative to the mass
center. The results of the calculation are compared to the well-known observed dependences.
[1] Kobelev V.N., Milovanov A.G. Sredstva vyvedeniya kosmicheskikh apparatov [Launch vehicles of spacecrafts]. Moscow, Restar Publ., 2009. 528 p.
[2] Lipnitskiy Yu.M., Krasil’shchikov A.V., Pokrovskiy A.N., Shmanenkov V.N. Nestatsionarnaya aerodinamika ballisticheskogo poleta [Non-steady aerodynamics of ballistic flight]. Moscow, Fizmatlit
Publ., 2003. 176 p.
[3] Belotserkovskiy S.M., Sripach B.K., Tabachnikov V.G. Krylo v nestatsionarnom potoke [The wing in unsteady flow]. Moscow, Nauka Publ., 1971. 768 p.
[4] Guzhenko G.A. The faked pattern method and its application in studying the airship rotational flight. Proc. TsAGI, 1934, iss. 182. 64 p. (in Russ.).
[5] Lipnitskiy Yu.M., Galaktionov A.Yu. Numerical simulation of unsteady aerodynamic characteristics of a blunted cone within the framework of the full Navier-Stokes equations. Kosm. i raketostr.
[Cosmonautics and Rocketry], 2006, no. 3, pp. 23-28 (in Russ.).
[6] Krasnov N.F. Aerodinamika tel vrashcheniya [Aerodynamics of revolved solids]. Moscow, Mashinostroenie Publ., 1964. 572 p.
[7] Krasnov N.F., Khlupnov A.I. Prikladnaya aerodinamika [Applied aerodynamics]. Moscow, Vyssh. shk. Publ., 1974. 732 p.
[8] Petrov K.P. Aerodinamika tel prosteyshey formy [Aerodynamics of the simplest form bodies]. Moscow, Faktorial Publ., 1998. 432 p.
[9] Useltom Bob L., Wallace Arthur R., Damping-in-pitch and Drag Characteristics of the Viking Configuration at Mach Number from 1.6 through 3. AEDC-TR-72-56, May, 1972. | {"url":"https://vestnikmach.bmstu.ru/eng/catalog/avroc/heattr/963.html","timestamp":"2024-11-12T07:48:33Z","content_type":"application/xhtml+xml","content_length":"11347","record_id":"<urn:uuid:4336c107-2ea2-4baa-8317-96375a645060>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00329.warc.gz"} |
6. Structures
Modern mathematics makes essential use of algebraic structures, which encapsulate patterns that can be instantiated in multiple settings. The subject provides various ways of defining such structures
and constructing particular instances.
Lean therefore provides corresponding ways of defining structures formally and working with them. You have already seen examples of algebraic structures in Lean, such as rings and lattices, which
were discussed in Chapter 2. This chapter will explain the mysterious square bracket annotations that you saw there, [Ring α] and [Lattice α]. It will also show you how to define and use algebraic
structures on your own.
For more technical detail, you can consult Theorem Proving in Lean, and a paper by Anne Baanen, Use and abuse of instance parameters in the Lean mathematical library.
6.1. Defining structures
In the broadest sense of the term, a structure is a specification of a collection of data, possibly with constraints that the data is required to satisfy. An instance of the structure is a particular
bundle of data satisfying the constraints. For example, we can specify that a point is a tuple of three real numbers:
structure Point where
x : ℝ
y : ℝ
z : ℝ
The @[ext] annotation tells Lean to automatically generate theorems that can be used to prove that two instances of a structure are equal when their components are equal, a property known as
#check Point.ext
example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by
repeat' assumption
We can then define particular instances of the Point structure. Lean provides multiple ways of doing that.
def myPoint1 : Point where
x := 2
y := -1
z := 4
def myPoint2 : Point :=
⟨2, -1, 4⟩
def myPoint3 :=
Point.mk 2 (-1) 4
In the first example, the fields of the structure are named explicitly. The function Point.mk referred to in the definition of myPoint3 is known as the constructor for the Point structure, because it
serves to construct elements. You can specify a different name if you want, like build.
structure Point' where build ::
x : ℝ
y : ℝ
z : ℝ
#check Point'.build 2 (-1) 4
The next two examples show how to define functions on structures. Whereas the second example makes the Point.mk constructor explicit, the first example uses an anonymous constructor for brevity. Lean
can infer the relevant constructor from the indicated type of add. It is conventional to put definitions and theorems associated with a structure like Point in a namespace with the same name. In the
example below, because we have opened the Point namespace, the full name of add is Point.add. When the namespace is not open, we have to use the full name. But remember that it is often convenient to
use anonymous projection notation, which allows us to write a.add b instead of Point.add a b. Lean interprets the former as the latter because a has type Point.
namespace Point
def add (a b : Point) : Point :=
⟨a.x + b.x, a.y + b.y, a.z + b.z⟩
def add' (a b : Point) : Point where
x := a.x + b.x
y := a.y + b.y
z := a.z + b.z
#check add myPoint1 myPoint2
#check myPoint1.add myPoint2
end Point
#check Point.add myPoint1 myPoint2
#check myPoint1.add myPoint2
Below we will continue to put definitions in the relevant namespace, but we will leave the namespacing commands out of the quoted snippets. To prove properties of the addition function, we can use rw
to expand the definition and ext to reduce an equation between two elements of the structure to equations between the components. Below we use the protected keyword so that the name of the theorem is
Point.add_comm, even when the namespace is open. This is helpful when we want to avoid ambiguity with a generic theorem like add_comm.
protected theorem add_comm (a b : Point) : add a b = add b a := by
rw [add, add]
ext <;> dsimp
repeat' apply add_comm
example (a b : Point) : add a b = add b a := by simp [add, add_comm]
Because Lean can unfold definitions and simplify projections internally, sometimes the equations we want hold definitionally.
theorem add_x (a b : Point) : (a.add b).x = a.x + b.x :=
It is also possible to define functions on structures using pattern matching, in a manner similar to the way we defined recursive functions in Section 5.2. The definitions addAlt and addAlt' below
are essentially the same; the only difference is that we use anonymous constructor notation in the second. Although it is sometimes convenient to define functions this way, and structural
eta-reduction makes this alternative definitionally equivalent, it can make things less convenient in later proofs. In particular, rw [addAlt] leaves us with a messier goal view containing a match
def addAlt : Point → Point → Point
| Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩
def addAlt' : Point → Point → Point
| ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩
theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by
theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by
rw [addAlt, addAlt]
-- the same proof still works, but the goal view here is harder to read
ext <;> dsimp
repeat' apply add_comm
Mathematical constructions often involve taking apart bundled information and putting it together again in different ways. It therefore makes sense that Lean and Mathlib offer so many ways of doing
this efficiently. As an exercise, try proving that Point.add is associative. Then define scalar multiplication for a point and show that it distributes over addition.
protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by
def smul (r : ℝ) (a : Point) : Point :=
theorem smul_distrib (r : ℝ) (a b : Point) :
(smul r a).add (smul r b) = smul r (a.add b) := by
Using structures is only the first step on the road to algebraic abstraction. We don’t yet have a way to link Point.add to the generic + symbol, or to connect Point.add_comm and Point.add_assoc to
the generic add_comm and add_assoc theorems. These tasks belong to the algebraic aspect of using structures, and we will explain how to carry them out in the next section. For now, just think of a
structure as a way of bundling together objects and information.
It is especially useful that a structure can specify not only data types but also constraints that the data must satisfy. In Lean, the latter are represented as fields of type Prop. For example, the
standard 2-simplex is defined to be the set of points \((x, y, z)\) satisfying \(x ≥ 0\), \(y ≥ 0\), \(z ≥ 0\), and \(x + y + z = 1\). If you are not familiar with the notion, you should draw a
picture, and convince yourself that this set is the equilateral triangle in three-space with vertices \((1, 0, 0)\), \((0, 1, 0)\), and \((0, 0, 1)\), together with its interior. We can represent it
in Lean as follows:
structure StandardTwoSimplex where
x : ℝ
y : ℝ
z : ℝ
x_nonneg : 0 ≤ x
y_nonneg : 0 ≤ y
z_nonneg : 0 ≤ z
sum_eq : x + y + z = 1
Notice that the last four fields refer to x, y, and z, that is, the first three fields. We can define a map from the two-simplex to itself that swaps x and y:
def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex
x := a.y
y := a.x
z := a.z
x_nonneg := a.y_nonneg
y_nonneg := a.x_nonneg
z_nonneg := a.z_nonneg
sum_eq := by rw [add_comm a.y a.x, a.sum_eq]
More interestingly, we can compute the midpoint of two points on the simplex. We have added the phrase noncomputable section at the beginning of this file in order to use division on the real
noncomputable section
def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex
x := (a.x + b.x) / 2
y := (a.y + b.y) / 2
z := (a.z + b.z) / 2
x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num)
y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num)
z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num)
sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq]
Here we have established x_nonneg, y_nonneg, and z_nonneg with concise proof terms, but establish sum_eq in tactic mode, using by.
Given a parameter \(\lambda\) satisfying \(0 \le \lambda \le 1\), we can take the weighted average \(\lambda a + (1 - \lambda) b\) of two points \(a\) and \(b\) in the standard 2-simplex. We
challenge you to define that function, in analogy to the midpoint function above.
def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1)
(a b : StandardTwoSimplex) : StandardTwoSimplex :=
Structures can depend on parameters. For example, we can generalize the standard 2-simplex to the standard \(n\)-simplex for any \(n\). At this stage, you don’t have to know anything about the type
Fin n except that it has \(n\) elements, and that Lean knows how to sum over it.
open BigOperators
structure StandardSimplex (n : ℕ) where
V : Fin n → ℝ
NonNeg : ∀ i : Fin n, 0 ≤ V i
sum_eq_one : (∑ i, V i) = 1
namespace StandardSimplex
def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n
V i := (a.V i + b.V i) / 2
NonNeg := by
intro i
apply div_nonneg
· linarith [a.NonNeg i, b.NonNeg i]
sum_eq_one := by
simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib,
a.sum_eq_one, b.sum_eq_one]
end StandardSimplex
As an exercise, see if you can define the weighted average of two points in the standard \(n\)-simplex. You can use Finset.sum_add_distrib and Finset.mul_sum to manipulate the relevant sums.
We have seen that structures can be used to bundle together data and properties. Interestingly, they can also be used to bundle together properties without the data. For example, the next structure,
IsLinear, bundles together the two components of linearity.
structure IsLinear (f : ℝ → ℝ) where
is_additive : ∀ x y, f (x + y) = f x + f y
preserves_mul : ∀ x c, f (c * x) = c * f x
variable (f : ℝ → ℝ) (linf : IsLinear f)
#check linf.is_additive
#check linf.preserves_mul
It is worth pointing out that structures are not the only way to bundle together data. The Point data structure can be defined using the generic type product, and IsLinear can be defined with a
simple and.
def Point'' :=
ℝ × ℝ × ℝ
def IsLinear' (f : ℝ → ℝ) :=
(∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x
Generic type constructions can even be used in place of structures with dependencies between their components. For example, the subtype construction combines a piece of data with a property. You can
think of the type PReal in the next example as being the type of positive real numbers. Any x : PReal has two components: the value, and the property of being positive. You can access these
components as x.val, which has type ℝ, and x.property, which represents the fact 0 < x.val.
def PReal :=
{ y : ℝ // 0 < y }
variable (x : PReal)
#check x.val
#check x.property
#check x.1
#check x.2
We could have used subtypes to define the standard 2-simplex, as well as the standard \(n\)-simplex for an arbitrary \(n\).
def StandardTwoSimplex' :=
{ p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 }
def StandardSimplex' (n : ℕ) :=
{ v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 }
Similarly, Sigma types are generalizations of ordered pairs, whereby the type of the second component depends on the type of the first.
def StdSimplex := Σ n : ℕ, StandardSimplex n
variable (s : StdSimplex)
#check s.fst
#check s.snd
#check s.1
#check s.2
Given s : StdSimplex, the first component s.fst is a natural number, and the second component is an element of the corresponding simplex StandardSimplex s.fst. The difference between a Sigma type and
a subtype is that the second component of a Sigma type is data rather than a proposition.
But even though we can use products, subtypes, and Sigma types instead of structures, using structures has a number of advantages. Defining a structure abstracts away the underlying representation
and provides custom names for the functions that access the components. This makes proofs more robust: proofs that rely only on the interface to a structure will generally continue to work when we
change the definition, as long as we redefine the old accessors in terms of the new definition. Moreover, as we are about to see, Lean provides support for weaving structures together into a rich,
interconnected hierarchy, and for managing the interactions between them.
6.2. Algebraic Structures
To clarify what we mean by the phrase algebraic structure, it will help to consider some examples.
1. A partially ordered set consists of a set \(P\) and a binary relation \(\le\) on \(P\) that is transitive and reflexive.
2. A group consists of a set \(G\) with an associative binary operation, an identity element \(1\), and a function \(g \mapsto g^{-1}\) that returns an inverse for each \(g\) in \(G\). A group is
abelian or commutative if the operation is commutative.
3. A lattice is a partially ordered set with meets and joins.
4. A ring consists of an (additively written) abelian group \((R, +, 0, x \mapsto -x)\) together with an associative multiplication operation \(\cdot\) and an identity \(1\), such that
multiplication distributes over addition. A ring is commutative if the multiplication is commutative.
5. An ordered ring \((R, +, 0, -, \cdot, 1, \le)\) consists of a ring together with a partial order on its elements, such that \(a \le b\) implies \(a + c \le b + c\) for every \(a\), \(b\), and \(c
\) in \(R\), and \(0 \le a\) and \(0 \le b\) implies \(0 \le a b\) for every \(a\) and \(b\) in \(R\).
6. A metric space consists of a set \(X\) and a function \(d : X \times X \to \mathbb{R}\) such that the following hold:
□ \(d(x, y) \ge 0\) for every \(x\) and \(y\) in \(X\).
□ \(d(x, y) = 0\) if and only if \(x = y\).
□ \(d(x, y) = d(y, x)\) for every \(x\) and \(y\) in \(X\).
□ \(d(x, z) \le d(x, y) + d(y, z)\) for every \(x\), \(y\), and \(z\) in \(X\).
7. A topological space consists of a set \(X\) and a collection \(\mathcal T\) of subsets of \(X\), called the open subsets of \(X\), such that the following hold:
□ The empty set and \(X\) are open.
□ The intersection of two open sets is open.
□ An arbitrary union of open sets is open.
In each of these examples, the elements of the structure belong to a set, the carrier set, that sometimes stands proxy for the entire structure. For example, when we say “let \(G\) be a group” and
then “let \(g \in G\),” we are using \(G\) to stand for both the structure and its carrier. Not every algebraic structure is associated with a single carrier set in this way. For example, a bipartite
graph involves a relation between two sets, as does a Galois connection, A category also involves two sets of interest, commonly called the objects and the morphisms.
The examples indicate some of the things that a proof assistant has to do in order to support algebraic reasoning. First, it needs to recognize concrete instances of structures. The number systems \
(\mathbb{Z}\), \(\mathbb{Q}\), and \(\mathbb{R}\) are all ordered rings, and we should be able to apply a generic theorem about ordered rings in any of these instances. Sometimes a concrete set may
be an instance of a structure in more than one way. For example, in addition to the usual topology on \(\mathbb{R}\), which forms the basis for real analysis, we can also consider the discrete
topology on \(\mathbb{R}\), in which every set is open.
Second, a proof assistant needs to support generic notation on structures. In Lean, the notation * is used for multiplication in all the usual number systems, as well as for multiplication in generic
groups and rings. When we use an expression like f x * y, Lean has to use information about the types of f, x, and y to determine which multiplication we have in mind.
Third, it needs to deal with the fact that structures can inherit definitions, theorems, and notation from other structures in various ways. Some structures extend others by adding more axioms. A
commutative ring is still a ring, so any definition that makes sense in a ring also makes sense in a commutative ring, and any theorem that holds in a ring also holds in a commutative ring. Some
structures extend others by adding more data. For example, the additive part of any ring is an additive group. The ring structure adds a multiplication and an identity, as well as axioms that govern
them and relate them to the additive part. Sometimes we can define one structure in terms of another. Any metric space has a canonical topology associated with it, the metric space topology, and
there are various topologies that can be associated with any linear ordering.
Finally, it is important to keep in mind that mathematics allows us to use functions and operations to define structures in the same way we use functions and operations to define numbers. Products
and powers of groups are again groups. For every \(n\), the integers modulo \(n\) form a ring, and for every \(k > 0\), the \(k \times k\) matrices of polynomials with coefficients in that ring again
form a ring. Thus we can calculate with structures just as easily as we can calculate with their elements. This means that algebraic structures lead dual lives in mathematics, as containers for
collections of objects and as objects in their own right. A proof assistant has to accommodate this dual role.
When dealing with elements of a type that has an algebraic structure associated with it, a proof assistant needs to recognize the structure and find the relevant definitions, theorems, and notation.
All this should sound like a lot of work, and it is. But Lean uses a small collection of fundamental mechanisms to carry out these tasks. The goal of this section is to explain these mechanisms and
show you how to use them.
The first ingredient is almost too obvious to mention: formally speaking, algebraic structures are structures in the sense of Section 6.1. An algebraic structure is a specification of a bundle of
data satisfying some axiomatic hypotheses, and we saw in Section 6.1 that this is exactly what the structure command is designed to accommodate. It’s a marriage made in heaven!
Given a data type α, we can define the group structure on α as follows.
structure Group₁ (α : Type*) where
mul : α → α → α
one : α
inv : α → α
mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z)
mul_one : ∀ x : α, mul x one = x
one_mul : ∀ x : α, mul one x = x
inv_mul_cancel : ∀ x : α, mul (inv x) x = one
Notice that the type α is a parameter in the definition of group₁. So you should think of an object struc : Group₁ α as being a group structure on α. We saw in Section 2.2 that the counterpart
mul_inv_cancel to inv_mul_cancel follows from the other group axioms, so there is no need to add it to the definition.
This definition of a group is similar to the definition of Group in Mathlib, and we have chosen the name Group₁ to distinguish our version. If you write #check Group and ctrl-click on the definition,
you will see that the Mathlib version of Group is defined to extend another structure; we will explain how to do that later. If you type #print Group you will also see that the Mathlib version of
Group has a number of extra fields. For reasons we will explain later, sometimes it is useful to add redundant information to a structure, so that there are additional fields for objects and
functions that can be defined from the core data. Don’t worry about that for now. Rest assured that our simplified version Group₁ is morally the same as the definition of a group that Mathlib uses.
It is sometimes useful to bundle the type together with the structure, and Mathlib also contains a definition of a GroupCat structure that is equivalent to the following:
structure Group₁Cat where
α : Type*
str : Group₁ α
The Mathlib version is found in Mathlib.Algebra.Category.GroupCat.Basic, and you can #check it if you add this to the imports at the beginning of the examples file.
For reasons that will become clearer below, it is more often useful to keep the type α separate from the structure Group α. We refer to the two objects together as a partially bundled structure,
since the representation combines most, but not all, of the components into one structure. It is common in Mathlib to use capital roman letters like G for a type when it is used as the carrier type
for a group.
Let’s construct a group, which is to say, an element of the Group₁ type. For any pair of types α and β, Mathlib defines the type Equiv α β of equivalences between α and β. Mathlib also defines the
suggestive notation α ≃ β for this type. An element f : α ≃ β is a bijection between α and β represented by four components: a function f.toFun from α to β, the inverse function f.invFun from β to α,
and two properties that specify these functions are indeed inverse to one another.
variable (α β γ : Type*)
variable (f : α ≃ β) (g : β ≃ γ)
#check Equiv α β
#check (f.toFun : α → β)
#check (f.invFun : β → α)
#check (f.right_inv : ∀ x : β, f (f.invFun x) = x)
#check (f.left_inv : ∀ x : α, f.invFun (f x) = x)
#check (Equiv.refl α : α ≃ α)
#check (f.symm : β ≃ α)
#check (f.trans g : α ≃ γ)
Notice the creative naming of the last three constructions. We think of the identity function Equiv.refl, the inverse operation Equiv.symm, and the composition operation Equiv.trans as explicit
evidence that the property of being in bijective correspondence is an equivalence relation.
Notice also that f.trans g requires composing the forward functions in reverse order. Mathlib has declared a coercion from Equiv α β to the function type α → β, so we can omit writing .toFun and have
Lean insert it for us.
example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) :=
example (x : α) : (f.trans g) x = g (f x) :=
example : (f.trans g : α → γ) = g ∘ f :=
Mathlib also defines the type perm α of equivalences between α and itself.
example (α : Type*) : Equiv.Perm α = (α ≃ α) :=
It should be clear that Equiv.Perm α forms a group under composition of equivalences. We orient things so that mul f g is equal to g.trans f, whose forward function is f ∘ g. In other words,
multiplication is what we ordinarily think of as composition of the bijections. Here we define this group:
def permGroup {α : Type*} : Group₁ (Equiv.Perm α)
mul f g := Equiv.trans g f
one := Equiv.refl α
inv := Equiv.symm
mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm
one_mul := Equiv.trans_refl
mul_one := Equiv.refl_trans
inv_mul_cancel := Equiv.self_trans_symm
In fact, Mathlib defines exactly this Group structure on Equiv.Perm α in the file GroupTheory.Perm.Basic. As always, you can hover over the theorems used in the definition of permGroup to see their
statements, and you can jump to their definitions in the original file to learn more about how they are implemented.
In ordinary mathematics, we generally think of notation as independent of structure. For example, we can consider groups \((G_1, \cdot, 1, \cdot^{-1})\), \((G_2, \circ, e, i(\cdot))\), and \((G_3, +,
0, -)\). In the first case, we write the binary operation as \(\cdot\), the identity at \(1\), and the inverse function as \(x \mapsto x^{-1}\). In the second and third cases, we use the notational
alternatives shown. When we formalize the notion of a group in Lean, however, the notation is more tightly linked to the structure. In Lean, the components of any Group are named mul, one, and inv,
and in a moment we will see how multiplicative notation is set up to refer to them. If we want to use additive notation, we instead use an isomorphic structure AddGroup (the structure underlying
additive groups). Its components are named add, zero, and neg, and the associated notation is what you would expect it to be.
Recall the type Point that we defined in Section 6.1, and the addition function that we defined there. These definitions are reproduced in the examples file that accompanies this section. As an
exercise, define an AddGroup₁ structure that is similar to the Group₁ structure we defined above, except that it uses the additive naming scheme just described. Define negation and a zero on the
Point data type, and define the AddGroup₁ structure on Point.
structure AddGroup₁ (α : Type*) where
(add : α → α → α)
-- fill in the rest
structure Point where
x : ℝ
y : ℝ
z : ℝ
namespace Point
def add (a b : Point) : Point :=
⟨a.x + b.x, a.y + b.y, a.z + b.z⟩
def neg (a : Point) : Point := sorry
def zero : Point := sorry
def addGroupPoint : AddGroup₁ Point := sorry
end Point
We are making progress. Now we know how to define algebraic structures in Lean, and we know how to define instances of those structures. But we also want to associate notation with structures so that
we can use it with each instance. Moreover, we want to arrange it so that we can define an operation on a structure and use it with any particular instance, and we want to arrange it so that we can
prove a theorem about a structure and use it with any instance.
In fact, Mathlib is already set up to use generic group notation, definitions, and theorems for Equiv.Perm α.
variable {α : Type*} (f g : Equiv.Perm α) (n : ℕ)
#check f * g
#check mul_assoc f g g⁻¹
-- group power, defined for any group
#check g ^ n
example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_inv_cancel, mul_one]
example : f * g * g⁻¹ = f :=
mul_inv_cancel_right f g
example {α : Type*} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f :=
mul_inv_cancel_right f g
You can check that this is not the case for the additive group structure on Point that we asked you to define above. Our task now is to understand that magic that goes on under the hood in order to
make the examples for Equiv.Perm α work the way they do.
The issue is that Lean needs to be able to find the relevant notation and the implicit group structure, using the information that is found in the expressions that we type. Similarly, when we write x
+ y with expressions x and y that have type ℝ, Lean needs to interpret the + symbol as the relevant addition function on the reals. It also has to recognize the type ℝ as an instance of a commutative
ring, so that all the definitions and theorems for a commutative ring are available. For another example, continuity is defined in Lean relative to any two topological spaces. When we have f : ℝ → ℂ
and we write Continuous f, Lean has to find the relevant topologies on ℝ and ℂ.
The magic is achieved with a combination of three things.
1. Logic. A definition that should be interpreted in any group takes, as arguments, the type of the group and the group structure as arguments. Similarly, a theorem about the elements of an
arbitrary group begins with universal quantifiers over the type of the group and the group structure.
2. Implicit arguments. The arguments for the type and the structure are generally left implicit, so that we do not have to write them or see them in the Lean information window. Lean fills the
information in for us silently.
3. Type class inference. Also known as class inference, this is a simple but powerful mechanism that enables us to register information for Lean to use later on. When Lean is called on to fill in
implicit arguments to a definition, theorem, or piece of notation, it can make use of information that has been registered.
Whereas an annotation (grp : Group G) tells Lean that it should expect to be given that argument explicitly and the annotation {grp : Group G} tells Lean that it should try to figure it out from
contextual cues in the expression, the annotation [grp : Group G] tells Lean that the corresponding argument should be synthesized using type class inference. Since the whole point to the use of such
arguments is that we generally do not need to refer to them explicitly, Lean allows us to write [Group G] and leave the name anonymous. You have probably already noticed that Lean chooses names like
_inst_1 automatically. When we use the anonymous square-bracket annotation with the variables command, then as long as the variables are still in scope, Lean automatically adds the argument [Group G]
to any definition or theorem that mentions G.
How do we register the information that Lean needs to use to carry out the search? Returning to our group example, we need only make two changes. First, instead of using the structure command to
define the group structure, we use the keyword class to indicate that it is a candidate for class inference. Second, instead of defining particular instances with def, we use the keyword instance to
register the particular instance with Lean. As with the names of class variables, we are allowed to leave the name of an instance definition anonymous, since in general we intend Lean to find it and
put it to use without troubling us with the details.
class Group₂ (α : Type*) where
mul : α → α → α
one : α
inv : α → α
mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z)
mul_one : ∀ x : α, mul x one = x
one_mul : ∀ x : α, mul one x = x
inv_mul_cancel : ∀ x : α, mul (inv x) x = one
instance {α : Type*} : Group₂ (Equiv.Perm α) where
mul f g := Equiv.trans g f
one := Equiv.refl α
inv := Equiv.symm
mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm
one_mul := Equiv.trans_refl
mul_one := Equiv.refl_trans
inv_mul_cancel := Equiv.self_trans_symm
The following illustrates their use.
#check Group₂.mul
def mySquare {α : Type*} [Group₂ α] (x : α) :=
Group₂.mul x x
#check mySquare
variable {β : Type*} (f g : Equiv.Perm β)
example : Group₂.mul f g = g.trans f :=
example : mySquare f = f.trans f :=
The #check command shows that Group₂.mul has an implicit argument [Group₂ α] that we expect to be found by class inference, where α is the type of the arguments to Group₂.mul. In other words, {α :
Type*} is the implicit argument for the type of the group elements and [Group₂ α] is the implicit argument for the group structure on α. Similarly, when we define a generic squaring function
my_square for Group₂, we use an implicit argument {α : Type*} for the type of the elements and an implicit argument [Group₂ α] for the Group₂ structure.
In the first example, when we write Group₂.mul f g, the type of f and g tells Lean that in the argument α to Group₂.mul has to be instantiated to Equiv.Perm β. That means that Lean has to find an
element of Group₂ (Equiv.Perm β). The previous instance declaration tells Lean exactly how to do that. Problem solved!
This simple mechanism for registering information so that Lean can find it when it needs it is remarkably useful. Here is one way it comes up. In Lean’s foundation, a data type α may be empty. In a
number of applications, however, it is useful to know that a type has at least one element. For example, the function List.headI, which returns the first element of a list, can return the default
value when the list is empty. To make that work, the Lean library defines a class Inhabited α, which does nothing more than store a default value. We can show that the Point type is an instance:
instance : Inhabited Point where default := ⟨0, 0, 0⟩
#check (default : Point)
example : ([] : List Point).headI = default :=
The class inference mechanism is also used for generic notation. The expression x + y is an abbreviation for Add.add x y where—you guessed it—Add α is a class that stores a binary function on α.
Writing x + y tells Lean to find a registered instance of [Add.add α] and use the corresponding function. Below, we register the addition function for Point.
instance : Add Point where add := Point.add
variable (x y : Point)
#check x + y
example : x + y = Point.add x y :=
In this way, we can assign the notation + to binary operations on other types as well.
But we can do even better. We have seen that * can be used in any group, + can be used in any additive group, and both can be used in any ring. When we define a new instance of a ring in Lean, we
don’t have to define + and * for that instance, because Lean knows that these are defined for every ring. We can use this method to specify notation for our Group₂ class:
instance hasMulGroup₂ {α : Type*} [Group₂ α] : Mul α :=
instance hasOneGroup₂ {α : Type*} [Group₂ α] : One α :=
instance hasInvGroup₂ {α : Type*} [Group₂ α] : Inv α :=
variable {α : Type*} (f g : Equiv.Perm α)
#check f * 1 * g⁻¹
def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) :=
In this case, we have to supply names for the instances, because Lean has a hard time coming up with good defaults. What makes this approach work is that Lean carries out a recursive search.
According to the instances we have declared, Lean can find an instance of Mul (Equiv.Perm α) by finding an instance of Group₂ (Equiv.Perm α), and it can find an instance of Group₂ (Equiv.Perm α)
because we have provided one. Lean is capable of finding these two facts and chaining them together.
The example we have just given is dangerous, because Lean’s library also has an instance of Group (Equiv.Perm α), and multiplication is defined on any group. So it is ambiguous as to which instance
is found. In fact, Lean favors more recent declarations unless you explicitly specify a different priority. Also, there is another way to tell Lean that one structure is an instance of another, using
the extends keyword. This is how Mathlib specifies that, for example, every commutative ring is a ring. You can find more information in Section 7 and in a section on class inference in Theorem
Proving in Lean.
In general, it is a bad idea to specify a value of * for an instance of an algebraic structure that already has the notation defined. Redefining the notion of Group in Lean is an artificial example.
In this case, however, both interpretations of the group notation unfold to Equiv.trans, Equiv.refl, and Equiv.symm, in the same way.
As a similarly artificial exercise, define a class AddGroup₂ in analogy to Group₂. Define the usual notation for addition, negation, and zero on any AddGroup₂ using the classes Add, Neg, and Zero.
Then show Point is an instance of AddGroup₂. Try it out and make sure that the additive group notation works for elements of Point.
class AddGroup₂ (α : Type*) where
add : α → α → α
-- fill in the rest
It is not a big problem that we have already declared instances Add, Neg, and Zero for Point above. Once again, the two ways of synthesizing the notation should come up with the same answer.
Class inference is subtle, and you have to be careful when using it, because it configures automation that invisibly governs the interpretation of the expressions we type. When used wisely, however,
class inference is a powerful tool. It is what makes algebraic reasoning possible in Lean.
6.3. Building the Gaussian Integers
We will now illustrate the use of the algebraic hierarchy in Lean by building an important mathematical object, the Gaussian integers, and showing that it is a Euclidean domain. In other words,
according to the terminology we have been using, we will define the Gaussian integers and show that they are an instance of the Euclidean domain structure.
In ordinary mathematical terms, the set of Gaussian integers \(\Bbb{Z}[i]\) is the set of complex numbers \(\{ a + b i \mid a, b \in \Bbb{Z}\}\). But rather than define them as a subset of the
complex numbers, our goal here is to define them as a data type in their own right. We do this by representing a Gaussian integer as a pair of integers, which we think of as the real and imaginary
structure GaussInt where
re : ℤ
im : ℤ
We first show that the Gaussian integers have the structure of a ring, with 0 defined to be ⟨0, 0⟩, 1 defined to be ⟨1, 0⟩, and addition defined pointwise. To work out the definition of
multiplication, remember that we want the element \(i\), represented by ⟨0, 1⟩, to be a square root of \(-1\). Thus we want
\[\begin{split}(a + bi) (c + di) & = ac + bci + adi + bd i^2 \\ & = (ac - bd) + (bc + ad)i.\end{split}\]
This explains the definition of Mul below.
instance : Zero GaussInt :=
⟨⟨0, 0⟩⟩
instance : One GaussInt :=
⟨⟨1, 0⟩⟩
instance : Add GaussInt :=
⟨fun x y ↦ ⟨x.re + y.re, x.im + y.im⟩⟩
instance : Neg GaussInt :=
⟨fun x ↦ ⟨-x.re, -x.im⟩⟩
instance : Mul GaussInt :=
⟨fun x y ↦ ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩
As noted in Section 6.1, it is a good idea to put all the definitions related to a data type in a namespace with the same name. Thus in the Lean files associated with this chapter, these definitions
are made in the GaussInt namespace.
Notice that here we are defining the interpretations of the notation 0, 1, +, -, and * directly, rather than naming them GaussInt.zero and the like and assigning the notation to those. It is often
useful to have an explicit name for the definitions, for example, to use with simp and rewrite.
theorem zero_def : (0 : GaussInt) = ⟨0, 0⟩ :=
theorem one_def : (1 : GaussInt) = ⟨1, 0⟩ :=
theorem add_def (x y : GaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ :=
theorem neg_def (x : GaussInt) : -x = ⟨-x.re, -x.im⟩ :=
theorem mul_def (x y : GaussInt) :
x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ :=
It is also useful to name the rules that compute the real and imaginary parts, and to declare them to the simplifier.
theorem zero_re : (0 : GaussInt).re = 0 :=
theorem zero_im : (0 : GaussInt).im = 0 :=
theorem one_re : (1 : GaussInt).re = 1 :=
theorem one_im : (1 : GaussInt).im = 0 :=
theorem add_re (x y : GaussInt) : (x + y).re = x.re + y.re :=
theorem add_im (x y : GaussInt) : (x + y).im = x.im + y.im :=
theorem neg_re (x : GaussInt) : (-x).re = -x.re :=
theorem neg_im (x : GaussInt) : (-x).im = -x.im :=
theorem mul_re (x y : GaussInt) : (x * y).re = x.re * y.re - x.im * y.im :=
theorem mul_im (x y : GaussInt) : (x * y).im = x.re * y.im + x.im * y.re :=
It is now surprisingly easy to show that the Gaussian integers are an instance of a commutative ring. We are putting the structure concept to good use. Each particular Gaussian integer is an instance
of the GaussInt structure, whereas the type GaussInt itself, together with the relevant operations, is an instance of the CommRing structure. The CommRing structure, in turn, extends the notational
structures Zero, One, Add, Neg, and Mul.
If you type instance : CommRing GaussInt := _, click on the light bulb that appears in VS Code, and then ask Lean to fill in a skeleton for the structure definition, you will see a scary number of
entries. Jumping to the definition of the structure, however, shows that many of the fields have default definitions that Lean will fill in for you automatically. The essential ones appear in the
definition below. A special case are nsmul and zsmul which should be ignored for now and will be explained in the next chapter. In each case, the relevant identity is proved by unfolding definitions,
using the ext tactic to reduce the identities to their real and imaginary components, simplifying, and, if necessary, carrying out the relevant ring calculation in the integers. Note that we could
easily avoid repeating all this code, but this is not the topic of the current discussion.
instance instCommRing : CommRing GaussInt where
zero := 0
one := 1
add := (· + ·)
neg x := -x
mul := (· * ·)
nsmul := nsmulRec
zsmul := zsmulRec
add_assoc := by
ext <;> simp <;> ring
zero_add := by
ext <;> simp
add_zero := by
ext <;> simp
neg_add_cancel := by
ext <;> simp
add_comm := by
ext <;> simp <;> ring
mul_assoc := by
ext <;> simp <;> ring
one_mul := by
ext <;> simp
mul_one := by
ext <;> simp
left_distrib := by
ext <;> simp <;> ring
right_distrib := by
ext <;> simp <;> ring
mul_comm := by
ext <;> simp <;> ring
zero_mul := by
ext <;> simp
mul_zero := by
ext <;> simp
Lean’s library defines the class of nontrivial types to be types with at least two distinct elements. In the context of a ring, this is equivalent to saying that the zero is not equal to the one.
Since some common theorems depend on that fact, we may as well establish it now.
instance : Nontrivial GaussInt := by
use 0, 1
rw [Ne, GaussInt.ext_iff]
We will now show that the Gaussian integers have an important additional property. A Euclidean domain is a ring \(R\) equipped with a norm function \(N : R \to \mathbb{N}\) with the following two
• For every \(a\) and \(b \ne 0\) in \(R\), there are \(q\) and \(r\) in \(R\) such that \(a = bq + r\) and either \(r = 0\) or N(r) < N(b).
• For every \(a\) and \(b \ne 0\), \(N(a) \le N(ab)\).
The ring of integers \(\Bbb{Z}\) with \(N(a) = |a|\) is an archetypal example of a Euclidean domain. In that case, we can take \(q\) to be the result of integer division of \(a\) by \(b\) and \(r\)
to be the remainder. These functions are defined in Lean so that the satisfy the following:
example (a b : ℤ) : a = b * (a / b) + a % b :=
Eq.symm (Int.ediv_add_emod a b)
example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b :=
Int.emod_nonneg a
example (a b : ℤ) : b ≠ 0 → a % b < |b| :=
Int.emod_lt a
In an arbitrary ring, an element \(a\) is said to be a unit if it divides \(1\). A nonzero element \(a\) is said to be irreducible if it cannot be written in the form \(a = bc\) where neither \(b\)
nor \(c\) is a unit. In the integers, every irreducible element \(a\) is prime, which is to say, whenever \(a\) divides a product \(bc\), it divides either \(b\) or \(c\). But in other rings this
property can fail. In the ring \(\Bbb{Z}[\sqrt{-5}]\), we have
\[6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}),\]
and the elements \(2\), \(3\), \(1 + \sqrt{-5}\), and \(1 - \sqrt{-5}\) are all irreducible, but they are not prime. For example, \(2\) divides the product \((1 + \sqrt{-5})(1 - \sqrt{-5})\), but it
does not divide either factor. In particular, we no longer have unique factorization: the number \(6\) can be factored into irreducible elements in more than one way.
In contrast, every Euclidean domain is a unique factorization domain, which implies that every irreducible element is prime. The axioms for a Euclidean domain imply that one can write any nonzero
element as a finite product of irreducible elements. They also imply that one can use the Euclidean algorithm to find a greatest common divisor of any two nonzero elements a and b, i.e.~an element
that is divisible by any other common divisor. This, in turn, implies that factorization into irreducible elements is unique up to multiplication by units.
We now show that the Gaussian integers are a Euclidean domain with the norm defined by \(N(a + bi) = (a + bi)(a - bi) = a^2 + b^2\). The Gaussian integer \(a - bi\) is called the conjugate of \(a +
bi\). It is not hard to check that for any complex numbers \(x\) and \(y\), we have \(N(xy) = N(x)N(y)\).
To see that this definition of the norm makes the Gaussian integers a Euclidean domain, only the first property is challenging. Suppose we want to write \(a + bi = (c + di) q + r\) for suitable \(q\)
and \(r\). Treating \(a + bi\) and \(c + di\) are complex numbers, carry out the division
\[\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c-di)} = \frac{ac + bd}{c^2 + d^2} + \frac{bc -ad}{c^2+d^2} i.\]
The real and imaginary parts might not be integers, but we can round them to the nearest integers \(u\) and \(v\). We can then express the right-hand side as \((u + vi) + (u' + v'i)\), where \(u' +
v'i\) is the part left over. Note that we have \(|u'| \le 1/2\) and \(|v'| \le 1/2\), and hence
\[N(u' + v' i) = (u')^2 + (v')^2 \le 1/4 + 1/4 \le 1/2.\]
Multiplying through by \(c + di\), we have
\[a + bi = (c + di) (u + vi) + (c + di) (u' + v'i).\]
Setting \(q = u + vi\) and \(r = (c + di) (u' + v'i)\), we have \(a + bi = (c + di) q + r\), and we only need to bound \(N(r)\):
\[N(r) = N(c + di)N(u' + v'i) \le N(c + di) \cdot 1/2 < N(c + di).\]
The argument we just carried out requires viewing the Gaussian integers as a subset of the complex numbers. One option for formalizing it in Lean is therefore to embed the Gaussian integers in the
complex numbers, embed the integers in the Gaussian integers, define the rounding function from the real numbers to the integers, and take great care to pass back and forth between these number
systems appropriately. In fact, this is exactly the approach that is followed in Mathlib, where the Gaussian integers themselves are constructed as a special case of a ring of quadratic integers. See
the file GaussianInt.lean.
Here we will instead carry out an argument that stays in the integers. This illustrates an choice one commonly faces when formalizing mathematics. Given an argument that requires concepts or
machinery that is not already in the library, one has two choices: either formalizes the concepts or machinery needed, or adapt the argument to make use of concepts and machinery you already have.
The first choice is generally a good investment of time when the results can be used in other contexts. Pragmatically speaking, however, sometimes seeking a more elementary proof is more efficient.
The usual quotient-remainder theorem for the integers says that for every \(a\) and nonzero \(b\), there are \(q\) and \(r\) such that \(a = b q + r\) and \(0 \le r < b\). Here we will make use of
the following variation, which says that there are \(q'\) and \(r'\) such that \(a = b q' + r'\) and \(|r'| \le b/2\). You can check that if the value of \(r\) in the first statement satisfies \(r \
le b/2\), we can take \(q' = q\) and \(r' = r\), and otherwise we can take \(q' = q + 1\) and \(r' = r - b\). We are grateful to Heather Macbeth for suggesting the following more elegant approach,
which avoids definition by cases. We simply add b / 2 to a before dividing and then subtract it from the remainder.
def div' (a b : ℤ) :=
(a + b / 2) / b
def mod' (a b : ℤ) :=
(a + b / 2) % b - b / 2
theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by
rw [div', mod']
linarith [Int.ediv_add_emod (a + b / 2) b]
theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : |mod' a b| ≤ b / 2 := by
rw [mod', abs_le]
· linarith [Int.emod_nonneg (a + b / 2) h.ne']
have := Int.emod_lt_of_pos (a + b / 2) h
have := Int.ediv_add_emod b 2
have := Int.emod_lt_of_pos b zero_lt_two
revert this; intro this -- FIXME, this should not be needed
Note the use of our old friend, linarith. We will also need to express mod' in terms of div'.
theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b]
We will use the fact that \(x^2 + y^2\) is equal to zero if and only if \(x\) and \(y\) are both zero. As an exercise, we ask you to prove that this holds in any ordered ring.
theorem sq_add_sq_eq_zero {α : Type*} [LinearOrderedRing α] (x y : α) :
x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by
We will put all the remaining definitions and theorems in this section in the GaussInt namespace. First, we define the norm function and ask you to establish some of its properties. The proofs are
all short.
def norm (x : GaussInt) :=
x.re ^ 2 + x.im ^ 2
theorem norm_nonneg (x : GaussInt) : 0 ≤ norm x := by
theorem norm_eq_zero (x : GaussInt) : norm x = 0 ↔ x = 0 := by
theorem norm_pos (x : GaussInt) : 0 < norm x ↔ x ≠ 0 := by
theorem norm_mul (x y : GaussInt) : norm (x * y) = norm x * norm y := by
Next we define the conjugate function:
def conj (x : GaussInt) : GaussInt :=
⟨x.re, -x.im⟩
theorem conj_re (x : GaussInt) : (conj x).re = x.re :=
theorem conj_im (x : GaussInt) : (conj x).im = -x.im :=
theorem norm_conj (x : GaussInt) : norm (conj x) = norm x := by simp [norm]
Finally, we define division for the Gaussian integers with the notation x / y, that rounds the complex quotient to the nearest Gaussian integer. We use our bespoke Int.div' for that purpose. As we
calculated above, if x is \(a + bi\) and y is \(c + di\), then the real and imaginary parts of x / y are the nearest integers to
\[\frac{ac + bd}{c^2 + d^2} \quad \text{and} \quad \frac{bc -ad}{c^2+d^2},\]
respectively. Here the numerators are the real and imaginary parts of \((a + bi) (c - di)\), and the denominators are both equal to the norm of \(c + di\).
instance : Div GaussInt :=
⟨fun x y ↦ ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩
Having defined x / y, We define x % y to be the remainder, x - (x / y) * y. As above, we record the definitions in the theorems div_def and mod_def so that we can use them with simp and rewrite.
instance : Mod GaussInt :=
⟨fun x y ↦ x - y * (x / y)⟩
theorem div_def (x y : GaussInt) :
x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ :=
theorem mod_def (x y : GaussInt) : x % y = x - y * (x / y) :=
These definitions immediately yield x = y * (x / y) + x % y for every x and y, so all we need to do is show that the norm of x % y is less than the norm of y when y is not zero.
We just defined the real and imaginary parts of x / y to be div' (x * conj y).re (norm y) and div' (x * conj y).im (norm y), respectively. Calculating, we have
(x % y) * conj y = (x - x / y * y) * conj y = x * conj y - x / y * (y * conj y)
The real and imaginary parts of the right-hand side are exactly mod' (x * conj y).re (norm y) and mod' (x * conj y).im (norm y). By the properties of div' and mod', these are guaranteed to be less
than or equal to norm y / 2. So we have
norm ((x % y) * conj y) ≤ (norm y / 2)^2 + (norm y / 2)^2 ≤ (norm y / 2) * norm y.
On the other hand, we have
norm ((x % y) * conj y) = norm (x % y) * norm (conj y) = norm (x % y) * norm y.
Dividing through by norm y we have norm (x % y) ≤ (norm y) / 2 < norm y, as required.
This messy calculation is carried out in the next proof. We encourage you to step through the details and see if you can find a nicer argument.
theorem norm_mod_lt (x : GaussInt) {y : GaussInt} (hy : y ≠ 0) :
(x % y).norm < y.norm := by
have norm_y_pos : 0 < norm y := by rwa [norm_pos]
have H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩
· ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring
have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y
· calc
norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj]
_ = |Int.mod' (x.re * y.re + x.im * y.im) (norm y)| ^ 2
+ |Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)| ^ 2 := by simp [H1, norm, sq_abs]
_ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos
_ = norm y / 2 * (norm y / 2 * 2) := by ring
_ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num
calc norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right H2 norm_y_pos
_ < norm y := by
apply Int.ediv_lt_of_lt_mul
· norm_num
· linarith
We are in the home stretch. Our norm function maps Gaussian integers to nonnegative integers. We need a function that maps Gaussian integers to natural numbers, and we obtain that by composing norm
with the function Int.natAbs, which maps integers to the natural numbers. The first of the next two lemmas establishes that mapping the norm to the natural numbers and back to the integers does not
change the value. The second one re-expresses the fact that the norm is decreasing.
theorem coe_natAbs_norm (x : GaussInt) : (x.norm.natAbs : ℤ) = x.norm :=
Int.natAbs_of_nonneg (norm_nonneg _)
theorem natAbs_norm_mod_lt (x y : GaussInt) (hy : y ≠ 0) :
(x % y).norm.natAbs < y.norm.natAbs := by
apply Int.ofNat_lt.1
simp only [Int.natCast_natAbs, abs_of_nonneg, norm_nonneg]
apply norm_mod_lt x hy
We also need to establish the second key property of the norm function on a Euclidean domain.
theorem not_norm_mul_left_lt_norm (x : GaussInt) {y : GaussInt} (hy : y ≠ 0) :
¬(norm (x * y)).natAbs < (norm x).natAbs := by
apply not_lt_of_ge
rw [norm_mul, Int.natAbs_mul]
apply le_mul_of_one_le_right (Nat.zero_le _)
apply Int.ofNat_le.1
rw [coe_natAbs_norm]
exact Int.add_one_le_of_lt ((norm_pos _).mpr hy)
We can now put it together to show that the Gaussian integers are an instance of a Euclidean domain. We use the quotient and remainder function we have defined. The Mathlib definition of a Euclidean
domain is more general than the one above in that it allows us to show that remainder decreases with respect to any well-founded measure. Comparing the values of a norm function that returns natural
numbers is just one instance of such a measure, and in that case, the required properties are the theorems natAbs_norm_mod_lt and not_norm_mul_left_lt_norm.
instance : EuclideanDomain GaussInt :=
{ GaussInt.instCommRing with
quotient := (· / ·)
remainder := (· % ·)
quotient_mul_add_remainder_eq :=
fun x y ↦ by simp only; rw [mod_def, add_comm] ; ring
quotient_zero := fun x ↦ by
simp [div_def, norm, Int.div']
r := (measure (Int.natAbs ∘ norm)).1
r_wellFounded := (measure (Int.natAbs ∘ norm)).2
remainder_lt := natAbs_norm_mod_lt
mul_left_not_lt := not_norm_mul_left_lt_norm }
An immediate payoff is that we now know that, in the Gaussian integers, the notions of being prime and being irreducible coincide.
example (x : GaussInt) : Irreducible x ↔ Prime x := | {"url":"https://leanprover-community.github.io/mathematics_in_lean/C06_Structures.html","timestamp":"2024-11-11T23:23:27Z","content_type":"text/html","content_length":"254187","record_id":"<urn:uuid:ba24d70b-4384-4e36-8113-13defa3b619e>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00371.warc.gz"} |
The Research Group of Mark van der Laan
This post is part of our Q&A series. A question from graduate students in our Spring 2021 offering of the new course “Targeted Learning in Practice” at UC Berkeley: Question: Hi Mark, Statistical
analyses of high-throughput sequencing data are often made difficult due to the presence of unmeasured sources of technical and biological variation. Examples of potentially unmeasured technical
factors are the time and date when individual samples were prepared for sequencing, as well as which lab personnel performed the experiment. | {"url":"https://vanderlaan-lab.org/","timestamp":"2024-11-05T12:52:56Z","content_type":"text/html","content_length":"22194","record_id":"<urn:uuid:2fe9661c-f116-49d1-865e-3b7d6daf689e>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00341.warc.gz"} |
Why is the Euclidean line the same as the real line?
This paper investigates whether the differential structure of spacetime follows from accepted laws of physics, or is a mathematical invention. In view of the results of [7], it suffices to consider
whether the assumed identity of the Euclidean line E - the line of the geometers of ancient Greece - and the real line R of modern analysis, follows from any known law of nature (i.e., one that can
be falsified empirically). Since the totality of empirical data is finite, one is forced to conclude that the completeness of R cannot be falsified empirically - and therefore, according to Popper's
criterion, the real line must be an invention, and not a discovery. It then becomes difficult to tell whether Newton's second law of motion (expressed as the differential equation f = mẍ) is an
invention or a discovery! Finally, some alternatives to the above analysis are briefly analysed.
• Discrete to continuous in physics
• Invention or discovery
• Mathematics
• Structure of the Euclidean line
ASJC Scopus subject areas
• General Physics and Astronomy
Dive into the research topics of 'Why is the Euclidean line the same as the real line?'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/why-is-the-euclidean-line-the-same-as-the-real-line","timestamp":"2024-11-02T08:01:30Z","content_type":"text/html","content_length":"55281","record_id":"<urn:uuid:32eb60fb-9c27-4cc7-9807-593c20ef1c33>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00894.warc.gz"} |
sq yd to sq ft and sq ft to sq yd
Seamlessly convert square yards to square feet and back with our Precision Area Converter at examples.com. Experience accurate and instant results every time.
sq yd to sq ft
Formula: Length in square foot = Length in square yard × 9
Square Yard Square Foot Exponential
1 9 9e+0
sq ft to sq yd
Formula: Length in square yard = Length in square foot ÷ 9
Square Foot Square Yard Exponential
1 0.111111111 1.11111111e-1
Area Converters to Square Yard (sq yd)
Area Converters to Square Foot (sq ft)
Conversion Factors:
• Square Yards to Square Feet: 1 square yard = 9 square feet
• Square Feet to Square Yards: 1 square foot = 1/9 square yard
How to Convert Square Yards to Square Feet:
To convert square yards to square feet, multiply the number of square yards by 9.
Square Feet=Square Yards×9
Example: Convert 5 square yards to square feet.
Square Feet=5×9=45 square feet
How to Convert Square Feet to Square Yards:
To convert square feet to square yards, divide the number of square feet by 9.
Square Yards=Square Feet/9
Example: Convert 18 square feet to square yards.
Square Yards=18/9=2 square yards
Square Yards to Square Feet Conversion Table
Square Yards Square Feet
1 sq yd 9 sq ft
2 sq yd 18 sq ft
3 sq yd 27 sq ft
4 sq yd 36 sq ft
5 sq yd 45 sq ft
6 sq yd 54 sq ft
7 sq yd 63 sq ft
8 sq yd 72 sq ft
9 sq yd 81 sq ft
10 sq yd 90 sq ft
20 sq yd 180 sq ft
30 sq yd 270 sq ft
40 sq yd 360 sq ft
50 sq yd 450 sq ft
60 sq yd 540 sq ft
70 sq yd 630 sq ft
80 sq yd 720 sq ft
90 sq yd 810 sq ft
100 sq yd 900 sq ft
sq yd to sq ft Conversion Chart
Square Feet to Square Yards Conversion Table
Square Feet Square Yards
1 sq ft 0.111 sq yd
2 sq ft 0.222 sq yd
3 sq ft 0.333 sq yd
4 sq ft 0.444 sq yd
5 sq ft 0.556 sq yd
6 sq ft 0.667 sq yd
7 sq ft 0.778 sq yd
8 sq ft 0.889 sq yd
9 sq ft 1.000 sq yd
10 sq ft 1.111 sq yd
20 sq ft 2.222 sq yd
30 sq ft 3.333 sq yd
40 sq ft 4.444 sq yd
50 sq ft 5.556 sq yd
60 sq ft 6.667 sq yd
70 sq ft 7.778 sq yd
80 sq ft 8.889 sq yd
90 sq ft 10.000 sq yd
100 sq ft 11.111 sq yd
sq ft to sq yd Conversion Chart
Difference Between Square Yards to Square Feet
Aspect Square Yards Square Feet
Unit Size Larger unit; covers a broader area. Smaller unit; covers a smaller area.
Conversion Factor 1 square yard = 9 square feet. 1 square foot = 0.111 square yards.
Usage in Contexts Commonly used in measuring larger areas like lawns. Often used for smaller areas like rooms.
Measurement Standard Typically used in countries using the imperial system. Widely used in the U.S. for commercial and residential purposes.
Mathematical Calculations Multiplying by 9 to convert to square feet. Dividing by 9 to convert to square yards.
Precision Less precise for small measurements due to larger size. More precise for small measurements.
Common Applications Used in sports fields, carpeting, and some real estate. Used in architecture, interior design, and floor plans.
Visualization Harder to visualize for indoor spaces. Easier to visualize for indoor applications.
1. Solved Examples on Converting Square Yards to Square Feet
Example 1: Converting 1 Square Yard to Square Feet
1 square yard=1×9 square feet
1 square yard is 9 square feet.
Example 2: Converting 5 Square Yards to Square Feet
5 square yards=5×9 square feet=45 square feet
5 square yards is 45 square feet.
Example 3: Converting 2.5 Square Yards to Square Feet
2.5 square yards=2.5×9 square feet =22.5 square feet
2.5 square yards is 22.5 square feet.
Example 4: Converting 10 Square Yards to Square Feet
10 square yards=10×9=90 square feet
10 square yards is 90 square feet.
Example 5: Converting 8 Square Yards to Square Feet
8 square yards=8×9=72 square feet
8 square yards is 72 square feet.
2. Solved Examples on Converting Square Feet to Square Yards
Example 1: Converting 9 Square Feet to Square Yards
Square Yards=9 square feet/9=1 square yard
9 square feet is 1 square yard.
Example 2: Converting 27 Square Feet to Square Yards
Square Yards=27 square feet/9=3 square yards
27 square feet is 3 square yards.
Example 3: Converting 45 Square Feet to Square Yards
Square Yards=45 square feet/9=5 square yards
45 square feet is 5 square yards.
Example 4: Converting 18 Square Feet to Square Yards
Square Yards=18 square feet/9=2 square yards
18 square feet is 2 square yards.
Example 5: Converting 90 Square Feet to Square Yards
Square Yards=90 square feet/9=10 square yards
90 square feet is 10 square yards.
What are some common uses for converting square yards to square feet?
Common uses include calculating the area for flooring materials, carpeting, landscaping projects, and large surface areas in construction projects.
How accurate is the conversion from square yards to square feet?
The conversion is mathematically exact. One square yard is precisely 9 square feet, so as long as the initial measurement is accurate, the conversion will also be accurate.
Are square yards more commonly used in certain industries than square feet?
Square yards are often used in the carpet and flooring industry, while square feet are more common in real estate, construction, and interior design.
How does converting square yards to square feet help in home renovations?
Converting square yards to square feet can help determine the amount of material needed for renovations, ensuring accurate calculations for flooring, paint, and other supplies.
What should be considered when converting irregularly shaped areas from square yards to square feet?
When converting irregularly shaped areas, ensure you accurately measure the total area in square yards first, then convert to square feet by multiplying by 9. Use tools like planimeters or GIS
software for precision.
Can you explain the difference between linear yards and square yards?
Linear yards measure length (1-dimensional), whereas square yards measure area (2-dimensional). Converting square yards to square feet involves area measurement, not just length.
How can conversion between square yards and square feet assist in real estate listings?
Real estate listings often use square feet to describe property size. Converting square yards to square feet can make listings clearer and more accessible to potential buyers familiar with square
foot measurements.
Are there any software tools that automatically convert square yards to square feet in design applications?
Yes, many design and architecture software tools like AutoCAD, SketchUp, and Revit can automatically convert units, including square yards to square feet, streamlining the design process. | {"url":"https://www.examples.com/maths/sq-yd-sq-ft","timestamp":"2024-11-09T06:55:27Z","content_type":"text/html","content_length":"113181","record_id":"<urn:uuid:1dd23c33-fcb9-41a6-8450-b97b16b8e48c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00686.warc.gz"} |
How To Write The Ordinal Numbers - OrdinalNumbers.com
Ordinal Numbers Sums – A vast array of sets can be enumerated using ordinal numbers as a tool. They can also be used to generalize ordinal quantities. 1st The ordinal number is among the fundamental
concepts in mathematics. It is a number that indicates where an object is in a list. The ordinal number is … Read more
Ordinal Numbers And Months Of The Year Puzzle
Ordinal Numbers And Months Of The Year Puzzle – It is possible to enumerate an unlimited number of sets by making use of ordinal numbers as an instrument. It is also possible to use them to
generalize ordinal numbers. 1st One of the fundamental ideas of math is the ordinal number. It is a number … Read more
Ordinal Numbers Posters
Ordinal Numbers Posters – You can list an unlimited amount of sets using ordinal figures as tool. These numbers can be utilized as a method to generalize ordinal figures. 1st The ordinal number is
among the foundational ideas in mathematics. It is a number that identifies the location of an object within the list. An … Read more
Ordinal Numbers Challenge
Ordinal Numbers Challenge – An unlimited number of sets can be listed by using ordinal numbers as tools. You can also use them to generalize ordinal number. 1st The ordinal number is among the
foundational ideas in mathematics. It is a number that identifies the position of an object within a list. Ordinal numbers are … Read more
Ordinal Numbering
Ordinal Numbering – An unlimited number of sets can be listed using ordinal numbers to serve as an instrument. They are also able to generalize ordinal numbers.But before you can utilize them, you
must comprehend what they exist and how they function. 1st The ordinal number is one of the foundational ideas in math. It … Read more
Article Before Ordinal Numbers
Article Before Ordinal Numbers – There are a myriad of sets that can be listed using ordinal numbers to serve as an instrument. They also can be used as a generalization of ordinal quantities. 1st
One of the basic concepts of mathematics is the ordinal number. It is a number that indicates the position of … Read more
How To Write Ordinal Numbers In Word
How To Write Ordinal Numbers In Word – A vast array of sets can be enumerated using ordinal numbers to serve as a tool. They can also be used as a method to generalize ordinal numbers. 1st One of the
most fundamental concepts of mathematics is the ordinal numbers. It is a number that indicates … Read more
Ordinal Numbers Display
Ordinal Numbers Display – There are a myriad of sets that can easily be enumerated with ordinal numerals to aid in the process of. They are also able to broaden ordinal numbers.But before you use
them, you need to know the reasons why they exist and how they operate. 1st The ordinal numbers are one … Read more
Months Of The Year And Ordinal Numbers Worksheet
Months Of The Year And Ordinal Numbers Worksheet – You can count the infinite amount of sets making use of ordinal numbers as tool. They also can help generalize ordinal quantities. 1st The basic
concept of math is the ordinal. It is a number that indicates the place of an object within an array of … Read more
Ordinal Numbers Ap Style
Ordinal Numbers Ap Style – An unlimited number of sets can be listed using ordinal numbers to serve as an instrument. It is also possible to use them to generalize ordinal number. 1st The basic
concept of mathematics is the ordinal. It is a number that indicates where an object is in a list of … Read more | {"url":"https://www.ordinalnumbers.com/tag/how-to-write-the-ordinal-numbers/","timestamp":"2024-11-02T11:09:21Z","content_type":"text/html","content_length":"101472","record_id":"<urn:uuid:c530dc23-cbea-4ce5-b7ab-effe58488aa1>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00691.warc.gz"} |
Program - Complex Network
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut
aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa
qui officia deserunt mollit anim id est laborum
morning (9.30-12.30)
Introduction to complex networks (Serrano): What is a complex system? Networks: a change of paradigm. Basic network representations: unweighted/weighted, undirected/directed, unipartite/bipartite,
singlelayered/multilayered. Mathematical and computational encodings of networks. Basic network metrics: global, local, and mesoscopic properties. Basic network models: Erdos-Renyi, Configuration
Model, Watts-Strogatz, Barabasi-Albert. Dynamical processes: the Voter model; epidemic spreading.
afternoon (14.30-17.30)
Implications of power laws on algorithms in complex networks (Litvak): Background: power laws as a mathematical model for network hubs, highest degrees. I. Centrality: how to find hubs quickly,
PageRank in power law networks. Connectivity: why power law networks are never disassortative, weighted triangles for detecting geometry.
morning (9.30-12.30)
Network geometry (Serrano): Distances in complex networks. Spatial random graphs. Hyperbolic geometry: the S^1/ H^2 model; network maps and embedding techniques. I. The problem of scales. Geometric
renormalization: coarse-graining and fine-graining; self-similarity of real multsicale networks; self-similar evolution of real networks. Scaled down and scaled up network replicas. II. The problem
of dimensions. Dimensionality of real networks, the S^D/ H^D+1 model. Dimensional reduction, multidimensional hyperbolic maps.
afternoon (14.30-17.30)
No lectures
morning (9.30-12.30)
Advanced network models: multilayer and higher-order networks (I) (Battiston): Multilayer networks: vectorial formalism, basic node, edge and local properties, shortest paths, correlations,
reducibility. Impact of multiplexity on dynamics (public goods game and other examples). Higher-order networks: basic ideas, structural analysis of higher-order networks with HGX (basic node and edge
properties, motif analysis, community detection, temporal correlations). Impact of non-pairwise interactions on dynamics (social dilemmas and other examples).
afternoon (14.30-17.30)
Advanced network models: multilayer and higher-order networks (II) (Ferraz de Arruda): Multilayer networks: tensorial formalism, spectral properties, assortativity. Higher-order networks: epidemic
spreading and social contagion processes, spectral properties, random walks.
evening (20.00)
social dinner
morning (9.30-12.30)
short talks by students
No lectures
morning (9.30-12.30)
Evolutionary game theory and human behaviour (Lenaerts): (1) A general introduction to game structures, solution and equilibria concepts: simultaneous, sequential and stochastic games, social
dilemmas, Nash and other equilibria, minimax and regret. (2) Evolutionary game theory in well-mixed and networked populations: simulations, replicator dynamics, Moran processes, evolutionary
stability, evolutionary robustness, five rules of cooperation and the impact of topology on dynamics. (3) Case studies related to cognitive capacities and delegation of decision-making, illustrated
with some behavioral experiment data.
afternoon (14.30-17.30)
Networks for Economics and Society (del Rio-Chanona): (1) Application of the network clustering algorithms known as Economic Complexity for analyzing national development and workers’ capabilities.
(2) Multilayered Systems: Explores the use of multilayer networks, in particular, the interplay between production and worker networks, with a focus on assortativity in the net-zero transition. (3)
Agent-Based Modeling: Details the integration of economic theory into network modeling. We discuss an agent-based model for the labor market dynamics during technological automation and displacement
of workers and another for analyzing the health-economy trade-off during the COVID-19 pandemic. | {"url":"https://ntmh.lakecomoschool.org/program/","timestamp":"2024-11-10T21:12:52Z","content_type":"text/html","content_length":"120919","record_id":"<urn:uuid:749c3d3c-bc63-41ee-84de-c7812a916eda>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00078.warc.gz"} |
xt Course 2023SP Stataties MAX 00 Lei Edinity Write your answer... | Filo
Question asked by Filo student
xt Course 2023SP Stataties MAX 00 Lei Edinity Write your answer in the box below. edfinity.com/assessments/630485452e5109007662408 04 (a) I am estimating the ability to throw a crumpled piece of
paper into a wastebasket. I am using this class as a sample. Attempt to toss the paper into the basket ten times. How many times did the crumpled paper fall into the basket? • Place a wastebasket in
a room away from any wall or piece of furniture, Stand approximately 8 to 10 feet from the basket. . From that position, throw a piece of crumpled paper into the basket. • Repeat the throw ten times.
Record how many times the paper ends up in the basket. DOLL I will place your result into a table with the other student results. b) I want to choose a target population whose throwing ability can be
described well with the paper-throwing statistics (average, median, standard deviation range...) that I received from the students in this class. Here are some candidate populations:
a. MAT 181 students that commute to class by car.
b. Students in my classes for this year and the last ten years.
c. Online Boston area students during the Pandemic.
d. Online Community College students that have live Webex or Zoom classes with required attendance.
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xt Course 2023SP Stataties MAX 00 Lei Edinity Write your answer in the box below. edfinity.com/assessments/630485452e5109007662408 04 (a) I am estimating the ability to throw a crumpled
piece of paper into a wastebasket. I am using this class as a sample. Attempt to toss the paper into the basket ten times. How many times did the crumpled paper fall into the basket? • Place
Question a wastebasket in a room away from any wall or piece of furniture, Stand approximately 8 to 10 feet from the basket. . From that position, throw a piece of crumpled paper into the basket. •
Text Repeat the throw ten times. Record how many times the paper ends up in the basket. DOLL I will place your result into a table with the other student results. b) I want to choose a target
population whose throwing ability can be described well with the paper-throwing statistics (average, median, standard deviation range...) that I received from the students in this class.
Here are some candidate populations:
Updated Jan 29, 2023
Topic The Sampling Distribution of the Sample Mean
Subject Statistics
Class High School | {"url":"https://askfilo.com/user-question-answers-statistics/xt-course-2023sp-stataties-max-00-lei-edinity-write-your-34313737373435","timestamp":"2024-11-05T15:25:04Z","content_type":"text/html","content_length":"122796","record_id":"<urn:uuid:2333afae-f1fd-448c-81e9-6f01a4b3d0b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00005.warc.gz"} |
Physics Union Mathematics
ISLE Curriculum Summary
ISLE achieves the goals by developing science and mathematic abilities
ISLE (Investigative Science Learning Environment) is a comprehensive learning system originally developed for introductory college-level physics (funded by NSF DUE-0241078 and DUE-0088906). It
incorporates the processes through which scientists acquire knowledge into students’ learning. It also purposefully utilizes a large array of multiple representations helping students move from
concrete representations to abstract mathematical representations. The logic of ISLE is based on the elements of scientific reasoning (inductive, analogical and hypothetico-deductive) and its
practical implementation is based on the science of learning: multiple intelligences and multiple representations, collegial work, learning communities, and formative assessment with constructive
feedback. Also ISLE’s library of tasks helps students develop various abilities used in the practice of science, mathematics and engineering, including the ability to represent knowledge in multiple
ways, design experiments to formulate and test hypothesis, account for anomalous data, evaluate reasoning and experimental results, and communicate. A part of ISLE resources is a website with more
than 200 videotaped experiments supported by questions that can be used for data collection, pattern recognition, model building and testing, and analysis of anomalous data (http://paer.rutgers.edu/
pt3). In addition, there is a validated set of rubrics that can help students self-assess their mastery of these abilities. (Examples of rubrics are provided below; all of them are available at http:
//paer.rutgers.edu/scientificabilities). These tasks and rubrics have been developed as a part of an NSF project. They have been used in introductory college classes, teacher preparation classes and
in professional development of middle school science teachers. ISLE’s summative assessments indicate that the curriculum is very effective in helping students master: (1) traditional physics content
(learning gains of .56 on the Force Concept Inventory, post-test scores of 73% on Conceptual Survey of Electricity and Magnetism), (2) problem solving skills (ISLE students scored on average 76%
correct when traditionally taught students scored 61% correct on the same 8 problems chosen by the professor who taught the traditional class), and (3) in helping them acquire scientific abilities.
The strengths of the ISLE system and the ALG curriculum is the repeated process though which students acquire knowledge and abilities. ISLE students begin each conceptual unit by observing simple,
carefully selected phenomena (called observational experiments). They look for patterns in their observations and produce qualitative explanations based on these patterns. To find these patterns they
analyze their observations using different representations, learning to reason in ways that are natural to their own learning styles. They then use their own explanations to make predictions about
the outcomes of new experiments (called testing experiments, see an example at the end of this section). Based on the outcomes of their testing experiments, confidence in their explanation increases,
or it needs revision, or it is rejected.
This process is then repeated only this time using quantitative mathematical representations—using “progressive differentiation” (e.g. from qualitative understanding to more precise quantitative
understanding of a particular phenomenon), which involves a simultaneous focus on the structure of knowledge to be mastered and the learning process of students. During this quantitative stage,
students solve context-rich problems using multiple representation techniques. Students learn to represent processes in multiple ways and to check for consistency of the representations—for example,
the consistency of a free-body diagram and the application of Newton’s second law in component form to an object involved in some process. In labs students design their own experiments to test
principles quantitatively and qualitatively and solve challenging experimental problems. Once students have developed enough conceptual knowledge, they can use that knowledge to understand and
analyze a variety of modern technology, including devices such as: motion detectors, global positioning software and hardware, fiber optics technology, cell phones, force probes, electronic scales,
analogue and digital galvanometers, magnetic field probes, spectrometers, and digital cameras. It is important to note that during all stages of this process students work cooperatively and learn to
come to a consensus in terms of what they observed, how to explain it, and how to test the explanations. Student reasoning is heavily scaffolded by the curriculum materials, which suggests materials
and questions for the analysis of observational experiments and testing experiments, guides students through the invention of new physical quantities, and provides data to help students find
relationships between the quantities.
In The Physics Active Learning Guide (Van Heuvelen & Etkina, 2006) is a set of activities that follow the ISLE philosophy and can be used in a college classroom. all activities are grouped into 4
categories in each chapter: qualitative concept building and testing, conceptual reasoning, quantitative concept building and testing, and quantitative reasoning. It is this repeated structure that
allows student to see how science ideas are built on evidence, tested by evidence, and applied to practical life. Additionally it builds mathematics knowledge by engaging students in data analysis,
pattern recognition, proportional reasoning, unit conversions, measurement, algebraic representations, linear functions, integers, estimation, significant figures, charts and graphing, algebraic
mean, vectors, trigonometry, rates, symbolic representations, etc.
ALG activities form a natural sequence of concept building and formative assessment at progressively more difficult levels, allowing them to be used at different levels of mathematical and physical
sophistication – starting with qualitative and quantitative concept building and testing in middle school, which involves use of appropriate pre-algebra and algebra reasoning skills and then moving
to reasoning that involves more complicated algebra-2 concepts.
ISLE naturally and strongly supports the perspectives on the 4 lenses of learning environments supported by the National Science Foundation. As we mentioned above, learning of ISLE students has been
studied at various levels – using standardized tests, traditional problems, and performance assessments. The results indicate that the students do learn physics content as well and better than
students taught though other reformed curricula, much better than students taught traditionally, and they acquire a vast array of scientific abilities.
The ISLE system and ALG activities have been successfully implemented in the calculus-based physics for regular engineering majors and engineers at-risk, in algebra-based courses for science majors,
in physics methods courses for future physics teachers, and in professional development workshops for middle school teachers. | {"url":"http://pum.islephysics.net/isle.php","timestamp":"2024-11-05T19:09:36Z","content_type":"text/html","content_length":"13005","record_id":"<urn:uuid:88b0085d-dba1-42de-9194-843ba1b513c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00852.warc.gz"} |
Turbulence and singularity of the Einstein equation with a negative cosmological constant studied from classical turbulence theory
Project/Area 16K13850
Research Category Grant-in-Aid for Challenging Exploratory Research
Allocation Type Multi-year Fund
Research Field Mathematical physics/Fundamental condensed matter physics
Research Kyoto University
Project Period 2016-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Budget Amount
*help Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords 古典乱流のカスケード / スケーリング則 / 偏微分方程式の爆発解 / 乱流理論 / 重力方程式
It has been known that a certain spherical symmetric Einstein equation with a negative cosmological constant has a turbulent solution by a numerical simulation. The equation has a
Outline of Final global conservation law. The turbulent solution successively generates small-scale activities by respecting a specific scaling law. The generation occurring in an accelerated
Research manner is considered to reach the infinitesimally small scale in a finite time.
Achievements We studied this turbulent solution and its singular behavior with the method of analyzing turbulence of the Navier-Stokes equations. In particular, we adopted the picture of the
energy cascade in the Navier-Stokes turbulence to the conservative quantity of the turbulent solution. So far we were not able to explain the exponent of the power-law scaling law
phenomenologicaly. Nevertheless, the cascade picture shed a new light to the small-scale generation mechanism of the turbulent solution.
(3 results)
Research Products
(2 results) | {"url":"https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16K13850/","timestamp":"2024-11-12T09:52:29Z","content_type":"text/html","content_length":"21113","record_id":"<urn:uuid:66c515d4-aa08-4a0e-9e36-08fbbee92b2f>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00306.warc.gz"} |
How To Find The Equation Of A Line Example - Graphworksheets.com
Finding The Equation Of A Graphed Line Worksheet – Line Graph Worksheets will help you understand how a line graph functions. There are many types of line graphs and each one has its own purpose. We
have worksheets that can be used to teach children how to draw, read, and interpret line graphs. Create a … Read more | {"url":"https://www.graphworksheets.com/tag/how-to-find-the-equation-of-a-line-example/","timestamp":"2024-11-11T03:27:26Z","content_type":"text/html","content_length":"46303","record_id":"<urn:uuid:b321cd81-93dd-437d-b3a7-b99ab04279de>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00809.warc.gz"} |
Adam Horvath's blog
The formula of a circle is quite straightforward ( r=sqrt(x^2+y^2) ) but it's not trivial how to draw a circle or calculate the trigonometric functions without advanced math. However, an interesting
finding from 1972 makes it really easy. Minsky discovered (by a mistake) that the following loop will draw an almost perfect circle on the screen: loop: x = x - epsilon * y y = y + epsilon * x #
NOTE: the x is the new x value here Turns out that epsilon in the equation is practically the rotation angle in radians, so the above loop will gradually rotate the x and y coordinates in a circle.
Calculating sinus If we can draw this circle, we can easily estimate the sinus values: for the current angle, which is basically the sum of epsilons so far, we have a height (y), which just needs to
be normalized to the 0-1 range to get the actual sinus for the angle. The smaller the steps (epsilon) are, the more accurate the formula will be. However, because it's not a p
One of the typical interview questions is the three way partitioning, also known as the Dutch national flag problem: given an array with three different values, sort it in a way that all values are
grouped together (like a three colored flag) in linear time without extra memory. The problem was first described by Edsger Dijkstra. As it's a typical sorting problem, any sorting would do but they
would need n*log(n) time. There are two good solutions to it: do a counting sort which requires two passes over the array; place the elements to their correct location using two pointers. The latter
uses a reader, lower, and an upper pointer to do the sorting. Reader and lower starts from one end of the array, upper pointer starts at the other end. The algorithms goes like this: - If it's a
small value, swap it with the lower pointer and step lower pointer one up - If it's a middle value, do nothing with it and step reader pointer one up - If it's a larger value, swap | {"url":"https://blog.teamleadnet.com/2013/09/","timestamp":"2024-11-04T21:39:32Z","content_type":"text/html","content_length":"94645","record_id":"<urn:uuid:da53bd33-0e89-4b46-95ed-ec0638576792>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00435.warc.gz"} |
Gradient | Andrew Gurung
A gradient is a vector that stores the partial derivatives of multi variable functions, often denoted by $abla$. It helps us calculate the slope at a specific point on a curve for functions with
multiple independent variables.
Find a Gradient
Consider a function with two variables (x and y): $f(x,y) = x^2 + y^3$
1) Find partial derivative with respect to x (Treat y as a constant like a random number 12)
$f'_x = \frac{\partial f}{\partial x} = 2x+0=2x$
2) Find partial derivative with respect to y (Treat x as a constant)
$f'_y =\frac{\partial f}{\partial y}= 0+3y^2=3y^2$
3) Store partial derivatives in a gradient
$abla f(x,y) = \begin{bmatrix}\frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y}\end{bmatrix} = \begin{bmatrix}2x \\ 3y^2\end{bmatrix}$
Properties of Gradients
There are two additional properties of gradients that are especially useful in deep learning. A gradient:
Always points in the direction of greatest increase of a function (explained here)
Is zero at a local maximum or local minimum
Directional Derivative
The directional derivative $abla_{\vec v} f$ is the rate at which the function $f(x,y)$ changes at a point $(x_1,y_1)$ in the direction ${\vec v}$.
Directional derivative is computed by taking the dot product of the gradient of $f$ and a unit vector ${\vec v}$
Note: Directional derivative of a function is a scalar while gradient is a vector.
Find Directional Derivative
Consider a function with two variables (x and y): $f(x,y) = x^2 + y^3$
${\vec v} = \begin{bmatrix}2 \\ 3\end{bmatrix}$
As described above, we take the dot product of the gradient and the directional vector:
$\begin{bmatrix}\frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y}\end{bmatrix} . \begin{bmatrix}2 \\ 3\end{bmatrix} ewline ewline ewline$
We can rewrite the dot product as:
$abla_{\vec v} f = 2\frac{\partial f}{\partial x} +3 \frac{\partial f}{\partial y} = 2(2x) + 3(3y^2)=4x+9y^2$
Hence, the directional derivative $abla_{\vec v} f$at co-ordinates $(5, 4)$ is: $abla_{\vec v} f = 4x+9y^2 = 4(5)+9(4)^2 = 164$
Link: - http://wiki.fast.ai/index.php/Calculus_for_Deep_Learning | {"url":"https://notes.andrewgurung.com/data-science/calculus/gradient","timestamp":"2024-11-06T10:38:03Z","content_type":"text/html","content_length":"401209","record_id":"<urn:uuid:2f3522ba-82cb-463b-9d37-14bb0419fdf3>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00447.warc.gz"} |
1. Measures
≪ 0. Overview of Measure Theory
Table of Contents
1.1. \(\sigma\)-Algebras ≫
In order to develop the idea of a derivative and an integral, we needed to use the theory of metric spaces (as well as the completeness of \(\R\)).
In much the same way, in order to extend the integral, we need to first lay the groundwork with measures and measurable spaces.
From a physical or intuitive perspective, measures rigorously generalise our notions of length, area, weight, etc., just like how metrics rigorously generalise our notions of distance.
One particular property of metrics is the triangle inequality, and that comes from our intuitive knowledge of how distances work. If you want to travel from point A to point B, taking a detour won’t
make that trip any shorter. In much the same way, we have some basic intuitive expectations of how measures should work. For now, I’ll use the “length” of an interval as our prototypical example: the
length of \([a, b]\) is just \(b-a\) when \(b\geq a\).
• The length of the empty set is zero.
• If \(I_1\) and \(I_2\) are disjoint intervals, the length of \(I_1\cup I_2\) is the sum of the lengths of \(I_1\) and \(I_2\).
It helps to extend this second property (AKA disjoint additivity) to countable collections of disjoint sets, but this has the consequence of so-called unmeasurable sets. The Vitali set is an infamous
example, and we’ll discuss this in more detail later in the chapter.
Measure theory ended up being highly applicable to probability theory, which lead to the development of things like stochastic calculus. This then blossomed into applied fields such as physics and
economics, thanks to the concept of Brownian motion.
This chapter will focus specifically on measures and measurable spaces (sets with a measure defined on them). We’ll develop their basic properties and observe some helpful ways to classify measures. | {"url":"https://hunterliu.xyz/math/notes/measure-theory/measures/","timestamp":"2024-11-07T23:40:46Z","content_type":"text/html","content_length":"4243","record_id":"<urn:uuid:66ea2f07-1772-4fad-a846-8c8d02e0f9e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00348.warc.gz"} |
From gigawatt to multi-gigawatt wind farms: wake effects, energy budgets and inertial gravity waves investigated by large-eddy simulations
Articles | Volume 8, issue 4
© Author(s) 2023. This work is distributed under the Creative Commons Attribution 4.0 License.
From gigawatt to multi-gigawatt wind farms: wake effects, energy budgets and inertial gravity waves investigated by large-eddy simulations
The size of newly installed offshore wind farms increases rapidly. Planned offshore wind farm clusters have a rated capacity of several gigawatts and a length of up to 100km. The flow through and
around wind farms of this scale can be significantly different than the flow through and around smaller wind farms on the sub-gigawatt scale. A good understanding of the involved flow physics is
vital for accurately predicting the wind farm power output as well as predicting the meteorological conditions in the wind farm wake. To date there is no study that directly compares small wind farms
(sub-gigawatt) with large wind farms (super-gigawatt) in terms of flow effects or power output. The aim of this study is to fill this gap by providing this direct comparison by performing large-eddy
simulations of a small wind farm (13km length) and a large wind farm (90km length) in a convective boundary layer, which is the most common boundary layer type in the North Sea.
The results show that there are significant differences in the flow field and the energy budgets of the small and large wind farm. The large wind farm triggers an inertial wave with a wind direction
amplitude of approximately 10^∘ and a wind speed amplitude of more than 1ms^−1. In a certain region in the far wake of a large wind farm the wind speed is greater than far upstream of the wind
farm, which can be beneficial for a downstream located wind farm. The inertial wave also exists for the small wind farm, but the amplitudes are approximately 4 times weaker and thus may be hardly
observable in real wind farm flows that are more heterogeneous. Regarding turbulence intensity, the wake of the large wind farm has the same length as the wake of the small wind farm and is only a
few kilometers long. Both wind farms trigger inertial gravity waves in the free atmosphere, whereas the amplitude is approximately twice as large for the large wind farm. The inertial gravity waves
induce streamwise pressure gradients inside the boundary layer, affecting the energy budgets of the wind farms. The most dominant energy source of the small wind farm is the horizontal advection of
kinetic energy, but for the large wind farm the vertical turbulent flux of kinetic energy is 5 times greater than the horizontal advection of kinetic energy. The energy input by the
gravity-wave-induced pressure gradient is greater for the small wind farm because the pressure gradient is greater. For the large wind farm, the energy input by the geostrophic forcing
(synoptic-scale pressure gradient) is significantly enhanced by the wind direction change that is related to the inertial oscillation. For both wind farms approximately 75% of the total available
energy is extracted by the wind turbines and 25% is dissipated.
Received: 06 Jul 2022 – Discussion started: 26 Jul 2022 – Revised: 04 Jan 2023 – Accepted: 12 Mar 2023 – Published: 13 Apr 2023
The size of newly installed offshore wind farms increases rapidly. The largest wind farm in operation Moray East (United Kingdom) has a rated capacity of 950MW and consists of 100 wind turbines (
Herzig, 2022). The largest wind farm under construction is Hollandse Kust Zuid (Netherlands), with a rated capacity of 1540MW. It consists of 140 wind turbines and has a length of approximately 15
km (Herzig, 2022). Offshore wind farms are often arranged in clusters, so that the cluster capacity can already be in the multi-gigawatt scale. One example is the planned wind farm cluster in Zone 3
in the German Bight with a planned capacity of 20GW and a length of approximately 100km (BSH, 2021).
The flow through and around wind farms of this scale can be significantly different than the flow through and around smaller wind farms on the sub-gigawatt scale, as recently published results show.
For example, large wind farms can cause a significant counterclockwise wind direction change in the wake and a vertical displacement of the inversion layer above the wind farm (Allaerts and Meyers,
2016; Lanzilao and Meyers, 2022; Maas and Raasch, 2022). A good understanding of the involved flow physics is vital for accurately predicting the wind farm power output as well as predicting the
meteorological conditions in the wind farm wake. The “improved understanding of atmospheric and wind power plant flow physics” is stated as one of the grand challenges in the science of wind energy
by Veers et al. (2019) because the involved scales range from microscale to mesoscale and interactions can be complex. The best numerical method for the investigation of these interactions that
considers all relevant physical processes but is still computationally feasible is large-eddy simulation (LES).
In recent years many LES studies investigated wind farm flows. The studies can be subdivided into three categories. The first category investigates infinitely large wind farms by using cyclic
boundary conditions in both lateral directions, e.g., Abkar and Porté-Agel (2013), Abkar and Porté-Agel (2014), Calaf et al. (2010), Calaf et al. (2011), Johnstone and Coleman (2012), Lu and
Porté-Agel (2011), Lu and Porté-Agel (2015), Meyers and Meneveau (2013), Porté-Agel et al. (2014) and VerHulst and Meneveau (2014). The second category investigates semi-infinite wind farms by using
cyclic boundary conditions only in the crosswise direction, e.g., Allaerts and Meyers (2016), Allaerts and Meyers (2017), Allaerts and Meyers (2018), Andersen et al. (2015), Centurelli et al. (2021),
Segalini and Chericoni (2021), Stevens et al. (2016), Wu and Porté-Agel (2017) and Zhang et al. (2019). The third category investigates wind farms that have a finite size in both lateral directions
which also include real wind farms, e.g., Dörenkämper et al. (2015), Ghaisas et al. (2017), Lanzilao and Meyers (2022), Maas and Raasch (2022), Nilsson et al. (2015), Porté-Agel et al. (2013), Witha
et al. (2014) and Wu and Porté-Agel (2015).
Typical wind farm lengths in the semi-infinite wind farm studies range from 5km (Centurelli et al., 2021) over 15km (Allaerts and Meyers, 2017) to 24km (Andersen et al., 2015). Typical wind farm
lengths in the finite-size wind farm studies range from 2km (Witha et al., 2014) over 15km (Lanzilao and Meyers, 2022) to approximately 100km (Maas and Raasch, 2022). Thus, most of the studies are
representative for existing, state-of-the-art wind farms and do not represent the spatial scales that future wind farm clusters will have. Specifically, there is no study that directly compares small
wind farms (10km scale) with large wind farms (100km scale) in terms of flow effects or power output, neither with LESs nor with simpler models.
The aim of this study is to provide this direct, systematic comparison by performing LESs of a small wind farm (13km length) and a large wind farm (90km length) with a semi-infinite wind farm
setup. The comparison focuses on the boundary layer flow inside the wind farm but also in the far wake and the overlying free atmosphere. A detailed energy budget analysis is made to identify the
dominant energy sources and sinks for small and large wind farms. The domain is more than 400km long to cover the far wake and has a height of 14km to cover wind-farm-induced gravity waves. The
boundary layer is filled with a turbine-wake-resolving grid resulting in more than 2 billion grid points in total.
The paper is structured as follows. The numerical model and the main and precursor simulations are described in Sect. 2. The simulation results are presented in Sect. 3, and Sect. 4 concludes and
discusses the results of the study.
2.1Numerical model
The simulations are carried out with the Parallelized Large-eddy Simulation Model (PALM; Maronga et al., 2020). PALM is developed at the Institute of Meteorology and Climatology of Leibniz
Universität Hannover, Germany. Several wind farm flow investigations have been successfully conducted with this code in the past (e.g., Witha et al., 2014; Dörenkämper et al., 2015; Maas and Raasch,
2022). PALM solves the non-hydrostatic, incompressible Navier–Stokes equations in Boussinesq-approximated form. The equations for the conservation of mass, momentum and internal energy then read as
$\begin{array}{}\text{(1)}& \frac{\partial {\stackrel{\mathrm{̃}}{u}}_{j}}{\partial {x}_{j}}=\mathrm{0}\phantom{\rule{0.125em}{0ex}},\text{(2)}& \begin{array}{rl}\frac{\partial {\stackrel{\mathrm{̃}}
{u}}_{i}}{\partial t}& =-\frac{\partial {\stackrel{\mathrm{̃}}{u}}_{i}{\stackrel{\mathrm{̃}}{u}}_{j}}{\partial {x}_{j}}-{\mathit{ϵ}}_{ijk}{f}_{j}{\stackrel{\mathrm{̃}}{u}}_{k}+{\mathit{ϵ}}_{i\mathrm{3}
j}{f}_{\mathrm{3}}{u}_{\mathrm{g},j}-\frac{\mathrm{1}}{{\mathit{\rho }}_{\mathrm{0}}}\frac{\partial {\mathit{\pi }}^{*}}{\partial {x}_{i}}\\ & +g\frac{\stackrel{\mathrm{̃}}{\mathit{\theta }}-〈\
stackrel{\mathrm{̃}}{\mathit{\theta }}〉}{〈\stackrel{\mathrm{̃}}{\mathit{\theta }}〉}{\mathit{\delta }}_{i\mathrm{3}}-\frac{\partial }{\partial {x}_{j}}\left(\stackrel{\mathrm{̃}}{{u}_{i}^{\prime \
prime }{u}_{j}^{\prime \prime }}-\frac{\mathrm{2}}{\mathrm{3}}e{\mathit{\delta }}_{ij}\right)+{d}_{i}\phantom{\rule{0.125em}{0ex}},\end{array}\text{(3)}& \frac{\partial \stackrel{\mathrm{̃}}{\mathit{\
theta }}}{\partial t}=-\frac{\partial {\stackrel{\mathrm{̃}}{u}}_{j}\stackrel{\mathrm{̃}}{\mathit{\theta }}}{\partial {x}_{j}}-\frac{\partial }{\partial {x}_{j}}\left(\stackrel{\mathrm{̃}}{{u}_{j}^{\
prime \prime }{\mathit{\theta }}^{\prime \prime }}\right)\phantom{\rule{0.125em}{0ex}},\end{array}$
where angular brackets indicate horizontal averaging and a double prime indicates subgrid-scale (SGS) quantities; a tilde denotes filtering over a grid volume; $i,j,k\in \mathit{\left\{}\mathrm{1},\
mathrm{2},\mathrm{3}\mathit{\right\}}$, u[i], u[j], u[k] are the velocity components in the respective directions (x[i], x[j], x[k]); θ is potential temperature; t is time; and ${f}_{i}=\left(\mathrm
{0},\mathrm{2}\mathrm{\Omega }\mathrm{cos}\left(\mathit{\varphi }\right),\mathrm{2}\mathrm{\Omega }\mathrm{sin}\left(\mathit{\varphi }\right)\right)$ is the Coriolis parameter with the Earth's
angular velocity $\mathrm{\Omega }=\mathrm{0.729}×{\mathrm{10}}^{-\mathrm{4}}$rads^−1 and the geographical latitude ϕ. The geostrophic wind speed components are u[g,j], and the basic state density
of dry air is ρ[0]. The modified perturbation pressure is ${\mathit{\pi }}^{*}=p+\frac{\mathrm{2}}{\mathrm{3}}{\mathit{\rho }}_{\mathrm{0}}e$, where p is the perturbation pressure and $e=\frac{\
mathrm{1}}{\mathrm{2}}\stackrel{\mathrm{̃}}{{u}_{i}^{\prime \prime }{u}_{i}^{\prime \prime }}$ is the SGS turbulence kinetic energy. The gravitational acceleration is g=9.81ms^−2, δ is the Kronecker
delta and d[i] represents the forces of the wind turbine actuator discs.
The SGS model uses a 1.5-order closure according to Deardorff (1980), modified by Moeng and Wyngaard (1988) and Saiki et al. (2000). The wind turbines are represented by an advanced actuator disc
model with rotation (ADM-R) that acts as an axial momentum sink and an angular momentum source (inducing wake rotation). The ADM-R is described in detail by Wu and Porté-Agel (2011) and was
implemented in PALM by Steinfeld et al. (2015). Additional information is also given by Maas and Raasch (2022). The wind turbines have a yaw controller that aligns the rotor axis with the wind
2.2Main simulations
The study consists of two simulations. The first simulation contains a small wind farm with ${N}_{x}×{N}_{y}=\mathrm{8}×\mathrm{8}=\mathrm{64}$ wind turbines resulting in a length of 13.44km. The
second simulation contains a large wind farm with ${N}_{x}×{N}_{y}=\mathrm{48}×\mathrm{8}=\mathrm{384}$ wind turbines resulting in a length of 90.24km (see Fig. 1). The wind farms extend over the
entire domain width, and cyclic boundary conditions are applied in the y direction, so that the wind farms are effectively infinitely large in this direction. This idealized setup has been used in
many other LES wind farm studies, e.g., Stevens et al. (2016), Allaerts and Meyers (2017) and Wu and Porté-Agel (2017). It simplifies the data analysis and allows us to focus only on streamwise
variations in the wind farm and the wake. The validity of the results for finite-size, real wind farms is discussed in Sect. 4.
The IEA 15MW wind turbine with a rotor diameter of D=240m and a rated power of 15MW is used (Gaertner et al., 2020). The hub height is set to 180m instead of 150m, so that the turbulent fluxes
at the rotor bottom are better resolved by the numerical grid. The wind turbines are arranged in a staggered configuration and have a streamwise and crosswise spacing of s=8D, resulting in an
installed capacity density of 4.07Wm^−2. The small wind farm has a length of 13.44km, which corresponds approximately to the length of the currently largest wind farm under construction, Hollandse
Kust Zuid. The large wind farm has a length of 90.24km, which corresponds approximately to the length of the planned wind farm cluster in Zone 3 in the German Bight. Note that the small wind farm is
already as long as the largest wind farms of most other LES studies, e.g., Wu and Porté-Agel (2017) (19.6km) and Allaerts and Meyers (2017) (15km).
The domain has a length of L[x]=409.6km to cover the far wake of the wind farms. The wind farms have a distance of 100km to the inflow boundary, so that the wind-farm-induced flow blockage is
covered. The domain width is L[y]=15.36km for the small and large wind farm case. A domain height of L[z]=14.0km is required to cover wind-farm-induced gravity waves. That the Boussinesq
approximation is still valid for such a large domain height is shown in Appendix A. To avoid reflection of the waves at the domain top, there is a Rayleigh damping layer above z[rd]=5km. The
Rayleigh damping factor increases from zero at the bottom of the damping layer to its maximum value of ${f}_{\mathrm{rdm}}=\mathrm{0.025}\left(\mathrm{\Delta }t{\right)}^{-\mathrm{1}}\approx \mathrm
{0.017}$s^−1 at the domain top according to this function (see Fig. 1):
$\begin{array}{}\text{(4)}& {f}_{\mathrm{rd}}\left(z\right)={f}_{\mathrm{rdm}}{\mathrm{sin}}^{\mathrm{2}}\left(\mathrm{0.5}\mathit{\pi }\frac{z-{z}_{\mathrm{rd}}}{{L}_{z}-{z}_{\mathrm{rd}}}\right).\
This sine wave profile leads to fewer reflections compared to a linear profile (Klemp and Lilly, 1978). The choice of these parameters is based on a set of test simulations with a larger grid spacing
that are performed to find parameters that result in a low reflectivity. The reflectivity is obtained by the method described by Allaerts and Meyers (2017), which is a modified version of the method
described by Taylor and Sarkar (2007). With the chosen parameters, less than 6% of the upwards propagating wave energy is reflected.
The domain is filled with an equidistant regular grid with a grid spacing of 20m, yielding a density of 12 grid points per rotor diameter. This is enough to resolve the most relevant eddies inside
the wind turbine wakes. Steinfeld et al. (2015) showed that even eight grid points per rotor diameter are sufficient to obtain a converged result for the mean wind speed profiles at a downstream
distance of 5D. Above 900m the grid is vertically stretched by 8% per grid point up to a maximum vertical grid spacing of 200m, which is enough for resolving the gravity waves with a vertical
wavelength of approximately 5km (see Table 1 in Sect. 3.4). The numerical grid has the same structure in both cases and contains ${n}_{x}×{n}_{y}×{n}_{z}=\mathrm{20480}×\mathrm{768}×\mathrm{128}\
approx \mathrm{2.01}×{\mathrm{10}}^{\mathrm{9}}$ grid points.
The flow field is initialized by the instantaneous flow field of the last time step of a precursor simulation. Details about the precursor simulation and the meteorological parameters are given in
the next section. The flow field is filled cyclically into the main domain because it is larger than the precursor domain. At the inflow, vertical velocity and temperature profiles averaged over the
last 2h of the precursor simulation are prescribed. The turbulent state of the inflow is maintained by a turbulence recycling method that maps the turbulent fluctuations from the recycling plane at
x=25km onto the inflow plane at x=0. Details of the recycling method are given in Maas and Raasch (2022). The large distance between inflow and recycling plane is chosen to cover elongated
convection rolls that appear in the convective boundary layer (CBL) and to cover at least twice the advection distance of the convective timescale ${U}_{\mathrm{g}}{z}_{i}/{w}_{*}=\mathrm{9.011}$ms
^−1600m$/\mathrm{0.49}$ms^−1≈11km, with the convective velocity scale ${w}_{*}={\left[\frac{g{z}_{i}}{\stackrel{\mathrm{‾}}{\mathit{\theta }}}〈\stackrel{\mathrm{‾}}{{w}^{\prime }{\mathit{\
theta }}^{\prime }}{〉}_{\mathrm{s}}\right]}^{\mathrm{1}/\mathrm{3}}$, where $\stackrel{\mathrm{‾}}{\mathit{\theta }}=\mathrm{280}$K and $〈\stackrel{\mathrm{‾}}{{w}^{\prime }{\mathit{\theta }}^{\
prime }}{〉}_{\mathrm{s}}$ is the horizontally averaged kinematic surface heat flux averaged over the last 4h of the precursor simulation. For the potential temperature, the absolute value is
recycled instead of the turbulent fluctuation, so that the inflow temperature increases according to the surface temperature. The otherwise constant inflow temperature profile would cause a
streamwise temperature gradient that triggers a thermal circulation inside the entire domain. The turbulent fluctuations are shifted in the y direction by +6.4km to avoid streamwise streaks in the
averaged velocity fields; for further details please refer to Maas and Raasch (2022) and Munters et al. (2016). Radiation boundary conditions as described by Miller and Thorpe (1981) and Orlanski (
1976) are used at the outflow plane. Hereby, the flow quantity q at the outflow boundary b is determined with the phase velocity $\stackrel{\mathrm{^}}{c}$ and the upstream derivative of the flow
$\begin{array}{}\text{(5)}& {q}_{b}^{t+\mathrm{\Delta }t}={q}_{b}^{t}-\left(\stackrel{\mathrm{^}}{c}\mathrm{\Delta }t/\mathrm{\Delta }x\right)\left({q}_{b}^{n}-{q}_{b-\mathrm{1}}^{n}\right)\phantom{\
The phase velocity $\stackrel{\mathrm{^}}{c}$ is set to the maximum possible phase velocity of $\mathrm{\Delta }x/\mathrm{\Delta }t$. The surface boundary conditions and other parameters are the same
as in the precursor simulation and are thus described in the next section. The physical simulation time of the main simulations is 20h, and the presented data are averaged over the last 4h.
2.3Precursor simulation
Initial and inflow profiles of both simulations are obtained by a precursor simulation without a wind farm. It has cyclic boundary conditions in both lateral directions and a domain size of ${L}_{x,\
mathrm{pre}}×{L}_{y,\mathrm{pre}}×{L}_{z,\mathrm{pre}}=\mathrm{15.36}×\mathrm{9.6}×\mathrm{14.0}$km^3. The number of vertical grid points, the vertical grid stretching and Rayleigh damping levels
are the same as in the main simulation. The initial horizontal velocity is set to the geostrophic wind $\left({U}_{\mathrm{g}},{V}_{\mathrm{g}}\right)=\left(\mathrm{9.011},-\mathrm{1.527}\right)$ms
^−1, resulting in a steady-state hub height wind speed of 9.0±0.02ms^−1 that is aligned with the x axis (±0.01^∘). The values for the geostrophic wind are obtained by iterative adjustments between
preliminary precursor simulations, of which two are needed to obtain the given accuracy. The latitude is ϕ=55^∘N. The initial potential temperature is set to 280K up to a height of 600m and has a
lapse rate of $\mathrm{\Gamma }=+\mathrm{3.5}$Kkm^−1 above. This lapse rate corresponds to the international standard atmosphere. The onset of turbulence is triggered by small random perturbations
in the horizontal velocity field below a height of 300m. A Dirichlet boundary condition is set for the surface temperature. Why a Dirichlet boundary condition is a good choice is explained in Maas
and Raasch (2022). A constant surface heating rate of ${\stackrel{\mathrm{˙}}{\mathit{\theta }}}_{\mathrm{0}}=+\mathrm{0.05}$Kh^−1 is applied, resulting in a Monin–Obukhov length of $L\approx -\
mathrm{400}$m, which is common value for convective boundary layers in the North Sea (Muñoz-Esparza et al., 2012). The resulting boundary layer height (height of maximum vertical potential
temperature gradient) of z[i]=600m is a small but still typical value for convective boundary layers over the North Sea (Maas and Raasch, 2022). Boundary layer growth is avoided by applying a
large-scale subsidence that acts only on the potential temperature field. The subsidence velocity is zero at the surface and increases linearly to its maximum value at z=600m and is constant above.
The maximum subsidence velocity is chosen in such a way that the temperature increase in the free atmosphere (FA) exactly matches the surface heating rate: ${w}_{\mathrm{sub}}={\stackrel{\mathrm{˙}}
{\mathit{\theta }}}_{\mathrm{0}}/\mathrm{\Gamma }\approx \mathrm{14.3}$mh^−1. The roughness length for momentum and heat is ${z}_{\mathrm{0}}={z}_{\mathrm{0},\mathrm{h}}=\mathrm{1}$mm, and a
constant flux layer is assumed between the surface and the lowest atmospheric grid level. At the domain top and bottom, a Neumann boundary condition for the perturbation pressure and Dirichlet
boundary conditions for the velocity components are used. For the potential temperature, a constant lapse rate is assumed at the domain top.
The physical simulation time of the precursor simulation is 48h, to obtain a steady-state mean flow; i.e., the hourly-averaged hub height wind speed changes less than 0.05ms^−1 within 8h. This
long simulation time is needed for the decay of an inertial oscillation in time that has a period of 14.6h. The inertial oscillation occurs because there is no equilibrium of forces in the boundary
layer (BL) at the beginning of the simulation.
3.1Mean flow at hub height
To make a first qualitative comparison between the small and the large wind farm case, the mean horizontal wind speed and streamlines of the mean flow at hub height are shown in Fig. 2 for both
cases. The most striking difference is the large modification of the wind direction that occurs for the large wind farm case. Inside the large wind farm the flow is deflected counterclockwise, but in
the wake the flow is deflected clockwise so that the wind direction reaches the inflow wind direction and even turns further clockwise. But also for the wind speed both cases show significant
differences. The wind speed reduction inside the large wind farm is significantly greater than inside the small wind farm, which is an expected result. Remarkable, however, is the fact that the wind
speed in the far wake of the large wind farm is significantly greater than the inflow wind speed.
To make a more quantitative comparison between the two cases, Fig. 3 shows the mean horizontal wind speed, wind direction and perturbation pressure at hub height along x for the small and large wind
farm. The quantities are averaged along y and a moving average with a window size of one turbine spacing is applied along x to smooth out turbine-related sharp gradients. It can be seen that upstream
of the wind farms the wind speed is reduced due to the blockage effect. The speed reduction 2.5D upstream of the first turbine row is 4.8% for the small wind farm and 7.9% for the large wind farm.
These values lie in the range of 1%–11%, reported by Wu and Porté-Agel (2017) for a 20km long wind farm under different FA stratifications. The blockage effect is caused by an increase in the
perturbation pressure of 4.8 and 8.5Pa relative to the pressure at the inflow for the small and large wind farm, respectively (see Fig. 3c). The perturbation pressure distribution is related to
gravity waves that form in the free atmosphere, as will be shown in Sect. 3.4. Inside the wind farm, the wind speed is further reduced due to momentum extraction by the wind turbines. For the small
wind farm, the wind speed decreases to 7.6ms^−1 at the wind farm trailing edge (TE). For the large wind farm, however, the wind speed reaches a minimum of 6.8ms^−1 approximately 40km downstream
of the leading edge (LE) and then increases again to 7.4ms^−1 at the wind farm TE. This acceleration is mainly caused by the large drop in the perturbation pressure of 30Pa from the wind farm LE
to TE. For the small wind farm this pressure drop is only approximately 7Pa. The acceleration is also caused by the wind direction change and thus a greater ageostrophic wind speed component that
results in a larger energy input by the geostrophic pressure gradient (Abkar and Porté-Agel, 2014). This will be shown in Sect. 3.5. In the wake of the large wind farm the wind speed increases
further and reaches a maximum of 10.1ms^−1, which is 12% more than the free-stream wind speed at the inflow. The maximum wind speed in the wake of the small wind farm exceeds the inflow wind speed
by only 2%. Further downstream the wind speed decreases again, indicating that it oscillates.
As shown in Fig. 3b, the wind direction is also significantly affected by the wind farms. Inside the wind farms the wind direction turns counterclockwise, reaching +2.3 and +10.1^∘ at the TE of the
small and large wind farm, respectively. Note that the wind direction already changes upstream of the wind farms, reaching +0.7 and +1.4^∘ at the LE of the small and large wind farm, respectively.
This wind direction change is caused by a reduction of the Coriolis force, which is a result of the reduced wind speed in and around the wind farms. For the large wind farm, the maximum deflection
angle of 10.4^∘ is reached inside the wind farm, at x≈180km. Further downstream the wind turns clockwise, reaches Ψ=0^∘ at x≈330km and turns further clockwise afterwards. For the small wind farm
the maximum deflection angle of 2.8^∘ is reached in the wake, at x≈140km. The wind direction is zero at x≈290km and reaches a minimum at x≈400km. Similar maximum deflection values of 2–3^∘ have
been reported in an LES study of Allaerts and Meyers (2016) for a 15km long wind farm in conventionally neutral boundary layers.
The sinusoidal shape of the wind speed and wind direction evolution suggests that it is related to an inertial oscillation or an inertial wave along x. The wind direction has a +90^∘ phase shift
relative to the wind speed; i.e., the wind direction is zero where the wind speed has a maximum. The inertial wave has a wavelength of
$\begin{array}{}\text{(6)}& {\mathit{\lambda }}_{I}\approx GT=\mathrm{9.14}\phantom{\rule{0.125em}{0ex}}\mathrm{m}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}\mathrm{14.6}\phantom{\rule
{0.125em}{0ex}}\mathrm{h}\approx \mathrm{480}\phantom{\rule{0.125em}{0ex}}\mathrm{km},\end{array}$
where G is the geostrophic wind speed and $T=\mathrm{12}\phantom{\rule{0.125em}{0ex}}\mathrm{h}/\mathrm{sin}\left(\mathit{\varphi }\right)=\mathrm{2}\mathit{\pi }/{f}_{\mathrm{3}}$ is the inertial
period (Stull, 1988, p. 639). Consequently, the distance between wind direction maximum and minimum should be half a wavelength (${\mathit{\lambda }}_{I}/\mathrm{2}=\mathrm{240}$km), which
corresponds well to the distance of 260km that can be measured in the wake of the small wind farm. To add further confidence to this result, an additional simulation with a latitude of 80^∘N
instead of 55^∘N is performed. The results are given in Appendix B and show that the wavelength decreases to λ[I]=400km due to the shorter inertial period at that latitude ($T=\mathrm{12}\phantom{\
rule{0.125em}{0ex}}\mathrm{h}/\mathrm{sin}\left({\mathrm{80}}^{\circ }\right)=\mathrm{12.1}$h).
The inertial wave can also be seen in the hodograph of the hub height wind velocity components u and v along x, which is shown in Fig. 4. The figure shows that the oscillation is triggered by a
reduction in u, followed by an increase in v. After the large perturbation by the wind farms, the hodograph approaches a circular path with clockwise direction. The center of these circular paths is
the steady-state velocity of the inflow and not the geostrophic wind velocity. This is consistent with the findings of Baas et al. (2012), who investigated inertial oscillations in the nocturnal BL
using the analytical model of van de Wiel et al. (2010) that accounts for frictional effects within the BL. The amplitude of the oscillations is 0.3ms^−1 for the small wind farm and 1.1ms^−1 for
the large wind farm at ${\mathit{\lambda }}_{I}/\mathrm{4}$ downstream of the respective wind farm trailing edge.
To investigate this effect in more detail, Fig. 5 shows the crosswise (perpendicular to streamlines) force components that act on the flow at hub height along x, averaged along y. Shown are the
vertical turbulent momentum flux divergence, the perturbation pressure gradient force and the geostrophic forcing (difference between geostrophic pressure gradient force and Coriolis force). Positive
values indicate a counterclockwise deflection, and negative values indicate a clockwise deflection. The analysis is made from a Lagrangian frame of reference; thus, advection terms are not included.
At the inflow all forces sum to zero and the mean flow is in a steady state. Due to the wind speed reduction upstream and inside the wind farms, the Coriolis force is reduced, so that the geostrophic
pressure gradient force predominates and tends to deflect the flow counterclockwise. The vertical momentum flux divergence, however, tends to deflect the flow clockwise, but this force is weaker, so
that the sum of these forces is still positive. Because the wind farms are infinite in the y direction, the gravity waves are uniform in the y direction, and thus the perturbation pressure gradient
force is parallel to the x axis and has no effect on the wind direction at first. However, due to the change in wind direction further downstream inside the large wind farm, the perturbation pressure
gradient force has a component perpendicular to the streamlines that tends to deflect the flow clockwise. At the end of the large wind farm the sum of all forces becomes negative, so that the flow
begins to turn clockwise. Because the wind speed increases to super-geostrophic values in the wake, the Coriolis force becomes greater than the geostrophic pressure gradient force so that the flow is
deflected clockwise. The most significant difference between the small and the large wind farm is that the speed deficit in the large wind farm is greater and lasts longer. This results in a greater
wind direction change and thus a greater inertial wave amplitude compared to the small wind farm. Whether a wind farm can trigger a significant inertial wave can be predicted by the Rossby number
that relates inertia to Coriolis forces:
$\begin{array}{}\text{(7)}& \mathit{Ro}=\frac{G}{{L}_{\mathrm{wf}}{f}_{\mathrm{3}}},\end{array}$
where L[wf] is the length of the wind farm. An inertial wave occurs if the Rossby number has the order of magnitude of 1 or smaller. Coriolis effects become more dominant for smaller Rossby numbers
so that the amplitude of the inertial wave is larger for the large wind farm (Ro=0.8) than for the small wind farm (Ro=5.0). That wind farms can trigger an inertial wave has not been reported by
any other study, although there are studies that investigate wind farms with a similar size compared to the small wind farm in this study, e.g., Allaerts and Meyers (2016) or Wu and Porté-Agel (2017)
. The reason is that the inertial wave is more than 20 times longer than the small wind farm and is thus usually not covered by the numerical domain of other studies.
3.2Turbulence at hub height
Figure 6 shows the total turbulence kinetic energy (TKE) and the turbulence intensity (TI) at hub height along x for the small and large wind farm case. Both quantities are averaged along y and
piecewise averaged along x, where the averaging windows have a size of one turbine spacing and are centered between the turbine rows. The TKE and TI are calculated as follows:
$\begin{array}{}\text{(8)}& \mathrm{TKE}=\frac{\mathrm{1}}{\mathrm{2}}\left(\stackrel{\mathrm{‾}}{{{u}^{\prime }}^{\mathrm{2}}}+\stackrel{\mathrm{‾}}{{{v}^{\prime }}^{\mathrm{2}}}+\stackrel{\mathrm
{‾}}{{{w}^{\prime }}^{\mathrm{2}}}\right)+\stackrel{\mathrm{‾}}{e}\phantom{\rule{0.125em}{0ex}},\text{(9)}& \mathrm{TI}=\frac{\sqrt{\frac{\mathrm{2}}{\mathrm{3}}\mathrm{TKE}}}{\stackrel{\mathrm{‾}}
where an overbar indicates a temporal average; a prime indicates the deviation from this average; $\stackrel{\mathrm{‾}}{{{u}^{\prime }}^{\mathrm{2}}}$, $\stackrel{\mathrm{‾}}{{{v}^{\prime }}^{\
mathrm{2}}}$ and $\stackrel{\mathrm{‾}}{{{w}^{\prime }}^{\mathrm{2}}}$ are resolved-scale variances; and $\stackrel{\mathrm{‾}}{e}$ is the SGS TKE. Upstream of the wind farms the ambient TKE is 0.22
m^2s^−2. Within four turbine rows the TKE reaches a plateau value of 0.85m^2s^−2 for the small wind farm and 0.80m^2s^−2 for the large wind farm. The TKE is greater in the small wind farm
because the wind speed is greater, and thus the turbines generate more TKE (see Fig. 3a). In the large wind farm the TKE decreases slightly to 0.76m^2s^−2 at the point where the minimum wind speed
occurs. Further downstream the TKE increases to its maximum value of 0.85m^2s^−2 at the TE.
The TI shows a slightly different behavior than the TKE. Due to the normalization by the wind speed, which decreases upstream of the wind farms, the TI increases upstream of the wind farms. It
increases from the ambient TI of 4.4% to 4.6% half a turbine spacing upstream of the LE. In the small wind farm, the TI reaches a plateau value of 9.9%. In the large wind farm the TI is greater
due to the smaller wind speed and reaches a maximum value of 10.5% at the point where the minimum wind speed occurs. Further downstream, the TI decreases and reaches 10.1% at the TE.
To compare the decay of the TI in the wake of the wind farms, the graphs in Fig. 6c are shifted so that the TEs of both wind farms coincide. It is remarkable that the decay of the TI in the wake of
the small and the large wind farm follows exactly the same curve. This curve can be approximated by the following exponential function:
$\begin{array}{}\text{(10)}& \mathrm{TI}\left(x\right)=a\mathrm{exp}\left(-b\left(x-{x}_{\mathrm{0}}{\right)}^{c}\right)+d,\end{array}$
with coefficients a=5.8%, b=0.28km^−c, c=0.68 and d=4.2%. Consequently, the wind farm size has no effect on the decay of the TI in wind farm wakes.
Further downstream, the TKE and the TI also show a slight oscillation as the wind speed and direction show (see Fig. 6a and b). However, the amplitude is much smaller than the TKE and TI levels that
occur inside the wind farms, and thus the oscillations are hardly visible.
3.3Boundary layer modification
The previous two sections focused on the flow at hub height. In this section it is shown how the wind farms modify the height and the internal structure of the BL.
The CBL is capped by an inversion layer (IL), which is displaced upwards due to the presence of the wind farms. The IL displacement δ is defined as the IL height z[i] relative to the IL height at the
$\begin{array}{}\text{(11)}& \mathit{\delta }\left(x\right)={z}_{i}\left(x\right)-{z}_{i}\left(x=\mathrm{0}\right),\end{array}$
where z[i] is defined as the height where the maximum vertical potential temperature gradient occurs. The IL displacement along x is shown in Fig. 7 for the small and large wind farm case. The IL
displacement begins already upstream of the wind farms and reaches +30 and +50m at the LE of the small and large wind farm, respectively. Note that these changes in IL height (+5% and +8%)
correspond well to the change in hub height wind speed (−5% and −8%; see Fig. 3) at the LE. This confirms that the IL displacement is a reaction of the flow to the speed reduction inside the
boundary layer that ensures a constant mass flux inside the boundary layer. This has also been stated by other studies (Allaerts and Meyers, 2017; Maas and Raasch, 2022).
The maximum displacement is +55m for the small wind farm and occurs near its TE. For the large wind farm the maximum displacement is +110m and occurs approximately 40km downstream of the LE. Thus,
the maximum displacements occur at the location of the minimum wind speed (see Fig. 3). In the wake of the wind farms the IL displacement becomes negative, due to the increasing wind speed inside the
boundary layer. For the small wind farm, the minimum displacement is $\mathit{\delta }\approx -\mathrm{20}$m and occurs at x≈290km, corresponding to the location at which the hub height wind speed
has a maximum and the wind direction is zero. The same holds for the large wind farm, except that the minimum displacement is $\mathit{\delta }\approx -\mathrm{55}$m and occurs at x≈330km.
Besides the top of the BL, the internal structure of the BL is also significantly modified by the wind farms. Figure 8 shows vertical profiles of the wind speed and direction at several streamwise
positions to demonstrate the development of the BL. As a reference, the inflow profiles are also shown. The second profile is located 2.5D upstream of the wind farm LEs. It shows that the speed
deficit, caused by the blockage effect, does not only occur at hub height but is rather constant over the entire BL. This is plausible because the speed reduction is caused by a positive streamwise
pressure gradient, which is approximately constant over the entire height of the BL. At the wind farm TE, the wind speed at hub height is significantly reduced. At the BL top, however, the wind speed
has increased from 9.0 to 9.6ms^−1 for the small wind farm and from 8.6 to 11.0ms^−1 for the large wind farm. Because turbulent momentum exchange is negligible at that height, these speed
differences are solely caused by a drop in the perturbation pressure. The drop in the perturbation pressure between these points is 7Pa for the small wind farm and 28Pa for the large wind farm (see
Fig. 3). Based on these pressure differences, Bernoulli's equation predicts these wind speed changes:
$\begin{array}{}\text{(12)}& \begin{array}{rl}{v}_{\mathrm{2}}& =\sqrt{\frac{\mathrm{2}}{\mathit{\rho }}\left({p}_{\mathrm{1}}-{p}_{\mathrm{2}}\right)+{u}_{\mathrm{1}}^{\mathrm{2}}}\\ & =\sqrt{\frac
{s}}^{-\mathrm{1}}\\ & \text{(small wind farm)}\\ & =\sqrt{\frac{\mathrm{2}}{\mathrm{1.17}\phantom{\rule{0.125em}{0ex}}\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{3}}}\mathrm{28}\
rule{0.125em}{0ex}}\mathrm{m}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}\\ & \text{(large wind farm)}\phantom{\rule{0.125em}{0ex}},\end{array}\end{array}$
which correspond to the observed wind speed changes. The pressure distribution in the BL is determined by gravity waves in the free atmosphere that are described in the next section. In the far wake,
one-quarter of the inertial wavelength (${\mathit{\lambda }}_{I}/\mathrm{4}=\mathrm{120}$km) downstream of the wind farm TEs, the wind speed in the bulk of the BL is supergeostrophic. At 300m
height the wind speed has increased to 9.2 and 10.1ms^−1 for the small and large wind farm, respectively.
The wind direction profiles of the small wind farm case show only small deviations of maximum ±3^∘ relative to the inflow profile. For the large wind farm case, however, the deviations can be as
large as ±10^∘. Because the profiles of the small and large wind farm case are qualitatively the same, only the large wind farm case is described in the following. At a distance of 2.5D upstream of
the large wind farm, the wind direction has turned to the left by 1.4^∘ at hub height and by 3.2^∘ near the BL top. At the TE the wind direction has turned to the left by 10.0^∘ up to a height of
≈600m. At the BL top the wind direction change is zero. One-quarter inertial wavelength downstream of the TE, the shape of the wind direction profile is nearly unchanged, but the wind direction has
turned back to the right by approximately 8^∘ relative to the profile at the TE. This also holds for the wind direction above the BL, indicating that there is also an inertial wave in the free
atmosphere. This effect will be investigated in the next section.
3.4Gravity waves
The displacement of the IL represents an obstacle for the flow in the overlying stably stratified free atmosphere and thus triggers atmospheric gravity waves. The gravity waves are investigated in
more detail in this section because they induce streamwise pressure gradients at the surface and thus also affect the flow inside the BL and the energy budgets in the wind farms. Due to the large
horizontal scales involved, Coriolis effects also affect the flow, so that the triggered gravity waves are not pure gravity waves but rather inertial gravity waves.
Figure 9 shows vertical cross sections of the horizontal wind speed and direction, the vertical wind speed, the perturbation pressure, and the potential temperature. The respective inflow profile is
subtracted from each quantity, so that only the deviations from the steady-state mean flow remain. All quantities are averaged in time and along y. The different quantities show the expected pattern
for stationary inertial gravity waves with upwards propagating energy; i.e., the phase lines are inclined upstream relative to the vertical. The phase relations between the quantities also correspond
to the expected relations for gravity waves; e.g., p and w are in phase and w and θ are 90^∘ out of phase (Durran, 1990, Fig. 4.1).
The shown wave fields are a superposition of waves with three different inclination angles α (see Table 1). The first type of waves occurs above the wind farm LE and TE and is only visible in the
vertical velocity field (see Fig. 9e and f). The phase lines are inclined by α[1]=60^∘ relative to the vertical. They are only visible in the vertical velocity field because the oscillation direction
is much more vertical than that of the other wave types. The second type of waves appears above the wind farm with phase lines inclined by α[2s]=83.7 and α[2l]=88.3^∘ for the small and large wind
farm, respectively. The third type of waves occurs above the wake and has phase lines that are inclined by α[3]=89.3^∘ (see dashed lines in Fig. 9a and b). The occurrence of these three different
wave types can be explained by the shape of the topography, which is in this case the inversion layer. The wave type one is triggered by the sharp increase and decrease in IL height at the wind farm
LE and TE (see Fig. 7). Wave types two and three, however, are triggered by the entire hill-like-shaped IL above the wind farm and the valley-like-shaped IL above the wake. The phase lines of wave
type two are not perfectly straight but have a slightly positive curvature. The reason might be that the shape of the IL above the wind farm is not sinusoidal but is rather flat. The curved phase
lines may also explain why the pressure distribution in the wind farm is not sinusoidal (as one could expect) but nearly linear (which is also true in the FA above the wind farm).
The amplitude of wave type one is approximately the same for the small and large wind farm case, while the amplitudes of wave types two and three are approximately 2 times greater for the large wind
farm case relative to the small wind farm case (see Fig. 9 and note the different color scale ranges). The reason is that the IL displacement is twice as large for the large wind farm than for the
small wind farm (see Fig. 7).
The wavelengths of the three different wave types are significantly different. For stationary waves, the horizontal wavelength can be calculated as the distance that an air parcel moves with the
background velocity U=U[g] during one oscillation period with oscillation frequency ω:
$\begin{array}{}\text{(13)}& {\mathit{\lambda }}_{x}=\frac{\mathrm{2}\mathit{\pi }}{\mathit{\omega }}U.\end{array}$
The oscillation frequency ω of an inertial gravity wave is given by the dispersion relation (Pedlosky, 2003, Eq. 11.33):
$\begin{array}{}\text{(14)}& \mathit{\omega }=\sqrt{{f}^{\mathrm{2}}{\mathrm{sin}}^{\mathrm{2}}\mathit{\alpha }+{N}^{\mathrm{2}}{\mathrm{cos}}^{\mathrm{2}}\mathit{\alpha }},\end{array}$
where $N=\sqrt{\frac{g}{{\mathit{\theta }}_{\mathrm{0}}}\mathrm{\Gamma }}=\mathrm{10.7}×{\mathrm{10}}^{-\mathrm{3}}$s^−1 is the Brunt–Väisälä frequency. Note that the oscillation frequency is higher
than for pure gravity waves because the Coriolis force acts as an additional restoring force. Equation (14) reduces to ω=N for pure vertical oscillating gravity waves (vertical phase lines) and to ω=
f for pure horizontal oscillating inertial waves (horizontal phase lines). The absolute wavelength λ, i.e., the wavelength in the direction of phase propagation, is then given by
$\begin{array}{}\text{(15)}& \mathit{\lambda }=\frac{\mathrm{2}\mathit{\pi }c}{\mathit{\omega }}=\frac{\mathrm{2}\mathit{\pi }U\mathrm{cos}\mathit{\alpha }}{\sqrt{{f}^{\mathrm{2}}{\mathrm{sin}}^{\
mathrm{2}}\mathit{\alpha }+{N}^{\mathrm{2}}{\mathrm{cos}}^{\mathrm{2}}\mathit{\alpha }}}=\frac{\mathrm{1}}{\sqrt{\mathrm{1}+\frac{{f}^{\mathrm{2}}{\mathrm{sin}}^{\mathrm{2}}\mathit{\alpha }}{{N}^{\
mathrm{2}}{\mathrm{cos}}^{\mathrm{2}}\mathit{\alpha }}}}\frac{\mathrm{2}\mathit{\pi }U}{N}\phantom{\rule{0.125em}{0ex}},\end{array}$
so that the absolute wavelength becomes smaller for a larger α. Note that for pure gravity waves, where the effect of f can be neglected, the absolute wavelength is independent of α and corresponds
to the Scorer length ${L}_{\mathrm{s}}=\mathrm{2}\mathit{\pi }U/N=\mathrm{5.3}$km. The vertical wavelength is given by
$\begin{array}{}\text{(16)}& {\mathit{\lambda }}_{z}=\frac{\mathit{\lambda }}{\mathrm{sin}\mathit{\alpha }}.\end{array}$
The inclination angles of each wave type are measured in a figure that is similar to Fig. 9 but uses equal scales for both axes (not shown). The calculated oscillation frequencies and wavelengths of
the three wave types are listed in Table 1.
The waves of type one have the smallest wavelength (10.6km). Their effect on the pressure and horizontal velocity field is negligible. The horizontal wavelengths of wave type two are 48 and 167km
for the small and large wind farm, respectively. Why do these wavelengths occur? The ratio of horizontal wavelength to the wind farm length is 3.6 for the small wind farm and 1.85 for the large wind
farm, so that the wind farm length is not a good measure to explain the wavelength. But the wavelength can be explained by the shape of the IL. The horizontal distance between the largest slope of
the IL (at the LE) and the location of the maximum displacement is 12 and 42km for the small and large wind farm, respectively (see Fig. 7). These distances correspond very well to ${\mathit{\lambda
}}_{x,\mathrm{2}}/\mathrm{4}$ of the waves above the wind farm.
The presented results generally correspond very well to the results of Allaerts and Meyers (2017), who investigated gravity waves above a 15km long wind farm, which approximately corresponds to the
length of the small wind farm in this study. One significant difference between the studies is the larger extent of the large wind farm in this study, causing inertial gravity waves due to Coriolis
effects that become dominant at that scale. The second significant difference is the weaker stratification of +1.0Kkm^−1 in their study compared to +3.5Kkm^−1 in this study. This leads to a
different Brunt–Väisälä frequency and thus a different Scorer length (which corresponds to the absolute wavelength of stationary pure gravity waves). Consequently, the wind farm in Allaerts and
Meyers (2017) has approximately the length of the Scorer length (${L}_{\mathrm{wf}}/{L}_{\mathrm{s}}=\mathrm{15}$km$/$12.8km≈1.2), whereas the small wind farm and large wind farm in this study are
${L}_{\mathrm{wf},\mathrm{s}}/{L}_{\mathrm{s}}=\mathrm{13.44}$km$/$5.3km≈2.5 and ${L}_{\mathrm{wf},\mathrm{l}}/{L}_{\mathrm{s}}=\mathrm{90.24}$km$/$5.3km≈17.0 times longer than the Scorer
length, respectively. Due to the large ratio of ${L}_{\mathrm{wf}}/{L}_{\mathrm{s}}$ in the large wind farm case, the waves at the wind farm LE and TE (type one) are separated by several wavelengths
and can thus be clearly distinguished from wave type two in this study. However, the less orderly shape of the w field in Allaerts and Meyers (2017) (their Fig. 12b) suggests that wave type one is
also present there.
The vertical structure of the gravity waves is shown by profiles of the wind speed and wind direction at different streamwise positions in Fig. 10. It can be seen that the amplitude of the waves is
approximately twice as large in the large wind farm case than in the small wind farm case, as already mentioned above. There is a phase shift of approximately 90^∘ between the wind speed and wind
direction. Inside the Rayleigh damping layer the wind speed variations decay within 3km, and the wind direction variations decay within 1km.
3.5Energy budget analysis
Wind turbines extract kinetic energy from the BL flow and convert it into electrical energy. Consequently, there is less energy available for wind turbines located in the wake of upstream wind
turbines. The energy extraction is considered by a velocity deficit zone in the wind turbine wake in classical wake models such as Jensen (1983). However, there are also sources of energy that add
new kinetic energy into the BL. As will be shown in this section, these sources depend on the above-discussed flow effects and significantly affect the wind turbine power, especially for the large
wind farm.
To analyze the different energy sources and sinks in the BL, an extensive energy budget analysis is presented in this section. The analysis is very similar to the energy budget analysis made by
Allaerts and Meyers (2017) for a 15km long wind farm. The energy budgets are calculated for three different control volumes. The control volume Ω[wt] envelops the wind turbine rotor, the control
volume Ω[bl] envelops the rest of the BL above Ω[wt] and the entire wind farm is enveloped by control volume Ω[wf], which is the sum of all Ω[wt] (see Fig. 11). The control volumes have a streamwise
length of one turbine spacing and are centered at the respective wind turbine hub. The bottom and top boundaries of Ω[wt] are $\left({z}_{\mathrm{b}},{z}_{\mathrm{t}}\right)=\left(\mathrm{50},\mathrm
{310}\right)$m, which is 1dz larger than the rotor diameter to cover the smeared forces of the wind turbine model. The bottom and top boundaries of Ω[bl] are $\left({z}_{\mathrm{b}},{z}_{\mathrm{t}}
\right)=\left(\mathrm{310}\phantom{\rule{0.125em}{0ex}}\mathrm{m},{z}_{i}\left(x\right)\right)$. In the y direction the control volumes are bounded by the cyclic domain boundaries.
The equation for the conservation of the resolved-scale kinetic energy can be obtained by multiplying PALM's momentum equation (Eq. 2) with u[i], averaging in time, assuming stationarity and
integrating over the control volume Ω:
$\begin{array}{}\text{(17)}& \begin{array}{rl}\mathrm{0}=& \underset{\mathcal{A}}{\underbrace{-\underset{\mathrm{\Omega }}{\int }\frac{\partial {\stackrel{\mathrm{‾}}{\stackrel{\mathrm{̃}}{u}}}_{j}\
phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{‾}}{E}}_{\mathrm{k}}}{\partial {x}_{j}}\mathrm{d}\mathrm{\Omega }}}\underset{\mathcal{P}}{\underbrace{-\underset{\mathrm{\Omega }}{\int }\frac{{\
stackrel{\mathrm{‾}}{\stackrel{\mathrm{̃}}{u}}}_{i}}{{\mathit{\rho }}_{\mathrm{0}}}\frac{\partial \stackrel{\mathrm{‾}}{{\mathit{\pi }}^{*}}}{\partial {x}_{i}}\mathrm{d}\mathrm{\Omega }}}\\ & \
underset{\mathcal{F}}{\underbrace{\begin{array}{c}-\underset{\mathrm{\Omega }}{\int }\frac{\partial }{\partial {x}_{j}}{\stackrel{\mathrm{‾}}{\stackrel{\mathrm{̃}}{u}}}_{i}\phantom{\rule{0.125em}
{0ex}}\stackrel{\mathrm{‾}}{{\stackrel{\mathrm{̃}}{u}}_{i}^{\prime }\phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{̃}}{u}}_{j}^{\prime }}\mathrm{d}\mathrm{\Omega }+\underset{\mathrm{\Omega }}{\int }\
frac{\partial }{\partial {x}_{j}}\stackrel{\mathrm{‾}}{{\stackrel{\mathrm{̃}}{u}}_{i}{\mathit{\tau }}_{ij}}\mathrm{d}\mathrm{\Omega }\\ -\underset{\mathrm{\Omega }}{\int }\frac{\partial }{\partial {x}
_{j}}\frac{\mathrm{1}}{\mathrm{2}}\stackrel{\mathrm{‾}}{{\stackrel{\mathrm{̃}}{u}}_{j}^{\prime }\phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{̃}}{u}}_{i}^{\prime }{\stackrel{\mathrm{̃}}{u}}_{i}^{\
prime }}\mathrm{d}\mathrm{\Omega }-\underset{\mathrm{\Omega }}{\int }\stackrel{\mathrm{‾}}{\frac{{\stackrel{\mathrm{̃}}{u}}_{i}^{\prime }}{{\mathit{\rho }}_{\mathrm{0}}}\frac{\partial {\mathit{\pi }}^
{*\prime }}{\partial {x}_{i}}}\mathrm{d}\mathrm{\Omega }\end{array}}}\\ & \underset{\mathcal{G}}{\underbrace{+\underset{\mathrm{\Omega }}{\int }\left({\stackrel{\mathrm{‾}}{\stackrel{\mathrm{̃}}{u}}}_
{\mathrm{2}}{f}_{\mathrm{3}}{u}_{\mathrm{g},\mathrm{1}}-{\stackrel{\mathrm{‾}}{\stackrel{\mathrm{̃}}{u}}}_{\mathrm{1}}{f}_{\mathrm{3}}{u}_{\mathrm{g},\mathrm{2}}\right)\mathrm{d}\mathrm{\Omega }}}\\ &
\underset{\mathcal{B}}{\underbrace{+\underset{\mathrm{\Omega }}{\int }\frac{g}{{\mathit{\theta }}_{\mathrm{0}}}\stackrel{\mathrm{‾}}{\left(\stackrel{\mathrm{̃}}{\mathit{\theta }}-{\mathit{\theta }}_{\
mathrm{0}}\right){\stackrel{\mathrm{̃}}{u}}_{\mathrm{3}}}\mathrm{d}\mathrm{\Omega }}}\\ & \underset{\mathcal{D}}{\underbrace{-\underset{\mathrm{\Omega }}{\int }\stackrel{\mathrm{‾}}{{\mathit{\tau }}_
{ij}\frac{\partial {\stackrel{\mathrm{̃}}{u}}_{i}}{\partial {x}_{j}}}\mathrm{d}\mathrm{\Omega }-\mathcal{R}}}\underset{\mathcal{W}}{\underbrace{+\underset{\mathrm{\Omega }}{\int }\stackrel{\mathrm{‾}}
{{\stackrel{\mathrm{̃}}{u}}_{i}{d}_{i}}\mathrm{d}\mathrm{\Omega }}}\phantom{\rule{0.125em}{0ex}}.\end{array}\end{array}$
Note that the mean kinetic energy (KE, ${\stackrel{\mathrm{‾}}{E}}_{\mathrm{k}}$) contains the kinetic energy of the mean flow (KEM) and the turbulence kinetic energy (TKE) of the resolved flow:
$\begin{array}{}\text{(18)}& {\stackrel{\mathrm{‾}}{E}}_{\mathrm{k}}=\frac{\mathrm{1}}{\mathrm{2}}\stackrel{\mathrm{‾}}{{\stackrel{\mathrm{̃}}{u}}_{i}{\stackrel{\mathrm{̃}}{u}}_{i}}=\frac{\mathrm{1}}{\
stackrel{\mathrm{̃}}{u}}_{i}^{\prime }{\stackrel{\mathrm{̃}}{u}}_{i}^{\prime }}.\end{array}$
The terms of Eq. (17) are categorized as follows:
• 𝒜 is the divergence of KE advection;
• 𝒫 is the energy input by mean perturbation pressure gradients;
• ℱ is the transport of KEM by resolved turbulent stresses (term 1), transport of KEM and TKE by SGS stresses (term 2), turbulent transport of resolved-scale TKE by velocity fluctuations (term 3),
and turbulent transport of KE by perturbation pressure fluctuations (term 4);
• 𝒢 is the energy input by geostrophic forcing;
• ℬ is the energy input by buoyancy forces;
• 𝒟 is the dissipation by SGS model and residual ℛ; and
• 𝒲 is the energy extraction by wind turbines.
Equation (17) has a positive residual ℛ because the magnitude of the calculated dissipation is underestimated, which has two reasons. First, the local velocity gradients are underestimated because
they are calculated with central differences. Second, the fifth-order upwind advection scheme of Wicker and Skamarock (2002) has numerical dissipation, suppressing the magnitude of the smallest
eddies, for which the gradients and the dissipation are highest (Maronga et al., 2013). The residual is subtracted from the (negative) dissipation term 𝒟 to compensate for the underestimated
magnitude of the calculated dissipation.
Instead of calculating terms 𝒜 and ℱ as a volume integral, they can also be calculated as a surface integral over the control volume surfaces (Gauss's theorem):
$\begin{array}{}\text{(19)}& \mathcal{A}=\underset{{\mathcal{A}}_{x}}{\underbrace{{\left[\underset{{\mathrm{\Gamma }}_{x}}{\int }\left(-{\stackrel{\mathrm{‾}}{\stackrel{\mathrm{̃}}{u}}}_{\mathrm{1}}\
phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{‾}}{E}}_{\mathrm{k}}\right)\mathrm{d}{\mathrm{\Gamma }}_{x}\right]}_{{x}_{l}}^{{x}_{r}}}}+\underset{{\mathcal{A}}_{z}}{\underbrace{{\left[\underset{{\
mathrm{\Gamma }}_{z}}{\int }\left(-{\stackrel{\mathrm{‾}}{\stackrel{\mathrm{̃}}{u}}}_{\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{‾}}{E}}_{\mathrm{k}}\right)\mathrm{d}{\mathrm{\Gamma }}
_{z}\right]}_{{z}_{\mathrm{b}}}^{{z}_{\mathrm{t}}}\phantom{\rule{0.125em}{0ex}},}}\text{(20)}& \begin{array}{rl}\mathcal{F}& =\underset{{\mathcal{F}}_{x}}{\underbrace{{\left[\underset{{\mathrm{\Gamma
}}_{x}}{\int }\left(-{\stackrel{\mathrm{‾}}{\stackrel{\mathrm{̃}}{u}}}_{i}\phantom{\rule{0.125em}{0ex}}\stackrel{\mathrm{‾}}{{\stackrel{\mathrm{̃}}{u}}_{i}^{\prime }\phantom{\rule{0.125em}{0ex}}{\
stackrel{\mathrm{̃}}{u}}_{\mathrm{1}}^{\prime }}+\stackrel{\mathrm{‾}}{{\stackrel{\mathrm{̃}}{u}}_{i}{\mathit{\tau }}_{i\mathrm{1}}}-\frac{\mathrm{1}}{\mathrm{2}}\stackrel{\mathrm{‾}}{{\stackrel{\
mathrm{̃}}{u}}_{\mathrm{1}}^{\prime }\phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{̃}}{u}}_{i}^{\prime }{\stackrel{\mathrm{̃}}{u}}_{i}^{\prime }}-\frac{\stackrel{\mathrm{‾}}{{\stackrel{\mathrm{̃}}{u}}_
{\mathrm{1}}^{\prime }{\mathit{\pi }}^{*\prime }}}{{\mathit{\rho }}_{\mathrm{0}}}\right)\mathrm{d}{\mathrm{\Gamma }}_{x}\right]}_{{x}_{l}}^{{x}_{r}}}}\\ & +\underset{{\mathcal{F}}_{z}}{\underbrace{{\
left[\underset{{\mathrm{\Gamma }}_{z}}{\int }\left(-{\stackrel{\mathrm{‾}}{\stackrel{\mathrm{̃}}{u}}}_{i}\phantom{\rule{0.125em}{0ex}}\stackrel{\mathrm{‾}}{{\stackrel{\mathrm{̃}}{u}}_{i}^{\prime }\
phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{̃}}{u}}_{\mathrm{3}}^{\prime }}+\stackrel{\mathrm{‾}}{{\stackrel{\mathrm{̃}}{u}}_{i}{\mathit{\tau }}_{i\mathrm{3}}}-\frac{\mathrm{1}}{\mathrm{2}}\stackrel
{\mathrm{‾}}{{\stackrel{\mathrm{̃}}{u}}_{\mathrm{3}}^{\prime }\phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{̃}}{u}}_{i}^{\prime }{\stackrel{\mathrm{̃}}{u}}_{i}^{\prime }}-\frac{\stackrel{\mathrm{‾}}
{{\stackrel{\mathrm{̃}}{u}}_{\mathrm{3}}^{\prime }{\mathit{\pi }}^{*\prime }}}{{\mathit{\rho }}_{\mathrm{0}}}\right)\mathrm{d}{\mathrm{\Gamma }}_{z}\right]}_{{z}_{\mathrm{b}}}^{{z}_{\mathrm{t}}}\
where 𝒜[x] and 𝒜[z] are the advection of KE through the left/right and bottom/top surfaces, respectively, and ℱ[x] and ℱ[z] are turbulent fluxes through the left/right and bottom/top surfaces,
3.5.1Energy budgets for the entire small and large wind farm
The energy budgets for a control volume that envelops the entire small/large wind farm are shown in Fig. 12. The budget terms of Eq. (17) are converted from Wρ^−1 to MW per turbine to make them more
meaningful. The air density is ρ=1.17kgm^−3.
With 5.6MW per turbine, the horizontal advection of kinetic energy (𝒜[x]) is the greatest energy source for the small wind farm. For the large wind farm, however, this source is only as large as 0.9
MW per turbine. This large difference is mainly the result of the fact that the large wind farm is 6 times longer than the small one, so that the influx of KE at the wind farm LE is distributed over
6 times more turbine rows. Additionally, the wind speed at the TE of the large wind farm is larger than at the TE of the small wind farm, so that more KE leaves the wind farm control volume (see
Figs. 3 and 13).
For both wind farms, approximately 40% of 𝒜[x] leaves the wind farm control volume again through vertical advection 𝒜[z]. KE is leaving the top of the control volume by a mean positive w, which is
the result of the turbine-induced flow deceleration and the requirement for mass flow conservation. This effect has also been described by Allaerts and Meyers (2017).
The horizontal turbulent fluxes ℱ[x] are a small energy sink of −0.3MW per turbine for the small wind farm. This sink is mainly caused by a net outflow of TKE in the first three turbine rows, where
the incoming flow contains less TKE than the outgoing flow (see Figs. 6a and 13). For the large wind farm ℱ[x] is negligible because the described effect spreads over 6 times more turbine rows.
The vertical turbulent flux of KE (ℱ[z]) is the greatest energy source for the large wind farm, contributing 4.4MW per turbine. For the small wind farm it is the third largest energy source with 2.9
MW per turbine. These results show that for large wind farms the vertical turbulent flux of KE is much more important than the horizontal advection (${\mathcal{F}}_{z}\approx \mathrm{5}×{\mathcal
{A}}_{x}$), whereas for small wind farms the horizontal advection of KE is more important (${\mathcal{A}}_{x}\approx \mathrm{2}×{\mathcal{F}}_{z}$).
The energy input by the geostrophic forcing (𝒢) is the fourth largest energy source for the small wind farm (1.9MW per turbine) but the second largest energy source for the large wind farm (2.5MW
per turbine). The 32% higher value for the large wind farm is the result of the wind direction change that is triggered by the wind farm itself (see Fig. 3). It causes the ageostrophic wind velocity
component to rise and thus leads to a higher energy input (see also Figs. 13 and 14). This effect has also been shown for infinitely large wind farms by Abkar and Porté-Agel (2014) and finite, large
wind farms by Maas and Raasch (2022).
The energy input by the mean perturbation pressure gradient (𝒫) is the second largest energy source for the small wind farm (3.5MW per turbine) and the third largest energy source for the large wind
farm (2.1MW per turbine). For the large wind farm 𝒫 is approximately 60% of 𝒫 for the small wind farm, although the difference in perturbation pressure between the LE and TE of the large wind farm
is approximately 4.3 times larger than that of the small wind farm ($\mathrm{30}\phantom{\rule{0.125em}{0ex}}\mathrm{Pa}/\mathrm{7}$Pa; see Fig. 3). However, this difference spreads over a 6 times
longer wind farm, so that the resulting pressure gradient is only 70% as large. The term 𝒫 also depends on the mean wind speed, which is generally smaller in the large wind farm, resulting in a
further reduction of 𝒫.
The production of KE by buoyancy (ℬ) is negligibly small for the small and large wind farm case. This is an expected result for the offshore-typical weakly unstable CBL with $L\approx -\mathrm{400}$
m. However, this term might be much larger for strong CBLs.
The total of all above named sources ($\mathcal{A}+\mathcal{F}+\mathcal{G}+\mathcal{P}+\mathcal{B}$) is 11.3MW per turbine for the small wind farm and 9.6MW per turbine for the large wind farm. For
the small wind farm 75% of this available power is used by the wind turbines ($\mathcal{W}=-\mathrm{8.5}$MW per turbine), and for the large wind farm it is 73% ($\mathcal{W}=-\mathrm{7.0}$MW per
turbine). The rest of the available energy is lost by dissipation (𝒟).
3.5.2Energy budgets in the turbine control volumes
The energy budgets inside the wind turbine control volumes Ω[wt] are shown in Fig. 13. In the first two turbine rows the horizontal advection of KE (𝒜[x]) is the dominant energy source. A large
amount of this KE, however, is lost by vertical advection of KE through the control volume top. This effect is caused by the fact that any horizontal convergence (flow deceleration with positive 𝒜[x]
) requires a vertical divergence (negative 𝒜[z]) so that the mass flux is conserved. Consequently, the shape of 𝒜[z] is qualitatively the vertically mirrored shape of 𝒜[x]. At row 21 of the large
wind farm the terms change sign because from there on the flow accelerates again (see Fig. 3). For the small wind farm this happens between the last two rows. From there on, more KE leaves the
control volume than KE enters the control volume in the streamwise direction. But 𝒜[z] is then positive, indicating that KE is transported into the wind farm from above by a negative mean vertical
velocity. The flow acceleration at the end of the wind farms is mainly caused by the negative perturbation pressure gradient that has the highest magnitude at the wind farm TE (see Fig. 3). The
energy input by the pressure gradient 𝒫 thus increases towards the TE of the large wind farm and reaches 5MW per turbine at the TE. The pressure distribution inside the wind farm is determined by
wave type two of the gravity waves (see Sect. 3.4 and Fig. 9). The flow acceleration near the TE of the wind farm and the related negative net advection of KE have also been reported by Allaerts and
Meyers (2017) for a 15km long wind farm in a conventionally neutral BL.
The horizontal turbulent fluxes are a weak energy sink ($\approx -\mathrm{1}$MW per turbine) in the first two rows because the outgoing flow contains more TKE than the incoming flow.
For both wind farms the vertical turbulent fluxes are zero at the first row. For the small wind farm they rise from 3MW in the middle of the wind farm to 4MW at the TE. For the large wind farm they
stay constantly at 4.5MW per turbine from approximately row 14, but from row 32 they start to rise again, reaching 5.5MW per turbine at the TE. The values of ℱ[z] are generally greater for the
large wind farm because there is more energy available in the upper part of the BL, which is mainly the result of the higher energy input by the geostrophic forcing for the large wind farm (see Fig.
14). From row 7 to the TE of the large wind farm the vertical turbulent fluxes are the greatest energy source of all terms.
For the small wind farm, the energy input by the geostrophic forcing is approximately constant at 2MW per turbine. For the large wind farm, however, it steadily rises from 2MW per turbine at the LE
to 3MW per turbine at the TE. As described in the last section, this effect is caused by the wind direction change along the wind farm that leads to a higher ageostrophic wind velocity component.
The wind turbines in the first two rows of the small and large wind farm extract approximately 10.0 and 9.0MW, respectively (remember the staggered turbine configuration). The wind turbine power is
constant at 8.0MW in the rest of the small wind farm. In the large wind farm, however, the turbine power slowly decreases to 6.5MW at row 24 and then increases to nearly 8.0MW at the last turbine
row. This power increase is the result of the wind speed increase in the second half of the wind farm that is related to the wind direction change and increase in 𝒢.
The energy dissipation is approximately constant at $\mathcal{D}=-\mathrm{3}$MW per turbine in the small wind farm and at $\mathcal{D}=-\mathrm{2.5}$MW in the large wind farm, except for the first
3 rows, where it is smaller. At the TE of the large wind farm 𝒟 is slightly higher than in the middle, which can be related to the higher TKE at that location (see Fig. 6).
3.5.3Energy budgets in the boundary layer control volumes
In the BL control volumes above the wind turbines the flow begins to accelerate earlier than inside the wind farm (row 4 of the small wind farm and row 14 of the large wind farm), as indicated by the
evolution of 𝒜[x] (see Fig. 14). The energy for this acceleration is provided by 𝒢 and 𝒫 in approximately equal parts (4MW per turbine) in the large wind farm, except towards the TE, where 𝒫
increases steeply due to a significant drop on perturbation pressure (see Fig. 3). For the small wind farm 𝒫 is 2 to 4 times larger than 𝒢, except at the first row, where they are equal.
In the small wind farm 𝒢 increases by only 10% from LE to TE, but in the large wind farm it increases by more than 100% (from 2.2 to 4.8MW per turbine). This is a much larger increase than in
the wind turbine control volume, although the wind direction change is the same at all heights (see Fig.8). However, the wind speed is much greater above the wind farm, resulting in a higher
ageostrophic wind velocity component and thus a higher 𝒢.
The vertical turbulent fluxes ℱ[z] have the same shape as but opposite sign to the turbine control volumes (see Fig. 13) because they transfer energy from the BL down into the wind farm. Their
magnitude is approximately 25% smaller in the turbine control volume than in the BL control volume because there is also a KE loss through the bottom of the turbine control volumes.
The aim of this LES study is to provide a systematic comparison between small and large wind farms, focusing on the flow effects and the energy budgets in and around the wind farms. The size of the
wind farms is chosen to be representative for current wind farm clusters (length of approximately 15km) and future wind farm clusters (length of approximately 90km).
The results show that there are significant differences in the flow field and the energy budgets of the small and large wind farm. The large wind farm triggers an inertial wave with a wind direction
amplitude of approximately 10^∘ and a wind speed amplitude of more than 1ms^−1. In a certain region in the far wake of a large wind farm the wind speed is greater than far upstream of the wind
farm. The inertial wave also exists for the small wind farm, but the amplitudes are approximately 4 times weaker and thus may be hardly observable in real wind farm flows that are more heterogeneous.
The decay of turbulence intensity in the wind farm wakes follows an exponential function and does not depend on the wind farm length. Thus, regarding turbulence, the wake of large wind farms has the
same length as that of small wind farms. The wind-farm-induced speed deficit causes an upward displacement of the IL, triggering inertial gravity waves above the small and large wind farm. Because
the inertial gravity waves have a substantial effect on the energy budgets in the wind farm, their existence should be proven by measurements in the future. However, this might be a difficult task
because the amplitudes in the vertical wind speed and pressure are very small (0.05ms^−1 and 20Pa).
The energy budget analysis shows that the dominant energy source in small wind farms is the advection of kinetic energy. For large wind farms, however, the advection is much less important and the
energy input by vertical turbulent fluxes becomes dominant. Due to the wind-farm-induced wind direction change and the related increase in the ageostrophic wind speed, the energy input by the
geostrophic forcing (synoptic-scale pressure gradient) can increase by more than 100%. This result shows that the presence of large offshore wind farm clusters will modify the offshore,
low-roughness BL towards a more onshore-typical, high-roughness BL. This leads to a faster wake recovery and allows for smaller turbine spacings. The energy budget analysis shows that the power
output of large wind farms depends on several different energy sources that are determined by the flow state inside and above the BL. Simple wake models do not take these different sources into
account and are expected to be inappropriate for accurate power predictions of large wind farms. Proving this hypothesis is an open research tasks.
The results in this study are based on very idealized simulation setups, assuming a homogeneous surface and a barotropic flow with constant geostrophic wind over a horizontal distance of 400km and a
constant lapse rate over a vertical distance of 5km. These idealized conditions rarely occur in reality. A deviation from these idealized conditions could distort and weaken the described effects.
Additionally, only one meteorological setup is used in this study. A change in BL height, stability or wind speed may affect the results significantly. Consequently, the presented results are a first
qualitative guess of what is different in large wind farms compared to small wind farms. Further research is needed to find out how sensitive the results are to the named assumptions and to changes
in the meteorological conditions and the turbine spacing. The largest deviation from reality is probably introduced by the assumption of an infinitely wide wind farm. The investigation of a
multi-gigawatt wind farm with a finite size in both lateral directions will be the subject of a follow-up study.
Appendix A:Validity of the Boussinesq approximation
The domain height in this study is much larger than in most large-eddy simulation studies that mainly cover the boundary layer. The incompressibility assumption requires the involved vertical length
scales to be much smaller than ${c}^{\mathrm{2}}/g\approx \mathrm{12}$km, where c is the speed of sound (Stull, 1988, p. 77). Therefore, the question of whether the Boussinesq approximation that
assumes a constant density is still valid for these simulations arises. To clarify this question, two additional test simulations were performed. One using the Boussinesq approximation and the other
using the anelastic approximation, for which the density can vary with height. The results are shown in Figs. A1–A3. The gravity waves are qualitatively the same in both cases (wavelength, angles of
the phase lines). But there are some quantitative differences at greater heights (e.g., 8km). At that height, the velocity and temperature amplitudes are greater and the pressure amplitudes are
smaller for the anelastic approximation. But these differences do not affect the results at hub height (wind speed, direction and perturbation pressure). Therefore, it is appropriate to use the
Boussinesq approximation for the simulations in this study.
Appendix B:Simulation with different latitude
Two additional large wind farm simulations with two different latitudes are performed to prove that the observed wave in the wake is an inertial wave. The domain length is increased further to 655.36
km to capture approximately one wavelength. The latitudes ϕ[1]=55^∘ (original simulation) and ϕ[2]=80^∘ (additional simulation) are used. The larger latitude should result in a shorter inertial
period ($T=\mathrm{12}\phantom{\rule{0.125em}{0ex}}\mathrm{h}/\mathrm{sin}\left(\mathrm{80}{}^{\circ }\right)=\mathrm{12.1}$h) and thus a shorter wavelength (${\mathit{\lambda }}_{I}\approx GT\
approx \mathrm{400}$km). This shorter wavelength can be observed in Fig. B1, confirming that the wind speed and direction oscillations in the wind farm wake are related to an inertial oscillation.
The author has declared that there are no competing interests.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was funded by the Federal Maritime and Hydrographic Agency (BSH) (grant no. 10044580) and supported by the North German Supercomputing Alliance (HLRN). Special thanks go to Siegfried Raasch
for guiding the manuscript preparation. I thank Dries Allaerts and Dieter Etling for informative discussions about gravity waves and Sukanta Basu for discussions about the validity of the Boussinesq
This work was funded by the Federal Maritime and Hydrographic Agency (BSH) (grant no. 10044580). The publication of this article was funded by the open-access fund of Leibniz Universität Hannover.
This paper was edited by Sara C. Pryor and reviewed by Dries Allaerts and two anonymous referees.
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6 Extracurriculars for High Schoolers Interested in Studying Math
What’s Covered:
After months of thinking about which classes you’ve enjoyed, your career goals, and your desired ROI, you’ve decided to study math in college. But how do you make it happen? How do you get into a
math program that will secure you the future you’ve been dreaming of?
Your academic record is important, of course, but you also need to consider other aspects of your profile, such as extracurricular activities. These activities show admissions officers that you’re
actively pursuing your passion for mathematics and developing your critical thinking skills outside of the classroom.
Read on to learn about the most impressive extracurriculars for aspiring mathematicians.
Extracurricular Activities for Aspiring Math Majors
1. Math Honor Society
The most obvious math extracurricular is Mu Alpha Theta. Mu Alpha Theta is an international math honor society that recognizes the achievements of students who excel in mathematics while inspiring
interest in the subject.
Students join Mu Alpha Theta through their school’s local chapter. To qualify for membership, students must have completed at least two years of college preparatory mathematics, such as algebra and
geometry, and have at least a 3.0 (on a 4-point scale) in math courses. Many school chapters will reach out to students who meet these qualifications, but you may also need to do some searching
Mu Alpha Theta is the kind of organization where students get out what they put into it. If you want to have a more active role in Mu Alpha Theta at a national or international level, the first step
is becoming more active within your chapter.
At a national level, each year Mu Alpha Theta holds an annual convention that features math-related events, activities, and competitions. The national board also hosts competitions like the
Mathematical Minutes Video Contest, the Log1 Contest, and the Rocket City Math League that all members have access to.
Lastly, the society offers scholarships and awards to noteworthy students. The most impressive scholarship offered by Mu Alpha Theta is the $4,000 Educational Foundation award.
The impact of Mu Alpha Theta on your college admissions chances depends heavily on your involvement. If you simply pay a membership fee and include the letters on your resume, you will not stand out.
On the other hand, if you compete in national competitions or receive a scholarship, that is meaningful and may affect your admissions chances.
2. Other Clubs
If Mu Alpha Theta doesn’t sound appealing to you, there are plenty of other math-related clubs that may strike your fancy. Remember that, even if a club isn’t directly related to math, it can be
valuable to you because of the skills you put to use. Important skills for future mathematicians involve problem-solving, creativity, and logical reasoning.
Your school might offer:
• Math Club
• Robotics Club
• Game Theory Club
• Mathletes
• Egg Drop
• Chess Club
• Puzzle Club
• Coding Club
If these types of clubs don’t exist at your school, we have plenty of guidance for you on how to start a club in high school. Steps in the process include finding a faculty adviser, planning a
budget, publicizing your club, and delegating duties and responsibilities.
3. Math Competitions
Like math clubs, math competitions offer a unique opportunity for future math majors to show off their abilities and gain prestigious recognition. The two most popular math competitions are the Math
Olympiad and the Math League, though there are many other math competitions for high schoolers.
Math Olympiad
Math Olympiad is a proof-based mathematical competition, where individual students’ ability to quickly solve math programs advances them through rounds of competition. The end goal is to compete on
the US Math Olympiad Team.
First, students across the country take the AMC 10/12 and AIME exams. High scores on these exams send 200-300 students to the two-day United States of America Mathematics Olympiad (USAMO), hosted by
the Mathematical Association of America (MAA). The top 12 students at the USAMO are then invited to attend the Mathematical Olympiad Program (MOP) during the summer. And, finally, six students from
MOP are selected to join the US Math Olympiad Team, which competes internationally.
Math Olympiad is an intense competition and advancement within the competition is very impressive during college admissions.
Math League
Math League is an organization that hosts various contests for students in elementary, middle, and high school. Students can participate individually or as a team, making Math League a more
collaborative competition than Math Olympiad.
Throughout the year, Math League hosts 6 competitions. Additionally, their contest offerings have expanded in recent years and they now offer course-specific contests, including the Algebra I Contest
(grades 6-9), the Geometry Contest (grades 7-10), and the Number Theory Contest (grades 8-12). Contests involve a set of 6 questions, and students must solve as many as possible in 30 minutes.
Math League is fun for students interested in collaboration and community. Successful Math League teams are often part of a school math club that facilitates practice sessions and mock contests.
4. Creative Competitions
While not as directly related to math, creative problem-solving competitions are invaluable for the future mathematician, while also being extremely impressive in college admissions. Three popular
creative problem-solving competitions are Destination Imagination, the Future Problem Solving Program, and Odyssey of the Mind.
Destination Imagination
Through the Destination Imagination (DI) competition, teams of students are presented with unique challenges that they must solve. Students typically rehearse their solutions for 2-4 months as they
prepare for their local tournament.
The challenges, which vary each year, touch on areas including technical, scientific, service learning, fine arts, and more. For example, an engineering challenge might involve building a bridge from
specific (weird) materials, testing the load it can bear, then presenting the bridge to judges as part of a creative story. Teams participate at the local, regional, affiliate, and global levels.
The Future Problem Solving Program
The Future Problem Solving is a collaborative program that encourages the development of critical and creative thinking in students. The program teaches a six-step decision-making model with the
ultimate goal of preparing students to face real-world problems.
FPSPI offers four competitions in the following categories: Global Issues Problem Solving, Community Problem Solving, Scenario Writing, and Scenario Performance.
Odyssey of the Mind
Like DI, Odyssey of the Mind is an international creative problem-solving program. Teams are provided a task problem that involves writing, design, construction, and theatrical performance. For
example, a challenge may ask students to apply artistic and technical knowledge to design mechanical dinosaurs and explain their backstory. For months, teams develop a solution to this task, then
they attend a competition to present their solution.
One interesting element of the competition is its “spontaneous portion” where, on competition day, students must also generate a solution to a problem they have not seen before.
5. Summer Programs
There are many summer programs for aspiring math majors. Some take place on college campuses, others in individual communities, and, nowadays, there are even fully virtual math summer programs. Find
just the program for you!
Some options:
6. Teaching/Mentoring
An important part of mastering a trade is being able to teach it to others. Moreover, teaching and mentoring roles look great during the college admissions process.
Some opportunities future math majors should look into include:
• Teaching math after school at a local community center
• Mentoring kids in math or chess club at your local elementary school
• Paid tutoring jobs (for example, Mathnasium)
• Working at math competitions, hackathons, or chess competitions
• Teaching at math summer camps
Most of these opportunities will be community-specific, so start by reaching out to institutions that exist within your community to see if they have any opportunities.
How Do Extracurriculars Impact Your College Chances?
Grades and test scores are important factors in the college admissions process, but admissions officers are also interested in who students are beyond the numbers. This is where factors like
extracurriculars, personal essays, and recommendations come into play. Through extracurriculars, students can show their specific interests and, more importantly, their dedication to specific
Because of the importance of dedication, our CollegeVine team recommends that students focus on 2-3 extracurricular activities that they care deeply about. If your extracurricular list shows breadth
rather than depth, your admissions officer might not understand how truly dedicated you are to the field of mathematics.
Additionally, admissions officers often group activities into one of the four tiers of extracurricular activities. The highest tiers—Tier 1 and Tier 2—have the most influence on college admissions
and are reserved for the rarest and most distinguished extracurriculars. Lower-tier activities—those in Tiers 3 and 4—are less well-known, less distinguished, and ultimately have less of an impact on
college admissions.
For example, an admissions officer is going to be more drawn to a student who advanced to the USAMO stage of the Math Olympiad and founded a math club at their high school—activities in Tiers 1 and
2—than a student who was a general member of Mu Alpha Theta—an activity in Tier 4.
As you choose your extracurriculars, think about what will stand out to admissions officers and what will showcase your dedication to the field of mathematics. Additionally, put your extracurriculars
into CollegeVine’s free chancing engine, which will tell you how specific extracurriculars will affect your admissions chances at specific colleges and universities. | {"url":"https://blog.collegevine.com/extracurriculars-for-high-schoolers-interested-in-studying-math/","timestamp":"2024-11-06T12:23:45Z","content_type":"text/html","content_length":"64303","record_id":"<urn:uuid:7e42dd87-7ae9-42c0-ad0c-139716c6f9b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00594.warc.gz"} |
equal temperament
Definitions for equal temperament
equal tem·per·a·ment
This dictionary definitions page includes all the possible meanings, example usage and translations of the word equal temperament.
Princeton's WordNet
1. equal temperamentnoun
the division of the scale based on an octave that is divided into twelve exactly equal semitones
"equal temperament is the system commonly used in keyboard instruments"
1. Equal temperament
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the
frequencies of any adjacent pair of notes is the same, which gives an equal perceived step size as pitch is perceived roughly as the logarithm of frequency.In classical music and Western music in
general, the most common tuning system since the 18th century has been twelve-tone equal temperament (also known as 12 equal temperament, 12-TET or 12-ET; informally abbreviated to twelve equal),
which divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2 (12√2 ≈ 1.05946). That resulting smallest interval, 1⁄12 the width
of an octave, is called a semitone or half step. In Western countries the term equal temperament, without qualification, generally means 12-TET. In modern times, 12-TET is usually tuned relative
to a standard pitch of 440 Hz, called A440, meaning one note, A, is tuned to 440 hertz and all other notes are defined as some multiple of semitones apart from it, either higher or lower in
frequency. However, the standard pitch has not always been 440 Hz; it has varied considerably and generally risen over the past few hundred years.Other equal temperaments divide the octave
differently. For example, some music has been written in 19-TET and 31-TET, while the Arab tone system uses 24-TET. Instead of dividing an octave, an equal temperament can also divide a different
interval, like the equal-tempered version of the Bohlen–Pierce scale, which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a "pseudo-octave" in that system,
into 13 equal parts. For tuning systems that divide the octave equally, but are not approximations of just intervals, the term equal division of the octave, or EDO can be used. Unfretted string
ensembles, which can adjust the tuning of all notes except for open strings, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for
acoustic reasons. Other instruments, such as some wind, keyboard, and fretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings. Some wind
instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles and vocal groups.
1. equal temperament
Equal temperament is a musical tuning system which divides any octave into a series of equal steps or intervals. This means that the frequency ratio between every adjacent pair of notes is
constant. It allows for music to be transposed into different keys without changing the relationship between notes. The most commonly used form of equal temperament in modern western music is
called twelve-tone equal temperament (12-TET), which divides the octave into 12 equal parts.
1. Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of
frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for every note in the system. In equal temperament tunings, an interval – usually the
octave – is divided into a series of equal steps. For classical music, the most common tuning system is twelve-tone equal temperament, inconsistently abbreviated as 12-TET, 12TET, 12tET, 12tet,
12-ET, 12ET, or 12et, which divides the octave into 12 parts, all of which are equal on a logarithmic scale. It is usually tuned relative to a standard pitch of 440 Hz, called A440. Other equal
temperaments exist, but in Western countries when people use the term equal temperament without qualification, they usually mean 12-TET. Equal temperaments may also divide some interval other
than the octave, a pseudo-octave, into a whole number of equal steps. An example is an equal-tempered Bohlen–Pierce scale. To avoid ambiguity, the term equal division of the octave, or EDO is
sometimes preferred. According to this naming system, 12-TET is called 12-EDO, 31-TET is called 31-EDO, and so on.
1. Chaldean Numerology
The numerical value of equal temperament in Chaldean Numerology is: 9
2. Pythagorean Numerology
The numerical value of equal temperament in Pythagorean Numerology is: 6
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Homework Assignment 5 Solution - Programming Help
Hand-in Instructions
This homework assignment includes two written problems and a programming problem in Java. Hand in all parts electronically to your Canvas assignments page. For each written question, submit a single
pdf file containing your solution. Handwritten submissions must be scanned. No photos or other file types allowed. For the programming question, submit a zip file containing all the Java code
necessary to run your program, whether you modified provided code or not.
Submit the following three files (with exactly these names):
For example, for someone with UW NetID crdyer@wisc.edu the first file name must be:
Late Policy
All assignments are due at 11:59 p.m. on the due date. One (1) day late, defined as a 24-hour period from the deadline (weekday or weekend), will result in 10% of the total points for the assignment
deducted. So, for example, if a 100-point assignment is due on a Wednesday and it is handed in between any time on Thursday, 10 points will be deducted. Two (2) days late, 25% off; three (3) days
late, 50% off. No homework can be turned in more than three (3) days late. Written questions and program submission have the same deadline. A total of three (3) free late days may be used throughout
the semester without penalty. Assignment grading questions must be discussed with a TA or grader within one week after the assignment is returned.
Collaboration Policy
You are to complete this assignment individually. However, you may discuss the general algorithms and ideas with classmates, TAs, peer mentors and instructor in order to help you answer the
questions. But we require you to:
• not explicitly tell each other the answers
• not to copy answers or code fragments from anyone or anywhere
• not to allow your answers to be copied
• not to get any code from the Web
CS 540 Fall 2019
Problem 1. Probabilities [15 points]
Suppose three documents 1, 2 and 3 contain only five words “ I”, “am”, “we”, “are” and “groot”. The following table summarizes the number of occurrences of each word in each document. Suppose one
document is chosen at random (each document with equal probability), then one word in that document is chosen at random (each word with equal probability within that document).
Document / Word I am we are groot
Note that the table can be used to compute conditional probabilities. For example, P(Word = groot | Document = 1) =
a. [2] What is the probability that the word is “groot”, i.e., P(Word = groot)?
b. [2] Suppose the randomly chosen word is “we”. What is the probability that it comes from document 1, i.e., P(Document = 1 | Word = we)?
c. [2] Suppose the randomly chosen word starts with an “a”. What is the probability that it comes from document 2, i.e., P(Document = 2 | Word = am or Word = are)?
Now suppose document 1 is chosen with probability P(Document = 1) = ^1[6], document 2 is chosen with probability P(Document = 2) = ^1[3], and document 3 is chosen with probability P(Document =
3. = ^1[2]. Then, one word in that document is chosen at random (each word with equal probability within that document).
d. [3] What is the probability that the word is “groot”, i.e., P(Word = groot)?
d. [3] Suppose the randomly chosen word is “we”. What is the probability that it comes from document 1, i.e., P(Document = 1 | Word = we)?
e. [3] Suppose the randomly chosen word starts with an “a”. What is the probability that it comes from document 2, i.e., P(Document = 2 | Word = am or Word = are)?
Show your calculations and round all final answers to 4 decimal places.
Problem 2. Using a Bayesian Network [20 points]
Use the Bayesian Network below containing 6 Boolean random variables to answer the following questions using inference by enumeration. For the problems, Spring = S, Rain = R, Worms = Wo, Wet = We,
Mowing = M, and Birds = B. Give your answers to 4 decimal places. Show your work.
a. [4] What is the probability that it rains? (P(R))
b. [4] What is the probability that the grass is wet? (P(We))
c. [4] Given that it is spring, how likely is there to be someone mowing? (P(M | S))
d. [4] Given that it is spring, how likely is there to be a bird on the lawn? (P(B | S))
e. [4] If there are birds on the lawn, but there are no worms out, how likely is it that it is spring? (P(S | B, ¬Wo))
The CPTs are defined as follows:
• P(R | S) = 0.7, P(R | ¬S) = 0.3
• P(Wo | R) = 0.7, P(Wo | ¬R) = 0.2
• P(We | R, Wo) = 0.12, P(We | R, ¬Wo) = 0.25, P(We | ¬R, Wo) = 0.1, P(We | ¬R, ¬Wo) = 0.08
• P(M | We) = 0.02, P(M | ¬We) = 0.42
• P(B | S, Wo) = 0.8, P(B | S, ¬Wo) = 0.4, P(B | ¬S, Wo) = 0.4, P(B | ¬S, ¬Wo) = 0.4
Problem 3. A Naïve Bayes Classifier for Sentiment Analysis [65 points]
One application of Naïve Bayes classifiers is sentiment analysis, which is a sub-field of AI that extracts affective states and subjective information from text. One common use of sentiment analysis
is to determine if a text document expresses negative or positive feelings. In this homework you are to implement a Naïve Bayes classifier for categorizing movie reviews as either POSITIVE or
NEGATIVE. The dataset provided consists of online movie reviews derived from an IMDb dataset: https://ai.stanford.edu/~amaas/data/sentiment/ that have been labeled based on the review scores. A
negative review has a score ≤ 4 out of 10, and a positive review has a score ≥ 7 out of 10. We have done some preprocessing on the original dataset to remove some noisy features. Each row in the
training set and test set files contains one review, where the first word in each line is the class label (1 represents POSITIVE and 0 represents NEGATIVE) and the remainder of the line is the review
Methods to Implement
We have provided code for you that will open a file, parse it, pass it to your classifier, and output the results. What you need to focus on is the implementation in the file
NaiveBayesClassifier.java If you open that file, you will see the following methods that you must implement:
• Map<Label, Integer> getDocumentsCountPerLabel (List<Instance> trainData)
This method counts the number of reviews per class label in the training set and returns a map that stores the (label, number of documents) key-value pair.
• Map<Label, Integer> getWordsCountPerLabel(List<Instance> trainData)
This method counts the number of words per label in the training set and returns a map that stores the (label, number of words) key-value pair.
• void train(List<Instance> trainData, int v)
This method trains your classifier using the given training data. The integer argument v is the size of the total vocabulary in your model. Store this argument as a field because you will need it in
computing the smoothed class-conditional probabilities. (See the section on Smoothing below.)
• ClassifyResult classify(List<String> words)
This method returns the classification result for a single movie review.
In addition, you will need to implement a k-fold cross validation function as defined in the
• double kFoldScore(Classifier clf, List<Instance> trainData, int k, int v)
This method takes a classifier clf and performs k-fold cross validation on trainData. The output is the average of the scores computed on each fold. You can assume that the number of instances in
trainData is always divisible by k, so the size
of each fold will the same. We will use the accuracy (number of correctly predicted instances / number of total instances) as the score metric.
We have also defined four class types to assist you in your implementation. The Instance class is a data structure holding the label and the movie review as a list of strings:
public class Instance {
public Label label;
public List<String> words;
The Label class is an enumeration of our class labels:
public enum Label {POSITIVE, NEGATIVE}
The ClassifyResult class is a data structure holding each label’s log probability (whose values are described in the log probabilities section below) and the predicted label:
public class ClassifyResult {
public Label label;
public Map<Label, Double> logProbPerLabel;
The only provided files you need edit are NaiveBayesClassifier.java and CrossValidation.java but you are allowed to add extra helper class files if you like. Do not include any package paths or
external libraries in your program. Your program is only required to handle binary classification problems.
There are two concepts we use here:
• Word token: an occurrence of a given word
• Word type: a unique word as a dictionary entry
For example, “the dog chases the cat” has 5 word tokens but 4 word types; there are two tokens of the word type “the”. Thus, when we say a word “token” in the discussion below, we mean the number of
words that occur and NOT the number of unique words. As another example, if a review is 15 words long, we would say that there are 15 word tokens. For example, if the word “lol” appeared 5 times, we
say there were 5 tokens of the word type “lol”.
The conditional probability P(w|l), where w represents some word type and l is a label, is a multinomial random variable. If there are |V| possible word types that might occur, imagine a |V|-sided
die. P(w |l) is the likelihood that this die lands with the w-side up. You will need to estimate two such distributions: P(w|Positive) and P(w|Negative).
^∑ ∈
CS 540 Fall 2019
One might consider estimating the value of P(w|Positive) by simply counting the number of tokens of type w and dividing by the total number of word tokens in all reviews in the training set labeled
as Positive, but this method is not good enough in general because of the “unseen event problem,” i.e., the possible presence of words in the test data that did not occur at all in the training data.
For example, in our classification task consider the word “foo”. Say “foo” does not appear in our training data but does occur in our test data. What probability would our classifier assign to P(foo|
Positive) and P(foo|Negative)? The probability would be 0, and because we are taking the sum of the logs of the conditional probabilities for each word and log 0 is undefined, the expression whose
maximum we are computing would be undefined.
What we do to get around this is pretend we actually did see some (possibly fractionally many) tokens of the word type “foo”. This goes by the name Laplace smoothing or add-δ smoothing, where δ is a
parameter. The conditional probability then is defined as:
where l ∈ {Positive, Negative}, and Cl(w) is the number of times the tokens of word type w appears in reviews labeled l in the training set. As above, |V| is the size of the total vocabulary we
assume we will encounter (i.e., the dictionary size). Thus, it forms a superset of the words used in the training and test sets. The value |V| will be passed to the train method of your classifier as
the argument int v. For this assignment, use the value δ = 1. With a little reflection, you will see that if we estimate our distributions in this way, we will have
( ∣ ) = 1. Use the equation above for P(w|l) to calculate the conditional probabilities in your implementation.
Log Probabilities
The second gotcha that any implementation of a Naïve Bayes classifier must contend with is underflow. Underflow can occur when we take the product of a number of very small floating-point values.
Fortunately, there is a workaround. Recall that a Naïve Bayes classifier computes
( ) = arg max � ( ) � ( ∣ )�
where l ∈ {Positive, Negative} and wi is the i^th word type in a review, numbered 1 to k. Because maximizing a formula is equivalent to maximizing the log value of that formula, f(w) computes the
same class as
( ) = arg max �log ( ) + � log ( ∣ )�
What this means for you is that in your implementation you should compute the g(w) formulation of the function above rather than the f(w) formulation. Use the Java function log(x) which
computes the natural logarithm of its input. This will result in code that avoids errors generated by multiplying very small numbers.
This is what you should return in the ClassifyResult class: logProbPerLabel.get(Label.POSITIVE) is the value log ( ) + ∑[ =1] log ( ∣ ) with l = Positive and logProbPerLabel.get(Label.NEGATIVE) with
l = Negative. The label returned in this class corresponds to the output of g(w). Break ties by classifying a review as Positive.
We will test your program on multiple training and test sets, using the following command line format:
java SentimentAnalysis <mode> <trainFilename> [<testFilename> | <K>]
where trainingFilename and testFilename are the names of the training set and test set files, respectively. mode is an integer from 0 to 3, controlling what the program will output. When mode is 0 or
1, there are only two arguments, mode and trainFilename; when the mode is 2 the third argument is testFilename; when mode is 3, the third argument is K, the number of folds used for cross validation.
The output for these four modes should be:
0. Prints the number of documents for each label in the training set
1. Prints the number of words for each label in the training set
2. For each instance in test set, prints a line displaying the predicted class and the log probabilities for both classes
0. Prints the accuracy score for K-fold cross validation
In order to facilitate debugging, we are providing sample training set and test set files called train.txt and test.txt in the zip file. In addition, we are providing the correct output for each of
the four modes based on these two datasets, in the files mode0.txt, …, mode3.txt
For example, the command
java SentimentAnalysis 2 train.txt test.txt
should train the classifier using the data in train.txt and print the predicted class for every review in test.txt
The command
java SentimentAnalysis 3 train.txt 5 should perform 5-fold cross-validation on train.txt
You are not responsible for handling parsing of the training and test sets, creating the classifier, or printing the results on the console. We have written the main class SentimentAnalysis for you,
which will load the data and pass it to the method you are implementing. Do NOT modify any IO code.
As part of our testing process, we will unzip the file you submit, remove any class files, call javac *.java to compile your code, and then call the main method SentimentAnalysis with parameters of
our choosing. Make sure your code runs and terminates in less than 20 seconds for each provided test case on the computers in the department because we will conduct our tests on these computers.
Hand in your modified versions of NaiveBayesClassifier.java and CrossValidation.java. Also submit any additional helper .java files required to run your program (including the provided ones in the
skeleton). Do not submit any other files including the data files (i.e., no .class or .txt files). All the .java files should be zipped as <wiscNetID>-HW5-P3.zip for submission to Canvas. | {"url":"https://www.edulissy.org/product/homework-assignment-5-solution-4/","timestamp":"2024-11-09T11:00:09Z","content_type":"text/html","content_length":"224329","record_id":"<urn:uuid:70012126-2445-4c60-ab60-65b68af5324c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00308.warc.gz"} |
Singular learning theory: connecting algebraic geometry and model selection in statistics
December 12 to December 16, 2011
at the
American Institute of Mathematics, Palo Alto, California
organized by
Russell Steele, Bernd Sturmfels, and Sumio Watanabe
This workshop, sponsored by AIM and the NSF, will be devoted to singular learning theory, the application of algebraic geometry to problems in statistical model selection and machine learning. The
intent of this workshop is to connect algebraic geometers and statisticians specializing in Bayesian model selection criteria in order to explore the relationship between analytical asymptotic
approximations from algebraic geometry and other commonly used methods in statistics (including traditional asymptotic and Monte Carlo approaches) for developing new model selection criteria. The
hope is to generate interest amongst both communities for collaborations that will spur new topics of research in both algebraic geometry and statistics.
Singular statistical learning is an approach for statistical learning and model selection that can be applied to singular parameter spaces, i.e. can be used for non-regular statistical models. The
methodology uses the method of resolution of singularities to generalize the criteria for regular statistical models to non-regular models. Examples of non-regular statistical models that have been
studied as part of singular learning theory include hidden Markov models, finite mixture models, and multi-layer neural network models. Although there exists a large body of recent published work in
this area, it is has not yet been integrated or even well-cited by the larger statistical community.
The workshop has three primary goals:
1. To introduce statisticians and computer scientists working the area of model selection to the topic of singular learning theory, in particular the application of the method of resolution of
singularities to model selection for non-regular statistical models.
2. To generate a list of open problems in algebraic geometry motivated by complex statistical models that cannot be covered by current results.
3. To collaboratively develop a set of core materials that will define the area of singular statistical learning that will be accessible to geometers and statisticians.
The main open problems the workshop will consider:
1. Exploring connections of Widely Applicable Information Criteria (WAIC) from singular learning theory to other model selection criteria, including the Deviance Information Criterion (DIC), regular
statistical versions of the AIC and BIC, and other criteria specific to particular non-regular statistical models (for example, the scan statistic from spatial statistics).
2. Identifying fundamental problems in algebraic geometry are related to generalizing these information criteria to model selection problems for Generalized Estimating Equations (GEE), which use
ideas from semi-parametric inference to obtain estimates of parameters without assuming a parametric form for the likelihood of the observed data.
3. Generalizing the singular learning theory information criteria be to statistical problems where some observations contain missing information and/or measurement error.
4. Establishing the finite sample properties of WAIC, in particular for problems where one can incorporate prior knowledge in a fully Bayesian modelling approach.
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the
workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the
field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel
working sessions.
The deadline to apply for support to participate in this workshop has passed.
For more information email workshops@aimath.org
Plain text announcement or brief announcement.
Go to the American Institute of Mathematics.
Go to the list of upcoming workshops. | {"url":"https://aimath.org/ARCC/workshops/modelselection.html","timestamp":"2024-11-11T07:28:00Z","content_type":"text/html","content_length":"6272","record_id":"<urn:uuid:a5c826a9-705f-4bd4-b4c7-3910180e21c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00590.warc.gz"} |
Welcome to the webpage for the Warwick Mathematics Postgraduate Seminar.
This term, all talks will be held in B3.02 at 12 noon on Wednesday (except when stated otherwise). The seminar will take a hybrid format so that students can join us virtually if they have (or
prefer) too. The link to join the seminar virtually is:
In addition, there will be `coffee afternoon' at 1 p.m immediately after the seminar, for those of us who still enjoy socialising.
Organisers: Lucas LavoyerLink opens in a new window and Sunny SoodLink opens in a new window.
Term 3 2021-22 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute.
Week 1: Wednesday 27th April
George KontogeorgiouLink opens in a new window - Trees of Intermediate GrowthLink opens in a new window
Trees of uniform exponential growth have been known since the dawn of time. Trees of uniform polynomial growth have been known since 2000, which is when Benjamini and Schramm constructed them. Both
the former and the latter include examples of unimodular trees.
In this talk I will explain why (very many) unimodular trees of uniform intermediate growth exist. This is joint work with Martin Winter, and answers a question by Itai Benjamini.
Week 2: Wednesday 4th May
Robin VisserLink opens in a new window - Cluster pictures for hyperelliptic curvesLink opens in a new window
Given a hyperelliptic curve $C : y^2 = f(x)$ over a number field $K$, we can study its reduction behaviour at odd primes using the machinery of cluster pictures, introduced by
Dokchitser--Dokchitser--Maistret--Morgan. This allows us to compute a wide range of arithmetic invariants for both $C$ and its Jacobian, including the conductor, minimal discriminant, Tamagawa
numbers, whether the curve is semistable, and much more! In this talk, we'll go through several examples of cluster pictures and will prove a couple of neat results about hyperelliptic curves along
the way.
Week 3: Wednesday 11th May
Osian ShelleyLink opens in a new window - On Tauberian Theorem for Generalised Signed MeasuresLink opens in a new window
According to Feller, Karamata's Tauberian theorem has a 'glorious history', even though it is often omitted from modern books on probability theory. In this talk, we outline the proof of this theorem
and show how it can extend to work for generalised signed measures. This extension has a surprising application in the theory of stochastic control. On route, we highlight the nuances between the
vague convergence of signed measures and the pointwise convergence of their distribution functions.
Week 4: Wednesday 18th May
Alice HodsonLink opens in a new window - A higher order virtual element method for the Cahn-Hilliard equationLink opens in a new window
In this talk we discuss nonconforming virtual element methods (VEMs) for fourth-order problems. At present, the available VEM literature on fourth-order problems only includes defining projection
operators based on the underlying variational problem. This approach involves constructing only one projection which depends on the local contribution to the bilinear form. Instead, we follow the
approach of defining a hierarchy of projection operators for the necessary derivatives with the starting point being a constraint least squares problem. By defining the projection operators in this
way, we show that we can directly apply our method to nonlinear fourth-order problems. This approach can also be easily included into existing software frameworks.
This talk showcases the application of our generalised method to the Cahn-Hilliard equation. As a consequence of our approach, we do not require any special treatment of the nonlinearity. Our method
is shown to converge with optimal order also in the higher order setting. The theoretical convergence result is verified numerically with standard benchmark tests from the literature.
Week 5: Wednesday 25th May
Philip HoldridgeLink opens in a new window - Random Diophantine Equations and the Prime Hasse PrincipleLink opens in a new window
If a homogeneous Diophantine equation has a nontrivial solution in the integers, then it also has a nontrivial solution in the reals and in the integers modulo $p^{n}$ for ever prime $p$ and $n>0$.
The Hasse Principle is said to hold for an equation if the converse also holds.
In their 2014 paper, Brüdern and Dietmann proved that for diagonal homogenous equations of degree $k$ in at least $3k+2$ variables, the Hasse Principle almost always holds, in the sense that if one
chooses the coefficients uniformly in a box of size $A$, then the probability that the Hasse Principle holds tends to 1 as $A$ tends to infinity.
In this seminar, we will outline the main ideas of their proof (which uses the Circle Method), and then talk about the analogous result for solubility of equations in the primes (solutions consisting
of prime numbers). It turns out that one of the main challenges of doing this is coming up with an analogue to the Hasse Principle, which I am calling a "Prime Hasse Principle".
Week 6: Wednesday 1st June
Paul PanteaLink opens in a new window - How to invert spheres and influence peopleLink opens in a new window
Spheres of negative dimension sound like something only mathematicians could come up with; and that's why nobody will talk to us at parties. But that's OK, because we have stable homotopy theory to
keep us entertained. In this talk, we will introduce some key ideas like categorification, stabilization, and spectra; and explain how formally inverting spheres leads us to the study of all possible
cohomology theories and to a vast generalization of commutative algebra. Finally, we will discuss how to incorporate group actions into this, and some in-progress effort to understand more things
about the group of order 2.
Week 7: Wednesday 8th June
Ruta SliazkaiteLink opens in a new window - Two extremes of Dehn functionsLink opens in a new window
In this talk I will discuss two extremes of Dehn functions. On one end we have every geometric group theorist’s favourite groups: hyperbolic groups, which have linear Dehn functions. On the other
end, we will look at a group $\Gamma$ with Ackermann Dehn function (a function that is not even primitive recursive). I will firstly give a brief introduction to hyperbolic groups and give the
definition of a Dehn function for a group. I will then show the construction of $\Gamma$ and give a sketch proof of the lower bound of its Dehn function. $\Gamma$ is an interesting example of a group
built as an HNN extension of a free-by-cyclic, one-relator, CAT(0) group $G$, relative to its free subgroup $H$. It shows that even groups that may seem ‘nice’ can have wild properties.
Week 8: Wednesday 15th June
Diego Martin DuroLink opens in a new window - The Knutson - Savitskii ConjectureLink opens in a new window
A complex representation of a group $G$ is a group homomorphism from $G$ to $GL_n(\mathbb{C})$ and a character is the trace of the of the matrix corresponding to an element of the group. About 40
years ago, Knutson conjectured that for every irreducible character, there is a generalised character such that their tensor product is the regular character. Savitskii disproved this in 1993 and
posed a new conjecture. In this talk, after a brief introduction to Character Theory, we will disprove Savitskii's Conjecture and discuss its relations to Kaplansky's Sixth Conjecture.
Week 9: Wednesday 22nd June
Chloé Postel-Vinay (University of Chicago)Link opens in a new window- A counter example to the periodic orbit conjectureLink opens in a new window
Let $M$ be a closed manifold, let $\phi_{t}$ be a flow on $M$ such that all of its orbits are periodic. A natural question is to ask whether or not the length of the orbits of $\phi$ is bounded (this
is the periodic orbit conjecture). It turns out it isn’t necessarily true when the dimension of $M$ is greater than or equal to 4. I will explain one of the first counterexamples to this conjecture,
given by Sullivan in dimension 5 in 1976.
Week 10: Wednesday 29th June
Thomas RichardsLink opens in a new window - Monodromy in Complex DynamicsLink opens in a new window
In this talk I will give an introduction to complex dynamics and discuss the monodromy problem in polynomial parameter space. This relates the topology of the parameter space of degree $d$
polynomials to the automorphisms of the space of one-sided infinite strings on $d$ symbols. I will discuss how this was proven and, time permitting, discuss the analogous problem in Hénon parameter
space, computational work to understand the higher dimensional problem, and interesting phenomena arising in the course of experimentation.
Term 2 2021-22 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute.
Week 1: Wednesday 12th January
Hollis WilliamsLink opens in a new window - The Penrose inequality for perturbations of Schwarzschild spacetimeLink opens in a new window
The Penrose inequality is a remarkable geometric inequality that relates the mass of a black hole spacetime to the total area of its black holes. Penrose suggested the inequality on physical grounds
in the 1960s, but a rigorous mathematical proof in the general case is still lacking. We present some new ideas towards a proof for the special case of perturbations of Schwarzschild spacetime using
an elliptic PDE called the Jang equation.
Week 2: Wednesday 19th January
Alvaro Gonzalez HernandezLink opens in a new window - An introduction to the Hasse principle through examplesLink opens in a new window
The Hasse principle asks a very important question in the study of Diophantine equations: does the existence of real and p-adic solutions imply the existence of rational solutions? In this talk I
will use examples of equations to motivate why this principle is useful and how it is linked to the geometry of the varieties defined by such equations. In particular, the connections between the
Hasse principle and the arithmetic structure of elliptic curves will be discussed.
If time permits, I will explain how to construct explicit counterexamples to the Hasse principle as the homogeneous spaces associated to elliptic curves with non-trivial Tate-Shaferevich groups.
Week 3: Wednesday 26th January
Diana MocanuLink opens in a new window - Elliptic Curves in CryptographyLink opens in a new window
Cryptography is the study of secure communication techniques used in increasingly many everyday tasks such as instant messaging and online transactions. Modern cryptography is heavily based on number
theory, with the latest research using elliptic curves to construct quantum-resistant cryptosystems. In this talk, we will review basic theory of elliptic curves and then we will see how to construct
two public key cryptosystems using it, discussing their security at the same time.
Week 4: Wednesday 2nd February
James Taylor (University of Oxford)Link opens in a new window - Representations of $GL_{2}$ and $p$-adic Symmetric SpacesLink opens in a new window
The general aim of representation theory is to classify all representations of an object up to equivalence, where the type of representation considered can vary depending on the context. In this
talk, I will discuss one way this can be achieved for abstract representations of $GL_{2}$ over a finite field, and smooth representations of $GL_{2}$ over a $p$-adic field. Both approaches involve
natural actions of $GL_{2}$ on some space, and these motivate studying actions of $GL_{2}$ (over a $p$-adic field) on certain rigid analytic spaces in order to better understand a larger class of
representations than just smooth. I will talk about the representations which arise from these constructions, and current work which attempts to better understand them.
Week 5: Wednesday 9th February
Marc Homs DonesLink opens in a new window - Workshop on iterative periodic functions and their generalizationLink opens in a new window
Can you find all continuous functions of the real line such that $f^{2}:=f \circ f = id$? And what about finding all periodic functions, i.e. $f^{n} = id$? Or even all solutions to $f^{n }= f^{k}$?
In this session, we will try to answer these questions together.
This is joint work with Armengol Gasull, presented in doi:10.3934/dcds.2020303 (Looking at it ahead of the presentation is considered cheating!).
Week 6: Wednesday 16th February
Steven GroenLink opens in a new window - Can you hear the shape of a curve?Link opens in a new window
In 1966, Mark Kac asked if one can determine the shape of a drum from the sound it makes. It turned out that this is in general not possible. In this talk, we approach a slightly twitched problem:
can we determine a curve over a finite field (up to isomorphism) from its number of points? Continuing the striking similarity between both questions, the answer is again no; we call curves with the
same point count isogenous. Instead we study 'doubly isogenous' curves, which are even more alike than isogenous curves. A natural question arises: are two doubly isogenous curves necessarily
isomorphic? We treat this question in great detail for a family of curves with prescribed automorphism groups.
This summarises a joint paper (https://arxiv.org/abs/2102.11419Link opens in a new window) with Vishal Arul, Jeremy Booher, Everett Howe, Wanlin Li, Vlad Matei, Rachel Pries and Caleb Springer.
Week 7: Wednesday 23rd February
Dimitris LygkonisLink opens in a new window - A tale about the many drunks of Flatland.Link opens in a new window
In this talk, we will attempt to shed light on a typical night of the many drunks of Flatland. We will show that although it can be pretty eventful, the many drunks are quite forgetful. Surprisingly,
using tools and ideas from functional analysis, quantum mechanics and random polymers can explain a lot about their behaviour.
This is joint work with N. Zygouras and can be found in arXiv:2109.06115 and arXiv:2202.08145.
Week 8: Wednesday 2nd March
Arshay ShethLink opens in a new window - A Historical Introduction to Transcendental Number TheoryLink opens in a new window
A complex number is said to be algebraic if it is the root of a non-zero polynomial with integer coefficients. A complex number that is not algebraic is called transcendental. Broadly speaking,
transcendental number theory is the study of transcendental numbers. In this talk, without assuming any previous background in number theory, we will give an introduction to this subject from a
historical point of view; we will begin by discussing some of the earliest results of the subject and end by discussing important open problems that remain shrouded in mystery to this day.
Week 9: Wednesday 9th March
Irene Gil FernándezLink opens in a new window- The Second Coming of the KrakenLink opens in a new window
In this talk we will see how a couple of krakens can be very useful to embed a pillar in a graph with large minimum degree (solving a conjecture of Thomassen, 1989). A pillar is a graph that consists
of two vertex-disjoint cycles of the same length, $s$ say, along with $s$ vertex-disjoint paths of the same length which connect matching vertices in order around the cycles. Despite the simplicity
of the structure of pillars and various developments of powerful embedding methods for paths and cycles in the past three decades, this innocent looking conjecture has seen no progress to date. In
this talk, we will try to give an idea of the proof of such embedding, which consists of building a pillar (algorithmically) in sublinear expanders. This is joint work with Hong Liu.
Week 10: Wednesday 16th March
Joshua Daniels-Holgate Link opens in a new window-Approximation of Mean Curvature Flow with Generic Singularities by Smooth Flows with SurgeryLink opens in a new window
We construct smooth flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of
Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is
then used to control these approximating flows. We demonstrate an application of this surgery flow by improving the entropy bound on the low-entropy Schoenflies conjecture.
Term 1 2021-22 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute.
Week 1: Wednesday 6th October
Hollis WilliamsLink opens in a new window - Noncommutative Geometry and CFTs
We give the abstract definition for QFTs and CFTs and then outline the applications of spectral triples (data sets which encode noncommutative geometries) in theoretical physics and field theory. We
discuss an interesting connection between spectral triples and two-dimensional superconformal field theories which is relevant for string theory.
After the seminar, the first year PhD students are invited to take part in a 'Treasure Hunt' around the campus. This will be a great opportunity to have a bit of mid-week fun, learn where the
important parts of the campus are and to get to know your fellow colleagues (in particular, who you'd rather not share an office with next year!).
In addition to this, the first year PhD students will also have to opportunity to take part in a `photoshoot' organised by the department. If you are interested, please make sure you are well dressed
for the occasion!
Week 2: Wednesday 13th October
Ryan Acosta BabbLink opens in a new window - Lᵖ Convergence of Series of Eigenfunctions for the Equilateral TriangleLink opens in a new window
The eigenfunctions of the Laplacian on a rectangle give rise to the familiar Fourier series, whose Lᵖ convergence is well known. We will use this result to obtain Lᵖ convergence of series of
trigonometric eigenfunctions of the Dirichlet Laplacian on an equilateral triangle. Along the way we will discuss some of the limitations of the argument owing to symmetry considerations.
Week 3: Wednesday 20th October
William O'ReganLink opens in a new window- Efficiently covering the Sierpinski carpet with tubesLink opens in a new window
We call a delta/2-neighbourhood of a line in R^d a tube of width delta. For a subset K of R^d it is an interesting problem to try and efficiently cover K with tubes as to try and minimise the total
width of the tubes used. If for every epsilon > 0 we are able to find a collection of tubes which cover K with their total width less than epsilon we say that K is tube-null. The notion of
tube-nullity has its roots in harmonic analysis, however, the notion is interesting in its own right. In the talk I will give an example of a set which is tube-null, the Sierpinski carpet, along with
a rough sketch of its proof. If time permits I will discuss some open problems in the area along with their progress.
Week 4: Wednesday 27th October
No talk this week- Non-academic careers for postgraduate mathematicians & researchersLink opens in a new window
What are the career options for Masters & PhD Mathematicians who decide not to pursue an academic career? Post graduate mathematicians have an analytical and problem-solving skill set that is in
demand, and there are a variety of very interesting career opportunities. In this interactive Q&A a panel of Warwick alumni will describe their career journey since graduation and share their hints
and tips to help you plan a non-academic career. The panel will feature:
Mattia Sanna (Data Scientist at Methods Analytics: Warwick PhD Computational Algebraic Number Theory 2020)
Chris Gamble (Applied Engineering co-lead, DeepMind: Warwick MORSE 2009, University of Oxford DPhil Machine Learning & Bayesian Statistics 2014)
Huan Wu (Project Leader at Numerical Modelling and Optimisation Section, TWI: Warwick PhD Mathematics & Statistics 2017 Atomistic-to-Continuum Coupling for Crystal Defects)
Zhana Kuncheva (Senior Scientist - Statistical Genetics at Silence Therapeutics plc): Warwick BSc MORSE, Imperial PhD Mathematics & Statistics, modelling populations of complex networks)
Week 5: Wednesday 3rd November
Patience AblettLink opens in a new window - Constructing Gorenstein curves in codimension fourLink opens in a new window
Projectively (or arithmetically) Gorenstein varieties are a frequently occurring subset of projective varieties, whose coordinate rings are Gorenstein. Whilst there exist concrete structure theorems
for such varieties in codimension three and below, the picture is less clear for codimension four. Recent work of Schenck, Stillman and Yuan outlines all possible Betti tables describing the minimal
free resolution of the coordinate ring for Gorenstein varieties of codimension and Castelnuovo-Mumford regularity four. We explain how to interpret these Betti tables as a recipe book for
constructing Gorenstein curves in $\mathbb{P}^5$, and give an example construction utilising the Tom and Jerry matrix formats of Brown, Kerber and Reid.
Week 6: Wednesday 10th November
Muhammad ManjiLink opens in a new window - The Bloch-Kato Conjecture and the method of Euler SystemsLink opens in a new window
The Bloch-Kato conjecture is a wide reaching conjecture in number theory relating in great generality algebraic objects (Selmer groups) and analytic objects (zeros of L-functions). It generalises
well known phenomena in number theory, most notably the Birch—Swinnerton-Dyer conjecture about elliptic curves; one of the Clay institute millennium problems. I hope to provide a low tech
introduction to the conjecture, defining the key concepts, and discuss important cases. If time permits, I will briefly discuss a modern approach to solving the conjecture for a range of cases using
Euler systems.
Week 7: Wednesday 17th November
Solly ColesLink opens in a new window - Knots in dynamics: Linking numbers for geodesic flowsLink opens in a new window
Knot theory is the study of topological characteristics of circles embedded in 3-dimensional space (knots). Often, invariants such as the linking number can be used to tell apart different
configurations of knots. In continuous-time dynamical systems, knots may arise as orbits of flows. In this talk I will discuss existing results for knots which come from dynamical systems, as well as
recent work on linking numbers for geodesic flows. If time permits, I will mention the more general case of Anosov flows.
Week 8: Wednesday 24th November
Nuno Arala SantosLink opens in a new window - Fourier Analysis methods in Number TheoryLink opens in a new window
We will explain the Fourier-analytic ideas behind the Hardy-Littlewood circle method and describe their role in the proof of Roth's Theorem on 3-term arithmetic progressions. We will also give a
rough sketch of the limitations that make these classical techniques unsuitable for tackling longer arithmetic progressions, and motivate the introduction by Gowers of the eponymous norms that led to
his celebrated new proof of Szemeredi's Theorem in 2001.
Week 9: Wednesday 1st December
Jakub TakacLink opens in a new window - The Mordor theorem for Orlicz spacesLink opens in a new window
Given a measurable space, one may consider the Lebesgue space $L^{p}$ consisting of all measurable functions $f$ for which $|f|^{p}$ is integrable. We shall define so-called Orlicz spaces, which
serve as a successful attempt at replacing the function $t \mapsto t^{p}$ in the definition of $L^{p}$ spaces with a more general Young function.
We shall explore some elementary properties of these Orlicz spaces, in particular their rearrangement-invariance. This leads to an axiomatic definition of a far more general object, a so-called
rearrangement-invariant (r.i. for short) function space. If time allows, we will discuss the relation of the class of all Orlicz spaces to the class of all r.i. spaces, in particular, we will present
the so-called Mordor theorem, a yet unpublished result which describes this relation in great detail.
A short artistic interlude will take place midway through the talk.
During and after the seminar, a photographer will be present to take semi-candid photographs of PhD students socialising and talking about maths. If you would like the chance to be the face of the
subsequent maths propaganda, you are more than welcome to come and have your picture taken.
Week 10: Wednesday 8th December
Katerina SanticolaLink opens in a new window - The Markoff Unicity Conjecture: when one door closes, another opens!Link opens in a new window
The Markoff Unicity Conjecture is a 108-year old conjecture about the solution set of the Diophantine equation $x^{2}+y^{2}+z^{2}=xyz$. The solutions, called Markoff numbers, turn up in a variety of
settings, from combinatorics, to number theory, to geometry and graph theory. In this talk, we will look at the translation of the conjecture to the world of hyperbolic geometry, arguing why this
approach fails to bring us closer to a proof of unicity. Then, we will look at a more promising translation to analytic number theory. Time permitting, we will go through the elementary proof that
the MUC holds for all prime powers. | {"url":"https://warwick.ac.uk/fac/sci/maths/research/events/seminars/areas/postgraduate/21-22/","timestamp":"2024-11-04T01:14:08Z","content_type":"text/html","content_length":"76677","record_id":"<urn:uuid:ccac10e7-c431-4f13-a4a1-e4030af89f25>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00419.warc.gz"} |
Summer Maths Camp | HEC Sydney
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Everything you need for mathematics coaching from Year 7 to HSC Advanced Maths, Extension 1 Maths, Extension 2 Maths.
Year 12 Advanced Trial Booster
• Structured VIDEO lessons.
• Comprehensive MATERIALS (Theory, reinforcement Practice Questions)
• 4 x Mock Trial Exams with Fully Worked MARKING Schemes
• Online 24/7 access to Fully-Worked SOLUTIONS and study RESOURCES.
• Hardcopy materials printed and delivered (Sydney Metro area only).
• Out-of-Class SUPPORT available via online Forum.
Course Outline:
Trial: Differentiation & its Applications; Integration & its Applications; Applications of Calculus to the Physical World; Quadratic Polynomials & the Parabola; Trigonometry; Probability &
Year 12 Extension 1 Trial Booster
• Structured VIDEO lessons.
• Comprehensive MATERIALS (Theory, reinforcement Practice Questions)
• 4 x Mock Trial Exams with Fully Worked MARKING Schemes
• Online 24/7 access to Fully-Worked SOLUTIONS and study RESOURCES.
• Hardcopy materials printed and delivered (Sydney Metro area only).
• Out-of-Class SUPPORT available via online Forum.
Course Outline:
Trial: Parameters, Locus & Parabola; Inverse Functions; Kinematics; Vectors; Statistical Analysis; Related Rates; Differential Equations.
Year 12 Extension 2 Trial Booster
• Structured VIDEO lessons.
• Comprehensive MATERIALS (Theory, reinforcement Practice Questions)
• 3 x Mock Trial Exams with Fully Worked MARKING Schemes
• Online 24/7 access to Fully-Worked SOLUTIONS and study RESOURCES.
• Hardcopy materials printed and delivered (Sydney Metro area only).
• Out-of-Class SUPPORT available via online Forum.
Course Outline:
Binomial Theorem
Year 12 Advanced HSC Booster
• 10 x structured VIDEO lessons.
• Comprehensive MATERIALS (Theory, reinforcement Practice Questions)
• 10 x Topic Tests with Fully Worked MARKING Schemes
• Online 24/7 access to Fully-Worked SOLUTIONS and study RESOURCES.
• Hardcopy materials printed and delivered (Sydney Metro area only).
• Out-of-Class SUPPORT available via online Forum.
Course Outline:
Revision: Radians, Trigonometric Ratios, Rules & Graphs; Trigonometric Identities, Proofs & Equations; Differential Calculus; Applications in Differential Calculus; Integral Calculus; Applications in
Integral Calculus; A.P., G.P. & Finance Maths; Rates of Change; Kinematics; Statistics.
Year 12 Extension 1 HSC Booster
• 10 x structured VIDEO lessons.
• Comprehensive MATERIALS (Theory, reinforcement Practice Questions)
• 10 x Topic Tests with Fully Worked MARKING Schemes
• Online 24/7 access to Fully-Worked SOLUTIONS and study RESOURCES.
• Hardcopy materials printed and delivered (Sydney Metro area only).
• Out-of-Class SUPPORT available via online Forum.
Course Outline:
Revision: Inequalities, Absolute Functions & Mathematical Induction; Trigonometry; Inverse Functions; Polynomials; Calculus Skills & Parabolas; Rates of Change; Kinematics & Projectile Motion;
Permulation, Combination, Probability & Binomial Probability; Vectors; Statistics.
Year 12 Extension 2 HSC Booster
• 10 x structured VIDEO lessons.
• Comprehensive MATERIALS (Theory, reinforcement Practice Questions)
• 10 x Topic Tests with Fully Worked MARKING Schemes
• Online 24/7 access to Fully-Worked SOLUTIONS and study RESOURCES.
• Hardcopy materials printed and delivered (Sydney Metro area only).
• Out-of-Class SUPPORT available via online Forum.
Course Outline:
bottom of page | {"url":"https://www.hecsydney.com/summermathscamp","timestamp":"2024-11-01T23:30:37Z","content_type":"text/html","content_length":"1051161","record_id":"<urn:uuid:52a9e3b4-f2f1-43e1-952c-d44cb7d76ac3>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00696.warc.gz"} |
Aaltodoc Repository :: Browsing by Author "von Hertzen, Raimo"
Browsing by Author "von Hertzen, Raimo"
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• Bridging plate theories and elasticity solutions
(PERGAMON-ELSEVIER SCIENCE LTD, 2017) Karttunen, Anssi; von Hertzen, Raimo; Reddy, JN; Romanoff, Jani; Department of Mechanical Engineering; Marine Technology; Solid Mechanics
In this work, we present an exact 3D plate solution in the conventional form of 2D plate theories without invoking any of the assumptions inherent to 2D plate formulations. We start by
formulating a rectangular plate problem by employing Saint Venant’s principle so that edge effects do not appear in the plate. Then the exact general 3D elasticity solution to the formulated
interior problem is examined. By expressing the solution in terms of mid-surface variables, exact 2D equations are obtained for the rectangular interior plate. It is found that the 2D
presentation includes the Kirchhoff, Mindlin and Levinson plate theories and their general solutions as special cases. The key feature of the formulated interior plate problem is that the
interior stresses of the plate act as surface tractions on the lateral plate edges and contribute to the total potential energy of the plate. We carry out a variational interior formulation of
the Levinson plate theory and take into account, as a novel contribution, the virtual work due to the interior stresses along the plate edges. Remarkably, this way the resulting equilibrium
equations become the same as in the case of a vectorial formulation. A gap in the conventional energy-based derivations of 2D engineering plate theories founded on interior kinematics is that the
edge work due to the interior stresses is not properly accounted for. This leads to artificial edge effects through higher-order stress resultants. Finally, a variety of numerical examples are
presented using the 3D elasticity solution.
• Exact elasticity-based finite element for circular plates
(Elsevier Limited, 2017) Karttunen, Anssi; von Hertzen, Raimo; Reddy, Junthula; Romanoff, Jani; Department of Mechanical Engineering; Marine Technology; Solid Mechanics
In this paper, a general elasticity solution for the axisymmetric bending of a linearly elastic annular plate is used to derive an exact finite element for such a plate. We start by formulating
an interior plate problem by employing Saint Venant’s principle so that edge effects do not appear in the plate. Then the elasticity solution to the formulated interior problem is presented in
terms of mid-surface variables so that it takes a form similar to conventional engineering plate theories. By using the mid-surface variables, the exact finite element is developed both by force-
and energy-based approaches. The central, nonstandard feature of the interior solution, and the finite element based on it, is that the interior stresses of the plate act as surface tractions on
the plate edges and contribute to the total potential energy of the plate. Finally, analytical and numerical examples are presented using the elasticity solution and the derived finite element.
• Exact theory for a linearly elastic interior beam
(2016-01-01) Karttunen, Anssi T.; von Hertzen, Raimo; Department of Mechanical Engineering
In this paper, an elasticity solution for a two-dimensional (2D) plane beam is derived and it is shown that the solution provides a complete framework for exact one-dimensional (1D) presentations
of plane beams. First, an interior solution representing a general state of any 2D linearly elastic isotropic plane beam under a uniform distributed load is obtained by employing a stress
function approach. The solution excludes the end effects of the beam and is valid sufficiently far away from the beam boundaries. Then, three kinematic variables defined at the central axis of
the plane beam are formed from the 2D displacement field. Using these central axis variables, the 2D interior elasticity solution is presented in a novel manner in the form of a 1D beam theory.
By applying the Clapeyron's theorem, it is shown that the stresses acting as surface tractions on the lateral end surfaces of the interior beam need to be taken into account in all energy-based
considerations related to the interior beam. Finally, exact1D rod and beam finite elements are developed by the aid of the axis variables from the 2D solution. (C) 2015 Elsevier Ltd. All rights
• Shear deformable plate elements based on exact elasticity solution
(Elsevier Limited, 2018-04-15) Karttunen, Anssi T.; von Hertzen, Raimo; Reddy, J. N.; Romanoff, Jani; Department of Mechanical Engineering; Texas A&M University
The 2-D approximation functions based on a general exact 3-D plate solution are used to derive locking-free, rectangular, 4-node Mindlin (i.e., first-order plate theory), Levinson (i.e., a
third-order plate theory), and Full Interior plate finite elements. The general plate solution is defined by a biharmonic mid-surface function, which is chosen for the thick plate elements to be
the same polynomial as used in the formulation of the well-known nonconforming thin Kirchhoff plate element. The displacement approximation that stems from the biharmonic polynomial satisfies the
static equilibrium equations of the 2-D plate theories at hand, the 3-D Navier equations of elasticity, and the Kirchhoff constraints. Weak form Galerkin method is used for the development of the
finite element model, and the matrices for linear bending, buckling and dynamic analyses are obtained through analytical integration. In linear buckling problems, the 2-D Full Interior and
Levinson plates perform particularly well when compared to 3-D elasticity solutions. Natural frequencies obtained suggest that the optimal value of the shear correction factor of the Mindlin
plate theory depends primarily on the boundary conditions imposed on the transverse deflection of the 3-D plate used to calibrate the shear correction factor.
• Variational formulation of the static Levinson beam theory
(2015-06) Karttunen, Anssi T.; von Hertzen, Raimo; Department of Mechanical Engineering
In this communication, we provide a consistent variational formulation for the static Levinson beam theory. First, the beam equations according to the vectorial formulation by Levinson are
reviewed briefly. By applying the Clapeyron's theorem, it is found that the stresses on the lateral end surfaces of the beam are an integral part of the theory. The variational formulation is
carried out by employing the principle of virtual displacements. As a novel contribution, the formulation includes the external virtual work done by the stresses on the end surfaces of the beam.
This external virtual work contributes to the boundary conditions in such a way that artificial end effects do not appear in the theory. The obtained beam equations are the same as the
vectorially derived Levinson equations. Finally, the exact Levinson beam finite element is developed. (C) 2015 Elsevier Ltd. All rights reserved. | {"url":"https://aaltodoc.aalto.fi/browse/author?value=von%20Hertzen,%20Raimo","timestamp":"2024-11-12T16:34:45Z","content_type":"text/html","content_length":"518125","record_id":"<urn:uuid:beeda092-2972-4503-9c55-ec4cffc0c602>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00010.warc.gz"} |
Dose finding is more general than dose escalation
Dose finding ≠ dose escalation
You’ll often hear Phase I dose-finding trials referred to as dose escalation studies. This is because simple dose-finding methods can only explore in one direction: they can only escalate.
Three-plus-three rule
The most common dose finding method is the 3+3 rule. There are countless variations on this theme, but the basic idea is that you give a dose of an experimental drug to three people. If all three are
OK, you go up a dose next time. If two out of three are OK, you give that dose again. If only one out of three is OK, you stop [1].
Deterministic thinking
The 3+3 algorithm implicitly assumes deterministic thinking, at least in part. The assumption is that if three out of three patients respond well, we know the dose is safe [2].
If you increase the dose level and the next three patients experience adverse events, you stop the trial. Why? Because you know that the new dose is dangerous, and you know the previous dose was
safe. You can only escalate because you assume you have complete knowledge based on three samples.
But if we treat three patients at a particular dose level and none have an adverse reaction we do not know for certain that the dose level is safe, though we may have sufficient confidence in its
safety to try the next dose level. Similarly, if we treat three patients at a dose and all have an adverse reaction, we do not know for certain that the dose is toxic.
Bayesian dose-finding
A Bayesian dose-finding method estimates toxicity probabilities given the data available. It might decide at one point that a dose appears safe, then reverse its decision later based on more data.
Similarly, it may reverse an initial assessment that a dose is unsafe.
A dose-finding method based on posterior probabilities of toxicity is not strictly a dose escalation method because it can explore in two directions. It may decide that the next dose level to explore
is higher or lower than the current level.
Starting at the lowest dose
In Phase I studies of chemotherapeutics, you conventionally start at the lowest dose. This makes sense. These are toxic agents, and you naturally want to start at a dose you have reason to believe
isn’t too toxic. (NB: I say “too toxic” because chemotherapy is toxic. You hope that it’s toxic to a tumor without being too toxic for the patient host.)
But on closer inspection maybe you shouldn’t start at the lowest dose. Suppose you want to test 100 mg, 200 mg, and 300 mg of some agent. Then 100 mg is the lowest dose, and it’s ethical to start at
100 mg. Now what if we add a dose of 50 mg to the possibilities? Did the 100 mg dose suddenly become unethical as a starting dose?
If you have reason to believe that 100 mg is a tolerable dose, why not start with that dose, even if you add a lower dose in case you’re wrong? This makes sense if you think of dose-finding, but not
if you think only in terms of dose escalation. If you can only escalate, then it’s impossible to ever give a dose below the starting dose.
More clinical trial posts
[1] I have heard, but I haven’t been able to confirm, that the 3+3 method has its origin in a method proposed by John Tukey during WWII for testing bombs. When testing a mechanical system, like a
bomb, there is much less uncertainty than when testing a drug in a human. In a mechanical setting, you may have a lot more confidence from three samples than you would in a medical setting.
[2] How do you explain the situation where one out of three has an adverse reaction? Is the dose safe or not? Here you naturally switch to probabilistic thinking because deterministic thinking leads
to a contradiction.
2 thoughts on “Dose finding ≠ dose escalation”
1. Dr. Cook,
Great article, I have a question for you, in the case of small populations, such as rare diseases or paediatrics, is there any way to apply Bayesian dose-finding?
2. Yes. Bayesian dose-finding methods are especially useful for rare diseases because a large portion of the population, maybe the entire population, is part of the study.
Conventional methods say to start with a sample in a clinical trial. The priority for treating this group is to learn as much as we ethically can, not to treat the patients in the trial as
effectively as possible. We learn effectively in the trial so we can treat effectively in the population. But when the trial group and the population are synonymous, we need to mix learning and
effective treatment, and Bayesian adaptive methods can do that. | {"url":"https://www.johndcook.com/blog/2019/02/12/dose-escalation/","timestamp":"2024-11-02T10:51:31Z","content_type":"text/html","content_length":"55406","record_id":"<urn:uuid:1d69efeb-968b-4599-af3b-1ac6e91d57aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00029.warc.gz"} |
I’ve been asking many of my grade 11 students what their ideal math lesson would look like. Not in terms of content, but in terms of process. I wanted to focus this question on math instead of
science because I didn’t want to confound typical learning activities with demonstrations and experiments. Most of the students cited very similar ideas, as follows: take up questions about homework
or last day’s work connect the new material to what they were working on last day possibly give some notes give (lots of) examples have them try some practice questions #1 above was universal, all
students started with this. | {"url":"https://www.physicsoflearning.com/tags/flow/","timestamp":"2024-11-09T01:19:28Z","content_type":"text/html","content_length":"37237","record_id":"<urn:uuid:4a1241e6-99a6-4f94-b717-ef237a2ecbff>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00542.warc.gz"} |
0 Times Table Chart - 10 Cute & Free Printables | SaturdayGift
0 Times Table Chart – 10 Cute & Free Printables
This post may contain affiliate links, which means I’ll receive a commission if you purchase through my links, at no extra cost to you. Please read full disclosure for more information.
Looking for a cute and free printable 0 times table chart? Here you can find 10 free printable multiplication charts for the table of 0 in black & white and pretty pastel colors.
Cute printable 0 Times Table Chart
These cute (and free!) printables and worksheets can help you teach your students or child the multiplication facts for the 0 times table.
They are perfect for kids, students, and anyone who wants to quickly learn the multiplication table of 0 and solve multiplication problems.
HOW TO PRINT: These printables are easy to print out and use – just follow these steps to get your worksheets printed:
1. Scroll down until you find the design you’d like to print out
2. Click on the instant download link under the image of the PDF
3. An ad might pop up, so close the ad, and the PDF will appear
4. Save the PDF to your computer and print them out.
Psst! The image has my logo on a banner, but no worries, the PDFs come without it.
Size: US Letter – but some can be easily resized and printed as A4.
Tip for printing: I always recommend saving the file to the computer first so that it will print out using the correct printer settings you have set up.
Cute Printable 0 Times Table Multiplication Charts
You can print as many as you like and use them as a great educational tool for homeschool or classroom and for parents to help their kids with their homework.
So here are the multiplication charts. You can find both black and white and colorful chart designs. Print your favorite design (black and white, simple design with a hint of color or the playing
card page), and start practicing your zero times tables.
I hope you like them and find them helpful!
All the tables, charts, and math worksheets are copyright protected ©SaturdayGift Ltd. Graphics purchased or licensed from other sources are also subject to copyright protection. For classroom and
personal use only, not to be hosted on any other website, sold, used for commercial purposes, or stored in an electronic retrieval system.
Fun 0 times multiplication tables
This fun zero times multiplication table comes in 4 cute colors. And it’s designed as a playing card.
1. Zero times table – Pink card
Table of 0 – pink playing card
DOWNLOAD: 0 Times Table – Pink Playing Card
Black & White 0 x tables
These black and white printable 0 times table sheets are perfect when you need to print a lot, and you want to save some ink. It’s still fun and cute even without the colors.
5. Zero times table – black and white
Black & White 0 Times Table
DOWNLOAD: Zero Times Table – Black & White
Free printable multiplication table of 0
These free printable 0 times table charts are perfect for the classroom, homeschooling, or parents who want to help their kids learn faster.
You can find this design in 3 different colors: pink, green, and grey.
7. Simple 0 times table – pink
Pretty pink zero times table
DOWNLOAD: Simple Pink 0 x Table
Other free printable math worksheets – multiplication table charts
Other Free printable multiplication times table charts
The times tables show multiplication equations, and you can find more information & details, math facts, and equation worksheet printables to learn and solve each multiplication table.
And each times table post has tips on how to remember each time table better: like when you multiply a one-digit number by 11. The answer is always the repetition of that same digit. Example: 8 x 11
= 88.
Knowing the times tables by heart will help later solve more complex math exercises like word problems.
Check out all the math printables
I offer a range of different math printables free for personal and classroom use.
Check out: All cute & free math printables
Psst! You might also like these Free Printable Multiplication Flash Cards
What are the 0 times tables?
Here you can find all the answers for 0 times tables up to 12 and 24. And the zero times table in words.
0 times table up to 12
0 x 1 = 0
0 x 2 = 0
0 x 3 = 0
0 x 4 = 0
0 x 5 = 0
0 x 6 = 0
0 x 7 = 0
0 x 8 = 0
0 x 9 = 0
0 x 10 = 0
0 x 11 = 0
0 x 12 = 0
0 times table up to 24
0 x 13 = 0
0 x 14 = 0
0 x 15 = 0
0 x 16 = 0
0 x 17 = 0
0 x 18 = 0
0 x 19 = 0
0 x 20 = 0
0 x 21 = 0
0 x 22 = 0
0 x 23 = 0
0 x 24 = 0
What is zero Times Table in words?
0 times table (up to 10) in words is as follows:
0 times 1 equals 0
0 times 2 Is equal to 0
0 times 3 equals 0
0 times 4 Is equal to 0
0 times 5 equals 0
0 times 6 Is equal to 0
0 times 7 equals 0
0 times 8 Is equal to 0
0 times 9 equals 0
0 times 10 Is equal to 0
You can say 0 times (number) is equal to (answer), or 0 times (number) equals (solution).
Tips for 0 times multiplication table – Is there a trick for 0 times tables?
The easiest way to memorize multiplication tables is to find patterns or skip counting/number line jumps.
But 0 times table is extra super easy to remember.
All numbers multiplied by 0 equal 0.
0 multiplied with any number equals 0.
• 2 x 0 = 0
• 7 x 0 = 0
• 0 x 3 = 0
• 0 x 173 = 0
• 0 x 23 434 = 0
So whatever digit you are multiplying with 0 – The answer is always zero.
Fun fact: Zero is an even number because it can be divided by 2, and an answer is a whole number.
How do you learn the zero times tables?
The best way to learn 0 times table (or any times table whatsoever) is practice. The three steps you can use to encourage your child are to view the table often, write the answers down and even read
aloud – and repeat.
Print out the practice sheets and do them over and over again. Hang one of the 0 times tables in the kids’ room.
Learn the few tips and tricks given above, and before you know it, you’ll master the 0 times multiplication table in no time.
I hope you enjoy these printable 0 times table worksheets. And remember to check out the multiplication chart 1 – 12 (12×12) and up to 100 (10 x table) and print them out as well. | {"url":"https://www.saturdaygift.com/0-times-table/","timestamp":"2024-11-08T12:43:53Z","content_type":"text/html","content_length":"329268","record_id":"<urn:uuid:9b378692-ada3-4690-9023-9975fa0a9d05>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00597.warc.gz"} |
Fifth Workshop on Compressible Multiphase Flows
In this workshop, we will address the modelling of multiphase flows. The goal is to share modelling methods, difficulties, (rigorous or more phenomenological) analysis, allowing the description of
multiphase flows with exchanges (mass transfer, energy exchange…) and apparition of shock waves. You can also find
Several talks are planned, with parts dedicated to open discussions. The main subjects will be a priori:
The talks will start the morning of Monday 5, and finish Wednesday 7 at noon.
The titles of the talks and the abstracts will be available soon.
A poster session will be organized. If you are interested in presenting your recent works in the topics of the workshop, we would be happy to welcome you. Please contact the organizers, sending a
title and an abstract of your poster.
The talks are given in room C8 of the first floor of the UFR (rez-de-chaussée de l'UFR):exact location of the room (Institut de Recherche Mathématique Avancée, Strasbourg)
RestaurantMeet Tuesday evening at 7:30pm at the restaurant La Victoire: 24 quai des Pêcheurs, Strasbourg. | {"url":"https://indico.math.cnrs.fr/event/9225/timetable/?print=1&view=standard","timestamp":"2024-11-10T11:37:55Z","content_type":"text/html","content_length":"57744","record_id":"<urn:uuid:04ae2fdd-998a-4f20-b9fe-d747421777e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00354.warc.gz"} |
Lesson 9
Comparing Graphs
• Let’s compare graphs of functions to learn about the situations they represent.
9.1: Population Growth
This graph shows the populations of Baltimore and Cleveland in the 20th century. \(B(t)\) is the population of Baltimore in year \(t\). \(C(t)\) is the population of Cleveland in year \(t\).
1. Estimate \(B(1930)\) and explain what it means in this situation.
2. Here are pairs of statements about the two populations. In each pair, which statement is true? Be prepared to explain how you know.
1. \(B(2000) > C(2000)\) or \(B(2000) < C(2000)\)
2. \(B(1900) = C(1900)\) or \(B(1900) > C(1900)\)
3. Were the two cities’ populations ever the same? If so, when?
9.2: Wired or Wireless?
\(H(t)\) is the percentage of homes in the United States that have a landline phone in year \(t\). \(C(t)\) is the percentage of homes with only a cell phone. Here are the graphs of \(H\) and \(C\).
1. Estimate \(H(2006)\) and \(C(2006)\). Explain what each value tells us about the phones.
2. What is the approximate solution to \(C(t)=20\)? Explain what the solution means in this situation.
3. Determine if each equation is true. Be prepared to explain how you know.
1. \(C(2011) = H(2011)\)
2. \(C(2015) = H(2015)\)
4. Between 2004 and 2015, did the percentage of homes with landlines decrease at the same rate at which the percentage of cell-phones-only homes increased? Explain or show your reasoning.
1. Explain why the statement \(C(t) + H(t) \leq 100\) is true in this situation.
2. What value does \(C(t) + H(t)\) appear to take between 2004 and 2017? How much does this value vary in that interval?
9.3: Audience of TV Shows
The number of people who watched a TV episode is a function of that show’s episode number. Here are three graphs of three functions—\(A, B\), and \(C\)—representing three different TV shows.
1. Match each description with a graph that could represent the situation described. One of the descriptions has no corresponding graph.
1. This show has a good core audience. They had a guest star in the fifth episode that brought in some new viewers, but most of them stopped watching after that.
2. This show is one of the most popular shows, and its audience keeps increasing.
3. This show has a small audience, but it’s improving, so more people are noticing.
4. This show started out huge. Even though it feels like it crashed, it still has more viewers than another show.
2. Which is greatest, \(A(7)\), \(B(7)\), or \(C(7)\)? Explain what the answer tells us about the shows.
3. Sketch a graph of the viewership of the fourth TV show that did not have a matching graph.
9.4: Functions $f$ and $g$
1. Here are graphs that represent two functions, \(f\) and \(g\).
Decide which function value is greater for each given input. Be prepared to explain your reasoning.
1. \(f(2)\) or \(g(2)\)
2. \(f(4)\) or \(g(4)\)
3. \(f(6)\) or \(g(6)\)
4. \(f(8)\) or \(g(8)\)
2. Is there a value of \(x\) at which the equation \(f(x)=g(x)\) is true? Explain your reasoning.
3. Identify at least two values of \(x\) at which the inequality \(f(x) < g(x)\) is true.
Graphs are very useful for comparing two or more functions. Here are graphs of functions \(C\) and \(T\), which give the populations (in millions) of California and Texas in year \(x\).
│ What can we tell about the │ How can we tell? │ How can we convey this │
│ populations? │ │ with function notation? │
│In the early 1900s, California had a smaller population than Texas. │The graph of \(C\) is below the graph of \(T\) when \(x\) is 1900. │\(C(1900) < T(1900)\) │
│Around 1935, the two states had the same population of about 5 million people. │The graphs intersect at about \((1935, 5)\). │\(C(1935) = 5\) and \(T(1935)=5\), │
│ │ │and \(C(1935)=T(1935)\) │
│After 1935, California has had more people than Texas. │When \(x\) is greater than 1935, the graph of \(C(x)\) is above that of \(T(x)\)│\(C(x) > T(x)\) for \(x>1935\) │
│ │. │ │
│Both populations have increased over time, with no periods of decline. │Both graphs slant upward from left to right. │ │
│From 1900 to 2010, the population of California has risen faster than that of │If we draw a line to connect the points for 1900 and 2010 on each graph, the │ │
│Texas. California had a greater average rate of change. │line for \(C\) has a greater slope than that for \(T\). │ │
• average rate of change
The average rate of change of a function \(f\) between inputs \(a\) and \(b\) is the change in the outputs divided by the change in the inputs: \(\frac{f(b)-f(a)}{b-a}\). It is the slope of the
line joining \((a,f(a))\) and \((b, f(b))\) on the graph.
• decreasing (function)
A function is decreasing if its outputs get smaller as the inputs get larger, resulting in a downward sloping graph as you move from left to right.
A function can also be decreasing just for a restricted range of inputs. For example the function \(f\) given by \(f(x) = 3 - x^2\), whose graph is shown, is decreasing for \(x \ge 0\) because
the graph slopes downward to the right of the vertical axis.
• horizontal intercept
The horizontal intercept of a graph is the point where the graph crosses the horizontal axis. If the axis is labeled with the variable \(x\), the horizontal intercept is also called the \(x\)
-intercept. The horizontal intercept of the graph of \(2x + 4y = 12\) is \((6,0)\).
The term is sometimes used to refer only to the \(x\)-coordinate of the point where the graph crosses the horizontal axis.
• increasing (function)
A function is increasing if its outputs get larger as the inputs get larger, resulting in an upward sloping graph as you move from left to right.
A function can also be increasing just for a restricted range of inputs. For example the function \(f\) given by \(f(x) = 3 - x^2\), whose graph is shown, is increasing for \(x \le 0\) because
the graph slopes upward to the left of the vertical axis.
• maximum
A maximum of a function is a value of the function that is greater than or equal to all the other values. The maximum of the graph of the function is the corresponding highest point on the graph.
• minimum
A minimum of a function is a value of the function that is less than or equal to all the other values. The minimum of the graph of the function is the corresponding lowest point on the graph.
• vertical intercept
The vertical intercept of a graph is the point where the graph crosses the vertical axis. If the axis is labeled with the variable \(y\), the vertical intercept is also called the \(y\)
Also, the term is sometimes used to mean just the \(y\)-coordinate of the point where the graph crosses the vertical axis. The vertical intercept of the graph of \(y = 3x - 5\) is \((0,\text-5)\)
, or just -5. | {"url":"https://curriculum.illustrativemathematics.org/HS/students/1/4/9/index.html","timestamp":"2024-11-11T07:17:49Z","content_type":"text/html","content_length":"123292","record_id":"<urn:uuid:cfb1f96f-a148-4678-918a-6c264634aafe>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00579.warc.gz"} |
sgejsv - Linux Manuals (3)
sgejsv (3) - Linux Manuals
sgejsv.f -
subroutine sgejsv (JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO)
Function/Subroutine Documentation
subroutine sgejsv (character*1JOBA, character*1JOBU, character*1JOBV, character*1JOBR, character*1JOBT, character*1JOBP, integerM, integerN, real, dimension( lda, * )A, integerLDA, real, dimension( n
)SVA, real, dimension( ldu, * )U, integerLDU, real, dimension( ldv, * )V, integerLDV, real, dimension( lwork )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)
SGEJSV computes the singular value decomposition (SVD) of a real M-by-N
matrix [A], where M >= N. The SVD of [A] is written as
[A] = [U] * [SIGMA] * [V]^t,
where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and
[V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are
the singular values of [A]. The columns of [U] and [V] are the left and
the right singular vectors of [A], respectively. The matrices [U] and [V]
are computed and stored in the arrays U and V, respectively. The diagonal
of [SIGMA] is computed and stored in the array SVA.
JOBA is CHARACTER*1
Specifies the level of accuracy:
= 'C': This option works well (high relative accuracy) if A = B * D,
with well-conditioned B and arbitrary diagonal matrix D.
The accuracy cannot be spoiled by COLUMN scaling. The
accuracy of the computed output depends on the condition of
B, and the procedure aims at the best theoretical accuracy.
The relative error max_{i=1:N}|d sigma_i| / sigma_i is
bounded by f(M,N)*epsilon* cond(B), independent of D.
The input matrix is preprocessed with the QRF with column
pivoting. This initial preprocessing and preconditioning by
a rank revealing QR factorization is common for all values of
JOBA. Additional actions are specified as follows:
= 'E': Computation as with 'C' with an additional estimate of the
condition number of B. It provides a realistic error bound.
= 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
D1, D2, and well-conditioned matrix C, this option gives
higher accuracy than the 'C' option. If the structure of the
input matrix is not known, and relative accuracy is
desirable, then this option is advisable. The input matrix A
is preprocessed with QR factorization with FULL (row and
column) pivoting.
= 'G' Computation as with 'F' with an additional estimate of the
condition number of B, where A=D*B. If A has heavily weighted
rows, then using this condition number gives too pessimistic
error bound.
= 'A': Small singular values are the noise and the matrix is treated
as numerically rank defficient. The error in the computed
singular values is bounded by f(m,n)*epsilon*||A||.
The computed SVD A = U * S * V^t restores A up to
This gives the procedure the licence to discard (set to zero)
all singular values below N*epsilon*||A||.
= 'R': Similar as in 'A'. Rank revealing property of the initial
QR factorization is used do reveal (using triangular factor)
a gap sigma_{r+1} < epsilon * sigma_r in which case the
numerical RANK is declared to be r. The SVD is computed with
absolute error bounds, but more accurately than with 'A'.
JOBU is CHARACTER*1
Specifies whether to compute the columns of U:
= 'U': N columns of U are returned in the array U.
= 'F': full set of M left sing. vectors is returned in the array U.
= 'W': U may be used as workspace of length M*N. See the description
of U.
= 'N': U is not computed.
JOBV is CHARACTER*1
Specifies whether to compute the matrix V:
= 'V': N columns of V are returned in the array V; Jacobi rotations
are not explicitly accumulated.
= 'J': N columns of V are returned in the array V, but they are
computed as the product of Jacobi rotations. This option is
allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
= 'W': V may be used as workspace of length N*N. See the description
of V.
= 'N': V is not computed.
JOBR is CHARACTER*1
Specifies the RANGE for the singular values. Issues the licence to
set to zero small positive singular values if they are outside
specified range. If A .NE. 0 is scaled so that the largest singular
value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
the licence to kill columns of A whose norm in c*A is less than
SQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
= 'N': Do not kill small columns of c*A. This option assumes that
BLAS and QR factorizations and triangular solvers are
implemented to work in that range. If the condition of A
is greater than BIG, use SGESVJ.
= 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
(roughly, as described above). This option is recommended.
For computing the singular values in the FULL range [SFMIN,BIG]
use SGESVJ.
JOBT is CHARACTER*1
If the matrix is square then the procedure may determine to use
transposed A if A^t seems to be better with respect to convergence.
If the matrix is not square, JOBT is ignored. This is subject to
changes in the future.
The decision is based on two values of entropy over the adjoint
orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).
= 'T': transpose if entropy test indicates possibly faster
convergence of Jacobi process if A^t is taken as input. If A is
replaced with A^t, then the row pivoting is included automatically.
= 'N': do not speculate.
This option can be used to compute only the singular values, or the
full SVD (U, SIGMA and V). For only one set of singular vectors
(U or V), the caller should provide both U and V, as one of the
matrices is used as workspace if the matrix A is transposed.
The implementer can easily remove this constraint and make the
code more complicated. See the descriptions of U and V.
JOBP is CHARACTER*1
Issues the licence to introduce structured perturbations to drown
denormalized numbers. This licence should be active if the
denormals are poorly implemented, causing slow computation,
especially in cases of fast convergence (!). For details see [1,2].
For the sake of simplicity, this perturbations are included only
when the full SVD or only the singular values are requested. The
implementer/user can easily add the perturbation for the cases of
computing one set of singular vectors.
= 'P': introduce perturbation
= 'N': do not perturb
M is INTEGER
The number of rows of the input matrix A. M >= 0.
N is INTEGER
The number of columns of the input matrix A. M >= N >= 0.
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
SVA is REAL array, dimension (N)
On exit,
- For WORK(1)/WORK(2) = ONE: The singular values of A. During the
computation SVA contains Euclidean column norms of the
iterated matrices in the array A.
- For WORK(1) .NE. WORK(2): The singular values of A are
(WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
sigma_max(A) overflows or if small singular values have been
saved from underflow by scaling the input matrix A.
- If JOBR='R' then some of the singular values may be returned
as exact zeros obtained by "set to zero" because they are
below the numerical rank threshold or are denormalized numbers.
U is REAL array, dimension ( LDU, N )
If JOBU = 'U', then U contains on exit the M-by-N matrix of
the left singular vectors.
If JOBU = 'F', then U contains on exit the M-by-M matrix of
the left singular vectors, including an ONB
of the orthogonal complement of the Range(A).
If JOBU = 'W' .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
then U is used as workspace if the procedure
replaces A with A^t. In that case, [V] is computed
in U as left singular vectors of A^t and then
copied back to the V array. This 'W' option is just
a reminder to the caller that in this case U is
reserved as workspace of length N*N.
If JOBU = 'N' U is not referenced.
LDU is INTEGER
The leading dimension of the array U, LDU >= 1.
IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
V is REAL array, dimension ( LDV, N )
If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
the right singular vectors;
If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
then V is used as workspace if the pprocedure
replaces A with A^t. In that case, [U] is computed
in V as right singular vectors of A^t and then
copied back to the U array. This 'W' option is just
a reminder to the caller that in this case V is
reserved as workspace of length N*N.
If JOBV = 'N' V is not referenced.
LDV is INTEGER
The leading dimension of the array V, LDV >= 1.
If JOBV = 'V' or 'J' or 'W', then LDV >= N.
WORK is REAL array, dimension at least LWORK.
On exit,
WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such
that SCALE*SVA(1:N) are the computed singular values
of A. (See the description of SVA().)
WORK(2) = See the description of WORK(1).
WORK(3) = SCONDA is an estimate for the condition number of
column equilibrated A. (If JOBA .EQ. 'E' or 'G')
SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1).
It is computed using SPOCON. It holds
N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
where R is the triangular factor from the QRF of A.
However, if R is truncated and the numerical rank is
determined to be strictly smaller than N, SCONDA is
returned as -1, thus indicating that the smallest
singular values might be lost.
If full SVD is needed, the following two condition numbers are
useful for the analysis of the algorithm. They are provied for
a developer/implementer who is familiar with the details of
the method.
WORK(4) = an estimate of the scaled condition number of the
triangular factor in the first QR factorization.
WORK(5) = an estimate of the scaled condition number of the
triangular factor in the second QR factorization.
The following two parameters are computed if JOBT .EQ. 'T'.
They are provided for a developer/implementer who is familiar
with the details of the method.
WORK(6) = the entropy of A^t*A :: this is the Shannon entropy
of diag(A^t*A) / Trace(A^t*A) taken as point in the
probability simplex.
WORK(7) = the entropy of A*A^t.
LWORK is INTEGER
Length of WORK to confirm proper allocation of work space.
LWORK depends on the job:
If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
-> .. no scaled condition estimate required (JOBE.EQ.'N'):
LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
->> For optimal performance (blocked code) the optimal value
is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal
block size for DGEQP3 and DGEQRF.
In general, optimal LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7).
-> .. an estimate of the scaled condition number of A is
required (JOBA='E', 'G'). In this case, LWORK is the maximum
of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
->> For optimal performance (blocked code) the optimal value
is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
In general, the optimal length LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF),
If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
-> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
-> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
where NB is the optimal block size for DGEQP3, DGEQRF, DGELQ,
DORMLQ. In general, the optimal length LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON),
N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
If SIGMA and the left singular vectors are needed
-> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
-> For optimal performance:
if JOBU.EQ.'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
if JOBU.EQ.'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR.
In general, the optimal length LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON),
2*N+LWORK(DGEQRF), N+LWORK(DORMQR)).
Here LWORK(DORMQR) equals N*NB (for JOBU.EQ.'U') or
M*NB (for JOBU.EQ.'F').
If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
-> if JOBV.EQ.'V'
the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).
-> if JOBV.EQ.'J' the minimal requirement is
LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
-> For optimal performance, LWORK should be additionally
larger than N+M*NB, where NB is the optimal block size
for DORMQR.
IWORK is INTEGER array, dimension M+3*N.
On exit,
IWORK(1) = the numerical rank determined after the initial
QR factorization with pivoting. See the descriptions
of JOBA and JOBR.
IWORK(2) = the number of the computed nonzero singular values
IWORK(3) = if nonzero, a warning message:
If IWORK(3).EQ.1 then some of the column norms of A
were denormalized floats. The requested high accuracy
is not warranted by the data.
INFO is INTEGER
< 0 : if INFO = -i, then the i-th argument had an illegal value.
= 0 : successfull exit;
> 0 : SGEJSV did not converge in the maximal allowed number
of sweeps. The computed values may be inaccurate.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Further Details:
SGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3,
SGEQRF, and SGELQF as preprocessors and preconditioners. Optionally, an
additional row pivoting can be used as a preprocessor, which in some
cases results in much higher accuracy. An example is matrix A with the
structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
diagonal matrices and C is well-conditioned matrix. In that case, complete
pivoting in the first QR factorizations provides accuracy dependent on the
condition number of C, and independent of D1, D2. Such higher accuracy is
not completely understood theoretically, but it works well in practice.
Further, if A can be written as A = B*D, with well-conditioned B and some
diagonal D, then the high accuracy is guaranteed, both theoretically and
in software, independent of D. For more details see [1], [2].
The computational range for the singular values can be the full range
( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
& LAPACK routines called by SGEJSV are implemented to work in that range.
If that is not the case, then the restriction for safe computation with
the singular values in the range of normalized IEEE numbers is that the
spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
overflow. This code (SGEJSV) is best used in this restricted range,
meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are
returned as zeros. See JOBR for details on this.
Further, this implementation is somewhat slower than the one described
in [1,2] due to replacement of some non-LAPACK components, and because
the choice of some tuning parameters in the iterative part (SGESVJ) is
left to the implementer on a particular machine.
The rank revealing QR factorization (in this code: SGEQP3) should be
implemented as in [3]. We have a new version of SGEQP3 under development
that is more robust than the current one in LAPACK, with a cleaner cut in
rank defficient cases. It will be available in the SIGMA library [4].
If M is much larger than N, it is obvious that the inital QRF with
column pivoting can be preprocessed by the QRF without pivoting. That
well known trick is not used in SGEJSV because in some cases heavy row
weighting can be treated with complete pivoting. The overhead in cases
M much larger than N is then only due to pivoting, but the benefits in
terms of accuracy have prevailed. The implementer/user can incorporate
this extra QRF step easily. The implementer can also improve data movement
(matrix transpose, matrix copy, matrix transposed copy) - this
implementation of SGEJSV uses only the simplest, naive data movement.
Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
[1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
LAPACK Working note 169.
[2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
LAPACK Working note 170.
[3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
factorization software - a case study.
ACM Trans. math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
LAPACK Working note 176.
[4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
QSVD, (H,K)-SVD computations.
Department of Mathematics, University of Zagreb, 2008.
Bugs, examples and comments:
Please report all bugs and send interesting examples and/or comments to drmac [at] math.hr. Thank you.
Definition at line 473 of file sgejsv.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/docs/linux/man/docs/linux/man/3-sgejsv/","timestamp":"2024-11-10T02:45:16Z","content_type":"text/html","content_length":"28930","record_id":"<urn:uuid:89906d36-859d-4ae8-8636-8f036f3a9da8>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00186.warc.gz"} |
Multiplication Charts 1-12 2024 - Multiplication Chart Printable
Multiplication Charts 1-12
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Here are some ideas to get the little one began: Multiplication Charts 1-12.
Multiplication Maps
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This session handles the basic principles of your multiplication table. In addition to learning the guidelines for multiplying, individuals will understand the thought of elements and patterning. By
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the end of the lesson, students should be able to recognize patterns in multiplication chart 1.
Different versions
As well as the standard multiplication graph or chart, individuals might need to produce a chart with increased variables or a lot fewer variables. To produce a multiplication graph with increased
aspects, pupils need to generate 12 desks, every single with a dozen lines and three columns. All 12 desks should in shape on one sheet of pieces of paper. Lines should be pulled having a ruler.
Graph paper is perfect for this undertaking. Students can use spreadsheet programs to make their own tables if graph paper is not an option.
Game suggestions
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Multiplication Chart 1. A few fun ideas are highlighted below. This game needs the individuals to be pairs and work on the same difficulty. Then, they will likely all hold up their charge cards and
talk about the answer for any minute. If they get it right, they win!
When you’re educating kids about multiplication, one of the best tools you are able to provide them with is really a computer multiplication chart. These printable linens can come in a number of
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Gallery of Multiplication Charts 1-12
Free Printable Multiplication Chart 1 12 Table PDF
5 Blank Multiplication Table 1 12 Printable Chart In PDF
Printable Multiplication Charts 1 12 PDF Free Memozor
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An explanation of the recent dark matter controversy
About two weeks ago, I
a recently published
by Chilean authors Moni Bidin
et al.,
claiming to have found no evidence for dark matter around our local neighbourhood in the Milky Way. Very soon afterwards, I
pointed out
a new
by Bovy and Tremaine that rebutted this claim. Although I suppose I was fairly quickly on the scene, I wasn't particularly happy with the way I had reported this argument: just mentioning that some
people disagree with each other without any explanation of the reasons is a bit closer to mere journalism than I was intending.
Since then, there have been several blog posts on this issue elsewhere — see
Peter Coles
Matt Strassler
Sean Carroll
among others — but to be honest I didn't think they actually did much more than report that people disagreed with each other either (although many of them did add some sociological comments). Which
is fine as far as it goes, but if you were looking for someone to explain the science to you, you might have been disappointed.
Jon Butterworth
at least made a commendable effort in his Guardian blog, but he was perhaps limited by space and an inability to include any mathematics. In the end he simply said that he believed the Bovy and
Tremaine result because Peter Coles believed it (and Peter Coles essentially told us to believe it because Scott Tremaine is an acknowledged world expert in galactic dynamics). Which is all very good
and probably correct, but not very enlightening!
So eventually I've decided to fill this gap in the market myself: this post will attempt to summarise and explain both papers to a non-expert (i.e. anyone who doesn't actually work in the field of
galactic dynamics). A major caveat before I start: I don't work in galactic dynamics either. I don't own, and had never read, the main reference work on this subject (by Binney and Tremaine; you see
why people might trust Tremaine to be right on this one!). In preparation for writing this post I borrowed a copy and read a small amount of it — enough to feel I understood what was going on — but
even so much of what I am about to write might be over-simplified, misrepresented or simply wrong. If you happen to spot any mistakes, please let me know.
Now that we've got that out of the way, let's start.
Most people who have read this far probably already know that our galaxy, the Milky Way, is a spiral galaxy, which means that the normal baryonic matter (which is mostly hydrogen and helium, in the
form of both stars and interstellar gas) is arranged in a relatively thin flat disk with spiral arms. There's a central bulge with older stars, and a supermassive black hole right at the very centre,
but that's not important for the moment. Here's a front-on picture of spiral galaxy NGC 5457; the Milky Way would look somewhat similar if we could get outside it to see it.
Figure 1: Spiral galaxy NGC 5457 viewed front-on.
However, we believe — for a whole variety of reasons — that this luminous matter is only a small fraction of the total mass of any galaxy. The remaining dark matter is non-baryonic, doesn't shine,
and interacts only very weakly with the baryonic matter, so it has only been detected so far through its gravitational influence on the dynamics of the matter that we do see. The dark matter is not
arranged in a disk like the baryonic matter; instead it forms a roughly spherical halo around the visible part of the galaxy and extending much farther out than it. This is a cartoon to illustrate a
side-on view of the Milky Way:
Figure 2: A cartoon image of the Milky Way viewed side-on. The dark matter halo is the grey shaded region and the main galactic disk is blue. The little circles dotted around are globular clusters of
stars and not dark matter. Not a very precise sketch! Image credit: David P. Bennett.
In what follows I will use a cylindrical coordinate system $(R,\phi,z)$, with the galactic centre at the origin. The Sun lies in the disk, about $R_0=8$ kiloparsecs from the centre of the galaxy
[a kiloparsec is
light years, or
. The total thickness of the disk at the solar location is about $2-3$ kpc. We don't actually know very much about the shape of the dark matter halo or about how the density of dark matter changes
with distance from the centre. There are several different models of the halo mass distribution but observations can't yet distinguish between them.
Moni Bidin
et al.
's method of determining the dark matter density at our location was to try measure the integrated surface density, defined by $$\Sigma(Z)\equiv\int_{-Z}^{Z}dz\;\rho(z)\,,$$ where $\rho(z)$ is the
total matter density at radius $R_0$ in the solar vicinity. Assuming symmetry on either side of the galactic plane means we need only bother with positive $Z$.
$\Sigma(Z)$ contains contributions from both baryonic and dark matter, but since almost all of the baryonic mass lies within the disk, at heights $Z>2$ kpc the baryonic contribution to the integrand
drops to essentially zero. If no dark matter were present, $\Sigma(Z)$ should therefore be a constant above this height. In the usual models of the dark matter halo, on the other hand, $\Sigma(Z)$
continues to increase linearly with height, with the slope dependent on the details of the model.
So the way to check for a contribution from dark matter is to measure $\Sigma(Z)$ at various heights and check to see if it keeps increasing at $Z>2$ kpc. It's particularly important to get out to
large values of $Z$ because otherwise the baryonic contribution is still non-zero and as we don't know exactly what non-zero value it should have the uncertainty introduced is too much to make any
useful statements.
The question is then how to measure $\Sigma(Z)$ at large enough heights. Well, the Poisson equation relates the mass density to derivatives of the gravitational potential, and as $\Sigma(Z)$ is an
integral over the mass density and the derivatives of the gravitational potential can be written in terms of the components of the force acting on objects at any location, we can obtain an equation
for $\Sigma(Z)$ in terms of these force components. I won't write it down here.
The components of the force at a location in the galaxy, denoted $F_R$ and $F_Z$, determine the dynamics of stars there. What Moni Bidin and collaborators did was to identify a group of 412 stars
that lay at large enough heights above the galactic plane that by observing their velocities they could calculate $F_R$ and $F_Z$, and then in turn calculate $\Sigma(Z)$. Treating the galaxy as a
virialised system in a steady state allows one to use the Jeans equation to describe the force components in terms of observed velocities of the tracer population of stars. The point of contention is
how they determined $F_R$ from the observed velocities.
Instead of using $F_R$, it is more convenient to describe things in terms of the
circular speed
$V_c(R)$, defined by $V_c^2=-RF_R$. The circular speed is essentially the rotational speed with which a star at distance $R$ from the centre of the galaxy would move if it were on a circular orbit
about the galactic centre. Actually stars aren't on circular orbits but elliptical ones, so they have a rotational velocity $V_\phi$ which is different from $V_c$. Individual stars all have different
orbits, so actually we need to deal with average quantities for the tracer population of 412 stars, in particular the
azimuthal (or rotational) velocity $\bar{V}_\phi(R,Z)$.
There is a difference between $\bar{V}_\phi$ and $V_c$ as well, called the 'asymmetric drift'. This is because of two factors. Firstly, stars on the outer part of an elliptical orbit have a smaller
rotational velocity than when they are on the section of the ellipse closer to the galactic centre. Since the density of stars decreases as you go away from the centre, more of the stars at the solar
radius are 'inner' stars on the outer part of their orbit than 'outer' stars on the inner part of their orbit. Secondly, the dispersion in the velocities of a population of 'inner' stars is greater
than that of 'outer' stars.
The radial Jeans equation then specifies the difference between $\bar{V}_\phi$ and $V_c$ in terms of quantities related to the density of the tracer stars and their velocity dispersion.
To make further progress in calculating $F_R$ as a function of $Z$ (and thus $\Sigma(Z)$) from the observed value of $\bar{V}_\phi$, several assumptions need to be made about how the velocity
dispersion and the tracer density change with radius and height above the disk. In fact, Moni Bidin
et al.
list a total of 11 assumptions they make, of which 10 are apparently more or less reasonable and seem to roughly agree with observations. The key assumption that Bovy and Tremaine claim is
reasonable is that the "rotation curve is locally flat", i.e. $$\frac{\partial \bar{V}_\phi}{\partial R}=0$$ at all relevant values of $Z$.
As I understand it, when people say the rotation curve is flat, they normally mean the
circular speed
doesn't change with radius, i.e. $\partial V_c/\partial R=0$. This has been measured and we know it is roughly true away from the very centre of the galaxy (in fact the flatness of the circular speed
curve is one reason why we need dark matter in models of galaxies). But this assumption is crucially different! Moni Bidin
et al
. later on in their paper relax this assumption and calculate how non-flat the
azimuthal velocity
curve would need to be in order for their data to be compatible with the usual models of the dark matter halo. They dismiss the value they obtain as impossible, and cite some papers they claim have
demonstrated that the curve is indeed essentially flat.
I skimmed through these references and as far as I can tell, they only ever refer to the
circular speed
curve, never to the
azimuthal velocity
curve. As far as I know, there are no measurements of how the azimuthal velocity curve behaves, only the circular speed curve. So this seems to be a simple error of reading comprehension! In fact
Bovy and Tremaine show that if the azimuthal velocity curve were indeed flat, the circular speed curve would have a strange behaviour, in contradiction with observations.
Here is a figure to demonstrate the effect of this assumption. It shows $\Sigma(R_0,Z)$ as a function of $Z$. The thick grey curves are theoretical, based on various models for the dark matter halo;
the one labelled VIS assumes that there is no dark matter halo at all. The lower, flat, black line and data points is what you get when you interpret the Moni Bidin observations assuming $\partial \
bar{V}_\phi/\partial R=0$ — consistent with no dark matter. Instead if you assume $\partial V_c/\partial R=0$ you get the upper solid black curve. Changing a few other assumptions produces the dashed
black curve or the shaded grey region. So basically consistent with many of the halo models!
Figure 3: Surface density integral as a function of height above the galactic plane. Taken from arXiv:1205.4033.
All in all this is really a bit of a shame for the Chileans. Firstly of course because they had a big
press conference
to announce their result, and now it turns out to be wrong. (Yes, I'm pretty convinced it is wrong.) Secondly because the reason it was wrong was because of a mistake that could really have been
spotted and corrected before the press release (and
have been spotted by the ApJ referee who appears to have focussed on various other less relevant things instead!). But mostly it is sad because their method is quite a good one: in fact interpreted
correctly it gives the best measurement of the dark matter density yet. They've obviously done good work identifying those 412 stars and measuring their velocities, they just somehow got the wrong
answer out at the end. In fact they even got the right answer in their paper but dismissed it as implausible!
I also sort of want to defend them against many of the comments I have seen online. Neither their paper nor their press conference really contain any evidence that they are the loony "dark matter
deniers" some people have painted them as. They very clearly stated their assumptions, and their reasons for believing them, and then the logical conclusions that followed. That the assumptions
turned out to be wrong is unfortunate, but far better to be methodical and clear in that manner than a lot of other rubbish you see on the arXiv. This allows science to progress faster!
If I were to draw one moral from this long story it would be that you shouldn't place too much faith in peer review by journal referees. Referees often don't read papers carefully enough, with the
result that publication ends up depending on their gut feeling about the result rather than the details of the science presented therein. Gut feelings are often wrong. Hence why
more transparency in the review process might be a better idea
I don't want to sound critical of these other bloggers: none of them work on galactic dynamics professionally so it's perfectly natural that they shouldn't write detailed posts about it. I decided to
write this mostly because it gave me a reason to learn a little bit about a new field, and to procrastinate from other stuff I was meant to do.
This method can only tell you about the dark matter density
exactly at our location
if you make some further assumptions about how this density varies with height and extrapolate to $Z\simeq0$. That's fine; if we could definitely say the existing halo models were wrong that would be
exciting enough even without being certain exactly what was happening at our location.
Binney and Tremaine provide a nice analogy: "there are more Japanese than Nepalese in Oxford in the summer, both because the population of Japan exceeds that of Nepal, and because Japanese have
larger travel budgets than Nepalese."
2 comments:
1. Following a conversation I had at lunch today about these papers, it struck me that rather than simply asserting that Moni Bidin et al (from now on, MB12) wrongly interpreted the papers they
cited, I should give you the evidence and let you judge for yourselves.
MB12 cite Xue et al. (arXiv version; journal version) and Fuchs et al. (arXiv version; journal version) as evidence that $\partial\bar{V}/\partial R$ is close to zero, and even quote numerical
values supposedly taken from these papers.
But Xue et al. specifically talk about the circular velocity (they call it $V_{cir}$ rather than my $V_c$), not $\bar{V}$ as you can see for yourself. The numerical value in Fuchs et al. that
MB12 quote actually refers to the value of $d\ln V_c/d\ln R$, not $\partial\bar{V}/\partial R$ (see their equation 17).
MB12 also claim that this paper, this paper and this paper provide some precedent for assuming the mean azimuthal velocity curve is flat, whereas as far as I can tell all three papers actually
only ever make any assumptions about the circular velocity curve (the latter two use the symbol $\Theta(R)$ rather than $V_c(R)$).
Anyway, that's my summary of the evidence. On this basis I don't see any alternative to the Bovy and Tremaine assertion that MB12 have misinterpreted the papers they cite. Have a look for
yourself and see if you agree.
2. Great readinng this | {"url":"https://blankonthemap.blogspot.com/2012/06/explanation-of-recent-dark-matter.html","timestamp":"2024-11-10T08:17:09Z","content_type":"application/xhtml+xml","content_length":"107345","record_id":"<urn:uuid:e4037f7b-2eb2-4f4f-baa1-743936d8397b>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00262.warc.gz"} |
How do I
How do I count grouped data in SQL?
How do I count grouped data in SQL?
SQL – count() with Group By clause The count() function is an aggregate function use to find the count of the rows that satisfy the fixed conditions. The count() function with the GROUP BY clause is
used to count the data which were grouped on a particular attribute of the table.
Can you use SUM and count together SQL?
SQL SUM() and COUNT() using variable SUM of values of a field or column of a SQL table, generated using SQL SUM() function can be stored in a variable or temporary column referred as alias. The same
approach can be used with SQL COUNT() function too.
How do I SUM by GROUP BY?
Use DataFrame. groupby(). sum() to group rows based on one or multiple columns and calculate sum agg function. groupby() function returns a DataFrameGroupBy object which contains an aggregate
function sum() to calculate a sum of a given column for each group.
How can I get grand total in SQL query?
The ‘simple’ grand total (CUBE or ROLLUP): In the ‘simple’ subtotal query using the CUBE or ROLLUP function will do the same thing: create one additional record – the “total” record. You’ll notice
that it shows this with a “NULL” in the ‘Assigned Site’ column.
Can we use count and GROUP BY together?
The use of COUNT() function in conjunction with GROUP BY is useful for characterizing our data under various groupings. A combination of same values (on a column) will be treated as an individual
Do I need to use GROUP BY with count?
We can use GROUP BY to group together rows that have the same value in the Animal column, while using COUNT() to find out how many ID’s we have in each group.
How do you write a count in SQL query?
SQL COUNT() Function
1. SQL COUNT(column_name) Syntax. The COUNT(column_name) function returns the number of values (NULL values will not be counted) of the specified column:
2. SQL COUNT(*) Syntax. The COUNT(*) function returns the number of records in a table:
3. SQL COUNT(DISTINCT column_name) Syntax.
Is count an aggregate function in SQL?
The COUNT operator is usually used in combination with a GROUP BY clause. It is one of the SQL “aggregate” functions, which include AVG (average) and SUM. This function will count the number of rows
and return that count as a column in the result set.
How do I sum aggregate in SQL?
The SQL Server SUM() function is an aggregate function that calculates the sum of all or distinct values in an expression. In this syntax: ALL instructs the SUM() function to return the sum of all
values including duplicates. ALL is used by default.
How do I group values in a column in SQL?
The SQL GROUP BY Statement The GROUP BY statement groups rows that have the same values into summary rows, like “find the number of customers in each country”. The GROUP BY statement is often used
with aggregate functions ( COUNT() , MAX() , MIN() , SUM() , AVG() ) to group the result-set by one or more columns.
How do I use subtotal and grand total in SQL?
In order to calculate a subtotal in SQL query, we can use the ROLLUP extension of the GROUP BY statement. The ROLLUP extension allows us to generate hierarchical subtotal rows according to its input
columns and it also adds a grand total row to the result set. | {"url":"https://www.resurrectionofgavinstonemovie.com/how-do-i-count-grouped-data-in-sql/","timestamp":"2024-11-03T14:06:20Z","content_type":"text/html","content_length":"34880","record_id":"<urn:uuid:4db7c1fd-ede8-42e6-a04e-127526a2fd75>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00731.warc.gz"} |