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Commentary on Improving Precision and Power in Randomized Trials for COVID-19 Treatments Using Covariate Adjustment, for Binary, Ordinal, and Time-to-Event Outcomes – Statistical Thinking
Commentary on Improving Precision and Power in Randomized Trials for COVID-19 Treatments Using Covariate Adjustment, for Binary, Ordinal, and Time-to-Event Outcomes
This is a commentary on the paper by Benkeser, Díaz, Luedtke, Segal, Scharfstein, and Rosenblum
Standard covariate adjustment as commonly used in randomized clinical trials recognizes which quantities are likely to be constant (relative treatment effects) and which quantities are likely to vary
(within-treatment-group outcomes and absolute treatment effects). Modern statistical modeling tools such as regression splines, penalized maximum likelihood estimation, Bayesian shrinkage priors,
semiparametric models, and hierarchical or serial correlation models allow for great flexibility in the covariate adjustment context. A large number of parametric model adaptations are available
without changing the meaning of model parameters or adding complexity to interpretation. Standard covariate adjustment provides an ideal basis for capturing evidence about heterogeneity of treatment
effects through formal interaction assessment. It is argued that absolute treatment effects (e.g., absolute risk reduction estimates) should be computed only on an individual patient basis and should
not be averaged, because of the predictably large variation in risk reduction across patient types. This is demonstrated with a large randomized trial. Quantifying treatment effects through average
absolute risk reductions hides many interesting phenomena, is inconsistent with individual patient decision making, is not demonstrated to add value, and provides less insights than standard
regression modeling. And since it is not likelihood based, focusing on average absolute treatment effects does not build a bridge to Bayesian or longitudinal frequentist models that are required to
take external information and various design complexities into account.
We usually treat individuals not populations. That being so, rational decision-making involves considering who to treat not whether the population as a whole should be treated. I think there are few
exceptions to this. One I can think of is water-fluoridation but in most cases we should be making decisions about individuals. In short, there may be reasons on occasion to use marginal models but
treating populations will rarely be one of them. — Stephen Senn, 2022
Benkeser, Díaz, Luedtke, Segal, Scharfstein, and Rosenblum^1 have written a paper aimed at improving precision and power in COVID-19 randomized trials. Some of the analytic results were available^2
and could have been compared to these. But here we want to raise the questions of whether ordinary covariate adjustment is more insightful, already solves the problems needing to be solved, and what
are the advantages of the authors’ multi-step approach. Benkeser et al focus on the following treatment effect estimands:
^1 Benkeser D, Díaz I, Luedtke A, Segal J, Scharfstein D, Rosenblum M (2020): Improving precision and power in randomized trials for COVID-19 treatments using covariate adjustment, for binary,
ordinal, and time-to-event outcomes. To appear in Biometrics, DOI:10.1111/biom.13377.
^2 Lasaffre E, Senn S (2003): A note on non-parametric ANCOVA for covariate adjustment in randomized clinical trials. Statistics in Medicine 22: 3583-3596.
• risk difference between treatment arms
• risk ratio
• odds ratio
• difference in mean scored ordinal outcomes
• Wilcoxon-Mann-Whitney statistic or probability index, also known as the concordance probability \(c\) for an ordinal outcome
• average log odds ratio for an ordinal outcome
• difference in restricted mean survival times for time-to-event outcomes
• difference in cumulative incidence for same
• relative risk for same
Taking for example the risk difference, one of the authors’ procedures is as follows for a trial of \(N\) subjects (both treatment arms combined).
• fit a binary logistic model containing treatment and baseline covariates
• for each of the \(N\) subjects compute the risk of the outcome using the subject’s covariates but setting treatment to control for all \(N\)
• for each of the subjects compute the risk of the outcome using the subject’s covariates but now setting treatment to the new treatment for all \(N\)
The paper in our humble opinion began on the wrong foot by taking as given that “the primary goal of covariate adjustment is to improve precision in estimating the marginal treatment effect.” Not
only are marginal effects distorted by subjects not being representative of the target population, but the primary goals of covariate adjustment are (1) in a linear model, to improve power and
precision by reducing residual variation, and (2) in a nonlinear model, to improve the fit of the model, which improves power, and to get a more appropriate treatment effect estimate in the face of
strong heterogeneity of outcomes within a single treatment arm. Whichever statistical model is used, a treatment effect estimand should have meaning outside of the study. We seek an estimand that has
the highest chance of applying to patients not in the clinical trial. When the statistical model is nonlinear (e.g., logistic or Cox proportional hazards models), the effect that is capable of being
constant is relative efficacy, e.g., conditional odds or conditional hazards ratio. Because of non-collapsibility of the odds and hazards ratio, the effect ratio that does not condition on covariates
will be tilted towards 1.0 when easily explainable outcome heterogeneity is not explained. And in the Cox model, failure to condition on covariates will result in a greater departure from the
proportional hazards assumption. Ordinary covariate adjustment works quite well and has been relied upon in countless randomized clinical trials. As will be discussed later, standard covariate
adjustment provides more transportable effect estimates. On the other hand, the average treatment effect advocated by Benkeser et al does not even transport to the original clinical trial, in the
sense that it is an average of unlikes and will apply neither to low-risk subjects nor to high-risk subjects. Importantly, this average difference is a strong function of not only the clinical
trial’s inclusion/exclusion criteria but also of the characteristics of subjects actually randomized, limiting its application to other populations. Note that the idea of making results from a
conditional model consistent with a marginal one was discussed in Lee and Nelder^3.
^3 Lee Y, Nelder JA (2004): Conditional and marginal models: Another view. Statistical Science 19:219-228.
^4 Senn SJ (2013): Being efficient about efficacy estimation. Statistics in Biopharmaceutical Research 5:204-210.
Benkeser et al (2020) are quite correct in stating that covariate adjustment is a much underutilized statistical method in randomized clinical trials (a point argued by many for many years^4). The
resulting loss of power and precision and inflation of sample size borders on scandalous. But we fear that the authors’ remedy will make clinical trialists less likely to account for covariates. This
is because the authors’ approach is overly complicated without a corresponding gain over traditional covariate adjustment, is harder to interpret, can provide misleading effect estimates by the
averaging of unlikes as discussed in detail below, is harder to pre-specify (pre-specification being a hallmark of rigorous randomized clinical trials), does not provide a basis for interaction
assessment (heterogeneity of treatment effect), and does not extend to longitudinal or clustered data. One can argue that what the authors proposed should not be called covariate adjustment in the
usual sense, but we will leave that for others to debate.
When we read a statistical methods paper, the first questions we ask ourselves are these: Does the paper solve a problem that needed to be solved? Are there other problems that would have been better
to solve? Did the authors conduct a proper comparative study to demonstrate the net benefit of the new method? Unfortunately, we do not view the authors’ paper favorably on these counts. Thinking of
all the many problems we have in COVID-19 therapeutic research alone, we have more pressing problems that talented statistical researchers such as Benkeser et al could have attacked instead. Some of
these problems from which to select include
1. How do we model outcomes in a way that recognizes that a treatment may not affect mortality by the same amount that it affects non-fatal outcomes?
2. What is the best statistical evidential measure for whether a mortality effect is consistent with the other effects?
3. What are the best ways to judge which treatment is better when results for different patient outcomes conflict (e.g., when a treatment slightly raises mortality but seems to cause a sharp
reduction in a disease severity measure)?
4. What is the best combination of statistical efficiency and clinical interpretation in constructing an outcome variable?
5. What is the information gain from a longitudinal ordinal response when compared to a time-to-event outcome?
6. How should one elicit prior distributions in a pandemic, or how should one form skeptical priors?
7. How should simultaneous studies inform each other during a pandemic?
8. What is the optimal number of parameters to devote to covariate adjustment and what is the best way to relax linearity assumptions when doing covariate modeling?
9. Is it better to use subject matter knowledge to pre-specify a rather small number of covariates, or should one use a large number of covariates with a ridge (\(L_{2}\)) penalty for their effects^
10. What is the best way to model treatment \(\times\) covariate interactions? Should Bayesian priors be put on the interaction effects, and how should such priors be elicited?
11. Can Bayesian models provide exact small-sample inference in the presence of missing covariate values?
12. What is the best adaptive design to use and how often should one analyze the data?
13. What is the best longitudinal model to use for ordinal outcomes, is a simple random effects model as good as a serial correlation model, what prior distributions work best for correlation-related
parameters, and should absorbing states be treated in a special way?
^5 Chen Q, Nian H, Zhu Y, Talbot HK, Griffin MR, Harrell, FE (2016): Too many covariates and too few cases? - a comparative study. Statistics in Medicine 35: 4546-4558.
To frame the discussion below, consider an alternative to the authors’ proposed methods: flexible parametric models that adjust for key pre-specified covariates without assuming that continuous
covariates operate linearly (e.g., continuous covariates are represented with regression splines in the model). Such a standard model has the following properties.
1. The parametric model parameterizes the treatment effect on a scale for which it is possible for the treatment effect to be constant (log odds, log hazard, etc.) and hence represented by a single
2. Because the treatment effect parameter has an unrestricted range, the need for interactions between treatment and covariates is minimized.
3. The model provides a basis for interactions that more likely represent biologic or pharmacologic effects than tricks to restrict the model’s probability estimates to \([0,1]\).
4. The model is readily extended to handle longitudinal binary and ordinal responses and multi-level models (e.g., days within patients within clinical centers).
5. Well studied multiple imputation procedures exist to handle missing covariates.
6. Full likelihood methods used in fitting the model elegantly handle various forms of censoring and truncation.
As shown by example here, standard covariate adjustment is more robust than Benkeser et al imply, and even when an ill-fitting model is used, the result may be more useful than marginal estimates.
More on Estimands
Benkeser et al’s primary estimand is the average difference in outcome probability in going from treatment A to treatment B. The authors have chosen a method of incorporating covariates that is
robust, but it is robust because the method averages out estimation errors into an oversimplified effect measure. In order to get proper individualized absolute risk reductions, the authors would
have to model covariates to the same standard sought by regular covariate adjustment. The authors’ method is flexible, but their estimand hides important data features. Standard regression models are
very likely to fit data well enough to provide excellent covariate adjustment, but an estimand that represents a differences in averages (e.g., an overall absolute risk reduction—ARR) is an example
of combining unlikes. ARR due to a treatment is a strong function of the base risk. For example, patients who are sicker at baseline have “more room to move” so have larger ARR than less sick
patients. ARR is an estimand that should be estimated only on a single-patient basis (see here, here, and here). In the binary outcome case, personalized ARR is a simple function of the odds ratio
and baseline risk in the absence of treatment interactions. When there is a treatment interaction on the logit scale, the difference in ARR that is estimated by the difference in two logistic model
is only slightly more complex.
The authors did not expose this issue to the readers. For example, one would find that a baseline-risk-specific version of their Figure 1 would reveal much variation in ARR.
As an example, the GUSTO I study^6 of thrombolytics in treatment of acute myocardial infarction with its sample size of 41,021 patients and overall 0.07 incidence of death has been analyzed in detail
, and various risk models have been developed from this randomized trial’s data. As shown here, for a treatment comparison of major interest, accelerated t-PA ($n=$10,348) vs. streptokinase (SK, $n=
$20,162), the average ARR for the probability of 30d mortality is 0.011. But this is misleading as it is dominated by a minority of high risk patients as shown in the figure described below. The
median ARR is 0.007 which is much more representative of what patients can expect. But far better is to provide individualized ARR estimates.
^6 GUSTO Investigators (1993): An international randomized trial comparing four thrombolytic strategies for acute myocardial infarction. New England Journal of Medicine 329: 673-682.
In the analysis of $n=$40,830 patients (2851 deaths) in GUSTO-I presented here, three treatments (two indicator variables) were allowed to interact with 6 covariates, with the continuous covariates
expanded into spline functions. The interaction likelihood ratio \(\chi^{2}\) test statistic was 16.6 on 20 d.f., \(p=0.68\) showing little evidence for interaction. Thus the assumption of constancy
of the treatment odds ratios is difficult to reject. Another way to allow a more flexible model fit is to penalize all the interaction terms down to what optimally cross-validates. Using a quadratic
(ridge) penalty, the optimum penalty was \(\infty\), again indicating no reason that the treatment odds ratio should not be considered to be a single number. Giving interactions one more benefit of
the doubt, penalized maximum likelihood estimates were obtained, penalizing all the covariate \(\times\) treatment interaction terms to effectively a single degree of freedom. This model was used to
estimate the distribution of individual patient ARR with t-PA shown in the figure below.
Distribution of ARR for 30d mortality with t-PA obtained after computing the difference in two predicted values for each patient (setting treatment to t-PA then to SK) using a penalized interaction
binary logistic model. Left arrow: median difference over patients; right arrow: mean ARR.
One sees the large amount of variation in ARR. Other results show large variation in risk ratios and minimal variation on ORs (and keep in mind that the optimal estimate of this OR variation is in
fact zero). The relationship between baseline risk under control therapy (SK) and the estimated ARR is shown below.
Per-patient ARR estimates vs. their baseline (SK) risk obtained from the same penalized interaction model. The darkness of points increases as the frequency of patients at the point increases, as
defined in the key to the right of the plot. About 3200 patients had baseline risk and risk reduction near zero. Were interactions forced to be zero (which would have been optimal by both a
likelihood ratio test and AIC), all the points would be on a single curve.
Through proper conditioning and avoidance of averaging of unlikes by estimating ARR for individual patients and not averaging these estimates over patients, it is seen that standard covariate
adjustment using well accepted relative treatment effect estimands is simpler, can fit data patterns better, requires only standard software, and is more insightful.
See Hoogland et al^7 for a detailed article on individualized treatment effect prediction.
^7 Hoogland J, IntHout J, Belias M, Rovers MM, Riley RD, Harrell FE, Moons KGM, Debray TPA, Reitsma JB (2021): A tutorial on individualized treatment effect prediction from randomized trials with a
binary endpoint. Accepted, Statistics in Medicine. Preprint
^8 Pearl J, Bareinboim E (2014): External validity: From do-calculus to transportability across populations. Statistical Science 29:579-595.
Others have claimed that our argument in favor of transportability of conditional estimands for treatment effects is incorrect, and that marginal estimands should form the basis for transportability
of findings to other patient populations, as advocated by Pearl and Bareinboim^8. The marginal estimand is not appropriate in our context for the following reasons:
• Pearl and Bareinboim developed their approach to apply to complex situations where a covariate may be related to the amount of treatment received. We are dealing only with exogeneous pre-existing
patient-level covariates here.
• The transport equation 3.4 of Pearl and Bareinboim requires the use of the covariate distribution in the target population. This distribution is not usually available when a clinical trial
finding is published or is evaluated by regulators.
• It is easier to transport an efficacy estimate to an individual patient (at least under the no-interaction assumption or if interactions are correctly modeled in the fully conditional model). The
one patient is the target and one only needs to know her/his covariate value, not the distribution of an entire target population.
• We are ultimately interested in individual patient decision making, not group decision making.
Lack of Comparisons
Benkeser et al showed large efficiency gains of their approach over ignoring covariates. But the authors did not compare the power for their approach vs. standard covariate adjustment. This leaves
readers wondering whether the new method is worth the trouble or is in fact less efficient than the standard.
New Odds Ratio Estimator
With the aim of dealing with non-proportional odds, the authors developed a weighted log odds ratio estimator. But the standard maximum likelihood estimator already solves this problem. As detailed
here and here, the Wilcoxon-Mann-Whitney concordance probability \(c\), i.e., the probability that a randomly chosen patient on treatment B has a higher level response \(Y\) than a randomly chosen
patient on treatment A, also called the probability index, is a simple function of the maximum likelihood estimate of the treatment regression coefficient whether or not proportional odds holds. The
conversion equation is \(c = \frac{\mathrm{OR}^{0.66}}{1 + \mathrm{OR}^{0.66}}\) where OR = \(\exp(\hat{\beta})\). This formula is accurate to an average absolute error for computing \(c\) from \(\
hat{\beta}\) of 0.002.
The equivalence of the OR and the Wilcoxon-Mann-Whitney estimand also makes the authors’ estimand 2 in Section 3.2 somewhat moot.
We note that overlap measures are not without problems^9.
^9 Senn SJ (2011): U is for unease: Reasons to mistrust overlap measures in clinical trials. Statistics in Biopharmaceutical Research 3:302-309.
Lack of a Likelihood
COVID-19 therapeutic research is an area where Bayesian methods are being used with increasing frequency. Frequentist methods that use a full likelihood approach provide an excellent bridge to
development of or comparison with Bayesian analogs that use the same likelihood and only need to add a prior. The authors’ methods are not likelihood based, so they do not provide a bridge to Bayes.
The proposed methods do not provide exact inference for small \(n\) as does Bayes, has no way of incorporating skepticism or external information about treatment effects, and has no way to quantify
evidence that is as actionable as posterior probabilities of (1) any efficacy and (2) clinically meaningful efficacy.
The authors briefly discuss missing ordinal outcomes. Instead of this being an issue of missingness, in many situations an ordinal outcome is interval censored. This is the case, for example, when a
range of ordinal values is not ascertained on a given day. To deal with interval censoring, a full likelihood approach is helpful, and the authors’ approach may not be extendible to deal with general
The lack of a likelihood also prevents the authors’ approach from dealing with variation across sites in a multi-site clinical trial through the use of random effects models, and extensions to
longitudinal outcomes are not provided. The flexibility of a Bayesian longitudinal proportional odds model that allows for general censoring and departures from the proportional odds assumption is
described here.
Technical Issues
The authors stated “By incorporating baseline variable information, covariate adjusted estimators often enjoy smaller variance …”. This must be clarified to apply to ARR estimates, but not in
general. In logistic and Cox models, for example, covariate adjustment increases treatment effect variance on the log ratio scale (see this and this for a literature review, and also the important
paper by Robinson and Jewell^10). Despite this increase, traditional covariate adjustment still results in increased Bayesian or frequentist power because the model being more correct (i.e., some of
the previously unexplained outcome heterogeneity is now explained) moves the treatment effect estimate farther from zero. The regression coefficient increases in absolute value faster than the
standard error increases, hence the power gain. For logistic and Cox models, variance reduction occurs only on probability-scale estimands.
^10 Robinson, LD, Jewell NP (1991): Some surprising results about covariate adjustment in logistic regression models. International Statistical Review 58:227-240.
In their Section 6 the authors recommended that when a utility function can be agreed upon, one should consider the difference in mean utilities between treatment arms when the outcome is ordinal.
Even though the difference in mean utilities can be a highly appropriate measure of treatment effectiveness, it is important to note that the patient-specific utilities are still discrete and are
likely to have a very strange, even bimodal, distribution. Hence utilities may be best modeled with a semiparametric model such as the proportional odds model.
The authors credited the proportional odds model to McCullagh^11 but this model was developed by Walker and Duncan^12 and other work even predates this.
^11 McCullagh, P (1980): Regression models for ordinal data. Journal of the Royal Statistical Society Series B 42:109-142.
^12 Walker, SH, Duncan, DB (1967): Estimation of the probability of an event as a function of several independent variables. Biometrika 54:167-178.
The authors discuss asymptotic accuracy of their method, which is of interest to statisticians but not practitioners who need to know the accuracy of the method in their (usually too small) sample.
The authors did not seem to have rigorously evaluated the accuracy of bootstrap confidence intervals. We have an example where none of the standard bootstraps provides sufficient accuracy for a
confidence interval for a standard logistic model odds ratio. Non-coverage probabilities are not close to 0.025 in either tail when attempting to compute a 0.95 interval. It is important to evaluate
the left and right non-coverage probabilities, as the confidence interval can be right on the average but wrong for both tails.
The use of categorization for continuous predictors (e.g., the treatment of age just before section 4.1.3) does not represent best statistical practice.
To be very picky, the authors’ (and so many other authors) use of the term “type I error” does not refer to the probability of an error but rather to the probability of making an assertion.
The paper advises readers to consider using variable selection algorithms. Stepwise variable selection brings a host of problems, typically ruins standard error estimates (Greenland^13), and is not
consistent with full pre-specification.
^13 Greenland, S (2000): When should epidemiologic regressions use random coefficients? Biometrics 56:915-921.
The idea to use information monitoring in forming stopping rules needs to be checked for consistency with optimum decision making, and it may be difficult to specify the information threshold.
Related to missing covariates, the recommendation to use single imputation and its implication to not use \(Y\) in the imputation process has been well studied and found to be lacking, especially
with regard to getting standard errors correct.
In the authors’ Supporting Information, the intuition for how covariate adjustment can lead to precision gains begins with a discussion of covariate imbalance. With the linear model, a random
marginal conditional imbalance term is identified thus becoming a conditional bias, which is then removed, adjusting the estimate if necessary and reducing the variance. It is the possibility that
the estimate may have to be adjusted that makes the variance for the conditional and the marginal estimates ‘correct’ given the model^14. Covariate adjustment produces conditionally and
unconditionally unbiased estimates. But imbalance is not the primary reason for doing covariate adjustment in randomized trials. Covariate adjustment is more about accounting for easily explainable
outcome heterogeneity^15 ^16. At any rate, apparent covariate imbalances may be offset by counterbalancing covariates one did not bother to analyze.
^14 Senn SJ (2019): The well-adjusted statistician: Analysis of covariance explained. https://www.appliedclinicaltrialsonline.com/view/well-adjusted-statistician-analysis-covariance-explained . A
valid standard error reflects how the estimate will vary over all randomisations. The adjustment that occurs as a result of fitting a covariate can be represented as \((Y_t - Y_c) - \beta(X_t - X_c)
\). Here \(Y\) and \(X\) are means of the outcome and the covariate respectively and \(t\) stands for treatment and \(c\) stands for control. For simplicity assume the uncorrected differences at
outcome and baseline have been standardised to have a variance of one. Other things being equal there cannot be a reduction in the residual variance by fitting the covariate unless \(\beta\) is large
which in turn implies that \(X_t - X_c\) and \(Y_t - Y_c\) must vary together. If you don’t adjust you are allowing for the fact that in the absence of any treatment effect \(Y_t - Y_c\) might differ
from 0 due to the fact that \(X_t - X_c\) differ randomly from 0 and this will affect the outcome. By adjusting you cash in the bonus by reducing the variance of the estimate. But this can only
happen because adjustment is possible.
^15 Lane PW, Nelder JA (1982): Analysis of covariance and standardization as instances of prediction. Biometrics 38: 613-621.
^16 Senn SJ (2013): Seven myths of randomization in clinical trials. Statistics in Medicine 32:1439-1450.
Methods that develop models from only one treatment arm are prone to overfitting, which entails fitting idiosyncratic associations in that arm in such a way that when the outcome comparison is made
with the other arm can result in bias in non-huge samples.
Omission of simulated samples with empty cells may slightly bias simulation results and is not necessary in ordinary proportional odds modeling.
Contrasted with the group sequential design outlined by the authors, a continuously sequential Bayesian design using a traditional proportional odds model for covariate adjustment is likely to
provide more understandable evidence and result in earlier stopping for efficacy, harm, or futility.
To add your comments, discussions, and questions go to datamethods.org here. See the end of this post for a discussion archive.
Grant Support
This work was supported by CONNECTS and by CTSA award No. UL1 TR002243 from the National Center for Advancing Translational Sciences. Its contents are solely the responsibility of the authors and do
not necessarily represent official views of the National Center for Advancing Translational Sciences or the National Institutes of Health. CONNECTS is supported by NIH NHLBI 1OT2HL156812-01, ACTIV
Integration of Host-targeting Therapies for COVID-19 Administrative Coordinating Center from the National Heart, Lung. and Blood Institute (NHLBI).
Discussion Archive (2021)
Lars v: Thank you for this very interesting read that leads to much thoughtful discussion. Here is some food-for-thought. Suppose was observe (X,A,Y) where X are baseline variables, A is a binary
treatment, and Y is a continuous outcome. Let us assume that E[Y|A=a,X=x] = cA + dX is a linear model so that we can identify conditional effects by parameters that one can estimate at the rate
square-root n (which is a big assumption). Consider, the marginal ATE effect parameter E_XE[Y|A=1,X] - E_XE[Y|A=0,X] . Under the linear model assumption, we actually have E_XE[Y|A=1,X] - E_XE[Y|A=
0,X] = E_X[E[Y|A=1,X] - E[Y|A=0,X] ] = E_X[c1 + dX -c0 - dX ] = E_X[c] = c. Thus, the marginal ATE equals the conditional treatment effect parameter! This is no coincidence. Most, if not all,
conditional treatment effect parameters based on parametric assumptions correspond with some nonparametric marginal treatment effect parameter (e.g. marginalized hazard ratios, odds ratios, etc). The
strong parametric assumptions allow one to turn marginal effects into conditional effects. Note that simply including an interaction between A and X already makes the identification of a conditional
effect parameter much more challenging. The benefit of estimating the marginal effect E_XE[Y|A=1,X] - E_XE[Y|A=0,X] instead is that our inference is non-parametrically correct even when the
conditional mean is not linear. Nonetheless, there is substantial work on estimating and obtaining inference for conditional treatment effects nonparametrically (e.g. CATE). Note that this is a
fairly difficult problem, as conditional treatment effect parameters are usually not square-root(n) estimable in the nonparametric setting.
As a note, this certainly motivates looking for marginal effect parameters that identify conditional effect parameters under stricter assumptions. If one is willing to believe that the assumptions
are true, one can always interpret it as a conditional effect. Also, pairing estimates and inference of conditional effects based on parametric models with those of marginal effects from
nonparametric models is a good way to obtain robust results that cover all bases. If the marginal effect and conditional effect are substantially different (assuming they identify the same parameter
under stricter assumptions), then this might lead one to conclude that the parametric model assumptions are violated.
Frank Harrell: It’s easier to manage discussions on datamethods.org where there is a link above to a topic already started for this area. On a quick read of your interesting comments I get the idea
that that thinking is restricted to linear models. If so, the scope may be too narrow. | {"url":"https://www.fharrell.com/post/ipp/","timestamp":"2024-11-13T09:02:38Z","content_type":"application/xhtml+xml","content_length":"147289","record_id":"<urn:uuid:d55af6ff-f59a-497c-bec5-bf3ff5d042df>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00226.warc.gz"} |
Phase Stability and Intercalant Ordering in V2O5 Cathodes
Shodor > NCSI > XSEDE EMPOWER > XSEDE EMPOWER Positions > Phase Stability and Intercalant Ordering in V2O5 Cathodes
Status Completed
Mentor Name Hartwin Peelaers
Mentor's XSEDE Research Allocation
Mentor Has Been
in XSEDE 4-5 years
Project Title Phase Stability and Intercalant Ordering in V2O5 Cathodes
We will use cluster expansions, constructed based on density functional theory (DFT) calculations, to investigate the phase diagram upon intercalation of V2O5, a potential cathode
Summary material for non-Li-ion batteries, i.e., batteries that use ions such as Na, Mg, Ca, etc. Cluster expansions are required to be able to run large-scale Monte Carlo simulations.
These will be used to introduce temperature to the DFT results, so that phase diagrams for different polymorphs and intercalant concentrations can be obtained. This will also allow
for voltage profiles to be calculated.
The student will create python scripts using the Atomic Simulation Environment (ASE) framework and the CLuster Expansion in Atomic Simulation Environment (CLEASE) python package to
construct cluster expansions for different polymorphs of V2O5 and for different intercalated ions. These expansions will be fitted using machine-learning approaches, and will be
Job Description based on density functional theory (DFT) calculations, as implemented in the VASP package. These calculations will be parallelized using MPI (both the VASP calculations, but also
the Monte Carlo simulations). Results will be stored in databases.
The student will learn to interact with high-performance computing resources, and will write code to perform all steps: underlying DFT calculations, fitting of cluster expansions
using machine-learning approaches, such as compressed sensing, creating Monte Carlo simulations based on the cluster expansions, and scripts to plot and analyze results.
Computational The DFT calculations will be performed on XSEDE (in particular on PSC Bridges-2), and will be controlled by scripts written in python (via ASE). Such calculations will typically
Resources require 1 node (128 cores on Bridges-2) for 4-6 hours. Fitting of the cluster expansions will be done on the student’s computer and/or the local cluster, and subsequent Monte
Carlo simulations will be done on XSEDE machines.
Contribution to This project will train an undergraduate student in the use of HPC resources to perform computational research on materials properties using first-principles codes. This will
Community increase the student’s knowledge and proficiency in HPC computing and the underlying physics and material science, which are valuable skills. The project scientific goals might
lead to better understanding of, and potential improved, battery cathodes for novel non-Li-ion batteries.
Position Type Apprentice
The student will learn how to code in python, will learn solid state physics, thermodynamics, and quantum mechanics (as needed), basic linux shell (cd, mkdir, cp, ls, etc.), and
interact with HPC resources (ssh, scp, SLURM, etc.). This learning will occur through 1-on-1 mentoring with the PI (through Zoom), combined with online tutorials. Initial testing
will be done on a known (and quick to calculate) model system (Au-Cu alloys), so that the student can gain confidence in the methodology before applying it to a completely unknown
Training Plan system (intercalated V2O5). Once initial results are obtained, the student will study these based on guiding questions. The more experienced the student becomes, the more open
questions and tasks will become. This will introduce the student gradually to more aspects of the scientific process (making hypothesis, designing calculations to test these,
analyze results, refine hypothesis, and so on). The student will also take part in regular group meetings to learn about other group members' research and discussions about that
Prerequisites/ Basic proficiency with linux, python, hpc environments.
Duration Summer
Start Date 05/24/2021
End Date 08/01/2021
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Status Completed
Mentor Name Hartwin Peelaers
Mentor's XSEDE Research Allocation
Mentor Has Been
in XSEDE 4-5 years
Project Title Phase Stability and Intercalant Ordering in V2O5 Cathodes
We will use cluster expansions, constructed based on density functional theory (DFT) calculations, to investigate the phase diagram upon intercalation of V2O5, a potential cathode
Summary material for non-Li-ion batteries, i.e., batteries that use ions such as Na, Mg, Ca, etc. Cluster expansions are required to be able to run large-scale Monte Carlo simulations.
These will be used to introduce temperature to the DFT results, so that phase diagrams for different polymorphs and intercalant concentrations can be obtained. This will also allow
for voltage profiles to be calculated.
The student will create python scripts using the Atomic Simulation Environment (ASE) framework and the CLuster Expansion in Atomic Simulation Environment (CLEASE) python package to
construct cluster expansions for different polymorphs of V2O5 and for different intercalated ions. These expansions will be fitted using machine-learning approaches, and will be
Job Description based on density functional theory (DFT) calculations, as implemented in the VASP package. These calculations will be parallelized using MPI (both the VASP calculations, but also
the Monte Carlo simulations). Results will be stored in databases.
The student will learn to interact with high-performance computing resources, and will write code to perform all steps: underlying DFT calculations, fitting of cluster expansions
using machine-learning approaches, such as compressed sensing, creating Monte Carlo simulations based on the cluster expansions, and scripts to plot and analyze results.
Computational The DFT calculations will be performed on XSEDE (in particular on PSC Bridges-2), and will be controlled by scripts written in python (via ASE). Such calculations will typically
Resources require 1 node (128 cores on Bridges-2) for 4-6 hours. Fitting of the cluster expansions will be done on the student’s computer and/or the local cluster, and subsequent Monte
Carlo simulations will be done on XSEDE machines.
Contribution to This project will train an undergraduate student in the use of HPC resources to perform computational research on materials properties using first-principles codes. This will
Community increase the student’s knowledge and proficiency in HPC computing and the underlying physics and material science, which are valuable skills. The project scientific goals might
lead to better understanding of, and potential improved, battery cathodes for novel non-Li-ion batteries.
Position Type Apprentice
The student will learn how to code in python, will learn solid state physics, thermodynamics, and quantum mechanics (as needed), basic linux shell (cd, mkdir, cp, ls, etc.), and
interact with HPC resources (ssh, scp, SLURM, etc.). This learning will occur through 1-on-1 mentoring with the PI (through Zoom), combined with online tutorials. Initial testing
will be done on a known (and quick to calculate) model system (Au-Cu alloys), so that the student can gain confidence in the methodology before applying it to a completely unknown
Training Plan system (intercalated V2O5). Once initial results are obtained, the student will study these based on guiding questions. The more experienced the student becomes, the more open
questions and tasks will become. This will introduce the student gradually to more aspects of the scientific process (making hypothesis, designing calculations to test these,
analyze results, refine hypothesis, and so on). The student will also take part in regular group meetings to learn about other group members' research and discussions about that
Prerequisites/ Basic proficiency with linux, python, hpc environments.
Duration Summer
Start Date 05/24/2021
End Date 08/01/2021 | {"url":"http://computationalscience.org/xsede-empower/positions/271","timestamp":"2024-11-14T15:12:31Z","content_type":"application/xhtml+xml","content_length":"13765","record_id":"<urn:uuid:754f5478-fe44-4e60-8fb4-734865996f1a>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00265.warc.gz"} |
17 Times Table Chart & Printable PDF – Times Table Club
17 Times Table Chart
17 Times table is the multiplication chart of the number 17 provided here for students, parents and teachers. These time tables charts are helpful in solving math questions quickly and easily.
How to read the 6 times table chart?
One times seventeen is 17, two times seventeen is 34, three times seventeen is 51, ect.
How to memorise the 17 times table chart orally?
Write the 17 times tables on a sheet of paper and read them aload repeatedly.
What is an example math question using the 17 times table chart?
Maths Question: A recipe calls for 17 grams of sugar per serving. If you want to make 5 servings of the dessert, how many grams of sugar will you need in total?
Solution: Using the 17 times table chart, the total number of grams of sugar needed for 5 servings: 17 × 5 = 85g of sugar
│Tables 2 to 20 │Multiplication Chart │ | {"url":"https://timestableclub.com/17-times-table-chart/","timestamp":"2024-11-03T04:16:12Z","content_type":"text/html","content_length":"35900","record_id":"<urn:uuid:3ec26fd7-6724-4fab-9e06-00f98f3aa117>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00884.warc.gz"} |
TI-Basic Developer
Conway's Game of Life
Displays Conway's Game of Life on the screen.
∟X - the list of X coordinates
∟Y - the list of Y coordinates
∟X, ∟Y, ∟P, ∟Q, S, X, Y
URL: United TI
:Repeat getKey or S<2
:DelVar ∟PDelVar∟Q1→S
:3=Ans or 3=Ans+max(∟X=X and ∟Y=Y
:If Ans:Then
Conway's Game of Life is a game designed to model how a simple environment evolves over time. Although it is called a game, it is not really a game, since it has no players and it operates on its
own. The basic functionality involves moving pieces around the screen, with a black pixel being considered alive and a white pixel being considered dead.
A piece can move in any of the eight directions, and the basic rules it follows are: a pixel stays alive if it has two or three live neighboring pixels, and a dead pixel becomes alive if it has
exactly three live neighboring pixels. The different patterns and shapes that emerge all depend on the pieces that you start out with.
In order to setup the pieces, we need to store the X and Y coordinates in the respective ∟X and ∟Y lists. There are a couple things you need to remember: the screen is 94x62, so those are the maximum
numbers you can use for coordinates; there needs to be a corresponding Y coordinate for every X coordinate, otherwise you will get a ERR:DIM MISMATCH error.
We use a stat plot for displaying the pieces because it has built-in functionality for working with lists, and storing the pieces in lists is more efficient than any of the other alternatives
available. In addition to the two main coordinate lists, we also create two other temporary lists (∟P and ∟Q).
These two lists are used for storing all of the different movements by the pieces. When we have finished checking all of the pieces to see if any of them should be moved (based on the game rules), we
update the coordinates lists, and store the new coordinates of the pieces to the ∟X and ∟Y lists. We repeat this over and over again for the life of the game.
Displaying and checking the pieces should take only a few seconds, but it all depends on the number of pieces that you choose to use; fewer pieces will be faster, and likewise more pieces will be
slower. When you are done using ∟X, ∟Y, ∟P, and ∟Q, you should clean them up at the end of your program. | {"url":"http://tibasicdev.wikidot.com/game-of-life","timestamp":"2024-11-03T23:37:07Z","content_type":"application/xhtml+xml","content_length":"29802","record_id":"<urn:uuid:13fbe507-64f2-4891-b99b-62d87210cdb7>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00071.warc.gz"} |
Integration of Differential Equations
CS 482 Lecture, Dr. Lawlor
In a class about simulations, you'd be correct in thinking we're going to design and analyze quite a few simulations. These are going to differ in really fundamental ways, but there are many broad
concepts that show up repeatedly in this field.
One of the central concepts in simulation is the idea of integrating differential equations, like the equations of motion for a simple Newtonian object, or the Navier-Stokes equations for fluid
Differential Equations
The usual Newtonian definition of velocity V is the time derivative of position P:
V = dP/dt
If we have a nice functional form for V or P, this is extremely mathematically useful, since we can use all the tools of calculus on a differential definition like this. If we don't have a nice
functional form, like with many real problems, we can't use calculus at all (this is also the case if we've forgotten calculus).
This is why we use computers, but computers don't do infinitesimals. There are a variety of ways to "discretize" this continuous differential equation into computational steps; but the simplest is to
replace the infinitesimal derivatives with a finite difference:
V = delta_P / delta_t
which means
V = (new_P - old_P) / delta_t
Given a set of positions, this finite difference lets us estimate the velocity between known positions. We can also solve for new_P, to get a classic equation known as an Euler timestep:
new_P = old_P + V * delta_t
In a programming language, we'd usually write this as:
P += V * delta_t;
One big problem here is to best approximate the underlying partial differential equation, "delta_t" should be as small as possible. Small enough, and roundoff begins to change your simulation
results. Also, making the timestep one-tenth as big needs ten times as many steps for the same simulation, so you quickly run into computational limitations: even a simple partial differential
equation can consume arbitrary computational power and still not be perfect!
The other problem is our finite difference equation isn't a very accurate approximation to the continuous equation, usually causing energy to bleed off the system. You can do better by taking the
average V, known as Leapfrog integration (more next week!).
3D Vectors
In the finite difference above, if P and V are vectors with xyz components, we can just update each component of the vector:
P.x += V.x * delta_t;
P.y += V.y * delta_t;
P.z += V.z * delta_t;
It's always possible to break down 3D computations into a set of (coupled) 1D computations, but this ".x .y .z" copy and paste is quite error prone--I always end up with ".x .y .y" somewhere, which
gives bizarre results and is very hard to locate and fix.
In the OpenGL shader language GLSL, or in C++ with a 3D vector class, you can avoid the copy and paste by defining P and V as 3D vectors, of type "vec3". You can then operate on 3D vectors just like
they were scalars:
P += V * delta_t;
JavaScript is missing the operator overloading that lets a class vec3 handle multiplication and addition nicely, but in PixAnvil I added some little helpers to the existing class vec3 for these
│ │ C++ or GLSL │ JavaScript/PixAnvil │ Operation │
│ Vector sum │ vec3 vecC = vecA+vecB; │ var vecC=vecA.p(vecB); │ plus │
│ Vector addition │ vecA += vecB; │ vecA.pe(vecB); │ plus-equal │
│ Vector * scalar │ vecA *= 2.0; │ vecA.te(2.0); │ times-equal │
Physical quantities have units, like meters or seconds or watts. Most programming languages don't know or care about the units, so you can easily make unit errors like:
float energy_in_joules = distance_in_feet * force_in_pounds;
Of course, foot * pound force gives you ft-lb of energy, not joules. But all the variables are "float", so the compiler happily lets these errors creep in.
The best way to avoid unit errors is to pick your units when you begin building the simulation, and stick with them throughout. MKS metric units (meters, kilograms, seconds) seem to be the most
common, although I've worked with everything from astrophysical simulations where "1.0" is the size of the observable universe, and molecular dynamics simulations where "1.0" is the size of an atom.
For 3D printed models, I've seen STL files where 1.0 means 1 centimeter, 1 millimeter, or even 1 inch.
It is possible to go back and figure out the effective units you're working in after the fact, but it's less effort and more reliable to figure them out as you go. Sometimes I'll work without units
just to try to capture the overall "feel" of a phenomena before actually matching it to reality, but it's usually easier if everything has units from the start.
Boundary Conditions
Often the trickiest part of a simulation is not "knowing how to simulate", it's "knowing when to stop". Since our computers are finite, we need to do something at the borders of the domain that we
can afford to simulate. That "something" can be very simple, but there are times when this boundary condition messes up the simulation in the interior.
For a particle simulation, a boundary condition might be something simple like:
if (particle is outside the boundary) {
change the velocity to make it move back inside
Simulating waves or fluid flow is especially tricky, since it's easy to get wind resistance or wave reflection off the boundary. One common trick is to make the boundaries periodic, essentially
wrapping the simulation around at the boundaries.
Conserved Quantities, like Energy
Energy is conserved in both the universe and our simulations mostly because:
1. If you keep adding energy, everything explodes (the world "ends in fire").
2. If you keep removing energy, everything freezes in place (the world "ends in ice").
3. If you don't add or remove energy, everything keeps working reasonably.
Many simulations calculate and monitor the total energy of everything in the simulation, as a way to verify that things are working correctly. This is a good way to detect even surprisingly subtle
flaws in the physics or numerics. Be aware that conserving energy is a goal, not an absolute requirement, and often tiny conservation errors from things like roundoff are difficult to avoid.
You can also lose or gain energy at the borders of the simulation, where things like the energy added to the wind as your object passes by, or the electrical energy input to the motor, aren't
accounted for. You can often fairly easily add these border energy terms back in, and sometimes they can be quite useful (e.g., noting that most of the drag energy loss in a boat is happening at the
Linear and angular momentum are also conserved quantities. Loss of momentum conservation tends to look weird, causing objects to fly off into the distance. | {"url":"https://www.cs.uaf.edu/2015/spring/cs482/lecture/01_15_integration.html","timestamp":"2024-11-08T14:41:36Z","content_type":"text/html","content_length":"10270","record_id":"<urn:uuid:33abbdc1-e0d4-4bb1-b6b1-71fc21c64096>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00597.warc.gz"} |
Efficiently searching a graph by a smell-oriented vertex process for Ann. Math. Artif. Intell.
Ann. Math. Artif. Intell.
Efficiently searching a graph by a smell-oriented vertex process
View publication
Efficient graph search is a central issue in many aspects of AI. In most of existing work there is a distinction between the active "searcher", which both executes the algorithm and holds the memory,
and the passive "searched graph", over which the searcher has no control at all. Large dynamic networks like the Internet, where the nodes are powerful computers and the links have narrow bandwidth
and are heavily-loaded, call for a different paradigm, in which most of the burden of computing and memorizing is moved from the searching agent to the nodes of the network. In this paper we suggest
a method for searching an undirected, connected graph using the Vertex-Ant-Walk method, where an a(ge)nt walks along the edges of a graph G, occasionally leaving "pheromone" traces at nodes, and
using those traces to guide its exploration. We show that the ant can cover the graph within time O(nd), where n is the number of vertices and d the diameter of G. The use of traces achieves a
trade-off between random and self-avoiding walks, as it dictates a lower priority for already-visited neighbors. Further properties of the suggested method are: (a) modularity: a group of searching
agents, each applying the same protocol, can cooperate on a mission of covering a graph with minimal explicit communication between them; (b) possible convergence to a limit cycle: a Hamiltonian path
in G (if one exists) is a possible limit cycle of the process. | {"url":"https://research.ibm.com/publications/efficiently-searching-a-graph-by-a-smell-oriented-vertex-process","timestamp":"2024-11-13T20:52:34Z","content_type":"text/html","content_length":"77416","record_id":"<urn:uuid:e0733b45-8925-449f-bc7e-d76d21f8d88d>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00345.warc.gz"} |
Troubleshooting Motors and Controls
4 Homework 1
Ken Dickson-Self
Use the following schematic for questions 1, 2 and 3.
1. Between which points would you expect to measure no voltage?
2. Between which points would you expect to measure source voltage?
3. The light bulb stops working. Using your meter, you read zero volts between points F and H and between E and J. You measure source voltage from I and B and from G and J. Where is the
problem located?
4. Draw a ladder diagram of the following circuit:
Wired Circuit
5. Using Ohm’s Law, fill in the missing values for the following circuit:
Series Circuit
R1 R2 R3 Total
E 12V
R 3Ω 6Ω 4Ω
6. Using Ohm’s Law, fill in the missing values for the following circuit:
Parallel circuit
R1 R2 R3 Total
E 12V
R 3Ω 6Ω 4Ω
7. Given the following schematic diagram with a control relay, siren and four switches, describe the behavior of this circuit in words.
Speaker Circuit
8. How much current is in a 12V circuit that has 24kΩ of resistance?
9. How much voltage is present in a circuit with 30mA of current and 800Ω of resistance?
10. What resistance is present in a circuit with 120V and 80µA of current?
11. How would you write 2.4MΩ in scientific notation?
12. How would you write 2.4µA in scientific notation?
13. How might you write 3.45 x 10^8A?
14. How might you write 8.34 x 10^-6Ω?
15. In the circuit below, draw in your meter to measure the following properties:
1. Resistance of R1
2. Voltage across R2
3. Current through R1, R2, and R3 | {"url":"https://openoregon.pressbooks.pub/motorcontrols/chapter/homework-1/","timestamp":"2024-11-08T10:45:37Z","content_type":"text/html","content_length":"71318","record_id":"<urn:uuid:49703734-7925-4399-a5ee-0d4a91181a4d>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00316.warc.gz"} |
On non fundamental group equivalent surfaces
In this paper we present an example of two polarized K3 surfaces which are not Fundamental Group Equivalent (their fundamental groups of the complement of the branch curves are not isomorphic;
denoted by FGE) but the fundamental groups of their related Galois covers are isomorphic. For each surface, we consider a generic projection to CP2 and a degenerations of the surface into a union of
planes - the "pillow" degeneration for the non prime surface and the "magician" degeneration for the prime surface. We compute the Braid Monodromy Factorization (BMF) of the branch curve of each
projected surface, using the related degenerations. By these factorizations, we compute the above fundamental groups. It is known that the two surfaces are not in the same component of the Hilbert
scheme of linearly embedded K3 surfaces. Here we prove that furthermore they are not FGE equivalent, and thus they are not of the same Braid Monodromy Type (BMT) (which implies that they are not a
projective deformation of each other).
• Branch curve
• Curves and singularities
• Fundamental group
• Generic projection
Dive into the research topics of 'On non fundamental group equivalent surfaces'. Together they form a unique fingerprint. | {"url":"https://cris.biu.ac.il/en/publications/on-non-fundamental-group-equivalent-surfaces-4","timestamp":"2024-11-07T10:23:11Z","content_type":"text/html","content_length":"54132","record_id":"<urn:uuid:1ae806d5-f4c8-4e2a-bc13-d458330233df>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00087.warc.gz"} |
Dissipation Time for Wound Roll
Published: August 26, 2014, By Kelly Robinson
I received an interesting question. A wound roll of biaxially oriented polypropylene (BOPP) has high static. To get rid of the static, we plan to put the roll on a concrete floor and let it sit for a
week. Won’t this get rid of the static?
The quick answer is no. Charge persists on wound rolls of polypropylene for weeks, months, or longer. The best solution is to improve static control so that no static is wound into the roll.
The longer answer is to analyze an idealized geometry. Static electricity has a long history of investigation. Workers contribute to our understanding by writing about their observations and
thoughts. We benefit from their work by applying their observations and thoughts to our applications. Let's analyze an idealized geometry and see how long it takes for static to dissipate. However,
what idealized problem makes sense?
The factory floor geometry in Figure 1 shows the wound roll sitting on the concrete floor. Static charges are distributed throughout the wound roll. Electric field lines that originate on the static
charges within the wound terminate on the grounded concrete floor. Most concrete floors built since about 1950 are well grounded. I have seen only one concrete aggregate floor that was insulating.
And, this floor dated back to ~1870 when the original factory building was built.
The factory floor geometry in Figure 1 is so complex that I’d use a computer running one of the powerful finite element method programs to find how long charge dissipation would take. A good
alternative is to analyze a model geometry that has two important features.
First, it must include the mechanism for charge dissipation; electrical conduction.
Second, it must be simple enough so that I can estimate the time needed for charge to dissipate.
One idealized geometry in Figure 2 that has these two features is a rectangular slab on a grounded plane. The slab models the charge roll. One compromise is that our idealized geometry is flatter
than the actual geometry. The height H of the slab is smaller than the length L and rolls usually have a width that is about the same as the roll diameter. So, our estimate of the charge dissipation
time for the model geometry may be somewhat different from the time needed for our roll. None-the-less, we should get a pretty good “ball park” estimate.
The electric fields within the slab are found using the Poisson equation in (3) where the permittivity ε is a material property and Q[V] is the volumetric charge density within the roll.
In our idealized geometry, the electric fields are vertical. That is, they have only a z component. So, (3) simplifies to (4)
Next, we’ll assume that our wound roll is homogeneous. This approximation is OK when the web has no coatings. In this case, the permittivity e is uniform (does not vary with z) and (4) simplifies to
Next, we’ll keep things simple and assume that the charge density Q[V] is uniform. This simplifying assumption is OK if charge on the web was uniform when the roll was wound. If not, this assumption
considers only the average charge. With this assumption, (5) simplifies to (6).
Now, we can integrate to find the electric field as a function z in (7).
Since there is no charge above the roll, the electric field at height H in (7) must be zero.
Our solution for the electric field is sensible. The field is proportional to the charge density Q[V]. More charge means higher fields. And, the field is maximum at the bottom of the roll that
touches the floor.
Now that we know the electric field within our idealized geometry, we can find the time needed for charge to dissipate. Moving charge is current. And, the current density J is proportional to the
electric field E. This approximation in (8) is Ohm’s law.
The constant of proportionality ρ[V] is the electrical resistivity of our material. The actual charge movement in most insulators varies strongly with electric field. So, this assumption leads to a
severe under estimate of the charge dissipation time. None-the-less, let’s use it and follow through with the analysis.
Conservation of charge in (9) requires that the volumetric charge density Q[V] of any little volume within the slab must go down when charge moves out of the volume.
In our idealized geometry, the electric fields have only a z component, and (9) simplifies to (10).
Using (7), (10) simplifies to (11).
The solution for (11) is well known. The charge density Q[V] in (12) decreases exponentially with time.
The charge relaxation time τ in (12) is given in (13).
Note that the charge relaxation time τ is independent of the height H and width W of the slab. The key result of our analysis is that the charge relaxation time is a material property.
For polypropylene, the dielectric constant is about 2.2. While the volumetric resistivity varies, it is on the order of 10^+17 W-cm. The charge relaxation time for polypropylene in (14) is about 5.4
Our analysis estimates that 95% of the initial charge should dissipate from our wound roll of polypropylene in about 3 time constants or about a day. My experience is that this analysis severely
underestimates the charge dissipation time by a factor of at least 100X. Charge persists on wound rolls of polypropylene for weeks or months. Ohmic conduction is a very poor approximation for charge
movement in polypropylene. african porn videos
I invite you to ask questions about this blog and to suggest future topics. My email address is: This email address is being protected from spambots. You need JavaScript enabled to view it.. | {"url":"https://mail.pffc-online.com/static-beat/12335-dissipation-time","timestamp":"2024-11-01T19:39:24Z","content_type":"application/xhtml+xml","content_length":"510846","record_id":"<urn:uuid:91ee0f2c-90bc-45a7-b305-86202d2053ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00682.warc.gz"} |
A uniform long tube is bent into a circle of radius R and it lies in v
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Python - Comprehension (Iteration)
Comprehension are a way to loop over elements of:
Comprehensions permit to build collections out of other collections.
Comprehensions allow one to express computations over the elements of a set, list, or dictionary without a traditional for-loop.
Use of comprehensions leads to more compact and more readable code, code that more clearly expresses the mathematical idea behind the computation being expressed.
Comprehensions are implemented using a function scope.
IF as a filter
An IF that filter a sequence
[ EXP for x in seq if COND ]
>>> list = [1, 2, 3]
>>> [x for x in list if x == 2]
IF as a ternary
An IF as a ternary operator to transform the element of the sequence
[ EXP IF condition then EXP else EXP for x in seq ]
>>> list = [1, 2, 3]
>>> [x if x==2 else 0 for x in list]
[0, 2, 0]
S = [2 * x for x in range(101) if x ** 2 > 3]
Count Number of Occurrence in a List
The following piece of code create a dictionary with as key the element and as value, the number of occurrence of the element in the list
>>> tokens
[1, 2, 3, 2]
>>> {x:tokens.count(x) for x in tokens}
{1: 1, 2: 2, 3: 1}
Documentation / Reference | {"url":"https://datacadamia.com/lang/python/grammar/comprehension","timestamp":"2024-11-15T04:48:05Z","content_type":"text/html","content_length":"150922","record_id":"<urn:uuid:0f8f4950-ee9d-48e7-83fc-76896003fd22>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00434.warc.gz"} |
18. Vibration
• Every moving system has some amount of vibration. This is a natural result of unbalanced components, rubbing, reversals, etc.
• Some vibrations are very predictable, and can be estimated mathematically, and their effects compensated for.
• Mathematically a vibration is a displacement/force/velocity/acceleration of small amplitude that will ‘shake’ and object.
18.1 Vibration Modeling
• The most significant vibration in engineered systems is periodic. In these systems there is often an approximate spring-mass-damper system that gives us a second order response to disturbances.
• In vibration modeling we typically assume that all components are linear. In a linear system the forcing (input) frequencies are directly related to response (output) frequencies.
• In non-linear vibration systems we end up with the frequency of the forcing function being transformed to other frequencies. This tends to make the vibrations seem less clear, and appear more
• There are a few types of descriptive terms for these systems,
Damping Factor: The damping factor will indicate if vibrations will tend to die off. If the damping factor is too low the vibrations may build continually until failure.
Forced Vibration: When a periodic excitation is applied to these systems they will tend to show a steady state response
Free Vibration: When displaced/disturbed and released there is an oscillation at a natural frequency for any system. This is one measure of a system, and is typically induced by displacing a system
and letting it go.
Natural Frequency: Each system will have one or more frequencies that it will prefer to vibrate at. When we excite a system at a natural frequency the system will resonate, and the response will
become the greatest.
Response: This is a measure of how a system behaves when it is disturbed. For example, this could be measured by looking at the position of a point on a mechanism.
Steady State Response: After a system settles down it will assume a regular periodic response, this is steady state. The steady state excludes the transient.
Transient Response: When a forcing function on a system changes, there will be a short lived response that tends to be somewhat irregular. The transient will eventually die off, and the system will
settle out to a steady state.
• These systems can be modeled a number of ways, but we typically start with a differential equation.
18.1.1 Differential Equations
• In modeling any linear system we are best to start by developing a differential equation for the system components.
• Consider the simple spring-mass-damper system shown below. A free body diagram can be drawn for the mass ‘M’, and a sum of forces can be written, and expanded with the values for the mechanical
• We may also consider a torsional vibration. We will assume that the vertical shaft has a stiffness of Ks and a damping coefficient of Kd. There is an applied torque ‘T’ and a moment of inertia ‘I’.
18.1.2 Modeling Mechanical Systems with Laplace Transforms
• Before doing any sort of analysis of a vibrating system, a system model must be developed. The obvious traditional approach is with differential equations.
18.1.3 Second Order Systems
• Basically these systems tend to vibrate simply. This vibration will often decay naturally. The contrast is the first order system that tends to move towards new equilibrium points without any sort
of resonance or vibration.
• These generally have an effects on the Bode plot that are very evident.
• Under the influence of damping, the natural frequency will shift slightly,
• To continue the example with numerical values
18.1.4 Phase Plane Analysis
• When doing analysis of a system that has both a steady state and transient response it can be handy to do a phase plane analysis to help separate out the components.
• To construct a phase plane graph we plot the value of a response variable against it’s first or second derivative.
• The shape of the graph exposes the phase between the displacement, and one of the derivatives. Here we see the system start to spiral out to an outer radius. The change in the radius of the spiral
is the transient, the final radius is the steady state. If the forcing function changed, the path would then shift to a new steady state position.
18.2 Control
• Vibrations are the natural result of many engineered systems.
• These vibrations can become significant when they shake carefully designed structures, or induce sounds in the air.
• As an engineer attempts to suppress or negate vibrations and sound, one of the most powerful weapons is a good analytical understanding of the phenomena.
18.2.1 Vibration Control
• If this displacement is induced by a machine in the next room, and it travels through the floor, we want to isolate the noise source for the high frequency (2000Hz) that will be noticeable as a
• Your boss asks you to design a mounting for the machine that gets rid of the high frequency whine.
• In this case the bode plot would reduce the noise by over 40dB for the high pitch sound. For the frequencies below 1000 Hz there would not be much reduction, but since this is 1/100th the frequency
of the original sound, it should not be a problem.
• We check the manuals and find the machine weighs 10000kg, and has 8 mounting points.
• The result would look something like,
• The force transmitted to the floor is,
• We can develop a table of gains and phase angles for the isolator
• Consider damping at various frequencies, but consider that with damping the isolation was reduced at the high frequency, but the resonance was also reduced.
18.3 Vibration Control
18.3.1 Isolation
• We have already done an example of using springs for damping, but, what are the practical options,
Springs: good for low frequencies because they are not massless as assumed. In fact, they have a damping ration of about 0.005. They are also well suited to harsh (long life) environments. They can
have problems with “rocking”.
Elastomeric Mounts: Rubber compounds best used in compression, also good for shear. Common ranges are 30-durometer for soft, low K rubber, to 80-durometer high K rubber (damping ratio about 0.05).
These materials are good for high frequencies.
Isolation pads: Cork, felt, fiberglass. Frequencies start at low values (18Hz and up for fiberglass), common damping coefficients for cork and felt are .06.
• Elastomers are not linear, so the spring constant will vary as they are loaded. Graphical solutions work well when finding spring constants. Some example curves are given below.
• A solution can be done entirely with graphical methods. Manufacturers will provide graphs for specific materials and thicknesses.
• Note: When designing you should always attempt to get the natural frequency at least three times lower than the frequency to be damped.
• The same type of design techniques can be done with cork. (Note: they would have similar graphs to those for elastomers)
• A set of specifications for an elastomer isolator are summarized below,
18.3.2 Inertial
Inertial Blocks: Increase the mass of the object to decrease vibration amplitude, and decrease natural frequency.
Absorption: A secondary mass is added to draw off, and hopefully cancel out, vibrations
18.3.3 Active
Active Systems: These systems are becoming very popular in new cars, etc. The example below uses a bellows with an adjustable pressure ‘P’. This pressure over area ‘A’ gives a spring constant ‘K’ and
height ‘h’. If the pressure in the bellows is adjusted by the addition of gas, the spring constant will rise, tending to damp out different vibration frequencies (remember the ideal gas law ‘PV=
• The ‘K’ values vary significantly for Elastomers and Isolation pads as the load varies, therefore graphical values are often required to find the spring constants.
• An example of a practical active vibration control system is piezo electric actuators mounted on the skin of an airplane wing. When unwanted vibrations occurred in the wings the actuators could
have voltages applied to counteract the vibrations (both axial and torsional). [Mechanical Engineering, 1995]
18.3.4 References
18.1 Mechanical Engineering, “Controlling Wing Flutter With Miniature Actuators”, Mechanical Engineering Magazine, ASME, 1995.
18.4 Vibration Measurement
Vibration Source: hammers can be used to generate impulse/step function responses. Load cells/vibrators can be used to excite frequency responses (Bode plots and phase shift plots)
Sensors: Velocity Pickups/Accelerometers: lightweight devices that are mounted on structures. They produce small voltages (approx. 10mV). Velocity meters are not as accurate as accelerometers.
Accelerometers are very common, and are used for vibrations above 1KHz. Many other sensors are possible.
Preamplifiers: Can power sensors, filter and amplify output.
Signal Processor: Many types used, from software packages, to older pen based plotters, or tape recorders
• Consider the example of an amplitude and phase plot measured for a real device,
18.5 Vibration Signals
• Some of the wave properties of interest,
• It is worth recalling the discussion of a signal spectrum
18.6 Vibration Transducers
18.6.1 Velocity Pickups
• Output voltage is proportional to velocity (V/(cm/s))
• These devices have low natural frequencies, and are used for signals with higher frequencies.
• well suited to measuring severe vibrations, but it may be affected by noise from AC sources.
• because signals are velocity, some form of integration must be done, making these devices bulky, and somewhat inaccurate
• There are two common methods for mounting velocity pickups,
Magnetic mounts allow fast and easy mounting, but the magnetic mount acts as a slight spring mass isolator, limiting the frequency range.
Stud mounted transducers have a thin layer of silicone grease to improve contact
18.6.2 Accelerometers
• Compared to velocity pickups
• electronic integrators can provide velocity and position
• The accelerometer is mounted with electrically isolated studs and washers, so that the sensor may be grounded at the amplifier to reduce electrical noise.
• Cables are fixed to the surface of the object close to the accelerometer, and are fixed to the surface as often as possible to prevent noise from the cable striking the surface.
• Background vibrations in factories are measured by attaching control electrodes to ‘non-vibrating’ surfaces. (The control vibrations should be less than 1/3 of the signal for the error to be less
than 12%)
• Piezoelectric accelerometers typically have parameters such as,
operate well below one forth of the natural frequency
• Accelerometer designs vary, so the manufacturers specifications should be followed during application.
• There is often a trade-off between wide frequency range and device sensitivity (high sensitivity requires greater mass)
• Two type of accelerometers are compression and shear types.
• Mass of the accelerometers should be less than a tenth of the measurement mass.
• Accelerometers can be linear up to 50,000 to 100,000 m/s**2 or up to 1,000,000 m/s**2 for high shock designs.
• Typically used for 10-10,000 Hz, but can be used up to 10KHz
• Temperature variations can reduce the accuracy of the sensors.
• These devices can be calibrated with shakers, for example a 1g shaker will hit a peak velocity of 9.81 m/s**2
18.6.3 Preamplifiers
• The input can be either current or voltage
• sensor signals often have very low values.
• the output of preamplifiers is typically voltage
• these devices can also provide isolation both to and from the sensor
• current amplifiers generally are more costly, but they are more immune to noise.
18.6.4 Modal Analysis
• Basically, excite a vibration, and measure how it is transmitted through a structure
18.7 Attenuating Vibrations
• Vibrations are basically the result of cyclic applications of forces. After the vibrations have been identified as frequencies, the phenomenon can be associated with physical design features.
18.7.1 Sources
• unbalanced rotating masses: these can be overcome by addition of counterweights
• rubbing will result in partial or full contact during some repetitive motions. Rubbing often worsens and leads to failure.
• misaligned couplings can lead to displacement or bending forces that induce vibrations.
• loose fittings will knock, rock, rub, etc.
• resonance caused by lack of damping.
• oil whirl and whip: oil films in hole shaft bearings can flow in a sporadic manner, causing the shaft to vibrate, sometimes catastrophically.
18.8 Resources
• There are a number of resources available to the student. In many cases older textbooks will contain valuable information.
18.9 Problems
Problem 18.1 A machine contains a 60Hz source of vibration that disturbs other machines in the same room. Find the spring coefficient, and natural frequency of an elastomer (with a damping
coefficient of .05) that will isolate the vibration source.
Problem 18.2 There is a large machine that weighs 1000 Kg, and has three legs. We will mount some elastomer under each leg. The graph below shows the characteristics of the isolator. From the graph
determine a spring constant (hint: a slope), and determine the natural frequency, and damping ration of the mount.
Problem 18.3 A piece of electronic equipment is to be isolated from a mounting panel which is vibrating at 8Hz. If 90% isolation is specified what static deflection would you expect?
Answer 18.3 0.042 m
Problem 18.4 A piece of mechanical equipment contains a 60Hz electric motor driving a reciprocating mechanism which generates motion excitation at 12Hz. The equipment has a total mass of 450 Kg and
is mounted on isolators. To establish some criteria regarding the actual isolation an accelerometer is mounted along the vertical axis of the machine. First, the static deflection is measured and
found to be 11mm. When the machine is switched off after operation, the output from the accelerometer is captured as a trace, on a storage oscilloscope. The response ratio between two adjacent
positive maxima on the trace (i.e., one cycle separation) is 1.65,
a) find the damped natural frequency of the equipment.
b) determine the percentage isolation (interpolating fig 9.8is adequate if you note your entry figures).
c) if the equipment was operated on a 50Hz supply (motor speed reduced by 17%), explain briefly what changes you would expect in the above results.
Answer 18.4 a) 29.77 rad/s, b) 72%, c) <60%.
Problem 18.5 It is required to provide 80% isolation for a machine using isolation pads conforming to density C specifications in the figure below (note the damping ratio is 0.05). Mounted equipment
will generate dynamic forces at 2600 rpm.
a) What would be the minimum mass of the machine if the total isolator pad area is limited to 750 cm2?
b) for the same pad area and a machine mass of 275 kg, what isolation efficiency would be afforded if density A material were used?
Problem 18.6 Using the data shown in the figure above a total of 4 isolators were selected to isolate a small piece of equipment having a mass of 10kg. The dominant forcing frequency is 60Hz and
damping can be neglected.
a) determine the isolation afforded by Isolator B.
b) What would be the result if Isolator C were substituted?
Problem 18.7 Using the figure above, select an isolator to assure 96% isolation efficiency at a forcing frequency of 112Hz (assume no damping). What would be the static deflection of the isolator
selected for a load/isolator of 15N
Answer 18.7 0.075cm
Problem 18.8 For a system weight/per isolator of 35N, use the figure above determine what isolation efficiency would be obtained if isolator D were used. System forcing frequency is 80Hz. Damping
ratio = 0.
Answer 18.8 95%
Problem 18.9 Cork isolation pads (damping ratio 0.05 use the figure above) are used at the 4 corners of the base of a small machine that weighs 2 KN and generates a forcing frequency of 105Hz. If the
pads are 10cm wide by 20cm long, what isolation efficiency would be afforded if they are made of a slab of density C?
Answer 18.9 95%
Problem 18.10 Four springs, each having a spring constant of 2400 N/m are placed at the 4 corners of a centrally loaded baseplate. What is the system weight if the static deflection is 10mm?
Answer 18.10 96N
Problem 18.11 An accelerometer is attached to a piece of equipment and connected via a preamplifier to an oscilloscope. The equipment is mounted on vibration isolators. Assuming a single d.o.f. model
which we can see is underdamped, what is the system damping ratio if the measured ratio of the amplitude response at two consecutive positive peaks on the decaying harmonic waveform is 1.8?
Problem 18.12 In the design of an accelerometer the requirement is for the maximum measurement error to be limited to 6% at 1/3 of the resonant frequency (i.e. magnification factor 1.06). What would
be the maximum damping ratio to meet these requirements?
Answer 18.12 0.48
Problem 18.13 Using an oscilloscope the ratio of two subsequent maxima was found to be 1.6. Determine the damping ratio.
Answer 18.13 0.075
Problem 18.14 A motor (mass M1) and controller (mass M2) are mounted on a heavy base (mass M3). Isolators are used to mount all masses as shown below. Dynamic forces generated by the motor are
forcing the system so it will be necessary to determine the forces on M2 and the support structure below M3.
a) show the lumped parameter model which represents the system.
b) using the Force/Current Mobility Analogue show the equivalent electrical circuit.
Problem 18.15 (A long problem) Find the force transmitted by the unbalanced load in the washing machine, to the floor. The rotating mass is 1kg at a distance of 20cm from center, turning at a speed
of 30rpm. The assembly consists of the upper drum and motor assembly with mass M2, is rested on a spring, that in turn rests on a large mass. This mass is suspended on a solid floor using a spring/
damper combination.
a) Develop the transfer function for the force applied by the eccentric mass to the ground.
b) Determine the input forcing function (from the eccentric mass)
c) Develop the time based reaction using Laplace transforms.
d) Use Fourier transforms to find the effect of the system in steady state.
e) Draw Bode plots for the system.
Problem 18.16 A machine stands on 6 legs in a corner of a room. in total the machine weighs 10,000 kg, and a vibrational force of 50N is applied at 120Hz by a rotating mass, and a force of 2 N is
applied at 60Hz by an AC motor. It has been decided that an isolator will be added to reduce the vibration passed to the floor. 6 isolators will be attached to the legs of the machine. The isolators
will be spring damper pairs connected in parallel. (Note: assume the floor movement is negligible). The spring constant is 100KN/m, and the damper is 200KNs/m.
a) Draw a Free Body Diagram of the system.
b) Develop a transfer function for the force input to the machine mass, to the force output applied to the floor.
c) Find the isolation for the two vibrations using the results in b).
d) Find a Laplace input function for the vibration, and determine what the Laplace output function will be.
e) Determine the time based response of the function in d).
f) Draw a Bode Plot for the transfer function in b).
g) Use the Bode plot in f) to find the steady state forces applied to the floor.
h) Use the Bode plot in f) to find the isolation of the vibrations.
i) Design an elastomeric isolator (instead of the spring-damper) to get 90% isolation for the 60Hz force.
Problem 18.17 Given the car wheel modeled below, relate a change in the height of the wheel to a change in the height of the car. The final result should be a Laplace transfer function of ‘ycar/
Problem 18.18 Find the time response ‘x(t)’ of a system with a transfer function G(s) that is excited by the force ‘F(t)’.
Problem 18.19 A large machine weighs 1000kg and vibrates at 20Hz, design an inertial damper.
Problem 18.20 A 10kg machine is set on isolation pads and vibrates at 60Hz, what should the natural frequency of an elastomer isolation pad be? If there are 3 pads, what should their spring constant
Problem 18.21 A machine stands on 6 legs in a corner of a room. in total the machine weighs 10,000 kg, and a vibrational force of 50N is applied at 120Hz by a rotating mass, and a force of 2 N is
applied at 60Hz by an AC motor. It has been decided that an isolator will be added to reduce the vibration passed to the floor. 6 isolators will be attached to the legs of the machine. The isolators
will be spring damper pairs connected in parallel. (Note: assume the floor movement is negligible). The spring constant is 100KN/m, and the damper is 200KNs/m.
a) Draw a Free Body Diagram of the system.
b) Develop a transfer function for the force input to the machine mass, to the force output applied to the floor.
c) Find the isolation for the two vibrations using the results in b).
d) Find a Laplace input function for the vibration, and determine what the Laplace output function will be.
e) Determine the time based response of the function in d).
f) Draw a Bode Plot for the transfer function in b).
g) Use the Bode plot in f) to find the steady state forces applied to the floor.
h) Use the Bode plot in f) to find the isolation of the vibrations.
i) Design an elastomeric isolator (instead of the spring-damper) to get 90% isolation for the 60Hz force.
18.10 Sound and Vibration Terms 18.11 References
18.2 Irwin, J.D., and Graf, E.R., Industrial Noise and Vibration Control, Prentice Hall Publishers, 1979. | {"url":"https://engineeronadisk.com/V3/engineeronadisk-125.html","timestamp":"2024-11-07T19:25:51Z","content_type":"text/html","content_length":"44334","record_id":"<urn:uuid:33f6382e-27c5-4c37-9fd1-dcafadf7aade>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00048.warc.gz"} |
Mbps to MB converter
This tool calculates the total number of Megabytes (MB) from Megabit-per-second (Mbps).
Mbps is the number of megabits that are transferred over a communication link every second. It is also called the Data rate, Speed, Throughput or Network bandwidth.
MB is a unit for data storage.
To find the total number of MB transferred, enter:
• Mbps
• Time in seconds/minutes/hours
Mb = (Mbps) * Time_in_seconds
Since 1 Megabyte = 8 Megabit,
MB = (Mbps) * (1/8) * Time_in_seconds
Example Calculation
5G wireless systems offers average download speeds of 100 Mb/s. Using the calculator on this page, 2000 Megabits or 250 Megabytes are transferred over a 20 second time interval.
The plot below shows this visually. The integrated area of the pink section represents the total data transferred over the time interval.
Note: 1 Megabyte = 8 Megabits. The amount of data transferred and stored is normally specified in Megabytes.
Bit vs. Byte: It’s important to differentiate between a bit and a byte. A bit is the smallest unit of data in computing, represented by a 0 or 1. A byte consists of 8 bits. Therefore, speeds
expressed in Mbps are not directly comparable to file sizes, which are usually measured in bytes (e.g., megabytes, MB). To convert from Mbps to megabytes per second (MB/s), we divide by 8, since
there are 8 bits in a byte.
Related Calculators
Related Posts
• The Netflix Bandwidth calculator uses the length of the show or movie and its quality to determine how much data is consumed.
• The streaming bandwidth calculator determines how much is used on a monthly basis so you can plan your bandwidth requirements. | {"url":"https://3roam.com/mbps-to-mb-converter/","timestamp":"2024-11-05T03:07:38Z","content_type":"text/html","content_length":"193661","record_id":"<urn:uuid:4959a240-ae1a-4fd3-b58f-0750ca05f267>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00614.warc.gz"} |
5.2.3: Diatomic Molecules of the First and Second Periods
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First Period Homonuclear Diatomic Molecules
In the first row of the periodic table, the valence atomic orbitals are \(1s\). There are two possible homonuclear diatomic molecules of the first period:
Dihydrogen, H[2 ]\([\sigma_g^2(1s)]\): This is the simplest diatomic molecule. It has only two molecular orbitals (\(\sigma_g\) and \(\sigma_u^{*}\)), two electrons, a bond order of 1, and is
diamagnetic. Its bond length is 74 pm. MO theory would lead us to expect bond order to decrease and bond length to increase if we either add or subtract one electron. The calculated bond length for
the \(\ce{H_2^{+}}\) ion is approximately 105 pm.^1
Dihelium, He[2][ ]\([\sigma_g^2\sigma_u^{*2}(1s)]\): This molecule has a bond order of zero due to equal number of electrons in bonding and antibonding orbitals. Like other nobel gases, He exists in
the atomic form and does not form bonds at ordinary temperatures and pressures.
Draw the complete molecular orbital diagrams for \(\ce{H2}\) and for \(\ce{He2}\). Include sketches of the atomic and molecular orbitals.
Figure for Exercise \(\PageIndex{1}\): Molecular orbital diagrams for dihydrogen and dihelium. Molecular orbital surfaces calculated using Spartan software. (CC-BY-NC-SA; Kathryn Haas)
A complete molecular orbital diagram includes all atomic orbitals and molecular orbitals, their symmetry labels, and electron filling.
Second Period Homonuclear Diatomic Molecules
The second period elements span from Li to Ne. The valence orbitals are 2s and 2p. In their molecular orbital diagrams, non-valence orbitals (1s in this case) are often disregarded in molecular
orbital diagrams.
Orbital mixing has significant consequences for the magnetic and spectroscopic properties of second period homonuclear diatomic molecules because it affects the order of filling of the \(\sigma_g(2p)
\) and \(\pi_u(2p)\) orbitals. Early in period 2 (up to and including nitrogen), the \(\pi_u(2p)\) orbitals are lower in energy than the \(\sigma_g(2p)\) (see Figure \(\PageIndex{1})\). However,
later in period 2, the \(\sigma_g(2p)\) orbitals are pulled to a lower energy. This lowering in energy of \(\sigma_g(2p)\) is not unique; all of the \(\sigma\) orbitals in the molecule are pulled to
lower energy due to the increasing positive charge of the nucleus. The \(\pi\) orbitals in the molecule are also affected, but to a much lesser extent than \(\sigma\) orbitals. The reason has to do
with the high penetration of \(s\) atomic orbitals compared to \(p\) atomic orbitals (recall our previous discussion on penetration and shielding, and its effect on periodic trends). The \(\sigma\)
molecular orbitals have more \(s\) character and thus their energy is more influenced by increasing nuclear charge. As nuclear charge increases, the energy of the \(\sigma_g(2p)\) orbital is lowered
significantly more than the energy of the \(\pi_u(2p)\) orbitals (Figure \(\PageIndex{1})\).
Figure \(\PageIndex{1}\): Molecular orbital energy-level diagrams for the diatomic molecules of the period 2 elements. Unlike earlier diagrams, only the valence molecular orbital energy levels for
the molecules are shown here (atomic orbitals not shown for simplicity). For Li[2] through N[2], the \( \sigma _g(2p) \) orbital is higher in energy than the \( \pi_u(2p) \) orbitals. In contrast,
from O[2] onward, the \( \sigma _g(2p) \) orbital is lower in energy than the \( \pi_u(2p) \) orbitals because the nuclear charge increases across the row. The experimental bond lengths correlate
with the calculated bond order. (CC-BY-NC-SA, Kathryn Haas)
Dilithium, Li[2][ ]\([\sigma_g^2(2s)]\): This molecule has a bond order of one and is observed experimentally in the gas phase to have one Li-Li bond.
Diberylium, Be[2][ ]\([\sigma_g^2\sigma_u^{*2}(2s)]\): This molecule has a bond order of zero due to the equal number of electrons in bonding and antibonding orbitals. Although Be[2] does not exist
under ordinary conditions, it can be produced in a laboratory and its bond length measured (Figure \(\PageIndex{2}\)). Although the bond is very weak, its bond length is surprisingly ordinary for a
covalent bond of the second period elements.^2
Diboron, B[2 ]\([\sigma_g^2\sigma_u^{*2}(2s)\pi_u^1\pi_u^1(2p)]\): The case of diboron is one that is much better described by molecular orbital theory than by Lewis structures or valence bond
theory. This molecule has a bond order of one. The molecular orbital description of diboron also predicts, accurately, that diboron is paramagnetic. The paramagnetism is a consequence of orbital
mixing, resulting in the \(\sigma_g\) orbital's being at a higher energy than the two degenerate \(\pi_u^*\) orbitals.
Dicarbon, C[2 ]\([\sigma_g^2\sigma_u^{*2}(2s)\pi_u^2\pi_u^2(2p)]\): This molecule has a bond order of two. Molecular orbital theory predicts two bonds with \(\pi\) symmetry, and no \(\sigma\)
bonding. C[2] is rare in nature because its allotrope, diamond, is much more stable.
Dinitrogen, N[2 ]\([\sigma_g^2\sigma_u^{*2}(2s)\pi_u^2\pi_u^2\sigma_g^2(2p)]\): This molecule is predicted to have a triple bond. This prediction is consistent with its short bond length and bond
dissociation energy. The energies of the \(\sigma_g(2p)\) and \(\pi_u(2p)\) orbitals are very close, and their relative energy levels have been a subject of some debate (see next section for
Dioxygen, O[2 ]\([\sigma_g^2\sigma_u^{*2}(2s)\sigma_g^2\pi_u^2\pi_u^2\pi_g^{*1}\pi_g^{*1}(2p)]\): This is another case where valence bond theory fails to predict actual properties. Molecular orbital
theory correctly predicts that dioxygen is paramagnetic, with a bond order of two. Here, the molecular orbital diagram returns to its "normal" order of orbitals where orbital mixing could be somewhat
ignored, and where \(\sigma_g(2p)\) is lower in energy than \(\pi_u(2p)\).
Difluorine, F[2 ]\([\sigma_g^2\sigma_u^{*2}(2s)\sigma_g^2\pi_u^2\pi_u^2\pi_g^{*2}\pi_g^{*2}(2p)]\): This molecule has a bond order of one and like oxygen, the \(\sigma_g(2p)\) is lower in energy than
Dineon, Ne[2][ ]\([\sigma_g^2\sigma_u^{*2}(2s)\sigma_g^2\pi_u^2\pi_u^2\pi_g^{*2}\pi_g^{*2}\sigma_u^{*2}(2p)]\): Like other noble gases, Ne exists in the atomic form and does not form bonds at
ordinary temperatures and pressures. Like Be[2], Ne[2] is an unstable species that has been created in extreme laboratory conditions and its bond length has been measured (Figure \(\PageIndex{2}\))
Draw the complete molecular orbital diagram for O[2]. Show calculation of its bond order and tell whether it is diamagnetic or paramagnetic.
O[2 ]is paramagnetic with a bond order of 2. Its \(\sigma_g(2p)\) molecular orbital is lower in energy that the set of \(\pi_{u}(2p)\) orbitals.
Bond order \(=\frac{1}{2}\left[\left(\begin{array}{c}\text { 8 electrons in} \\ \text { valence bonding orbitals }\end{array}\right)-\left(\begin{array}{c}\text {4 electrons in} \\ \text {
valence antibonding orbitals }\end{array}\right)\right]\)
Figure for Exercise \(\PageIndex{2}\). Molecular orbital diagram of O[2]. (CC-BY-NC-SA, Kathryn Haas)
Use a qualitative molecular orbital energy-level diagram to predict the electron configuration, the bond order, and the number of unpaired electrons in the peroxide ion (O[2]^2^−).
This diagram looks similar to that of \(\ce{O2}\), except that there are two additional electrons.
\( \left ( \sigma _{g}(2s) \right )^{2}\left ( \sigma_u ^{\star }(2s) \right )^{2}\left ( \sigma _g(2p) \right )^{2}\left ( \pi _{u}(2p) \right )^{4}\left ( \pi _g(2p) \right )^{4} \); bond order
of 1; no unpaired electrons.
Bond Lengths in Homonuclear Diatomic Molecules
The trends in experimental bond lengths are predicted by molecular orbital theory, specifically by the calculated bond order. The values of bond order and experimental bond lengths for the second
period diatomic molecules are given in Figure \(\PageIndex{1}\), and shown in graphical format on the plot in Figure \(\PageIndex{2}\). From the plot, we can see that bond length correlates well with
bond order, with a minimum bond length occurring where the bond order is greatest (\(\ce{N2}\)). The shortest bond distance is at \(\ce{N2}\) due to its high bond order of 3. From \(\ce{N2}\) to \(\
ce{F2}\) the bond distance increases despite the fact that atomic radius decreases.
Figure \(\PageIndex{2}\): Overlayed plots of bond length (black squares), bond order (pink circles), and atomic radius (teal triangles) versus atomic number for second period homonuclear diatomic
molecules. (CC-BY-NC-SA; Kathryn Haas)
1. NIST, Calculated Geometries available for H[2]^+ (Hydrogen cation) 2Σg+ D∞h, available at https://cccbdb.nist.gov/diatomicexpbondx.asp
2. Merritt, J. M.; Bondybey, V. E.; Heaven, M. C., Beryllium Dimer-Caught in the Act of Bonding. Science 2009, 324 (5934), 1548-1551. | {"url":"https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Chemistry_(LibreTexts)/05:_Molecular_Orbitals/5.02:_Homonuclear_Diatomic_Molecules/5.2.03:_Diatomic_Molecules_of_the_First_and_Second_Periods","timestamp":"2024-11-12T07:43:26Z","content_type":"text/html","content_length":"140034","record_id":"<urn:uuid:11f37601-fdff-409d-b96b-97f886d8fb69>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00281.warc.gz"} |
Perform a Continuous Morlet Wavelet Transform
morlet {dplR} R Documentation
Perform a Continuous Morlet Wavelet Transform
This function performs a continuous wavelet transform on a time series.
morlet(y1, x1 = seq_along(y1), p2 = NULL, dj = 0.25, siglvl = 0.95)
y1 numeric vector. Series to be transformed.
x1 numeric. A vector of values giving the years for the plot. Must be the same length as length(y1).
p2 numeric. The number of power of two to be computed for the wavelet transform. Calculated from length of y1 if NULL.
dj numeric. Sub-octaves per octave calculated.
siglvl numeric. Level for the significance test.
This performs a continuous wavelet transform of a time series. This function is typically invoked with wavelet.plot.
A list containing:
y numeric. The original time series.
x numeric. The time values.
wave complex. The wavelet transform.
coi numeric. The cone of influence.
period numeric. The period.
Scale numeric. The scale.
Signif numeric. The significant values.
Power numeric. The squared power.
This is a port of Torrence’s IDL code, which can be accessed through the Internet Archive Wayback Machine.
Andy Bunn. Patched and improved by Mikko Korpela.
Torrence, C. and Compo, G. P. (1998) A practical guide to wavelet analysis. Bulletin of the American Meteorological Society, 79(1), 61–78.
See Also
ca533.rwi <- detrend(rwl = ca533, method = "ModNegExp")
ca533.crn <- chron(ca533.rwi, prewhiten = FALSE)
Years <- time(ca533.crn)
CAMstd <- ca533.crn[, 1]
out.wave <- morlet(y1 = CAMstd, x1 = Years, dj = 0.1, siglvl = 0.99)
version 1.7.7 | {"url":"https://search.r-project.org/CRAN/refmans/dplR/html/morlet.html","timestamp":"2024-11-04T21:04:48Z","content_type":"text/html","content_length":"4654","record_id":"<urn:uuid:4366aba0-697d-4b41-a697-1f3b8d7caeb2>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00462.warc.gz"} |
Quantifying Daseinisation Using Shannon Entropy
5c65bfb2-2986-4459-89a9-f1524e844c8d20210322045602452wseamdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON SYSTEMS1109-277710.37394/23202http://wseas.org/wseas/cms.action?id=
4067220202022020201910.37394/23202.2020.19http://wseas.org/wseas/cms.action?id=23194RomanZapatrinThe State Russian Museum, Department of Information Technologies, St.Petersburg, RUSSIATopos formalism
for quantum mechanics is interpreted in a broader, information retrieval, perspective. Contexts, its basic components, are treated as sources of information. Their interplay, called daseinisation,
defined in purely logical terms, is reformulated in terms of two relations: exclusion and preclusion of queries. Then, broadening these options, daseinisation becomes a characteristic of proximity of
contexts; to quantify it numerically, Shannon entropy is used.3420203420208285https://www.wseas.org/multimedia/journals/systems/2020/a245103-055.pdf10.37394/23202.2020.19.12http://www.wseas.org/
multimedia/journals/systems/2020/a245103-055.pdfA. Doering, C.J. Isham. A Topos Foundation for Theories of Physics: I. Formal Languages for Physics (2007). [quant-ph/0703060]A. Doering, C.J. Isham. A
Topos Foundation for Theories of Physics II. Daseinisation and the Liberation of Quantum Theory (2007).[arxiv:quant-ph/0703062]C. Flori, Review of the Topos Approach to Quantum Theory,
arXiv:1106.5660 [math-ph]C. Flori, Lectures on Topos Quantum Theory,arXiv:1207.1744 [math-ph]M. Melucci, An investigation of quantum interference in information retrieval, In: Proceedings of the
Information Retrieval Facility Conference (IRFC), 2010 | {"url":"https://wseas.com/journals/systems/2020/10.37394_23202.2020.19.12.xml","timestamp":"2024-11-12T23:32:49Z","content_type":"application/xml","content_length":"4470","record_id":"<urn:uuid:93689a35-e69e-4bb5-9560-2f3418aee649>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00807.warc.gz"} |
Using multiple optimizers inside a loop with Lightning
I am working on a big project which for which I need to call manual_backward and optimizer.step inside a loop for every batch.
Here is some reference code for a training_function that works, and another that doesn’t:
def loss_fn_working(self, batch: Any, batch_idx: int):
env = self.envs[self.p]
actions = None
prev_log_rewards = torch.empty(env.done.shape[0]).type_as(env.state)
prev_forward_logprob = None
loss = torch.tensor(0.0, requires_grad=True)
TERM = env.terminal_index
while not torch.all(env.done):
active = ~env.done
forward_logprob, back_logprob = self.forward(env)
log_rewards = -self.get_rewards()
if actions is not None:
error = log_rewards - prev_log_rewards[active]
error += back_logprob.gather(1, actions[actions != TERM, None]).squeeze(1)
error += prev_forward_logprob[active, -1]
error -= forward_logprob[:, -1].detach()
error -= (
.gather(1, actions[actions != TERM, None])
loss = loss + F.huber_loss(
loss = loss * log_rewards.softmax(0)
loss = loss.mean(0)
actions = self.sample_actions(forward_logprob, active, TERM)
# save previous log-probs and log-rewards
if prev_forward_logprob is None:
prev_forward_logprob = torch.empty_like(forward_logprob)
prev_forward_logprob[active] = forward_logprob
prev_log_rewards[active] = log_rewards
return loss, log_rewards
def calculate_loss(
prefix, # Added for debugging
error = torch.tensor(0.0, requires_grad=True) + log_rewards - prev_log_rewards # [B]
error = error + (back_log_prob).gather(1, actions.unsqueeze(1)).squeeze(1) # P_B(s|s')
error = error - stop_prob.detach() # P(s_f|s')
if prev_stop_prob is not None and prev_forward_log_prob is not None:
error = error + prev_stop_prob.detach() # P(s_f|s)
error = error - (prev_forward_log_prob).gather(
1, actions.unsqueeze(1)
loss = loss + F.huber_loss( # accumulate losses
loss = loss * log_rewards.softmax(0)
return loss.mean(0)
def loss_fn_not_working(self, batch, batch_size, prefix, batch_idx):
gfn_opt, rep_opt = self.optimizers()
# some code here
losses = []
rep_losses = []
prev_forward_log_prob = None
prev_stop_prob = torch.zeros(batch_size, device='cuda')
loss = torch.tensor(0.0, requires_grad=True, device='cuda')
active = torch.ones((batch_size,), dtype=bool, device='cuda')
graph = torch.diag_embed(torch.ones(batch_size, self.n_dim)).cuda()
while active.any():
graph_hat = graph[active].clone()
adj_mat = graph_hat.clone()
rep_loss, latent_var = self.rep_model(torch.cat((adj_mat, next_id.unsqueeze(-1)), axis = -1))
rep_loss_tensor = torch.tensor(0.0, requires_grad=True) + rep_loss
forward_log_prob, Fs_masked, back_log_prob, next_prob, stop_prob = (
with torch.no_grad():
actions = self.sample_actions(Fs_masked)
graph = self.update_graph(actions)
log_rewards = -self.energy_model(graph_hat, batch, False, self.current_epoch)
if counter==0:
loss = self.calculate_loss(loss, log_rewards, prev_log_rewards, back_log_prob, actions, stop_prob, prefix)
loss = self.calculate_loss(loss, log_rewards, prev_log_rewards, back_log_prob, actions, stop_prob, prefix, prev_stop_prob[active], prev_forward_log_prob[active])
if prefix == 'train':
self.manual_backward(rep_loss_tensor, retain_graph=True)
self.clip_gradients(gfn_opt, gradient_clip_val=0.5, gradient_clip_algorithm="norm") # NEEDED??
with torch.no_grad():
active[indices_to_deactivate] = False #active updated appropriately
indices = indices[~current_stop]
# active_indices = ~current_stop # Not being used?
next_id = F.one_hot(indices, num_classes=self.n_dim)
prev_log_rewards = log_rewards[~current_stop]
counter += 1
if prev_forward_log_prob is None:
prev_forward_log_prob = torch.empty_like(forward_log_prob)
prev_forward_log_prob[active] = forward_log_prob[~current_stop]
prev_stop_prob[active] = stop_prob[~current_stop]
return losses, graph, log_rewards, counter, rep_losses
Here, the main variable of importance is prev_forward_log_prob in loss_fn_not_working. The loss is being calculated using calculate_loss() function.
I have kept manual_optimization as True.
When using loss_fn_not_working, and keeping retain_graph as false for loss, I get the following error:
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.
If I do keep retain_graph as True for loss (i.e. the loss for the second optimizer), I get the following error instead:
one of the variables needed for gradient computation has been modified by an inplace operation: [torch.cuda.FloatTensor [64, 10]], which is output 0 of AsStridedBackward0, is at version 3; expected version 1 instead.
If I use loss_fn_working, there is no problem. So, I understand that the problem arises when using backward calls inside the loop. I am not really making any in-place operations, so how can I make
the 2nd loss_fn work? I tried cloning the relevant variables but it doesn’t work until we detach them, which I can’t do. | {"url":"https://lightning.ai/forums/t/using-multiple-optimizers-inside-a-loop-with-lightning/7774","timestamp":"2024-11-03T23:13:43Z","content_type":"text/html","content_length":"24123","record_id":"<urn:uuid:d7d669c7-425b-40a8-9661-a32fe897b4d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00577.warc.gz"} |
Optimizing CUDA Recurrent Neural Networks with TorchScript
This week, we officially released PyTorch 1.1, a large feature update to PyTorch 1.0. One of the new features we’ve added is better support for fast, custom Recurrent Neural Networks (fastrnns) with
TorchScript (the PyTorch JIT) (https://pytorch.org/docs/stable/jit.html).
RNNs are popular models that have shown good performance on a variety of NLP tasks that come in different shapes and sizes. PyTorch implements a number of the most popular ones, the Elman RNN, GRU,
and LSTM as well as multi-layered and bidirectional variants.
However, many users want to implement their own custom RNNs, taking ideas from recent literature. Applying Layer Normalization to LSTMs is one such use case. Because the PyTorch CUDA LSTM
implementation uses a fused kernel, it is difficult to insert normalizations or even modify the base LSTM implementation. Many users have turned to writing custom implementations using standard
PyTorch operators, but such code suffers from high overhead: most PyTorch operations launch at least one kernel on the GPU and RNNs generally run many operations due to their recurrent nature.
However, we can apply TorchScript to fuse operations and optimize our code automatically, launching fewer, more optimized kernels on the GPU.
Our goal is for users to be able to write fast, custom RNNs in TorchScript without writing specialized CUDA kernels to achieve similar performance. In this post, we’ll provide a tutorial for how to
write your own fast RNNs with TorchScript. To better understand the optimizations TorchScript applies, we’ll examine how those work on a standard LSTM implementation but most of the optimizations can
be applied to general RNNs.
Writing custom RNNs
To get started, you can use this file as a template to write your own custom RNNs.
We are constantly improving our infrastructure on trying to make the performance better. If you want to gain the speed/optimizations that TorchScript currently provides (like operator fusion, batch
matrix multiplications, etc.), here are some guidelines to follow. The next section explains the optimizations in depth.
1. If the customized operations are all element-wise, that’s great because you can get the benefits of the PyTorch JIT’s operator fusion automatically!
2. If you have more complex operations (e.g. reduce ops mixed with element-wise ops), consider grouping the reduce operations and element-wise ops separately in order to fuse the element-wise
operations into a single fusion group.
3. If you want to know about what has been fused in your custom RNN, you can inspect the operation’s optimized graph by using graph_for . Using LSTMCell as an example:
# get inputs and states for LSTMCell
inputs = get_lstm_inputs()
# instantiate a ScriptModule
cell = LSTMCell(input_size, hidden_size)
# print the optimized graph using graph_for
out = cell(inputs)
This will generate the optimized TorchScript graph (a.k.a PyTorch JIT IR) for the specialized inputs that you provides:
graph(%x : Float(*, *),
%hx : Float(*, *),
%cx : Float(*, *),
%w_ih : Float(*, *),
%w_hh : Float(*, *),
%b_ih : Float(*),
%b_hh : Float(*)):
%hy : Float(*, *), %cy : Float(*, *) = prim::DifferentiableGraph_0(%cx, %b_hh, %b_ih, %hx, %w_hh, %x, %w_ih)
%30 : (Float(*, *), Float(*, *)) = prim::TupleConstruct(%hy, %cy)
return (%30)
with prim::DifferentiableGraph_0 = graph(%13 : Float(*, *),
%29 : Float(*),
%33 : Float(*),
%40 : Float(*, *),
%43 : Float(*, *),
%45 : Float(*, *),
%48 : Float(*, *)):
%49 : Float(*, *) = aten::t(%48)
%47 : Float(*, *) = aten::mm(%45, %49)
%44 : Float(*, *) = aten::t(%43)
%42 : Float(*, *) = aten::mm(%40, %44)
...some broadcast sizes operations...
%hy : Float(*, *), %287 : Float(*, *), %cy : Float(*, *), %outgate.1 : Float(*, *), %cellgate.1 : Float(*, *), %forgetgate.1 : Float(*, *), %ingate.1 : Float(*, *) = prim::FusionGroup_0(%13, %346, %345, %344, %343)
...some broadcast sizes operations...
return (%hy, %cy, %49, %44, %196, %199, %340, %192, %325, %185, %ingate.1, %forgetgate.1, %cellgate.1, %outgate.1, %395, %396, %287)
with prim::FusionGroup_0 = graph(%13 : Float(*, *),
%71 : Tensor,
%76 : Tensor,
%81 : Tensor,
%86 : Tensor):
...some chunks, constants, and add operations...
%ingate.1 : Float(*, *) = aten::sigmoid(%38)
%forgetgate.1 : Float(*, *) = aten::sigmoid(%34)
%cellgate.1 : Float(*, *) = aten::tanh(%30)
%outgate.1 : Float(*, *) = aten::sigmoid(%26)
%14 : Float(*, *) = aten::mul(%forgetgate.1, %13)
%11 : Float(*, *) = aten::mul(%ingate.1, %cellgate.1)
%cy : Float(*, *) = aten::add(%14, %11, %69)
%4 : Float(*, *) = aten::tanh(%cy)
%hy : Float(*, *) = aten::mul(%outgate.1, %4)
return (%hy, %4, %cy, %outgate.1, %cellgate.1, %forgetgate.1, %ingate.1)
From the above graph we can see that it has a prim::FusionGroup_0 subgraph that is fusing all element-wise operations in LSTMCell (transpose and matrix multiplication are not element-wise ops). Some
graph nodes might be hard to understand in the first place but we will explain some of them in the optimization section, we also omitted some long verbose operators in this post that is there just
for correctness.
Variable-length sequences best practices
TorchScript does not support PackedSequence. Generally, when one is handling variable-length sequences, it is best to pad them into a single tensor and send that tensor through a TorchScript LSTM.
Here’s an example:
sequences = [...] # List[Tensor], each Tensor is T' x C
padded = torch.utils.rnn.pad_sequence(sequences)
lengths = [seq.size(0) for seq in sequences]
padded # T x N x C, where N is batch size and T is the max of all T'
model = LSTM(...)
output, hiddens = model(padded)
output # T x N x C
Of course, output may have some garbage data in the padded regions; use lengths to keep track of which part you don’t need.
We will now explain the optimizations performed by the PyTorch JIT to speed up custom RNNs. We will use a simple custom LSTM model in TorchScript to illustrate the optimizations, but many of these
are general and apply to other RNNs.
To illustrate the optimizations we did and how we get benefits from those optimizations, we will run a simple custom LSTM model written in TorchScript (you can refer the code in the custom_lstm.py or
the below code snippets) and time our changes.
We set up the environment in a machine equipped with 2 Intel Xeon chip and one Nvidia P100, with cuDNN v7.3, CUDA 9.2 installed. The basic set up for the LSTM model is as follows:
input_size = 512
hidden_size = 512
mini_batch = 64
numLayers = 1
seq_length = 100
The most important thing PyTorch JIT did is to compile the python program to a PyTorch JIT IR, which is an intermediate representation used to model the program’s graph structure. This IR can then
benefit from whole program optimization, hardware acceleration and overall has the potential to provide large computation gains. In this example, we run the initial TorchScript model with only
compiler optimization passes that are provided by the JIT, including common subexpression elimination, constant pooling, constant propagation, dead code elimination and some peephole optimizations.
We run the model training for 100 times after warm up and average the training time. The initial results for model forward time is around 27ms and backward time is around 64ms, which is a bit far
away from what PyTorch cuDNN LSTM provided. Next we will explain the major optimizations we did on how we improve the performance on training or inferencing, starting with LSTMCell and LSTMLayer, and
some misc optimizations.
LSTM Cell (forward)
Almost all the computations in an LSTM happen in the LSTMCell, so it’s important for us to take a look at the computations it contains and how can we improve their speed. Below is a sample LSTMCell
implementation in TorchScript:
class LSTMCell(jit.ScriptModule):
def __init__(self, input_size, hidden_size):
super(LSTMCell, self).__init__()
self.input_size = input_size
self.hidden_size = hidden_size
self.weight_ih = Parameter(torch.randn(4 * hidden_size, input_size))
self.weight_hh = Parameter(torch.randn(4 * hidden_size, hidden_size))
self.bias_ih = Parameter(torch.randn(4 * hidden_size))
self.bias_hh = Parameter(torch.randn(4 * hidden_size))
def forward(self, input, state):
# type: (Tensor, Tuple[Tensor, Tensor]) -> Tuple[Tensor, Tuple[Tensor, Tensor]]
hx, cx = state
gates = (torch.mm(input, self.weight_ih.t()) + self.bias_ih +
torch.mm(hx, self.weight_hh.t()) + self.bias_hh)
ingate, forgetgate, cellgate, outgate = gates.chunk(4, 1)
ingate = torch.sigmoid(ingate)
forgetgate = torch.sigmoid(forgetgate)
cellgate = torch.tanh(cellgate)
outgate = torch.sigmoid(outgate)
cy = (forgetgate * cx) + (ingate * cellgate)
hy = outgate * torch.tanh(cy)
return hy, (hy, cy)
This graph representation (IR) that TorchScript generated enables several optimizations and scalable computations. In addition to the typical compiler optimizations that we could do (CSE, constant
propagation, etc. ) we can also run other IR transformations to make our code run faster.
• Element-wise operator fusion. PyTorch JIT will automatically fuse element-wise ops, so when you have adjacent operators that are all element-wise, JIT will automatically group all those
operations together into a single FusionGroup, this FusionGroup can then be launched with a single GPU/CPU kernel and performed in one pass. This avoids expensive memory reads and writes for each
• Reordering chunks and pointwise ops to enable more fusion. An LSTM cell adds gates together (a pointwise operation), and then chunks the gates into four pieces: the ifco gates. Then, it performs
pointwise operations on the ifco gates like above. This leads to two fusion groups in practice: one fusion group for the element-wise ops pre-chunk, and one group for the element-wise ops
post-chunk. The interesting thing to note here is that pointwise operations commute with torch.chunk: Instead of performing pointwise ops on some input tensors and chunking the output, we can
chunk the input tensors and then perform the same pointwise ops on the output tensors. By moving the chunk to before the first fusion group, we can merge the first and second fusion groups into
one big group.
• Tensor creation on the CPU is expensive, but there is ongoing work to make it faster. At this point, a LSTMCell runs three CUDA kernels: two gemm kernels and one for the single pointwise group.
One of the things we noticed was that there was a large gap between the finish of the second gemm and the start of the single pointwise group. This gap was a period of time when the GPU was
idling around and not doing anything. Looking into it more, we discovered that the problem was that torch.chunk constructs new tensors and that tensor construction was not as fast as it could be.
Instead of constructing new Tensor objects, we taught the fusion compiler how to manipulate a data pointer and strides to do the torch.chunk before sending it into the fused kernel, shrinking the
amount of idle time between the second gemm and the launch of the element-wise fusion group. This give us around 1.2x increase speed up on the LSTM forward pass.
By doing the above tricks, we are able to fuse the almost all LSTMCell forward graph (except the two gemm kernels) into a single fusion group, which corresponds to the prim::FusionGroup_0 in the
above IR graph. It will then be launched into a single fused kernel for execution. With these optimizations the model performance improves significantly with average forward time reduced by around
17ms (1.7x speedup) to 10ms, and average backward time reduce by 37ms to 27ms (1.37x speed up).
LSTM Layer (forward)
class LSTMLayer(jit.ScriptModule):
def __init__(self, cell, *cell_args):
super(LSTMLayer, self).__init__()
self.cell = cell(*cell_args)
def forward(self, input, state):
# type: (Tensor, Tuple[Tensor, Tensor]) -> Tuple[Tensor, Tuple[Tensor, Tensor]]
inputs = input.unbind(0)
outputs = torch.jit.annotate(List[Tensor], [])
for i in range(len(inputs)):
out, state = self.cell(inputs[i], state)
outputs += [out]
return torch.stack(outputs), state
We did several tricks on the IR we generated for TorchScript LSTM to boost the performance, some example optimizations we did:
• Loop Unrolling: We automatically unroll loops in the code (for big loops, we unroll a small subset of it), which then empowers us to do further optimizations on the for loops control flow. For
example, the fuser can fuse together operations across iterations of the loop body, which results in a good performance improvement for control flow intensive models like LSTMs.
• Batch Matrix Multiplication: For RNNs where the input is pre-multiplied (i.e. the model has a lot of matrix multiplies with the same LHS or RHS), we can efficiently batch those operations
together into a single matrix multiply while chunking the outputs to achieve equivalent semantics.
By applying these techniques, we reduced our time in the forward pass by an additional 1.6ms to 8.4ms (1.2x speed up) and timing in backward by 7ms to around 20ms (1.35x speed up).
LSTM Layer (backward)
• “Tree” Batch Matrix Muplication: It is often the case that a single weight is reused multiple times in the LSTM backward graph, forming a tree where the leaves are matrix multiplies and nodes are
adds. These nodes can be combined together by concatenating the LHSs and RHSs in different dimensions, then computed as a single matrix multiplication. The formula of equivalence can be denoted
as follows:
$L1 * R1 + L2 * R2 = torch.cat((L1, L2), dim=1) * torch.cat((R1, R2), dim=0)$
• Autograd is a critical component of what makes PyTorch such an elegant ML framework. As such, we carried this through to PyTorch JIT, but using a new Automatic Differentiation (AD) mechanism that
works on the IR level. JIT automatic differentiation will slice the forward graph into symbolically differentiable subgraphs, and generate backwards nodes for those subgraphs. Taking the above IR
as an example, we group the graph nodes into a single prim::DifferentiableGraph_0 for the operations that has AD formulas. For operations that have not been added to AD formulas, we will fall
back to Autograd during execution.
• Optimizing the backwards path is hard, and the implicit broadcasting semantics make the optimization of automatic differentiation harder. PyTorch makes it convenient to write tensor operations
without worrying about the shapes by broadcasting the tensors for you. For performance, the painful point in backward is that we need to have a summation for such kind of broadcastable
operations. This results in the derivative of every broadcastable op being followed by a summation. Since we cannot currently fuse reduce operations, this causes FusionGroups to break into
multiple small groups leading to bad performance. To deal with this, refer to this great post written by Thomas Viehmann.
Misc Optimizations
• In addition to the steps laid about above, we also eliminated overhead between CUDA kernel launches and unnecessary tensor allocations. One example is when you do a tensor device look up. This
can provide some poor performance initially with a lot of unnecessary allocations. When we remove these this results in a reduction from milliseconds to nanoseconds between kernel launches.
• Lastly, there might be normalization applied in the custom LSTMCell like LayerNorm. Since LayerNorm and other normalization ops contains reduce operations, it is hard to fuse it in its entirety.
Instead, we automatically decompose Layernorm to a statistics computation (reduce operations) + element-wise transformations, and then fuse those element-wise parts together. As of this post,
there are some limitations on our auto differentiation and graph fuser infrastructure which limits the current support to inference mode only. We plan to add backward support in a future release.
With the above optimizations on operation fusion, loop unrolling, batch matrix multiplication and some misc optimizations, we can see a clear performance increase on our custom TorchScript LSTM
forward and backward from the following figure:
There are a number of additional optimizations that we did not cover in this post. In addition to the ones laid out in this post, we now see that our custom LSTM forward pass is on par with cuDNN. We
are also working on optimizing backward more and expect to see improvements in future releases. Besides the speed that TorchScript provides, we introduced a much more flexible API that enable you to
hand draft a lot more custom RNNs, which cuDNN could not provide. | {"url":"https://pytorch.org/blog/optimizing-cuda-rnn-with-torchscript/","timestamp":"2024-11-12T05:36:09Z","content_type":"text/html","content_length":"57804","record_id":"<urn:uuid:826fd175-7e49-442f-885e-f0cdbbba0d71>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00346.warc.gz"} |
How to Generate random numbers in JavaScript - Source Freeze
In JavaScript, generating random numbers is a common requirement for various tasks, ranging from game development to statistical simulations. The ability to create random numbers is crucial, and
JavaScript provides us with methods to accomplish this efficiently. In this blog, we’ll learn how to generate random numbers in JavaScript
How to Generate random numbers in javascript
The Math.random() function in JavaScript is foundational for generating pseudo-random floating-point numbers. It returns a floating-point number between 0 (inclusive) and 1 (exclusive), where 0 is
inclusive and 1 is exclusive.
Consider this basic example:
const randomNumber = Math.random();
Here, Math.random() provides us with a random decimal number. It’s crucial to understand that this method doesn’t accept any arguments; it solely generates a random decimal between 0 (inclusive) and
1 (exclusive) every time it’s called.
Using Math.random() to Generate Specific Ranges
Now, if we want to create random numbers within a particular range, such as between 10 and 50, we need to manipulate the output of Math.random() using simple arithmetic.
For instance:
function getRandomInRange(min, max) {
return Math.random() * (max - min) + min;
const randomInRange = getRandomInRange(10, 50);
In this example, getRandomInRange() accepts a min and max value and generates a random number within that range.
Here’s some of the outputs we’ll get in the range on 10 & 50:
When generating random numbers in JavaScript, one common issue is the output of numbers with numerous decimal places, making the result less manageable or suitable for specific applications. For
instance, using Math.random() to create a range might produce lengthy decimal values.
Consider this code:
function getRandomInRange(min, max) {
return (Math.random() * (max - min) + min).toFixed(2);
const randomInRange = getRandomInRange(10, 50);
Here, we utilize the toFixed() method to limit the output to two decimal places. This action addresses the problem of excessive decimals, ensuring that the generated random number will always have
precisely two decimal points.
Here’s the output:
In this blog, we explored JavaScript’s capabilities in generating random numbers. We learned the foundational method, Math.random(), which allows us to produce unpredictability in our applications.
Alongside we also learnt about the toFixed() method to round up the unnecessary decimal places.
Also read:
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How To Perform A Non-Parametric Partial Correlation In SPSS
Partial correlations are great in that you can perform a correlation between two continuous variables whilst controlling for various confounders. However, the partial correlation option in SPSS is
defaulted to performing a Pearson’s partial correlation which assumes normality of the two variables of interest.
But what if you want to perform a Spearman’s partial correlation on non-normally distributed data?
If you go to Analyze > Correlate > Partial … you will see that there is no option to select a Spearman correlation. There is, however, a way around this using a little coding.
In this guide, I will explain how to perform a non-parametric, partial correlation in SPSS.
The required dataset
To be able to conduct a Spearman partial correlation in SPSS, you need a dataset, of course. For our example, we have the age and weight of 20 volunteers, as well as gender. What we want to test is
if there is a correlation between age and weight, after controlling for gender.
Creating the script
For this to work, you need to enter a small piece of script into the SPSS Syntax Editor. Open up the Syntax Editor by going to File > New > Syntax.
Next, copy and paste the following code:
/MISSING = LISTWISE
/MATRIX OUT(*).
RECODE rowtype_ ('RHO'='CORR') .
/significance = twotail
/MISSING = LISTWISE
/MATRIX IN(*).
You now need to add the appropriate variables next to the NONPAR CORR and PARTIAL CORR sections.
So, next to the NONPAR CORR enter all of the variables that will be involved in the partial correlation. In our example, this would be Age, Weight and Gender.
For the PARTIAL CORR line you need to enter the two variables of interest in the correlation followed by a BY then the variables you want to control for. Make sure all of the variables you enter
match the ones in your file correctly, otherwise the script will fail.
Here is what our example will look like:
NONPAR CORR Age Weight Gender
/MISSING = LISTWISE
/MATRIX OUT(*).
RECODE rowtype_ ('RHO'='CORR') .
PARTIAL CORR Age Weight BY Gender
/significance = twotail
/MISSING = LISTWISE
/MATRIX IN(*).
And here is what it looks like in the Syntax Editor:
Running the script
The script itself is separated into 3 parts: NONPAR CORR, RECODE and PARTIAL CORR.
The first is to perform a Spearman bivariate correlation for all variables and to add the Spearman rank correlation coefficients into a new file.
RECODE converts the row type from a Spearman (RHO) to a Pearson (CORR).
Finally, PARTIAL CORR performs the partial correlation on the desired variables by using the newly created Spearman correlation coefficients from the NONPAR CORR script.
Here is how to run the script:
1. To run the script, go to the Syntax Editor and with the NONPAR CORR section selected, hit the green play button.
This will give you an output for the Spearman’s rho between the variables. If you go to the SPSS Output file you will see:
You will also notice that a new SPSS data file has been created and is now open. This is usually named ‘Untitled’, or something similar. Within this file, you will see the Spearman’s rho values and
n numbers for each correlation.
2. You next need to go back to the Syntax Editor window and run the RECODE part of the script. Make sure you select the new dataset as the active worksheet for this, as you want to perform the RECODE
on the new sheet. You can toggle between datasets by clicking on the drop-down menu next to Active:. In this case, we select the Unnamed sheet:
Click the green play button again to run the RECODE script on this.
3. Finally, still in the Syntax window, select the PARTIAL CORR code and run this on the same Unnamed dataset. This will perform the final partial correlation.
The output
By looking in the output file, you should now see a Partial Corr box which contains the partial correlation coefficients and P values for the test:
You will see in this example that the non-parametric partial correlation for age with weight, after controlling for gender, has a coefficient value of ‘0.383’ and has a significant value of ‘0.105’.
Therefore, there is not a significant correlation for age and weight after accounting for gender.
43 COMMENTS
1. Hi Steven,
Thank you so much for this guide. One quick question: when I run it, my degrees of freedom are off, which impacts my p values. I think my degrees of freedom are being based on the variables in
the correlation matrix, not the actual number of cases. For example, although I have a sample size of 100, my df when I run the syntax ends up being 26. Although I am controlling for 6 variables
and am not sure exactly what the df should be, 26 doesn’t seem right to me. From your screenshots, it doesn’t seem like you had this problem. Do you have any thoughts as to why this would happen?
Thanks so much again!
□ Hi Emma,
So sorry for the delay.
So do you have 100 cases and all of these have matching data for the variables being controlled for, ie there are no missing data points?
2. Dear Steven, I just wonder how to cite this method?
□ Hi Juan,
I based this guide on the one produced by IBM. In that, they quote a reference:
Conover, W.J. (1999), “Practical Nonparametric Statistics (3rd Ed.). New York: Wiley, (p. 327-328).
This may be a good place to start?
I hope that helps,
Best wishes,
3. Hi Steven
I’m using the SAS instead.
Obtained from: https://en.wikipedia.org/wiki/Partial_regression_plot
1) Computing the residuals of regressing the response variable against the independent variables but
omitting Xi
2) Computing the residuals from regressing Xi against the remaining independent variables
3) Plotting the residuals from (1) against the residuals from (2).
Example of SAS code: (I wish to acknowledge the contribution of Mr. Lin (Robbin@TMU), for his assistance in the longitudinal stats class)
SAS code:
proc import datafile=”C:\Users\User\Desktop\working.xls” out=ddd replace dbms=excel;
proc print;run;quit;
*1. Computing the residuals of regressing the response variable against the independent variables but omitting Xi;
proc reg data=ddd;
model var1=ctrl1 ctrl2;
output out=out1 residual=r1;
proc print data=out1;run;quit;
*2.Computing the residuals from regressing Xi against the remaining independent variables;
proc reg data=ddd;
model var2=crtl1 ctrl2;
output out=out2 residual=r2;
proc print data=out2;run;quit;
*3.Plotting the residuals from (1) against the residuals from (2).;
data out1 ;set out1; _n+1;run;
data out2;set out2;_n+1;run;
data out3;merge out1 out2; by _n;run;
proc sgplot data=out3;
scatter x=r1 y=r2;
4. Hi Steven
This is the SPSS syntax for the non-parametric partial corr the syntax example from SPSS forum (https://developer.ibm.com/answers/questions/223269/plotting-a-partial-corr-using-pairwise-exclusion
The SPSS syntax as follows:
* Encoding: UTF-8.
NONPAR CORR var1 var2 ctrlvar1 ctrlvar2
/MISSING = LISTWISE
/MATRIX OUT(*).
RECODE rowtype_ (‘RHO’=’CORR’) .
PARTIAL CORR var1 var2 BY ctrlvar1 ctrlvar2
/significance = twotail
/MISSING = LISTWISE
/MATRIX IN(*).
/MISSING LISTWISE
/STATISTICS COEFF OUTS R ANOVA
/CRITERIA=PIN(.05) POUT(.10)
/DEPENDENT var1
/METHOD=ENTER ctrlvar1 ctrlvar2
/SAVE ZRESID.
/MISSING LISTWISE
/STATISTICS COEFF OUTS R ANOVA
/CRITERIA=PIN(.05) POUT(.10)
/DEPENDENT var2
/METHOD=ENTER ctrlvar1 ctrlvar2
/SAVE ZRESID.
/SCATTERPLOT(BIVAR)=RES_1 WITH RES_2
Please feel free to comment on this syntax. Much obliged.
Best wishes
Larry Lai
□ Hi Larry,
Thanks for sharing. Did this work for you? The syntax looks like it is doing a regression, similar to how I described, and plotting the residuals this way.
Best wishes,
5. Hi Steven
Thanks for sharing. By the way, how to plot a Non-Parametric Partial Correlation In SPSS?
Thanks for considering my request.
□ Hi Larry,
Thanks for your comment!
Plotting the results is something I have found quite difficult myself. But I am yet to find a conclusive answer.
I have seen others which plot the results via a regression:
What you can do in SPSS is plot these through a linear regression. Go to: Analyze -> Regression -> Linear Regression Put one of the variables of interest in the Dependent window and the other
in the block below, along with any covariates you wish to control for. Then click the Plots button and tick the option for ‘Produce all partial plots’. Then run the test. One of the graphs
produced will be the graph you are after. Hope that helps!
Whether this is the correct way, however, I am not so sure – sorry.
If you do find out, please come back and share!
Best wishes,
6. Hi Steven,
Thank you for this useful guide!
I worked with this syntax but I get this warning:
“The MATRIX subcommand on the PARTIAL CORR command specifies an input file which does not contain a correlation matrix for the current splitfile group. Within cell matrices are not acceptable. A
correlation matrix has a row type of “CORR”.”
can you help me out ?
7. Unfortunately, one can not meaningfully apply the partial correlation formulas from parametric (usually Pearson’s) correlation to Spearman’s Rank correlation. You can apply the formulas as you
have above, but the formulas were not developed for Spearman’s and the answers you get back are not meaningful partial correlations as they are with Pearson’s, so the Spearman’s the partial
correlations are meaningless and can not be interpreted. This might not stop people doing it, but their resulting conclusions are fatally flawed.
However, you can use Kendall’s Tau correlation for nonparametric correlation, and apply the same parametric partial correlation formula to get meaningful answers. Be aware though that Kendall’s
Tau has a different meaning to Pearson’s r in explaining the correlation relationship. Unfortunately, there’s no easy way to apply significance testing to partial correlations based upon
Kendall’s Tau since the underlying sample distribution is not defined (as it is for Pearson’s).
So if you want partial correlations for nonparametric data, use Kendall’s Tau rather than Sprearman’s r.
8. Hello! Thanks for this wonderful guide. Do you have any suggestions for how to plot the results of the nonparametric partial correlation on a graph? I cannot figure this out or find anything
9. Super helpful! Can you please tell me how to flag significant correlations on the output?
10. Thank you so much for a very helpful post and also helpful comments and replys above.
Is it possible to enter more than two variables at the same time (before BY)? Or do I have to repeat it for every depending variable I want to test?
□ Hello Ingrid,
Many thanks for your comments and kind words. I presume you can enter more than 2 variables before the ‘BY’. The results should then display a grid table so you can look at all your
correlations within the same output.
Let me know if there is an issue with this however.
☆ Thank you for the reply. It worked and gave a grid table as you said. If you want to test the correlation between one (undepenent) variable and all the others (depentend) varaibles, you
can place that one first or last and write WITH between in the partial recode.
Example: PARTIAL CORR Age WITH Weight Pain BY Gender
Then the output will show only Age as a horisontal collum and Weight and Pain in the vertical collums.
○ Excellent, glad it worked for you. And thank you very much for the additional tip 🙂 greatly appreciated
11. can i know why the partial coefficient value is higher than the spearman’s rho value? shouldn’t it be lower?
□ Hello,
The correlations can increase or decrease depending upon the relationship your covariates have on the variables you are interested in. There is a discussion on this on ResearchGate which may
be useful to see:
Hope that helps!
12. Hi Steven,
I am running your script, but having an error below. The problem is the new “unnamed” or “unknown” data sheet does not exist or I can’t find it. What to do?
DATASET ACTIVATE DataSet1.
RECODE ROWTYPE_ (‘RHO’=’CORR’).
Error # 4631 in column 8. Text: ROWTYPE_
On the RECODE command, the list of variables to be recoded includes the name
of a nonexistent variable.
Execution of this command stops.
□ Hello Terhi,
So sorry for the late reply. Did you manage to sort this? It seems like the new results are not being opened in a new datasheet. Have you ensured the ‘/MATRIX OUT(*).’ part of the code is
included before you run the RECODE part of the code.
13. Hi Steve,
Great post. Do you know how to compute 95% confidence intervals for Spearman’s partial correlations using the syntax? My reviewers are requesting confidence intervals for all point-estimates in
accordance with APA.
□ Hi Rose,
Thanks for the comment. Unfortunately I do not know how to report 95% CI for this. Upon reading around this it seems quite a few people are asking the same thing. I have found a link to this
website however, http://vassarstats.net/rho.html, which computes 95% CI from the r and n values. May be of use for you?
□ Hi Rose:
This reply may have come a little late for you. As I posted below, there is no such thing as partial correlations for Spearman’s rho. Therefore, compute Kendall’s Tau, where you can calculate
meaningful partial correlations. However, even for Kendall’s Tau, there is no defined sampling distribution, and so CIs can not be calculated. You can move forward in a few ways: First
carefully review your data to be sure that Pearson’s r can not be used. Pearson’s r is pretty robust, and unless your data are very skewed from normal, you might be able to proceed (don’t get
distracted by the type of data you collecting, you can apply Pearson’s r even to categorical data). If the first approach does not work, try a data transformation to make your data
sufficiently normal to apply Pearson’s r (Spearman’s itself is a kind of rank data transformation). Finally, if you end up using Kendall’s Tau you might be able to apply bootstrap methods to
develop a sampling distribution to create CIs around the partial correlations. This is a last resort for most people, and I’ve rarely seen this done.
☆ Thanks for the advice Ian, really appreciate it!
Best wishes,
14. Hi, Steven
Thank you so much for providing this. I read the IBM instructions for syntax and was totally bamboozled; your tutorial and example was very easy to follow and was immensely helpful!
FYI, I am using SPSS V24 on a Mac. When I ran the second part of the syntax [RECODE rowtype_ (‘RHO’=’CORR’) .] I received a warning message. I removed the space between “rowtype_” and “(‘rho’=
’corr’) and re-ran without any further problems [ie. RECODE rowtype_(‘RHO’=’CORR’) .].
□ Hi Marie,
Thanks very much for the feedback, very much appreciated. Also, thanks for providing details for the Mac users. Unfortunately I am just on Windows at the minute so I cannot provide too much
information on that system, but maybe in the future I can expand :).
Best wishes,
□ Hi,
I’m on a Mac also and I found the warning message disappeared if I ensured I had clicked at the top of the syntax (so that the procedure was run from the right place and not halfway down the
☆ Hi Rose,
Thanks very much for this, I was having exactly the same problems as you (on a Mac) and found that clicking the top of the syntax sorted this. Your info sharing and advice has made a
happy student!
15. Hi Steven,
This is really helpul, but can you control for more than one variable? E.g., 3 variables (1 continuous and 2 categorical).
□ Hi Omar,
Thanks for the feedback. Yes, you can control for more than one variable. However, the more variables you are controlling for the less reliable the test may become because you may over-fit
your analysis. If you have a large enough sample size then it should be okay. One rule, called the One in Ten rule (https://en.wikipedia.org/wiki/One_in_ten_rule), is suggested for regression
analysis and could be kept in mind when doing a partial correlation. Briefly, for every control (or predictor) variable you use there must be at least 10 samples in the analysis.
Hope that helps!
☆ Great! So, in this case I would need to do something like Age Weight BY Gender By SES BY Ethnicity, right?
○ When more than one control variable is entered then only one ‘BY’ is required. So:
Age Weight BY Gender SES Ethnicity
This will control for ‘Gender’, ‘SES’, and ‘Ethnicity’.
Hope that helps 🙂
16. Also, are you sure partial correlations can be run by categorical variable? I thought it was only to control for a continuous variable.
□ As far as I am aware, you can control for dichotomous variables (e.g. gender). However, I am no stats expert!
17. Hey Steven,
I am trying to run your syntax, but my output says:
“The input matrix file does not contain a ROWTYPE_ variable or the variable has been misspecified.”
Could you help me out?
Thank you!
□ Hi Jess,
Sorry for the late response. The error you are getting, when do you get this? Is this for the first (nonpar corr), second (recode) or third (partial corr) part of the script?
☆ Hi Steven,
I am also experiencing this error. I get it at the second [RECODE rowtype_ (‘RHO’=’CORR’)] part of the script.
The exact error message is as Jess stated, “The input matrix file does not contain a ROWTYPE_ variable or the variable has been misspecified.”
Thanks in advance
○ Hi Lauren,
I think this error is because you may be running the RECODE part of the script using original datasheet. Have you changed the ‘Active’ sheet to the newly created ‘unnamed’ one before
running the RECODE part? (See point 2 in the guide above).
I am in the process of creating a screencast video that will hopefully help.
Let me know if this works 🙂
18. This was exactly what I needed, thank you so much! I agree with Rachael that is really clearly described.
□ Thank you very much Suzanne for the comment. I am glad it helped you out too 🙂
19. This has been so helpful. Thank you. Really clear and easy to follow.
□ Thanks Rachael, I really appreciate your comment. I am glad it helped you out 🙂 | {"url":"https://toptipbio.com/spearman-partial-correlation-spss/","timestamp":"2024-11-11T05:20:08Z","content_type":"text/html","content_length":"289637","record_id":"<urn:uuid:78d5db8a-588e-4b79-b95d-6226534d18d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00665.warc.gz"} |
Unleashing the Power of PCA: Understanding and Interpreting High-Dimensional Datasets - Adventures in Machine Learning
Introduction to Principal Components Analysis (PCA)
Principal Components Analysis (PCA) is an unsupervised machine learning technique used to identify patterns in data that are not easily noticeable through visual inspection. It achieves this by
transforming the data into a set of new variables known as principal components.
Each principal component represents a linear combination of the original variables. PCA is a powerful tool that can help researchers gain a deeper understanding of the relationships between different
variables in the data.
By analyzing the variation explained by each principal component, researchers can identify which variables have the most significant impact on the overall variance of the data. In this article, we
will walk you through the process of preparing a dataset for PCA, and explain the importance of understanding variation explained by each principal component.
Preparing the Dataset for PCA
1) Importing the USArrests dataset and defining columns to use for PCA
To demonstrate the process of preparing a dataset for PCA, we will use the USArrests dataset, which contains data on crime rates in different states of the United States. The first step is to import
the dataset and define the columns we want to use for PCA.
In this case, we will use all four variables: Murder, Assault, UrbanPop, and Rape. The code to import the dataset and define the columns would look like this:
import pandas as pd
usarrests = pd.read_csv('USArrests.csv')
cols_to_use = ['Murder', 'Assault', 'UrbanPop', 'Rape']
data = usarrests[cols_to_use]
2) Creating a scaled version of the dataset using StandardScaler
The next step is to create a scaled version of the dataset using StandardScaler. StandardScaler scales each variable in the dataset to have a mean of 0 and a standard deviation of 1.
Scaling the variables is important because PCA is sensitive to the relative magnitude of the variables. If we do not scale the variables, a variable with a larger magnitude will have a more
significant impact on the principal components than a variable with a smaller magnitude, even if the smaller variable is more important in the data.
The code to scale the dataset using StandardScaler would look like this:
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaled_data = pd.DataFrame(scaler.fit_transform(data), columns=data.columns)
In this article, we have discussed the basics of Principal Components Analysis (PCA) as an unsupervised machine learning technique. We have also explained the importance of understanding the
variation explained by each principal component.
Additionally, we have demonstrated the process of preparing a dataset for PCA by importing the USArrests dataset and creating a scaled version of the dataset using StandardScaler. By using PCA,
researchers can reduce the dimensionality of the data without losing too much information.
They can identify the variables that have the most significant impact on the overall variance of the data and gain insights into the relationships between different variables. PCA is a powerful tool
that can help researchers make more informed decisions based on data analysis.
3) Performing PCA
Once the dataset is prepared and scaled, we can now perform PCA. In this article, we will use the PCA() function from the sklearn package to perform PCA on the USArrests dataset.
The PCA() function requires us to specify the number of principal components we want to extract. In many cases, we may not know how many principal components we want to extract, but there is a method
we can use to determine the number of components.
This method is called the scree plot. To extract the principal components using PCA(), the code would look like this:
from sklearn.decomposition import PCA
pca = PCA(n_components=4)
principal_components = pca.fit_transform(scaled_data)
In this case, we have specified the number of components as 4.
This means that we want to extract 4 principal components from the data.
4) Creating a Scree Plot
To determine the optimal number of principal components to use, we can create a scree plot. A scree plot is a line plot that shows the percentage of total variance explained by each principal
To calculate the percentage of total variance explained by each principal component, we can use the explained_variance_ratio_ attribute of the PCA object. This attribute returns an array of values
representing the percentage of variance explained by each component.
import matplotlib.pyplot as plt
percent_variance = pca.explained_variance_ratio_
scree_plot_data = {'PC{}'.format(i): [percent_variance[i-1]*100] for i in range(1,len(percent_variance)+1)}
scree_plot_df = pd.DataFrame(data=scree_plot_data)
plt.ylabel('Percent Variance Explained')
plt.xlabel('Principal Component')
The resulting scree plot displays the percentage of variance explained by each principal component on the y-axis, and the principal component values on the x-axis. The plot helps us visualize how
much variance each principal component contributes relative to the others.
From the scree plot, we can observe the elbow point on the plot, representing the point at which adding more principal components does not result in a significant increase in variance explained. In
this case, it appears that the optimal number of principal components is two.
Using just two components instead of all four could improve the performance of models that use this data, as it would reduce the dimensionality of the data while still retaining most of the
In this article, we have seen how the sklearn package can be used to perform PCA on a dataset. We have also learned about the scree plot and how it can be used to determine the optimal number of
principal components to use.
By using PCA and visualizing the results using a scree plot, researchers can gain valuable insights into the data and make more informed decisions. PCA and scree plots are powerful tools that can
help us extract meaningful information from high-dimensional datasets, and are an essential part of any data analysis toolkit.
5) Interpretation of Scree Plot Results
After creating a scree plot, it is essential to understand how to interpret the results. The scree plot helps us determine the appropriate number of principal components to use in the analysis.
Explanation of what the percentage of variance explained by each principal component means
Each principal component derived from PCA explains a specific amount of variation in the original dataset. The sum of the variances explained by all the principal components is equal to the total
variance of the data.
The percentage of variance explained by a principal component is obtained by dividing the variance explained by that principal component by the total variance of the data and converting it to a
percentage. For example, if a principal component explains 20% of the total variance of the data, it means that 20% of the total variability in the data can be explained by that principal component.
The remaining 80% of the variability is due to other factors not captured by that principal component. The percentage of variance explained by a principal component is a crucial piece of information
in interpreting the scree plot.
It helps us determine the relative importance of each principal component in the dataset.
Displaying exact percentage of variance explained by each principal component
To determine the exact percentage of variance explained by each principal component, we can use the explained_variance_ratio_ attribute of the PCA object. As mentioned earlier, this attribute returns
an array of values representing the percentage of variance explained by each component.
percent_variance = pca.explained_variance_ratio_
for i in range(len(percent_variance)):
print('PC{} explains {:.2f}% of variance in the data'.format(i+1, percent_variance[i]*100))
This code snippet will display the exact percentage of variance explained by each principal component in the data.
Interpreting the Scree Plot
After creating a scree plot and determining the optimal number of principal components, we need to consider how to interpret each principal component. Each principal component represents a linear
combination of the original variables.
Thus, it is essential to look at the loadings of each variable on each principal component to interpret their individual meanings. Loadings represent the correlation between each variable and the
principal component.
Loadings can be positive or negative, indicating the direction and strength of the correlation. To calculate the loading of each variable on each principal component, we can use the components_
attribute of the PCA object.
This attribute returns a matrix with dimensions (n_components, n_features) containing the loadings.
loadings = pd.DataFrame(pca.components_.T * np.sqrt(pca.explained_variance_), columns=['PC{}'.format(i) for i in range(1,5)], index=data.columns)
This code will display the loadings of each variable on each principal component. Interpretations of the loadings will be specific to the problem being studied.
In this article, we have seen how to interpret the results of a scree plot by understanding the percentage of variance explained by each principal component. We have also seen how to display the
exact percentage of variance explained by each principal component.
By using the loadings, we gain a deeper understanding of the relationship between the original variables and each principal component. This information can be used to make more informed decisions,
develop hypotheses for further analysis, and support the development of predictive models.
The scree plot, combined with the loadings, is a powerful tool for interpreting the results of PCA and gaining insights into high-dimensional datasets. In conclusion, Principal Component Analysis
(PCA) is a powerful unsupervised machine learning technique that helps to identify patterns in high-dimensional datasets.
To prepare a dataset for PCA, it’s important to import the dataset, define the columns to use, and scale the variables. Creating a scree plot helps to determine the optimal number of principal
components to use and interpret each principal component’s loadings.
Proper interpretation and use of PCA can lead to more informed decision-making, hypothesis development, and improved predictive models. By understanding the percentage of variance explained by each
principal component and the loadings of each variable, researchers gain deeper insights into relationships within their data.
For any data analyst or researcher, PCA is a powerful tool and understanding its use can improve the quality of their results. | {"url":"https://www.adventuresinmachinelearning.com/unleashing-the-power-of-pca-understanding-and-interpreting-high-dimensional-datasets/","timestamp":"2024-11-07T00:09:22Z","content_type":"text/html","content_length":"80715","record_id":"<urn:uuid:b1f68f51-45fd-4a69-bd81-bbdc61e6e38a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00490.warc.gz"} |
Colorado State University Algebra Discussion - Custom Scholars
Colorado State University Algebra Discussion
This week, your task is to create a relation involving two variables from your daily life, and then discuss whether or not your relation is a function.
Your task for this discussion is as follows:
1. Fill in the table with what you are typically doing during each of the following times.
Time (T) Activity (A)
4 AM
8 AM
12 PM
4 PM
8 PM
12 AM
1. Represent your relation (T, A) in an alternative manner other than the table created in part (a). (i.e., as ordered pairs or with a mapping diagram.)
2. Determine whether or not your relation is a function. Why or why not?
3. In your responses to peers, comment on whether or not you think their relations are also functions.(see attached peers)
Here is my list of everyday activities. Judging from this schedule, I need to
get out more!
Time (t)
Activity (A)
This table can also be represented as a set of ordered pairs as follows, given
that time (t) is the input (independent) and activity (A) is the output
{(4AM, Sleeping), (8AM, Working), (12PM, Eating Lunch), (
Mapped out in a diagram, it is also clear to see that each input (t) has one
output (A):
4 AM
12 PM
Eating Lunch
В РМ
12 AM
Based on the Abramson (2015, Section 3.1) definition of a function where
each input has exactly one output, this relation is a function which
represents Activity as a function of Time. It can be written as A = f(t).
Conversely, since there are more than one of the same output (sleeping and
working) that can be mapped back to different times, then this is not a one-
to-one function. Likewise, if Activity was the input and Time was the
output, then the relation would not be a function because there would be
multiple outputs for one or more of the inputs.
The following table represents the time slots for my activities.
Time (T) Activity (A)
4 AM Sleeping
8 AM
12 PM
4 PM
8 PM
12 PM Sleeping
As you can see the activities revolve around sleeping, eating, and working
which is productive but a bit of an eye opener. The independent variable in
this situation is Time (T) and the dependent variable is Activity (A). As seen
below the input and outputs can be illustrated in a diagram.
activas a
12 PM
4 PM
12 AM
We can also view this as order pairs as seen below.
{(4 AM, SLEEPING), (8 AM, WORK).(12 PM EATING),
Based on the diagram, you can easily see that each independent variable
has exactly one output. It is stated by Abramson (2015), “A function is a
relation in which each possible input value leads to exactly one output
value.” This verifies that this is indeed a function because it follows the
definition of a function. Therefor the function would be represented by
A= f(T). You can take it a step further and say that it is not a one-to-
one function because the outputs correspond to more than one input. | {"url":"https://customscholars.com/colorado-state-university-algebra-discussion/","timestamp":"2024-11-06T18:53:23Z","content_type":"text/html","content_length":"54364","record_id":"<urn:uuid:1ac18449-0ef7-4dbb-b70a-54be465e11a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00505.warc.gz"} |
Math 340 Home
Textbook Contents
Online Homework Home
Linear Equations
We solved exact equations by assuming there was a solution and working backwards. But there will be a solution to any equation we can solve, and it is a theorem that all (reasonable) first order
equations can be solved. So why aren't all equations exact? Consider the following example. $$ (x^2+y)+(x-\sin(y))\frac{dy}{dx}&=0\qquad\text{is exact} \\ \frac{x^2+y}{x}+\frac{x-\sin(y)}{x}\frac{dy}
{dx}&=0\qquad\text{is not exact} $$ There is no real difference between the two equations, we have just divided through by $x$ to obtain the second equation from the first. The equations have the
same solutions. But when we test for exactness, we find that only the first equation is exact. The reason most first order equations aren't exact is that some key factor has been divided out. One way
to solve the equation would be to find the factor, called an integrating factor, and put it back in. Unfortunately, finding integrating factors can be extremely difficult. There is one situation
where it is fairly straightforward, when the equation is linear. A first order differential equation is said to be
if it can be written in the form $$ y' + p(x)y = q(x) $$ For a linear equation, we find an integrating factor as follows $$ \mu(x) = e^{\int p(x)\,dx} $$ If we multiply through the linear equation by
$\mu(x)$ we obtain $$ \mu(x)y'(x) + \mu(x)p(x)y(x) = \mu(x)q(x) $$ Now $\mu'(x) = \mu(x)p(x)$ from the way we defined $\mu(x)$, so we can now write our equation as $$ \mu(x)y'(x) + \mu'(x)y(x) = \mu
(x)q(x) $$ or (applying the product rule in reverse) $$ \frac{d}{dx}(\mu(x)y(x)) = \mu(x)q(x) $$ Now we integrate both sides to obtain $$ \mu(x)y(x) = \int \mu(x)q(x)\,dx $$ so $$ y(x) = \frac{1}{\mu
(x)}\int \mu(x)q(x)\,dx $$ WARNING: You can't cancel the $\mu(x)$ inside the integral by the $\mu(x)$ outside the integral. This formula is confusing because we are using $x$ as both the dummy
variable of integration and as the independent variable of the resulting function. Because of this it is often preferable to write the solution as $$ y(x)=\frac{1}{\mu(x)}\left(\int_a^x \mu(s)q(s)
\,ds + C\right) \tag 1$$ In this form we use $s$ as the variable of integration and use $x$ as the upper limit of integration to show we want to treat the result as a function of $x$. The $a$ as the
lower limit of integration is an arbitrary constant as is $C$, the constant of integration. This seems to give us two arbitrary constants, but we really only have one "degree of freedom." That is,
for any choice of $a$, we can choose $C$ to give us any solution to the equation. This is easiest to understand when we deal with a particular example. Example: $y'+2y=e^x$ $$ \mu(x)&=e^{\int2\,dx}=e
^{2x} \\ y(x)&=\frac{\int_a^x e^{2s}e^s\,ds+C}{e^{2x}}=\frac{\int_a^x e^{3s}\,ds+C}{e^{2x}} \\ y(x)&=\frac{e^{3x}/3-e^{3a}/3+C}{e^{2x}} \\ y(x)&=e^x/3+\tilde{C}e^{-2x}\qquad(\tilde{C}=C-e^{3a}/3) $$
As you see, the choice of $a$ doesn't matter, it all gets sucked into the arbitrary constant in the end. Linear equations won't have singular solutions, so the general solution gives all the
solutions. You may be wondering about why I left out the constant of integration when computing $\mu(x)$. I could have included a constant of integration, which would give me an infinite family of
integrating factors. But I don't need an infinite family of integrating factors to solve the problem; I only need one integrating factor. So I can ignore that constant. We are now ready to give the
paradigm. While it is possible to just memorize the formula for the solution, I prefer to work through the whole process. While that takes longer, I find I am less likely to make mistakes if I go
through the details. You are welcome to use the paradigm or the formula (1) depending on which you find works best for you.
Solve $$dy/dx+2y/x=4$$
Step 1:
Find the integrating factor $$ \mu(x)=e^{\int 2/x\,dx}=e^{2\log(x)}=x^2 $$
Step 2:
Multiply through by the integrating factor $$ x^2\frac{dy}{dx}+2xy=4x^2 $$
Step 3:
Recognize the left hand side as the derivative of $\mu y$. $$ \frac{d}{dx}(x^2y)=4x^2 $$
Step 4:
Integrate both sides $$ x^2y=\int 4x^2\,dx=(4/3)x^3+C $$
Step 5:
Solve for $y$. $$ y(x)=(4/3)x+Cx^{-2} $$ EXAMPLE: Solve the initial value problem $$dy/dx+2xy=1, y(1)=2$$. FIRST: Find the general solution. Step 1: $$\displaystyle\mu(x)=e^{\int 2x\,dx}=e^{x^2}$$
Step 2: $$\displaystyle e^{x^2}dy/dx+2xe^{x^2}y=e^{x^2}$$ Step 3: $$\displaystyle \frac{d}{dx}(e^{x^2}y)=e^{x^2}$$ Step 4: $$\displaystyle e^{x^2}y=\int e^{x^2}\,dx=???$$ Unfortunately, I don't know
the indefinite integral of $e^{x^2}$. So I leave the integral as a definite integral with $x$ as the upper limit to give me a function of $x$ and $1$ as the lower limit because the initial value is
given at $1$ (you'll see why that matters in a second). And I won't forget the constant of integration. Step 4: (Take Two) $$\displaystyle e^{x^2}y=\int_1^x e^{s^2}\,ds+C$$ Step 5: $$\displaystyle y
(x)=e^{-x^2}\int_1^x e^{s^2}\,ds+Ce^{-x^2}$$ SECOND: Solve the initial value problem by plugging in. $$ y(1)=e^{-1^2}\int_1^1 e^{s^2}\,ds+Ce^{-1^2}&{\buildrel \text{set}\over =} 2 \\ C&=2e $$ While I
don't know the indefinite integral of $e^{s^2}$, I do know that the integral of anything from $1$ to $1$ is $0$. That is the advantage of choosing the lower limit to be the same as the place where
the initial value is given. So the final answer is $$y(x)=e^{-x^2}\int_1^x e^{s^2}\,ds+2e^{1-x^2}$$ You can generate
additional examples of initial value problems for first order linear equations here.
If you have any problems with this page, please contact bennett@math.ksu.edu.
©2010, 2014 Andrew G. Bennett | {"url":"https://onlinehw.math.ksu.edu/math340book/chap1/linear.php","timestamp":"2024-11-09T06:09:39Z","content_type":"text/html","content_length":"14270","record_id":"<urn:uuid:d9fdce96-fc6f-4632-98e7-3b04b0c1ab94>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00220.warc.gz"} |
5 Multiplication Worksheets
Math, especially multiplication, creates the foundation of countless academic techniques and real-world applications. Yet, for many learners, understanding multiplication can pose a challenge. To
address this difficulty, educators and moms and dads have actually welcomed an effective tool: 5 Multiplication Worksheets.
Intro to 5 Multiplication Worksheets
5 Multiplication Worksheets
5 Multiplication Worksheets -
Interactive Worksheet 3 Digit Multiplication Interactive Worksheet Two Minute Timed Test 3 Multiply Divide Interactive Worksheet Multi Digit Multiplication Division Worksheet Multiplication Practice
Part 2 Write the Missing Number Interactive Worksheet
Grade 5 multiplication worksheets Multiply by 10 100 or 1 000 with missing factors Multiplying in parts distributive property Multiply 1 digit by 3 digit numbers mentally Multiply in columns up to
2x4 digits and 3x3 digits Mixed 4 operations word problems
Importance of Multiplication Method Comprehending multiplication is critical, laying a solid structure for sophisticated mathematical concepts. 5 Multiplication Worksheets provide structured and
targeted method, cultivating a deeper understanding of this fundamental arithmetic operation.
Advancement of 5 Multiplication Worksheets
Multiplication 5 Times Table Worksheets 101 Activity
Multiplication 5 Times Table Worksheets 101 Activity
These free 5 multiplication facts table worksheets for printing or downloading in PDF format are specially aimed at primary school students You can also make a multiplication worksheet yourself using
the worksheet generator These worksheets are randomly generated and therefore provide endless amounts of exercise material for at home or in
Math explained in easy language plus puzzles games quizzes videos and worksheets For K 12 kids teachers and parents Multiplication Worksheets Worksheets Multiplication Mixed Tables Worksheets
Worksheet Number Range Online Primer 1 to 4 Primer Plus 2 to 6 Up To Ten 2 to 10 Getting Tougher 2 to 12 Intermediate 3
From standard pen-and-paper exercises to digitized interactive styles, 5 Multiplication Worksheets have developed, dealing with varied learning designs and preferences.
Types of 5 Multiplication Worksheets
Basic Multiplication Sheets Easy workouts focusing on multiplication tables, assisting students develop a strong math base.
Word Trouble Worksheets
Real-life circumstances incorporated into problems, improving essential reasoning and application skills.
Timed Multiplication Drills Examinations made to improve rate and precision, helping in quick mental math.
Benefits of Using 5 Multiplication Worksheets
Printable Multiplication Worksheets For Grade 5 Free Printable
Printable Multiplication Worksheets For Grade 5 Free Printable
These multiplication worksheets include some repetition of course as there is only one thing to multiply by Once students practice a few times these facts will probably get stuck in their heads for
life Some of the later versions include a range of focus numbers In those cases each question will randomly have one of the focus numbers in
Benefits of Grade 5 Multiplication Worksheets The benefits of 5th grade multiplication worksheets are that it encourages students to solve diverse problems so that students understand the concepts in
an engaging and interesting way The visuals in these worksheets can help students visualize concepts and get a crystal clear understanding of
Improved Mathematical Abilities
Consistent method hones multiplication proficiency, boosting overall math abilities.
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Word troubles in worksheets establish logical thinking and technique application.
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Worksheets accommodate specific understanding rates, cultivating a comfy and adaptable understanding atmosphere.
How to Create Engaging 5 Multiplication Worksheets
Incorporating Visuals and Shades Lively visuals and shades record focus, making worksheets aesthetically appealing and involving.
Including Real-Life Scenarios
Associating multiplication to day-to-day scenarios adds relevance and practicality to workouts.
Customizing Worksheets to Different Skill Degrees Personalizing worksheets based on varying proficiency levels makes sure comprehensive discovering. Interactive and Online Multiplication Resources
Digital Multiplication Tools and Gamings Technology-based sources supply interactive understanding experiences, making multiplication appealing and enjoyable. Interactive Web Sites and Apps On-line
systems supply diverse and obtainable multiplication method, supplementing traditional worksheets. Personalizing Worksheets for Different Knowing Styles Aesthetic Students Visual help and layouts aid
comprehension for students inclined toward visual knowing. Auditory Learners Verbal multiplication troubles or mnemonics cater to learners who understand concepts with auditory means. Kinesthetic
Students Hands-on tasks and manipulatives sustain kinesthetic students in recognizing multiplication. Tips for Effective Execution in Understanding Consistency in Practice Routine practice
strengthens multiplication abilities, advertising retention and fluency. Balancing Repeating and Selection A mix of repeated exercises and diverse trouble layouts maintains passion and comprehension.
Providing Positive Comments Feedback aids in identifying locations of renovation, motivating continued progress. Difficulties in Multiplication Practice and Solutions Inspiration and Engagement
Obstacles Monotonous drills can bring about disinterest; ingenious strategies can reignite motivation. Overcoming Concern of Math Negative understandings around math can hinder development;
developing a positive knowing setting is vital. Influence of 5 Multiplication Worksheets on Academic Performance Researches and Research Study Searchings For Research study shows a positive
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Final thought
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internet resources, these worksheets not only improve multiplication abilities yet also advertise important thinking and problem-solving capacities.
Printable multiplication worksheets For Grade 5 Times Tables worksheets Printable
5 Times Table
Check more of 5 Multiplication Worksheets below
Multiplication Worksheets Grade 5 Free Printable
Grade 5 Multiplication Worksheets
Free 5 Times Table Worksheets Activity Shelter
0 5 Multiplication Worksheets 100 multiplication Facts Worksheet 0 5 Worksheetsnew 2012 12 02
Multiplication Practice Sheets Printable Worksheets Multiplication Worksheets Pdf Grade 234
Multiplication Worksheets K5 Learning
Grade 5 multiplication worksheets Multiply by 10 100 or 1 000 with missing factors Multiplying in parts distributive property Multiply 1 digit by 3 digit numbers mentally Multiply in columns up to
2x4 digits and 3x3 digits Mixed 4 operations word problems
Multiplying by 5 worksheets K5 Learning
The first worksheet is a table of all multiplication facts 1 12 with five as a factor 5 times table Worksheet 1 49 questions Worksheet 2 Worksheet 3 100 questions Worksheet 4 Worksheet 5 3 More
Grade 5 multiplication worksheets Multiply by 10 100 or 1 000 with missing factors Multiplying in parts distributive property Multiply 1 digit by 3 digit numbers mentally Multiply in columns up to
2x4 digits and 3x3 digits Mixed 4 operations word problems
The first worksheet is a table of all multiplication facts 1 12 with five as a factor 5 times table Worksheet 1 49 questions Worksheet 2 Worksheet 3 100 questions Worksheet 4 Worksheet 5 3 More
0 5 Multiplication Worksheets 100 multiplication Facts Worksheet 0 5 Worksheetsnew 2012 12 02
Grade 5 Multiplication Worksheets
Multiplication Practice Sheets Printable Worksheets Multiplication Worksheets Pdf Grade 234
Free Multiplication Worksheets Grade 5
Times Tables Worksheets 2 3 4 5 6 7 8 9 10 11 And 12 Eleven Worksheets FREE
Times Tables Worksheets 2 3 4 5 6 7 8 9 10 11 And 12 Eleven Worksheets FREE
Multiplication Worksheets Year 5 PrintableMultiplication
FAQs (Frequently Asked Questions).
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Yes, worksheets can be customized to various age and skill degrees, making them versatile for different learners.
Exactly how often should students practice using 5 Multiplication Worksheets?
Constant technique is crucial. Regular sessions, preferably a couple of times a week, can yield substantial renovation.
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Worksheets are a valuable device however ought to be supplemented with diverse learning techniques for detailed ability growth.
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How can parents sustain their children's multiplication technique in your home?
Motivating constant method, offering aid, and creating a favorable knowing environment are beneficial steps. | {"url":"https://crown-darts.com/en/5-multiplication-worksheets.html","timestamp":"2024-11-06T12:26:52Z","content_type":"text/html","content_length":"27752","record_id":"<urn:uuid:8f18d292-e5bb-4bcc-90dc-133ba87ffea2>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00417.warc.gz"} |
5.2 Interior Angles of a Polygon | Education Auditorium
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Chapter 5 Angles and Trigonometry
5.2 Interior Angles of a Polygon
The sum of the interior angles of a polygon = (n - 2) x 180°, where n is the number of sides. In this section, we learn about angle sum of a polygon, angles in a regular polygon.
If you're a student in a school/college that's based in England, you might be entitled for full access to our eLearning platform where you can access to all our tutorial videos and practice
Please check with the head of Maths or deputy headteacher at your school/college to request access if they've already registered to our free pilot subscription. They can always reach us at the below
Note: Due to our safeguard and chilled protection policy; we wouldn't be able to respond to students enquiries directly.
The sum of the interior angles of a polygon  x , where  is the number of sides. In this section, we learn about angle sum of a polygon, angles in a regular polygon.
Using the interior angles of polygons to solve problems.
Interior Angles of a Polygon - Angle sum of a polygon, angles in a regular polygon.
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Does contracting a Fractional CTO make fiscal sense | Fractional Advantage
Does contracting a Fractional CTO make fiscal sense
Comparing the costs of FTE and Fractional CTO
With the recent hype around fractional roles, lots of Founders and CEOs have likely wondered if hiring a Fractional role makes sense financially. Specifically, I'll talk about a Fractional CTO. Does
it make financial sense to contract with a Fractional CTO?
TLDR: If you are time-deprived as most Founders and CEO, the short story is it does. And it's no surprise that I would say this, since that is what I do. If you have time, keep reading and I'll
explain why.
Before we get into the financial aspect, let's address a few other things that are equally, if not more, important. As the Chief Technology Officer, is an important role for a software companies.
A lot of startups and small companies end up naming engineer 1 (or close to 1) as CTO. While that is not necessarily bad, you end up with someone that doesn't have the breadth of knowledge most
Fractional CTOs have.
This is one reason many Fractional CTOs offer services to mentor and coach existing leadership to make them more successful in their role. Fractional CTOs can offer a depth and breath of experience
that current leaders likely lack and you can leverage them as a trusted advisor to support your current leaders. A fractional CTO can provide valuable guidance and coaching to the existing technology
team, and help them improve their skills and performance. They can also help the business attract and retain top talent while fostering a culture of innovation and excellence.
Filling a gap
While a CTO is an important role for any software organization, many startups and small companies have avoided hiring one. One of the major drivers may be cost. CTOs are not cheap and committing to
an FTE is often more than a startup or small company can afford. It's equally possible the workload in the early stages does not support the need for a full time CTO.
A Fractional CTO can offer the same or better quality of service as a full-time CTO at a fraction of the cost. They can also help optimize the technology budget and help reduce unnecessary expenses.
A Fractional CTO can be utilized for an individual project or as on as-needed service. Perhaps to implement some maturation activity or to evaluate the technology choices. A Fractional CTO can adapt
to the company's needs for strategic leadership activities. As much, or as little, as the company requires.
There are a few types of risk hiring a Fractional CTO can help. The easy one is you are only committing agreed upon hours a Fractional CTO works for you. Most Fractional CTO contract give you the
ability, with ample notice, to stop the contract.
A Fractional CTO brings a diverse and extensive set of skills and expertise to the table. These skills can help the business leverage the latest technologies and best practices. They can also help
the business access a wider network of resources and partners. Thus, allowing a company to avoid some of the early pitfalls that an engineering department may stumble on.
Hiring a Fraction CTO usually is much faster than recruiting a capable FTE.
A fractional CTO can help the business scale quickly and efficiently, and support the growth and expansion of the business. They can also help the business avoid common pitfalls and risks that come
with scaling.
A fractional CTO can adapt to the changing needs and goals of the business, and provide flexible and scalable solutions. They can also help the business cope with unexpected challenges and
opportunities and realize faster time to implementation.
The elementary thing to say is x hours from a Fractional CTO is less than FTE CTO but lets try to quantify that a bit more.
I ran some numbers to get a better idea of the true cost differential. And of course there is not straightforward and easy answer to that. I've calculated these numbers by leaning to the conservative
As you can see from the image above, I conservatively used two different approaches to calculations to make a basis of what a FTE CTO costs per hour. I came up with $207 to $240 an hour. This is
without calculating equity, which at the CTO level is very often part of the conversation. Also there are some places both by geography and vertical where that number can be many multiples of this.
For our sample, I'll use a number in-between. I'll use $225. There are 2080 hours in a year. Because the estimates took vacations and stuff into account, for an FTE, we can multiply 225 * 2080 to get
Now let's take a hit at what a Fraction CTO makes. The numbers I see are anywhere from $200 to $500. Again, this can vary by geography and vertical. I know many Fractional CTO that go from $200 to
$300 an hour. Remember, with a Fractional CTO, they have to pay ALL the taxes and benefits themselves. But for the sake of comparison, I'll use the $300 number (I'm not billing that much).
So looking at the two just by hourly, it seems the Fractional CTO is not a cost savings after all $300 is more than $225. However, that doesn't consider something incredibly important. You only use
the Fractional CTO for a fraction of the time. Most fractions roles are lower than twenty hours a week. Let's see what a couple of different uses calculate out to.
Recall we are using $468,000 as the total cost of an FTE and $300 for Fractional FTE. I'll do the calculations based on all twelve months to keep it simple. And even though it's a forgone conclusion,
you would not use a Fractional CTO for 40 hours a week, I'll calculate up to.
• 5 hours a week for a year: $78,000
• 10 hours a week for a year: $156,000
• 15 hours a week for a year: $234,000
• 20 hours a week for a year: $312,000
• 25 hours a week for a year: $390,000
• 30 hours a week for a year: $468,000
• 35 hours a week for a year: $546.000
• 40 hours a week for a year: $624,000
As you can see, at 30 hours a week for twelve months, at $300 an hour, is the breakeven point.
In reality, you most Fractional CTO would take some time off. Even without counting that, or equity, the thirty hours a week for a year would likely get you more value because of all the things I
talked about above.
For long-term contracts, most of the Fractional CTO I've talked with get 20 hours a week.
For short-term focused things, most would get up to 30 a week.
Now, let's run those numbers with a little lower rate of $225 an hour.
• 5 hours a week for a year: $58,500
• 10 hours a week for a year: $117,000
• 15 hours a week for a year: $175,500
• 20 hours a week for a year: $243,000
• 25 hours a week for a year: $292,500
• 30 hours a week for a year: $351,000
• 35 hours a week for a year: $409.500
• 40 hours a week for a year: $468,000
This should look familiar, as it is what the FTE numbers were. So if you hire a Fractional CTO for $225 and hour, not only is the money a wash, you get more for your money. | {"url":"https://fractionaladvantage.com/does-contracting-a-fractional-cto-make-fiscal-sense","timestamp":"2024-11-07T16:53:54Z","content_type":"text/html","content_length":"192258","record_id":"<urn:uuid:497cf0cc-f5b4-4725-a0f2-4db7178af338>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00386.warc.gz"} |
python – Absolute value for column in Python
Sure! Here are 8 examples of how to calculate the absolute value for a column in Python, along with step-by-step explanations of each code snippet:
Example 1: Using the abs() function with a list comprehension
numbers = [1, -2, 3, -4, 5]
absolute_values = [abs(num) for num in numbers]
– Create a list of numbers.
– Use a list comprehension to iterate over each number in the list and apply the abs() function to get the absolute value.
– Store the absolute values in a new list called `absolute_values` and print it.
Example 2: Using the abs() function with map()
numbers = [1, -2, 3, -4, 5]
absolute_values = list(map(abs, numbers))
– Create a list of numbers.
– Use the map() function to apply the abs() function to each element in the `numbers` list.
– Convert the map object to a list using the list() function and store the absolute values in a new list called `absolute_values` and print it.
Example 3: Using numpy’s absolute function
import numpy as np
numbers = np.array([1, -2, 3, -4, 5])
absolute_values = np.absolute(numbers)
– Import the `numpy` library as `np`.
– Create a numpy array of numbers.
– Use the `np.absolute()` function to get the absolute values of each element in the numpy array.
– Store the absolute values in a new numpy array called `absolute_values` and print it.
Example 4: Using pandas’ abs() method on a DataFrame column
import pandas as pd
data = {'A': [1, -2, 3, -4, 5]}
df = pd.DataFrame(data)
df['absolute_values'] = df['A'].abs()
– Import the `pandas` library as `pd`.
– Create a dictionary (`data`) with a key ‘A’ and a list of numbers as its value.
– Create a DataFrame (`df`) from the dictionary.
– Use the `abs()` method on the ‘A’ column to get the absolute values and assign it to a new column called ‘absolute_values’.
– Print the updated DataFrame.
Example 5: Using pandas’ apply() function with lambda function
import pandas as pd
data = {'A': [1, -2, 3, -4, 5]}
df = pd.DataFrame(data)
df['absolute_values'] = df['A'].apply(lambda x: abs(x))
– Import the `pandas` library as `pd`.
– Create a dictionary (`data`) with a key ‘A’ and a list of numbers as its value.
– Create a DataFrame (`df`) from the dictionary.
– Use the `apply()` function on the ‘A’ column to apply a lambda function that calculates the absolute value of each element.
– Assign the absolute values to a new column called ‘absolute_values’.
– Print the updated DataFrame.
Example 6: Using a for loop to calculate absolute values and store in a new list
numbers = [1, -2, 3, -4, 5]
absolute_values = []
for num in numbers:
– Create a list of numbers.
– Initialize an empty list called `absolute_values`.
– Use a for loop to iterate over each number in the `numbers` list.
– Append the absolute value of each number to the `absolute_values` list using the `append()` method.
– Print the `absolute_values` list.
Example 7: Using a list comprehension with if-else condition
numbers = [1, -2, 3, -4, 5]
absolute_values = [num if num > 0 else -num for num in numbers]
– Create a list of numbers.
– Use a list comprehension to iterate over each number in the list.
– Use an if-else condition inside the list comprehension to determine if the number is positive or negative.
– If the number is positive, keep it as is. If it is negative, convert it to a positive by multiplying it with -1.
– Store the resulting values in a new list called `absolute_values` and print it.
Example 8: Using a numpy array and np.where() function
import numpy as np
numbers = np.array([1, -2, 3, -4, 5])
absolute_values = np.where(numbers < 0, -numbers, numbers)
– Import the `numpy` library as `np`.
– Create a numpy array of numbers.
– Use the `np.where()` function to apply a condition (numbers < 0) and return the absolute values if the condition is true (-numbers), or return the numbers themselves if the condition is false.
– Store the resulting values in a new numpy array called `absolute_values` and print it.
I hope these examples help you understand how to calculate the absolute value for a column in Python. Let me know if you have any further questions! | {"url":"https://pythonkb.com/python-absolute-value-for-column-in-python/","timestamp":"2024-11-06T13:32:08Z","content_type":"text/html","content_length":"74858","record_id":"<urn:uuid:f4324867-bd95-400f-9eb5-a1194d5a155b>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00344.warc.gz"} |
Brain Teaser
If we know the volumetric flow rates of the water going into the two branches, then what would be the velocity of the water coming out?
Four words with the letters SNT, in that order, have all their other letters removed. You must use the letters below to fill in the blanks. _ _ S _ N T _ _ _ S _ _ N T S _ N _ _ T S _ _ _ N T Letters
to use: A, A, E, E, E, I, I, L, L, Q, S, S, U, U
A 25 ft ladder (assume it is not an extension ladder) is placed with its foot 7 ft away from a building. If the top of the ladder slips down 4 ft, how many feet will the bottom slide out?
John the distribution manager for a leading mobile phone company has a dilemma. The warehouse has 10 unlabelled rows of pallets, each row contains thousands of phones destined for different…
Doug had forgotten the 5 digit code to his briefcase. However, he did remember five clues: The fifth number plus the third number equals fourteen; The fourth number is one…
Challenge your brain with this interesting brain teaser. Given a number of statement find the solution to the question asked.
A team of 6 girls and 4 boys put together a 2200-piece jigsaw puzzle in 4 hours. The same jigsaw puzzle was put together in 8 hours by a team of two girls and five boys.
Who are better at putting jigsaw puzzles together, boys or girls, and by how much?
Insert +, -, x or / in suitable places on the left hand side of = so as to make the above equation true:
Can you figure out a way for four people to cross a rickety bridge in exactly (or under) 17 minutes?
Discuss. No electoral system will ever produce the right leaders. | {"url":"https://www.engineeringdaily.net/tag/brain-teasers/","timestamp":"2024-11-06T23:19:05Z","content_type":"text/html","content_length":"94164","record_id":"<urn:uuid:ab68aa6c-6e27-4831-9d22-081c92defbc6>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00667.warc.gz"} |
Amortization table
Calculate your repayment schedule with the tool below. This allows you to calculate and simulate your monthly payment, whether it’s a fixed or variable home loan. It shows you how much you will repay
in capital and interest, and what repayments you can expect. The repayment schedule shows both annual and monthly payments.
Our simulator can also calculate the repayment of a variable home loan. It shows you the worst-case scenario, but you can adjust the interest rates for other calculations. With a variable loan, you
can see the minimum and maximum repayments.
Of course, you can also simply calculate the repayment schedule of a classic fixed-rate mortgage loan. Discover all the possibilities of our handy repayment schedule calculator now!
Always get the best mortgage loan?
Would you like to be kept informed about opportunities to save money on your loan? Leave your details here and we will contact you when refinancing becomes interesting for your situation.
Noted! We’ll keep you posted when a review might be of interest to you.
Your simulation
Based on your data, you pay:
x.xx% monthly
+ x,xx% total interest
+ x,xx% total to be paid to the bank
Choose your scenario
Display style
Repayment table | {"url":"https://www.hypotheekwinkel.be/en/simulation/amortization-table/","timestamp":"2024-11-10T05:22:57Z","content_type":"text/html","content_length":"94384","record_id":"<urn:uuid:ca1ab1a3-e3b5-4b63-bd7f-868939f62e27>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00157.warc.gz"} |
Lesson 11
What Is the Same?
11.1: Find the Right Hands (5 minutes)
In this activity, students get their first formal introduction to the idea of mirror orientation, sometimes called “handedness” because left and right hands are reflections of each other. The easiest
way to decide which are the right hands is to hold one’s hands up and rotate them until they match a particular figure (or don’t). This prepares them for a discussion about whether figures with
different mirror orientation are the same or not.
Arrange students in groups of 2, and provide access to geometry toolkits. Give 2 minutes of quiet work time, followed by time for sharing with a partner and a whole-class discussion.
Show students this image or hold up both hands and point out that our hands are mirror images of each other. These are hands shown from the back. If needed, clarify for students that all of the hands
in the task are shown from the back.
Student Facing
A person’s hands are mirror images of each other. In the diagram, a left hand is labeled. Shade all of the right hands.
Activity Synthesis
Ask students to think about the ways in which the left and right hands are the same, and the ways in which they are different.
Some ways that they are the same include:
• The side lengths and angles on the left and right hands match up with one another.
• If a left hand is flipped, it can match it up perfectly with a right hand (and vice versa).
Some ways that they are different include:
• They can not be lined up with one another without flipping one of the hands over.
• It is not possible to make a physical left and right hand line up with one another, except as “mirror images.”
11.2: Are They the Same? (15 minutes)
In previous work, students learned to identify translations, rotations, and reflections. They started to study what happens to different shapes when these transformations are applied. They used
sequences of translations, rotations, and reflections to build new shapes and to study complex configurations in order to compare, for example, vertical angles made by a pair of intersecting lines.
Starting in this lesson, rigid transformations are used to formalize what it means for two shapes to be the same, a notion which students have studied and applied since the early grades of elementary
In this activity, students express what it means for two shapes to be the same by considering carefully chosen examples. Students work to decide whether or not the different pairs of shapes are the
same. Then the class discusses their findings and comes to a consensus for what it means for two shapes to be the same: the word “same” is replaced by “congruent” moving forward.
There may be discussion where a reflection is required to match one shape with the other. Students may disagree about whether or not these should be considered the same and discussion should be
encouraged. This activity encourages MP3 as students need to explain why they believe that a pair of figures is the same or is not the same.
Monitor for students who use these methods to decide whether or not the shapes are the same and invite them to share during the discussion:
• Observation (this is often sufficient to decide that they are not the same): Encourage students to articulate what feature(s) of the shapes help them to decide that they are not the same.
• Measuring side lengths using a ruler or angles using a protractor: Then use differences among these measurements to argue that two shapes are not the same.
• Cutting out one shape and trying to move it on top of the other: A variant of this would be to separate the two images and then try to put one on top of the other or use tracing paper to trace
one of the shapes. This is a version of applying transformations studied extensively prior to this lesson.
Give 5 minutes of quiet work time followed by a whole-class discussion. Provide access to geometry toolkits.
Action and Expression: Develop Expression and Communication. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they
explain their ideas. For example, “This pair of shapes is/is not the same because...” or “If I translate/rotate/reflect, then….”
Supports accessibility for: Language; Organization
Conversing, Representing: MLR2 Collect and Display. As students work on comparing shapes, circulate and listen to students talk. Record common or important phrases (e.g., side length, rotated,
reflected, etc.), together with helpful sketches or diagrams on a display. Pay particular attention to how students are using transformational language while determining whether the shapes are the
same. Scribe students’ words and sketches on a visual display to refer back to during whole-class discussions throughout this lesson and the rest of the unit. This will help students use mathematical
language during their group and whole-class discussions.
Design Principle(s): Support sense-making
Student Facing
For each pair of shapes, decide whether or not they are the same.
Anticipated Misconceptions
Students may think all of the shapes are the same because they are the same general shape at first glance. Ask these students to look for any differences they can find among the pairs of shapes.
Activity Synthesis
For each pair of shapes, poll the class. Count how many students decided each pair was the same or not the same. Then for each pair of shapes, select at least one student to defend their reasoning.
(If there is unanimous agreement over any of the pairs of shapes, these can be dealt with quickly, but allow the class to hear at least one argument for each pair of shapes.)
Sequence these explanations in the order suggested in the Activity Narrative: general observations, taking measurements, and applying rigid transformations with the aid of tracing paper.
The most general and precise of these criteria is the third which is the foundation for the mathematical definition of congruence: The other two are consequences. The moves allowed by rigid
transformations do not change the shape, size, side lengths, or angle measures.
There may be disagreement about whether or not to include reflections when deciding if two shapes are the same. Here are some reasons to include reflections:
• A shape and its reflected image can be matched up perfectly (using a reflection).
• Corresponding angles and side lengths of a shape and its reflected image are the same.
And here are some reasons against including reflections:
• A left foot and a right foot (for example) do not work exactly the same way. If we literally had two left feet it would be difficult to function normally!
• Translations and rotations can be enacted, for example, by putting one sheet of tracing paper on top of another and physically translating or rotating it. For a reflection the typical way to do
this is to lift one of the sheets and flip it over.
If this disagreement doesn't come up, ask students to think about why someone might conclude that the pair of figures in C were not the same. Explain to students that people in the world can mean
many things when they say two things are "the same." In mathematics there is often a need to be more precise, and one kind of "the same" is congruent. (Two figures are congruent if one is a
reflection of the other, but one could, if one wanted, define a different term, a different kind of "the same," where flipping was not allowed!)
Explain that Figure A is congruent to Figure B if there is a sequence of translations, rotations, and reflections which make Figure A match up exactly with Figure B.
Combining this with the earlier discussion a few general observations about congruent figures include
• Corresponding sides of congruent figures are congruent.
• Corresponding angles of congruent figures are congruent.
• The area of congruent figures are equal.
What can be “different” about two congruent figures? The location (they don't have to be on top of each other) and the orientation (requiring a reflection to move one to the other) can be different.
11.3: Area, Perimeter, and Congruence (10 minutes)
Sometimes people characterize congruence as “same size, same shape.” The problem with this is that it isn’t clear what we mean by “same shape.” All of the figures in this activity have the same shape
because they are all rectangles, but they are not all congruent. Students examine a set of rectangles and classify them according to their area and perimeter. Then they identify which ones are
congruent. Because congruent shapes have the same side lengths, congruent rectangles have the same perimeter. But rectangles with the same perimeter are not always congruent. Congruent shapes,
including rectangles, also have the same area. But rectangles with the same area are not always congruent. Highlighting important features, like perimeter and area, which can be used to quickly
establish that two shapes are not congruent develops MP7, identifying fundamental properties shared by any pair of congruent shapes.
Tell students that they will investigate further how finding the area and perimeter of a shape can help show that two figures are not congruent. It may have been a while since students have thought
about the terms area and perimeter. If necessary, to remind students what these words mean and how they can be computed, display a rectangle like this one for all to see. Ask students to explain what
perimeter means and how they can find the perimeter and area of this rectangle.
Arrange students in groups of 2. Provide access to geometry toolkits (colored pencils are specifically called for). Give 2 minutes for quiet work time followed by sharing with a partner and a
whole-class discussion.
Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. For example, provide students with a subset of the
rectangles to start with and introduce the remaining remaining rectangles once students have completed their initial set of comparisons.
Supports accessibility for: Conceptual processing; Organization
Student Facing
1. Which of these rectangles have the same area as Rectangle R but different perimeter?
2. Which rectangles have the same perimeter as Rectangle R but different area?
3. Which have the same area and the same perimeter as Rectangle R?
4. Use materials from the geometry tool kit to decide which rectangles are congruent. Shade congruent rectangles with the same color.
Student Facing
Are you ready for more?
In square \(ABCD\), points \(E\), \(F\), \(G\), and \(H\) are midpoints of their respective sides. What fraction of square \(ABCD\) is shaded? Explain your reasoning.
Anticipated Misconceptions
Watch for students who think about the final question in terms of “same shape and size.” Remind them of the definition of congruence introduced in the last activity.
Activity Synthesis
Invite students who used the language of transformations to answer the final question to describe how they determined that a pair of rectangles are congruent.
Perimeter and area are two different ways to measure the size of a shape. Ask the students:
• "Do congruent rectangles have the same perimeter? Explain your reasoning." (Yes. Rigid motions do not change distances, and so congruent rectangles have the same perimeter.)
• "Do congruent rectangles have the same area? Explain your reasoning." (Yes. Rigid motions do not change area or rigid motions do not change distances and so do not change the length times the
width in a rectangle.)
• "Are rectangles with the same perimeter always congruent?" (No. Rectangles D and F have the same perimeter but they are not congruent.)
• "Are rectangles with the same area always congruent?" (No. Rectangles B and C have the same area but are not congruent.)
One important take away from this lesson is that measuring perimeter and area is a good method to show that two shapes are not congruent if these measurements differ. When the measurements are the
same, more work is needed to decide whether or not two shapes are congruent.
A risk of using rectangles is that students may reach the erroneous conclusion that if two figures have both the same area and the same perimeter, then they are congruent. If this comes up, challenge
students to think of two shapes that have the same area and the same perimeter, but are not congruent. Here is an example:
Writing, Speaking: MLR1 Stronger and Clearer Each Time. Use this routine with to give students a structured opportunity to revise their written strategies for deciding which rectangles are congruent.
Give students time to meet with 2–3 partners to share and get feedback on their responses. Display prompts for feedback that will help individuals strengthen their ideas and clarify their language.
For example, “How was a sequence of transformations used to…?”, “What properties do the shapes share?”, and “What was different and what was the same about each pair?” Students can borrow ideas and
language from each partner to strengthen their final product.
Design Principle(s): Optimize output (for explanation)
Lesson Synthesis
Ask students to state their best definition of congruent. (Two shapes are congruent when there is a sequence of translations, rotations, and reflections that take one shape to the other.)
Some important concepts to discuss:
• "How can you check if two shapes are congruent?" (For rectangles, the side lengths are enough to tell. For more complex shapes, experimenting with transformations is needed.)
• "Are a shape and its mirror image congruent?" (Yes, because a reflection takes a shape to its mirror image.)
• "What are some ways to know that two shapes are not congruent?" (Two shapes are not congruent if they have different areas, side lengths, or angles.)
• "What are some properties that are shared by congruent shapes?" (They have the same number of sides, same length sides, same angles, same area.)
11.4: Cool-down - Mirror Images (5 minutes)
Student Facing
Congruent is a new term for an idea we have already been using. We say that two figures are congruent if one can be lined up exactly with the other by a sequence of rigid transformations. For
example, triangle \(EFD\) is congruent to triangle \(ABC\) because they can be matched up by reflecting triangle \(ABC\) across \(AC\) followed by the translation shown by the arrow. Notice that all
corresponding angles and side lengths are equal.
Here are some other facts about congruent figures:
• We don’t need to check all the measurements to prove two figures are congruent; we just have to find a sequence of rigid transformations that match up the figures.
• A figure that looks like a mirror image of another figure can be congruent to it. This means there must be a reflection in the sequence of transformations that matches up the figures.
• Since two congruent polygons have the same area and the same perimeter, one way to show that two polygons are not congruent is to show that they have a different perimeter or area. | {"url":"https://im.kendallhunt.com/MS/teachers/3/1/11/index.html","timestamp":"2024-11-06T00:52:04Z","content_type":"text/html","content_length":"116085","record_id":"<urn:uuid:2ea70d70-e594-422e-a5d0-07d0b7d6c4c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00601.warc.gz"} |
The Tortoise and Hare Algorithm – A Winning Formula for Efficient Problem Solving
Efficient problem solving techniques are indispensable for modern-day programmers and developers. One such algorithm that has gained significant popularity is the Tortoise and Hare Algorithm. In this
blog post, we will explore the concept of the Tortoise and Hare Algorithm, understand its execution, and delve into its various applications in problem-solving.
Understanding the Tortoise and Hare Algorithm
The Tortoise and Hare Algorithm, also known as Floyd’s Cycle-Finding Algorithm, is a standard technique used to detect cycles or loops in a sequence of elements. It provides an efficient way of
solving problems that involve analyzing repeated patterns or iteratively traversing data structures.
The algorithm utilizes two pointers – the “tortoise” and the “hare” – to determine if there is a cycle in the given sequence. The tortoise moves forward one step at a time, while the hare moves two
steps at a time. By the time the hare catches up to the tortoise, if the sequence has a cycle, they will meet; otherwise, the hare will reach the end of the sequence.
Application of the Tortoise and Hare Algorithm in Problem Solving
The Tortoise and Hare Algorithm offers several advantages when it comes to efficient problem solving. It can be applied in various real-life scenarios, such as:
• Detecting cycles in linked lists or arrays
• Finding the duplicate element in an array
• Identifying the starting point of a loop in a linked list
Compared to other problem-solving approaches, the Tortoise and Hare Algorithm provides a more optimal solution in terms of time and space complexity. It eliminates the need for additional storage or
multiple iterations, enabling developers to solve problems more effectively.
Implementing the Tortoise and Hare Algorithm
When implementing the Tortoise and Hare Algorithm, it’s important to consider a few key factors to ensure successful execution:
• Choosing the right data structure: Depending on the problem at hand, select the appropriate data structure, such as a linked list or an array, that can be traversed efficiently.
• Initializations and conditions: Set initial values for the tortoise and hare pointers. Determine the conditions for the algorithm to terminate, such as reaching the end of the sequence or finding
a repeating element.
• Variable manipulation: Manipulate the pointers’ positions based on the problem requirements, considering the appropriate steps to move forward or backward.
To help you better understand the implementation of the algorithm, here are a few examples of programming code:
def find_duplicate(nums): tortoise = nums[0] hare = nums[0]
while True: tortoise = nums[tortoise] hare = nums[nums[hare]]
if tortoise == hare: break
ptr1 = nums[0] ptr2 = tortoise
while ptr1 != ptr2: ptr1 = nums[ptr1] ptr2 = nums[ptr2]
return ptr1
The Tortoise and Hare Algorithm, with its efficiency and accuracy, has become an integral part of problem-solving processes. By leveraging this algorithm, programmers can detect cycles, find
duplicate elements, and identify loop starting points in a variety of real-life scenarios. With proper implementation and understanding, the Tortoise and Hare Algorithm can significantly enhance the
efficiency of problem-solving endeavors.
So, the next time you encounter a problem with a repeating pattern or need to traverse a data structure effectively, consider incorporating the Tortoise and Hare Algorithm into your solutions. Happy | {"url":"https://skillapp.co/blog/the-tortoise-and-hare-algorithm-a-winning-formula-for-efficient-problem-solving/","timestamp":"2024-11-11T06:57:30Z","content_type":"text/html","content_length":"108862","record_id":"<urn:uuid:7fe689b5-b3c3-4eb7-aa56-9c6fddfa8c0f>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00154.warc.gz"} |
What statistical analysis should I use? Statistical analyses using SPSS
This page shows how to perform a number of statistical tests using SPSS. Each section gives a brief description of the aim of the statistical test, when it is used, an example showing the SPSS
commands and SPSS (often abbreviated) output with a brief interpretation of the output. You can see the page Choosing the Correct Statistical Test for a table that shows an overview of when each
test is appropriate to use. In deciding which test is appropriate to use, it is important to consider the type of variables that you have (i.e., whether your variables are categorical, ordinal
or interval and whether they are normally distributed), see What is the difference between categorical, ordinal and interval variables? for more information on this.
About the hsb data file
Most of the examples in this page will use a data file called hsb2, high school and beyond. This data file contains 200 observations from a sample of high school students with demographic
information about the students, such as their gender (female), socio-economic status (ses) and ethnic background (race). It also contains a number of scores on standardized tests, including tests
of reading (read), writing (write), mathematics (math) and social studies (socst). You can get the hsb data file by clicking on hsb2.
One sample t-test
A one sample t-test allows us to test whether a sample mean (of a normally distributed interval variable) significantly differs from a hypothesized value. For example, using the hsb2 data file,
say we wish to test whether the average writing score (write) differs significantly from 50. We can do this as shown below.
/testval = 50
/variable = write.
The mean of the variable write for this particular sample of students is 52.775, which is statistically significantly different from the test value of 50. We would conclude that this group of
students has a significantly higher mean on the writing test than 50.
One sample median test
A one sample median test allows us to test whether a sample median differs significantly from a hypothesized value. We will use the same variable, write, as we did in the one sample t-test
example above, but we do not need to assume that it is interval and normally distributed (we only need to assume that write is an ordinal variable).
/onesample test (write) wilcoxon(testvalue = 50).
Binomial test
A one sample binomial test allows us to test whether the proportion of successes on a two-level categorical dependent variable significantly differs from a hypothesized value. For example, using
the hsb2 data file, say we wish to test whether the proportion of females (female) differs significantly from 50%, i.e., from .5. We can do this as shown below.
npar tests
/binomial (.5) = female.
The results indicate that there is no statistically significant difference (p = .229). In other words, the proportion of females in this sample does not significantly differ from the
hypothesized value of 50%.
Chi-square goodness of fit
A chi-square goodness of fit test allows us to test whether the observed proportions for a categorical variable differ from hypothesized proportions. For example, let’s suppose that we believe
that the general population consists of 10% Hispanic, 10% Asian, 10% African American and 70% White folks. We want to test whether the observed proportions from our sample differ significantly
from these hypothesized proportions.
npar test
/chisquare = race
/expected = 10 10 10 70.
These results show that racial composition in our sample does not differ significantly from the hypothesized values that we supplied (chi-square with three degrees of freedom = 5.029, p = .170).
Two independent samples t-test
An independent samples t-test is used when you want to compare the means of a normally distributed interval dependent variable for two independent groups. For example, using the hsb2 data file,
say we wish to test whether the mean for write is the same for males and females.
t-test groups = female(0 1)
/variables = write.
Because the standard deviations for the two groups are similar (10.3 and 8.1), we will use the “equal variances assumed” test. The results indicate that there is a statistically significant
difference between the mean writing score for males and females (t = -3.734, p = .000). In other words, females have a statistically significantly higher mean score on writing (54.99) than males
See also
Wilcoxon-Mann-Whitney test
The Wilcoxon-Mann-Whitney test is a non-parametric analog to the independent samples t-test and can be used when you do not assume that the dependent variable is a normally distributed interval
variable (you only assume that the variable is at least ordinal). You will notice that the SPSS syntax for the Wilcoxon-Mann-Whitney test is almost identical to that of the independent samples
t-test. We will use the same data file (the hsb2 data file) and the same variables in this example as we did in the independent t-test example above and will not assume that write, our dependent
variable, is normally distributed.
npar test
/m-w = write by female(0 1).
The results suggest that there is a statistically significant difference between the underlying distributions of the write scores of males and the write scores of females (z = -3.329, p = 0.001).
See also
Chi-square test
A chi-square test is used when you want to see if there is a relationship between two categorical variables. In SPSS, the chisq option is used on the statistics subcommand of the crosstabs
command to obtain the test statistic and its associated p-value. Using the hsb2 data file, let’s see if there is a relationship between the type of school attended (schtyp) and students’ gender
(female). Remember that the chi-square test assumes that the expected value for each cell is five or higher. This assumption is easily met in the examples below. However, if this assumption is
not met in your data, please see the section on Fisher’s exact test below.
/tables = schtyp by female
/statistic = chisq.
These results indicate that there is no statistically significant relationship between the type of school attended and gender (chi-square with one degree of freedom = 0.047, p = 0.828).
Let’s look at another example, this time looking at the linear relationship between gender (female) and socio-economic status (ses). The point of this example is that one (or both) variables may
have more than two levels, and that the variables do not have to have the same number of levels. In this example, female has two levels (male and female) and ses has three levels (low, medium
and high).
/tables = female by ses
/statistic = chisq.
Again we find that there is no statistically significant relationship between the variables (chi-square with two degrees of freedom = 4.577, p = 0.101).
See also
Fisher’s exact test
The Fisher’s exact test is used when you want to conduct a chi-square test but one or more of your cells has an expected frequency of five or less. Remember that the chi-square test assumes that
each cell has an expected frequency of five or more, but the Fisher’s exact test has no such assumption and can be used regardless of how small the expected frequency is. In SPSS unless you have
the SPSS Exact Test Module, you can only perform a Fisher’s exact test on a 2×2 table, and these results are presented by default. Please see the results from the chi squared example above.
One-way ANOVA
A one-way analysis of variance (ANOVA) is used when you have a categorical independent variable (with two or more categories) and a normally distributed interval dependent variable and you wish
to test for differences in the means of the dependent variable broken down by the levels of the independent variable. For example, using the hsb2 data file, say we wish to test whether the mean
of write differs between the three program types (prog). The command for this test would be:
oneway write by prog.
The mean of the dependent variable differs significantly among the levels of program type. However, we do not know if the difference is between only two of the levels or all three of the
levels. (The F test for the Model is the same as the F test for prog because prog was the only variable entered into the model. If other variables had also been entered, the F test for the
Model would have been different from prog.) To see the mean of write for each level of program type,
means tables = write by prog.
From this we can see that the students in the academic program have the highest mean writing score, while students in the vocational program have the lowest.
See also
Kruskal Wallis test
The Kruskal Wallis test is used when you have one independent variable with two or more levels and an ordinal dependent variable. In other words, it is the non-parametric version of ANOVA and a
generalized form of the Mann-Whitney test method since it permits two or more groups. We will use the same data file as the one way ANOVA example above (the hsb2 data file) and the same
variables as in the example above, but we will not assume that write is a normally distributed interval variable.
npar tests
/k-w = write by prog (1,3).
If some of the scores receive tied ranks, then a correction factor is used, yielding a slightly different value of chi-squared. With or without ties, the results indicate that there is a
statistically significant difference among the three type of programs.
Paired t-test
A paired (samples) t-test is used when you have two related observations (i.e., two observations per subject) and you want to see if the means on these two normally distributed interval variables
differ from one another. For example, using the hsb2 data file we will test whether the mean of read is equal to the mean of write.
t-test pairs = read with write (paired).
These results indicate that the mean of read is not statistically significantly different from the mean of write (t = -0.867, p = 0.387).
Wilcoxon signed rank sum test
The Wilcoxon signed rank sum test is the non-parametric version of a paired samples t-test. You use the Wilcoxon signed rank sum test when you do not wish to assume that the difference between
the two variables is interval and normally distributed (but you do assume the difference is ordinal). We will use the same example as above, but we will not assume that the difference between
read and write is interval and normally distributed.
npar test
/wilcoxon = write with read (paired).
The results suggest that there is not a statistically significant difference between read and write.
If you believe the differences between read and write were not ordinal but could merely be classified as positive and negative, then you may want to consider a sign test in lieu of sign rank
test. Again, we will use the same variables in this example and assume that this difference is not ordinal.
npar test
/sign = read with write (paired).
We conclude that no statistically significant difference was found (p=.556).
McNemar test
You would perform McNemar’s test if you were interested in the marginal frequencies of two binary outcomes. These binary outcomes may be the same outcome variable on matched pairs (like a
case-control study) or two outcome variables from a single group. Continuing with the hsb2 dataset used in several above examples, let us create two binary outcomes in our dataset: himath and
hiread. These outcomes can be considered in a two-way contingency table. The null hypothesis is that the proportion of students in the himath group is the same as the proportion of students in
hiread group (i.e., that the contingency table is symmetric).
compute himath = (math>60).
compute hiread = (read>60).
/tables=himath BY hiread
McNemar’s chi-square statistic suggests that there is not a statistically significant difference in the proportion of students in the himath group and the proportion of students in the hiread
One-way repeated measures ANOVA
You would perform a one-way repeated measures analysis of variance if you had one categorical independent variable and a normally distributed interval dependent variable that was repeated at
least twice for each subject. This is the equivalent of the paired samples t-test, but allows for two or more levels of the categorical variable. This tests whether the mean of the dependent
variable differs by the categorical variable. We have an example data set called rb4wide, which is used in Kirk’s book Experimental Design. In this data set, y is the dependent variable, a is
the repeated measure and s is the variable that indicates the subject number.
glm y1 y2 y3 y4
/wsfactor a(4).
a at the .05 level.
See also
Repeated measures logistic regression
If you have a binary outcome measured repeatedly for each subject and you wish to run a logistic regression that accounts for the effect of multiple measures from single subjects, you can perform
a repeated measures logistic regression. In SPSS, this can be done using the GENLIN command and indicating binomial as the probability distribution and logit as the link function to be used in
the model. The exercise data file contains 3 pulse measurements from each of 30 people assigned to 2 different diet regiments and 3 different exercise regiments. If we define a “high” pulse as
being over 100, we can then predict the probability of a high pulse using diet regiment.
GET FILE='C:mydatahttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/exercise.sav'.
GENLIN highpulse (REFERENCE=LAST)
BY diet (order = DESCENDING)
/MODEL diet
/REPEATED SUBJECT=id CORRTYPE = EXCHANGEABLE.
These results indicate that diet is not statistically significant (Wald Chi-Square = 1.562, p = 0.211).
Factorial ANOVA
A factorial ANOVA has two or more categorical independent variables (either with or without the interactions) and a single normally distributed interval dependent variable. For example, using
the hsb2 data file we will look at writing scores (write) as the dependent variable and gender (female) and socio-economic status (ses) as independent variables, and we will include an
interaction of female by ses. Note that in SPSS, you do not need to have the interaction term(s) in your data set. Rather, you can have SPSS create it/them temporarily by placing an asterisk
between the variables that will make up the interaction term(s).
glm write by female ses.
These results indicate that the overall model is statistically significant (F = 5.666, p = 0.00). The variables female and ses are also statistically significant (F = 16.595, p = 0.000 and F =
6.611, p = 0.002, respectively). However, that interaction between female and ses is not statistically significant (F = 0.133, p = 0.875).
See also
Friedman test
You perform a Friedman test when you have one within-subjects independent variable with two or more levels and a dependent variable that is not interval and normally distributed (but at least
ordinal). We will use this test to determine if there is a difference in the reading, writing and math scores. The null hypothesis in this test is that the distribution of the ranks of each
type of score (i.e., reading, writing and math) are the same. To conduct a Friedman test, the data need to be in a long format. SPSS handles this for you, but in other statistical packages you
will have to reshape the data before you can conduct this test.
npar tests
/friedman = read write math.
Friedman’s chi-square has a value of 0.645 and a p-value of 0.724 and is not statistically significant. Hence, there is no evidence that the distributions of the three types of scores are
Ordered logistic regression
Ordered logistic regression is used when the dependent variable is ordered, but not continuous. For example, using the hsb2 data file we will create an ordered variable called write3. This
variable will have the values 1, 2 and 3, indicating a low, medium or high writing score. We do not generally recommend categorizing a continuous variable in this way; we are simply creating a
variable to use for this example. We will use gender (female), reading score (read) and social studies score (socst) as predictor variables in this model. We will use a logit link and on the
print subcommand we have requested the parameter estimates, the (model) summary statistics and the test of the parallel lines assumption.
if write ge 30 and write le 48 write3 = 1.
if write ge 49 and write le 57 write3 = 2.
if write ge 58 and write le 70 write3 = 3.
plum write3 with female read socst
/link = logit
/print = parameter summary tparallel.
The results indicate that the overall model is statistically significant (p < .000), as are each of the predictor variables (p < .000). There are two thresholds for this model because there are
three levels of the outcome variable. We also see that the test of the proportional odds assumption is non-significant (p = .563). One of the assumptions underlying ordinal logistic (and
ordinal probit) regression is that the relationship between each pair of outcome groups is the same. In other words, ordinal logistic regression assumes that the coefficients that describe the
relationship between, say, the lowest versus all higher categories of the response variable are the same as those that describe the relationship between the next lowest category and all higher
categories, etc. This is called the proportional odds assumption or the parallel regression assumption. Because the relationship between all pairs of groups is the same, there is only one set
of coefficients (only one model). If this was not the case, we would need different models (such as a generalized ordered logit model) to describe the relationship between each pair of outcome
See also
Factorial logistic regression
A factorial logistic regression is used when you have two or more categorical independent variables but a dichotomous dependent variable. For example, using the hsb2 data file we will use female
as our dependent variable, because it is the only dichotomous variable in our data set; certainly not because it common practice to use gender as an outcome variable. We will use type of program
(prog) and school type (schtyp) as our predictor variables. Because prog is a categorical variable (it has three levels), we need to create dummy codes for it. SPSS will do this for you by
making dummy codes for all variables listed after the keyword with. SPSS will also create the interaction term; simply list the two variables that will make up the interaction separated by the
keyword by.
logistic regression female with prog schtyp prog by schtyp
/contrast(prog) = indicator(1).
The results indicate that the overall model is not statistically significant (LR chi2 = 3.147, p = 0.677). Furthermore, none of the coefficients are statistically significant either. This shows
that the overall effect of prog is not significant.
See also
A correlation is useful when you want to see the relationship between two (or more) normally distributed interval variables. For example, using the hsb2 data file we can run a correlation
between two continuous variables, read and write.
/variables = read write.
In the second example, we will run a correlation between a dichotomous variable, female, and a continuous variable, write. Although it is assumed that the variables are interval and normally
distributed, we can include dummy variables when performing correlations.
/variables = female write.
In the first example above, we see that the correlation between read and write is 0.597. By squaring the correlation and then multiplying by 100, you can determine what percentage of the
variability is shared. Let’s round 0.597 to be 0.6, which when squared would be .36, multiplied by 100 would be 36%. Hence read shares about 36% of its variability with write. In the output
for the second example, we can see the correlation between write and female is 0.256. Squaring this number yields .065536, meaning that female shares approximately 6.5% of its variability with
See also
Simple linear regression
Simple linear regression allows us to look at the linear relationship between one normally distributed interval predictor and one normally distributed interval outcome variable. For example,
using the hsb2 data file, say we wish to look at the relationship between writing scores (write) and reading scores (read); in other words, predicting write from read.
regression variables = write read
/dependent = write
/method = enter.
We see that the relationship between write and read is positive (.552) and based on the t-value (10.47) and p-value (0.000), we would conclude this relationship is statistically significant.
Hence, we would say there is a statistically significant positive linear relationship between reading and writing.
See also
Non-parametric correlation
A Spearman correlation is used when one or both of the variables are not assumed to be normally distributed and interval (but are assumed to be ordinal). The values of the variables are converted
in ranks and then correlated. In our example, we will look for a relationship between read and write. We will not assume that both of these variables are normal and interval.
nonpar corr
/variables = read write
/print = spearman.
The results suggest that the relationship between read and write (rho = 0.617, p = 0.000) is statistically significant.
Simple logistic regression
Logistic regression assumes that the outcome variable is binary (i.e., coded as 0 and 1). We have only one variable in the hsb2 data file that is coded 0 and 1, and that is female. We
understand that female is a silly outcome variable (it would make more sense to use it as a predictor variable), but we can use female as the outcome variable to illustrate how the code for this
command is structured and how to interpret the output. The first variable listed after the logistic command is the outcome (or dependent) variable, and all of the rest of the variables are
predictor (or independent) variables. In our example, female will be the outcome variable, and read will be the predictor variable. As with OLS regression, the predictor variables must be
either dichotomous or continuous; they cannot be categorical.
logistic regression female with read.
The results indicate that reading score (read) is not a statistically significant predictor of gender (i.e., being female), Wald = .562, p = 0.453. Likewise, the test of the overall model is not
statistically significant, LR chi-squared – 0.56, p = 0.453.
See also
Multiple regression
Multiple regression is very similar to simple regression, except that in multiple regression you have more than one predictor variable in the equation. For example, using the hsb2 data file we
will predict writing score from gender (female), reading, math, science and social studies (socst) scores.
regression variable = write female read math science socst
/dependent = write
/method = enter.
The results indicate that the overall model is statistically significant (F = 58.60, p = 0.000). Furthermore, all of the predictor variables are statistically significant except for read.
See also
Analysis of covariance
Analysis of covariance is like ANOVA, except in addition to the categorical predictors you also have continuous predictors as well. For example, the one way ANOVA example used write as the
dependent variable and prog as the independent variable. Let’s add read as a continuous variable to this model, as shown below.
glm write with read by prog.
read), writing scores still significantly differ by program type (prog), F = 5.867, p = 0.003.
See also
Multiple logistic regression
Multiple logistic regression is like simple logistic regression, except that there are two or more predictors. The predictors can be interval variables or dummy variables, but cannot be
categorical variables. If you have categorical predictors, they should be coded into one or more dummy variables. We have only one variable in our data set that is coded 0 and 1, and that is
female. We understand that female is a silly outcome variable (it would make more sense to use it as a predictor variable), but we can use female as the outcome variable to illustrate how the
code for this command is structured and how to interpret the output. The first variable listed after the logistic regression command is the outcome (or dependent) variable, and all of the rest
of the variables are predictor (or independent) variables (listed after the keyword with). In our example, female will be the outcome variable, and read and write will be the predictor
logistic regression female with read write.
These results show that both read and write are significant predictors of female.
See also
Discriminant analysis
Discriminant analysis is used when you have one or more normally distributed interval independent variables and a categorical dependent variable. It is a multivariate technique that considers
the latent dimensions in the independent variables for predicting group membership in the categorical dependent variable. For example, using the hsb2 data file, say we wish to use read, write
and math scores to predict the type of program a student belongs to (prog).
discriminate groups = prog(1, 3)
/variables = read write math.
Clearly, the SPSS output for this procedure is quite lengthy, and it is beyond the scope of this page to explain all of it. However, the main point is that two canonical variables are identified
by the analysis, the first of which seems to be more related to program type than the second.
See also
One-way MANOVA
MANOVA (multivariate analysis of variance) is like ANOVA, except that there are two or more dependent variables. In a one-way MANOVA, there is one categorical independent variable and two or more
dependent variables. For example, using the hsb2 data file, say we wish to examine the differences in read, write and math broken down by program type (prog).
glm read write math by prog.
The students in the different programs differ in their joint distribution of read, write and math.
See also
Multivariate multiple regression
Multivariate multiple regression is used when you have two or more dependent variables that are to be predicted from two or more independent variables. In our example using the hsb2 data file,
we will predict write and read from female, math, science and social studies (socst) scores.
glm write read with female math science socst.
These results show that all of the variables in the model have a statistically significant relationship with the joint distribution of write and read.
Canonical correlation
Canonical correlation is a multivariate technique used to examine the relationship between two groups of variables. For each set of variables, it creates latent variables and looks at the
relationships among the latent variables. It assumes that all variables in the model are interval and normally distributed. SPSS requires that each of the two groups of variables be separated by
the keyword with. There need not be an equal number of variables in the two groups (before and after the with).
manova read write with math science
* * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * *
EFFECT .. WITHIN CELLS Regression
Multivariate Tests of Significance (S = 2, M = -1/2, N = 97 )
Test Name Value Approx. F Hypoth. DF Error DF Sig. of F
Pillais .59783 41.99694 4.00 394.00 .000
Hotellings 1.48369 72.32964 4.00 390.00 .000
Wilks .40249 56.47060 4.00 392.00 .000
Roys .59728
Note.. F statistic for WILKS' Lambda is exact.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
EFFECT .. WITHIN CELLS Regression (Cont.)
Univariate F-tests with (2,197) D. F.
Variable Sq. Mul. R Adj. R-sq. Hypoth. MS Error MS F
READ .51356 .50862 5371.66966 51.65523 103.99081
WRITE .43565 .42992 3894.42594 51.21839 76.03569
Variable Sig. of F
READ .000
WRITE .000
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Raw canonical coefficients for DEPENDENT variables
Function No.
Variable 1
READ .063
WRITE .049
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Standardized canonical coefficients for DEPENDENT variables
Function No.
Variable 1
READ .649
WRITE .467
* * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * *
Correlations between DEPENDENT and canonical variables
Function No.
Variable 1
READ .927
WRITE .854
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Variance in dependent variables explained by canonical variables
CAN. VAR. Pct Var DE Cum Pct DE Pct Var CO Cum Pct CO
1 79.441 79.441 47.449 47.449
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Raw canonical coefficients for COVARIATES
Function No.
COVARIATE 1
MATH .067
SCIENCE .048
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Standardized canonical coefficients for COVARIATES
CAN. VAR.
COVARIATE 1
MATH .628
SCIENCE .478
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Correlations between COVARIATES and canonical variables
CAN. VAR.
Covariate 1
MATH .929
SCIENCE .873
* * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * *
Variance in covariates explained by canonical variables
CAN. VAR. Pct Var DE Cum Pct DE Pct Var CO Cum Pct CO
1 48.544 48.544 81.275 81.275
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Regression analysis for WITHIN CELLS error term
--- Individual Univariate .9500 confidence intervals
Dependent variable .. READ reading score
COVARIATE B Beta Std. Err. t-Value Sig. of t
MATH .48129 .43977 .070 6.868 .000
SCIENCE .36532 .35278 .066 5.509 .000
COVARIATE Lower -95% CL- Upper
MATH .343 .619
SCIENCE .235 .496
Dependent variable .. WRITE writing score
COVARIATE B Beta Std. Err. t-Value Sig. of t
MATH .43290 .42787 .070 6.203 .000
SCIENCE .28775 .30057 .066 4.358 .000
COVARIATE Lower -95% CL- Upper
MATH .295 .571
SCIENCE .158 .418
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
* * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * *
EFFECT .. CONSTANT
Multivariate Tests of Significance (S = 1, M = 0, N = 97 )
Test Name Value Exact F Hypoth. DF Error DF Sig. of F
Pillais .11544 12.78959 2.00 196.00 .000
Hotellings .13051 12.78959 2.00 196.00 .000
Wilks .88456 12.78959 2.00 196.00 .000
Roys .11544
Note.. F statistics are exact.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
EFFECT .. CONSTANT (Cont.)
Univariate F-tests with (1,197) D. F.
Variable Hypoth. SS Error SS Hypoth. MS Error MS F Sig. of F
READ 336.96220 10176.0807 336.96220 51.65523 6.52329 .011
WRITE 1209.88188 10090.0231 1209.88188 51.21839 23.62202 .000
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
EFFECT .. CONSTANT (Cont.)
Raw discriminant function coefficients
Function No.
Variable 1
READ .041
WRITE .124
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Standardized discriminant function coefficients
Function No.
Variable 1
READ .293
WRITE .889
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Estimates of effects for canonical variables
Canonical Variable
Parameter 1
1 2.196
* * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * *
EFFECT .. CONSTANT (Cont.)
Correlations between DEPENDENT and canonical variables
Canonical Variable
Variable 1
READ .504
WRITE .959
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
The output above shows the linear combinations corresponding to the first canonical correlation. At the bottom of the output are the two canonical correlations. These results indicate that the
first canonical correlation is .7728. The F-test in this output tests the hypothesis that the first canonical correlation is equal to zero. Clearly, F = 56.4706 is statistically significant.
However, the second canonical correlation of .0235 is not statistically significantly different from zero (F = 0.1087, p = 0.7420).
Factor analysis
Factor analysis is a form of exploratory multivariate analysis that is used to either reduce the number of variables in a model or to detect relationships among variables. All variables involved
in the factor analysis need to be interval and are assumed to be normally distributed. The goal of the analysis is to try to identify factors which underlie the variables. There may be fewer
factors than variables, but there may not be more factors than variables. For our example using the hsb2 data file, let’s suppose that we think that there are some common factors underlying the
various test scores. We will include subcommands for varimax rotation and a plot of the eigenvalues. We will use a principal components extraction and will retain two factors. (Using these
options will make our results compatible with those from SAS and Stata and are not necessarily the options that you will want to use.)
/variables read write math science socst
/criteria factors(2)
/extraction pc
/rotation varimax
/plot eigen.
Communality (which is the opposite of uniqueness) is the proportion of variance of the variable (i.e., read) that is accounted for by all of the factors taken together, and a very low communality
can indicate that a variable may not belong with any of the factors. The scree plot may be useful in determining how many factors to retain. From the component matrix table, we can see that all
five of the test scores load onto the first factor, while all five tend to load not so heavily on the second factor. The purpose of rotating the factors is to get the variables to load either
very high or very low on each factor. In this example, because all of the variables loaded onto factor 1 and not on factor 2, the rotation did not aid in the interpretation. Instead, it made the
results even more difficult to interpret.
See also | {"url":"https://stats.oarc.ucla.edu/spss/whatstat/what-statistical-analysis-should-i-usestatistical-analyses-using-spss/","timestamp":"2024-11-08T09:01:27Z","content_type":"text/html","content_length":"97892","record_id":"<urn:uuid:22a60465-6487-4906-9af1-3cd84abf2d5c>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00146.warc.gz"} |
Anova Calculator
The ANOVA test Calculator is used to calculate statistical analysis on multiple data sets to determine if there are any statistically significant differences between the means of three or more
independent groups.
What is ANOVA?
ANOVA stands for Analysis of Variance. It is a statistical method used to compare means among multiple groups to determine if they have statistically significant differences. It evaluates whether the
differences between the group means are greater than the differences within each group.
The basic concept of ANOVA is to divide the total variance observed in the data into two categories:
• Variance within groups
• Variance between groups
When the variance between groups is greater than within groups, it indicates real differences among the groups being analyzed.
ANOVA Formula
The ANOVA formula is systematically organized in the table. This ANOVA table can be utilized as follows:
│Source of Variation│Sum of Squares │Degrees of Freedom│Mean Squares │F Value │
│Within Groups │SSW = Σ (n[i ]- 1) S[i]^2 │df[w] = k-1 │MSW = SSW/ df[w] │F = MSB/ MSW│
│Between Groups │SSB=Σ n[i ](x̄[i ]− x̄)^2 │df[1] = k - 1 │MSB = SSB / (k - 1)│f=MSB/MSE │
│Error │SSE = (X - x̄[i]) │df[2] = N - k │MSE = SSE / (N - k)│ │
│Total │SST = SSB + SSE │df[3] = N - 1 │ │ │
Let's redefine the variables:
• F: The ANOVA coefficient (F-statistics)
• MSB: Mean sum of squares between groups
• MSW: Mean sum of squares within groups
• MSE: Mean sum of squares due to error
• SST: Total sum of squares
• p: Number of populations
• n: Number of samples in each population
• SSW: Sum of squares within groups
• SSB: Sum of squares between groups
• SSE: Sum of squares due to error
• s: Standard deviation of samples
• N: Total number of observations
Types of ANOVA Test
The ANOVA test has three basic types, which are discussed below:
This test is used when comparing the means of three or more groups based on one factor. This test is straightforward and useful in understanding the impact of a single independent variable on a
dependent variable.
This can be used when examining the effect of two factors on the dependent variable.
It is used when the same subjects are tested under different conditions or over different time points.
How to Calculate ANOVA?
This section will demonstrate the calculation of a one-way ANOVA test with the help of an example:
A researcher wants to compare the effectiveness of three diets on weight loss. The weights (in pounds) lost by participants after following the diets for a month are recorded as follows:
│Diet A│Diet B│Diet C│Standard Deviation │
│8 │6 │10 │Diet A = 1.5811 │
│12 │7 │15 │Diet B = 1.1402 │
│10 │5 │12 │Diet C = 1.9235 │
│9 │4 │11 │ │
│11 │6 │13 │ │
Make a one-way ANOVA table for the data and compute the SSB, SSW, MSB, MSW, and F-statistic using the defining formulas.
XA = 8+12+10+9+11/ 5 = 10
XB = 6+7+5+4+6/ 5=5.6
Xc = 10+15+12+11+13/ 5=12.2
x̄total = (8+12+10+9+11) +(6+7+5+4+6) + (10+15+12+11+13) / 15 =139/15 = 9.27
SSB = 5× (10 − 9.267)2 +5× (5.6 − 9.267)2 +5× (12.2 − 9.267)2
SSW= (5−1) × (1.5811)2 +(5−1) × (1.1402)2 +(5−1) × (1.9235)2
SST = SSB + SSE
SST = 29.999 + 112.933
SST = 142.932
MSB=29.999/ 3−1
MSB=29.999/ 2
MSB = 56.467
MSW= SSW / / N−K
MSW=56.467/ 15−3
MSW= 56.467/ 12
MSW= 2.5
F = 56.467 / 2.5
F = 22.587
Make a Decision
• If the F-statistic is greater than the critical value is less than α then reject the null hypothesis, indicating a significant difference among the group means.
• If not, fail to reject the null hypothesis, indicating no significant difference among the group means.
Using an ANOVA Calculator can greatly simplify the process by efficiently calculating the variance between and within groups, ensuring accuracy and efficiency in your statistical analysis.
Frequently Asked Questions
1. How to calculate F statistics from the ANOVA table?
In an ANOVA table, the F-statistic is calculated by dividing the mean sum of squares (MSB) by the error mean sum of squares (MSE).
F = MSB/MSE.
2. What is a small sample size for ANOVA?
The minimum sample size (n) for ANOVA depends on the number of groups in the data. For example, if you have 2–9 groups, the sample size for each group should be at least 15.
3. What are the four assumptions of ANOVA?
The four assumptions of Analysis of Variance (ANOVA) are:
• Interval data: The dependent data must be measured at an interval scale.
• Normality: The population distribution must be normal.
• Homogeneity of variance: The variance among the groups should be approximately equal.
• Independence: The observations should be independent of each other. | {"url":"https://www.criticalvaluecalculator.com/anova-calculator","timestamp":"2024-11-07T07:24:50Z","content_type":"text/html","content_length":"61055","record_id":"<urn:uuid:d9aa8dc6-9672-4d50-a6c0-0d89a4b03b0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00877.warc.gz"} |
$300 x .06 x .5 years= $9
Your yearly office supply budget is $1,200. You spend $350 each year on paper. What percent of your budget do you spend on paper? (Round off your answer to two places.)
Your yearly office supply budget is $1,200. You spend $350 each year on paper. What percent of your budget do you spend on paper? (Round off your answer to two places.)
You're planning to buy a new computer, monitor, and printer for your office. You've received the following prices from two stores: Store 1 Computer: $2,500 Monitor: $595 Printer: $398 Store 2
Computer: $2,350 Monitor: $435 Printer: $349 Find the total package price from each store and then calculate how much you'll save if you purchase the less expensive package
what is your question then?
You're planning to buy a new computer, monitor, and printer for your office. You've received the following prices from two stores: Store 1 Computer: $2,500 Monitor: $595 Printer: $398 Store 2
Computer: $2,350 Monitor: $435 Printer: $349 Find the total package price from each store and then calculate how much you'll save if you purchase the less expensive package
what is your question then?
Sales for the first four months of the year were as follows: January—$5,282 February—$4,698 March—$3,029 April—$6,390 Find the average monthly sales. (Round off your answer to the nearest dollar.)
Sorry, but can you provide more clarification regarding your question. Thanks
While shopping, you see a jacket marked down 20%. If the original price of the jacket is $175, what is the sale price of the jacket?
$140 is the sale price of the jacket. If the original price of the jacket is $175 at 20% off.
The present value of the money in your savings account is $420, and you're receiving 3% annual interest compounded monthly. What is the future value ... | {"url":"https://weegy.com/Home.aspx?Id=ArchivePage&SpModeType=1&SpAccountId=&SpRow=401001&SpLevel=4","timestamp":"2024-11-12T13:53:55Z","content_type":"application/xhtml+xml","content_length":"182152","record_id":"<urn:uuid:50acb524-98bc-40b1-b2c0-be9202947dec>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00518.warc.gz"} |
kth row of pascal's triangle python
2. For example, when k = 3, the row is [1,3,3,1]. Embed Embed this gist in your website. This is a common interview test of companies like Amazon or Google. The first row is 0 1 0 whereas only 1
acquire a space in pascal's triangle, 0s are invisible. def mk_row(triangle, row_number): """ function creating a row of a pascals triangle parameters: Implementations should support up to row 53.
The sum of all the elements of a row is twice the sum of all the elements of its preceding row. Better Solution: Let’s have a look on pascal’s triangle pattern . The Pascal Triangle. In a Pascal's
Triangle the rows and columns are numbered from 0 just like a Python list so we don't even have to bother about adding or subtracting 1. The next row value would be the binomial coefficient with the
same n-value (the row index value) but incrementing the k-value by 1, until the k-value is equal to the row … saavi2701 / kth row of pascals triangle. Given an integer rowIndex, return the rowIndex
th row of the Pascal's triangle. This problem is related to Pascal's Triangle which gets all rows of Pascal's triangle. In this function, we will initialize the top row first, using the trow
variable. Pascal's Triangle II 【题目】 Given a non-negative index k where k ≤ 33, return the kth index row of the Pascal's triangle. The first line of the loop is yield row, so the first time you
call next() you get the value (1,).. Then, the next row down is the 1 st 1^\text{st} 1 st row, and so on. Example 1: Input: rowIndex = 3 Output: [1,3,3,1] Example 2: If we look at row 5 (1 5 10 10
51), we can see that 5 and 10 are divisible by 5. The triangle is as shown below. Pascal's triangle can be derived using binomial theorem. We also initialize variable y=0. row = mk_row
(triangle,row_number) triangle.append(row) return triangle Now the only function that is missing is the function, that creates a new row of a triangle assuming you know the row number and you
calculated already the above rows. 1 … Pascal queries related to “making pascals triangle python” generating pascal's triangle in python; python program to create a function that prints Pascal's
triangle. Looking at the layout above it becomes obvious that what we need is a list of lists. Row by Row. 2) Prime Numbers in the Triangle: Another pattern visible in the triangle deals with prime
numbers. Share Copy sharable link for this gist. Print each row with each value separated by a single space. Algorithm. I think you are trying to code the formula nCk = (n-1)C(k-1) + (n-1)Ck. # In
Pascal's triangle, each … # Note that the row index starts from 0. If a row starts with a prime number or is a prime numbered row, all the numbers that are in that row (not counting the 1’s) are
divisible by that prime. Als leerervaring voor Python probeer ik mijn eigen versie van de driehoek van Pascal te coderen. An equation to determine what the nth line of Pascal's triangle could
therefore be n = 11 to the power of n-1. In Pascal's triangle, each number is the sum of the two numbers directly above it. Given a non-negative index k where k ≤ 33, return the _k_th index row of
the Pascal's triangle.. def nextrow(lst): lag = 0 for element in lst: yield lag + element lag = element yield element row = [1] for number in range(12): row = nextrow(row) print row In following good
computer science tradition, we'll define the first row to be row 0. For a given integer , print the first rows of Pascal's Triangle. Follow up: Could you optimize your algorithm to use only O(k)
extra space? Python Functions: Exercise-13 with Solution. Pascal's triangle is an arithmetic and geometric figure often associated with the name of Blaise Pascal, but also studied centuries earlier
in India, Persia, China and elsewhere.. Its first few rows look like this: 1 1 1 1 2 1 1 3 3 1 where each element of each row is either 1 or the sum of the two elements right above it. Note: Could
you optimize your algorithm to use only O(k) extra space? write a program that outputs the Nth row of the triangle, where N is given as input integer. Using the above formula you would get 161051.
Analysis. Created Jul 24, 2020. Embed. Each element in the triangle has a coordinate, given by the row it is on and its position in the row (which you could call its column). Remember: every time a
yield statement is encountered, execution pauses. Pascals Triangle. row of pascal's triangle in python; pascals triangle code python; paskal triangle python; pascal triangle program in python; Write
a Python function that prints the first n rows of Pascal’s triangle. However, the tests for this kata enumerate the rows starting with row n = 1 at the top (the 1st row). The value at the row and
column of the triangle is equal to where indexing starts from . If we have the a row of Pascal's triangle, we can easily compute the next row by each pair of adjacent values. Note that the row index
starts from 0. Again, the sum of third row is 1+2+1 =4, and that of second row is 1+1 =2, and so on. sum of elements in i th row 0th row 1 1 -> 2 0 1st row 1 1 2 -> 2 1 2nd row 1 2 1 4 -> 2 2 3rd row
1 3 3 1 8 -> 2 3 4th row 1 4 6 4 1 16 -> 2 4 5th row 1 5 10 10 5 1 32 -> 2 5 6th row 1 6 15 20 15 6 1 64 -> 2 6 7th row 1 7 21 35 35 21 7 1 128 -> 2 7 8th row … (n = 5, k = 3) I also highlighted the
entries below these 4 that you can calculate, using the Pascal triangle algorithm. In Pascal's triangle, each number is the sum of the two numbers directly above it. When the get_pascals_triangle_row
(row_number) is called with row_number = 1 it should return [1, 1]. Additional clarification: The topmost row in Pascal's triangle is the 0 th 0^\text{th} 0 th row. Sample Pascal's triangle : Each
number is the two numbers above it added together. In this challenge you have to implement an algorithm that returns the n-th row of the Pascal's Triangle. For a given non-negative row index, the
first row value will be the binomial coefficient where n is the row index value and k is 0). This highlights how the calculation is done. Second row is acquired by adding (0+1) and (1+0). Sample
Solution:- Python Code : Pascal’s Triangle using Python. Star 0 Fork 0; Star Code Revisions 1. Your function get_pascals_triangle_row(row_number) should return the row with the row_number of Pascal's
Triangle. The process continues till the required level is achieved. Note : Pascal's triangle is an arithmetic and geometric figure first imagined by Blaise Pascal. You are not, in fact, using
recursion at all in your answer. # Given a non-negative index k where k ≤ 33, return the kth index row of the Pascal's triangle. Both numbers are the same. The leftmost element in each row of
Pascal's triangle is the 0 th 0^\text{th} 0 th element. 4. Pascal Triangle in Python- “Algorithm” Now let us discuss the algorithm of printing the pascal triangle in Python After evaluating the above
image of pascal triangle we deduce the following points to frame the code 1. Pascal's triangle is a mathematical array of binomial coefficients. Write a Python function that that prints out the first
n rows of Pascal's triangle. Below is the first eight rows of Pascal's triangle with 4 successive entries in the 5 th row highlighted. Primes in Pascal triangle : First you have to understand how the
Pascal's Triangle works: Each row starts and ends with the number 1. Example: Input : k = 3 Return : [1,3,3,1] Java Solution of Kth Row of Pascal's Triangle In fact, if Pascal’s triangle was expanded
further past Row 5, you would see that the sum of the numbers of any nth row would equal to 2^n. Although the algorithm is very simple, the iterative approach to constructing Pascal's triangle can be
classified as dynamic programming because we construct each row based on the previous row. You need, therefore, to call combination from within itself (with a guard for the "end" conditions: nC0 =
nCn = 1):. The output is sandwiched between two zeroes. The first thing rows_of_pascals_triangle() does is declare the variable row and assign to it the tuple (1,) - the first row of Pascal’s
Triangle.. Next it enters an infinite loop ( cue theme from The Twilight Zone ). This leads to the number 35 in the 8 th row. Kth Row of Pascal's Triangle Solution Java Given an index k, return the
kth row of Pascal’s triangle. Note that the row index starts from 0. For example, sum of second row is 1+1= 2, and that of first is 1. For example, givenk= 3, Return[1,3,3,1].. Problem: Pascal’s
triangle is a useful recursive definition that tells us the coefficients in the expansion of the polynomial (x + a)^n. Two nested loops must be used to print pattern in 2-D format. The line following
has 2 ones. Uses the combinatorics property of the Triangle: For any NUMBER in position INDEX at row ROW: NUMBER = C(ROW, INDEX) A hash map stores the values of the combinatorics already calculated,
so the recursive function speeds up a little. Given an index k, return the kth row of the Pascal's triangle. This is the second line. Notice that the row index starts from 0. (n + k = 8) Pascal's
Triangle in a left aligned form. The 6th line of the triangle is 1 5 10 10 5 1. Example: When the get_pascals_triangle_row(row_number) is called with row_number = 0 it should return [1]. Triangle de
Pascal en Python avec tableaux 2D J'essaie d'écrire un code python qui itère sur un tableau 2-D, la liste externe doit contenir des lignes et les listes internes doivent contenir les éléments des
nombres dans le triangle de Pascal. From the wikipedia page the kata author cites: "The rows of Pascal's triangle (sequence A007318 in OEIS) are conventionally enumerated starting with row n = 0 at
the top (the 0th row)." This works till you get to the 6th line. Given an index k, return the kth row of the Pascal's triangle. What would you like to do? Briefly explaining the triangle, the first
line is 1. These values are the binomial coefficients. In this problem, only one row is required to return. This major property is utilized to write the code in C program for Pascal’s triangle.
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return the row and of... Is the sum of the Pascal 's triangle n + k = 8 ) given an k. That that prints out the first row to be row 0 triangle which all. Numbers directly above it becomes obvious that
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row_number of Pascal triangle. Till you get to the number 1, givenk= 3, the and. Row 0 companies like Amazon or Google derived using binomial theorem it becomes obvious that what we is. And column of
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triangle python row first, the. | {"url":"http://cwlinux.com/carte-grise-tlymyym/6776cb-kth-row-of-pascal%27s-triangle-python","timestamp":"2024-11-06T02:09:14Z","content_type":"text/html","content_length":"33442","record_id":"<urn:uuid:ea3fcd9a-7921-499b-9f9a-1d2deb04bdbd>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00556.warc.gz"} |
Hint: Assume the function = $f\left( x \right)={{\log }_{\text{e}}}\left( x \right)$.Since the function to be calculated consists of a small change, convert the given logarithmic function in the form
of \[{{\log }_{\text{e}}}\left( x+dx \right)\] where ‘x’ represents the original value and ‘dx’ represents the small change in the original value. In this question; x= 4 and dx=0.01
Complete step-by-step answer:
Then, differentiate the function with respect to x and substitute the values of ‘x’ and ‘dx’ to get the solution.
Let a function \[f\left( x \right)=y={{\log }_{\text{e}}}\left( x \right)......(1)\]
Differentiate both sides of equation (1) with respect to ‘x’, we get:
\[f'\left( x \right)=\dfrac{dy}{dx}\]
Therefore, we can write: \[dy=f'\left( x \right)dx\]
So, from equation (1), we can say: \[dy=f'\left( {{\log }_{\text{e}}}\left( x \right) \right)dx\]
Since, \[f'\left( {{\log }_{\text{e}}}\left( x \right) \right)=\dfrac{1}{x}\]
So, we can write: \[dy=\dfrac{1}{x}dx......(2)\]
Now by increasing f(x) by an element ‘dx’, we get:
\[f\left( x+dx \right)={{\log }_{\text{e}}}\left( x+dx \right)\]
Comparing with equation (1), we can write:
\[y+dy={{\log }_{\text{e}}}\left( x+dx \right)\]
Therefore, \[dy={{\log }_{\text{e}}}\left( x+dx \right)-y\]
\[\Rightarrow dy={{\log }_{\text{e}}}\left( x+dx \right)-{{\log }_{\text{e}}}\left( x \right)......(3)\]
Substitute the value of ‘dy’ from equation (2) in equation (4):
\[\dfrac{1}{x}dx={{\log }_{\text{e}}}\left( x+dx \right)-{{\log }_{\text{e}}}\left( x \right)......(4)\]
Now, compare equation (4) with the function given in the question i.e. \[{{\log }_{\text{e}}}\left( 4.01 \right)\]
Consider x = 4 and dx = 0.01 and put the values in equation (4).
We get:
\[\left( \dfrac{1}{4}\times 0.01 \right)={{\log }_{\text{e}}}\left( 4.01 \right)-{{\log }_{\text{e}}}\left( 4 \right)\]
Therefore, \[{{\log }_{\text{e}}}\left( 4.01 \right)={{\log }_{\text{e}}}\left( 4 \right)+\left( \dfrac{1}{4}\times 0.01 \right)\]
& \Rightarrow {{\log }_{\text{e}}}\left( 4.01 \right)=1.3868-0.0025 \\
& \Rightarrow {{\log }_{\text{e}}}\left( 4.01 \right)=1.3893 \\
So, the correct answer is “Option C”.
Note: As we know that, \[\begin{align}
& \underset{\Delta x\to 0}{\mathop{\lim }}\,\dfrac{\Delta y}{\Delta x}=\dfrac{dy}{dx}=f'\left( x \right) \\
& \Delta y=f'\left( x \right)\Delta x \\
& dy=f'\left( x \right)dx \\
\end{align}\] \[\underset{\Delta x\to 0}{\mathop{\lim }}\,\dfrac{\Delta y}{\Delta x}=\dfrac{dy}{dx}=f'\left( x \right)\]
Therefore, \[\Delta y=f'\left( x \right)\Delta x\]
Also \[dy=f'\left( x \right)dx\]
Hence, we can use differentials to calculate small changes in the dependent variable (dy) of a function corresponding to small changes in the independent variable f(x). | {"url":"https://www.vedantu.com/question-answer/if-log-eleft-4-right13868-then-log-eleft-401-class-11-maths-cbse-5f5cf99e9427543f91faffe6","timestamp":"2024-11-09T13:53:30Z","content_type":"text/html","content_length":"167756","record_id":"<urn:uuid:51a34671-005c-433e-a64a-6899d8c929ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00108.warc.gz"} |
A classification system that has a significant amount of ov… | Exam Equip
A clаssificаtiоn system thаt has a significant amоunt оf overlap is able give correctional administrators guidance about appropriate treatment for offenders.
The stаndаrd deductiоn аmоunt varies by filing status.
Fоr fi ling stаtus purpоses, the tаxpаyer's marital status is determined at what pоint during the year?
Mоnthly rentаl rаtes were recоrded frоm а random sample of 40 one-bedroom apartments in Beaverton, Oregon. Use the following summary statistics to answer the question below.
Summary Statistics Variable N Mean SE Mean StDev Minimum Median Maximum One Bedroom Monthly Rent 40 1510.48 86.31 545.84 1065 1399.5 4547 Construct a 90% confidence interval to estimate the mean
monthly rental rate in the population of all one-bedroom apartments in Beaverton, Oregon. The t* multiplier for a 90% interval with 39 degrees of freedom is 1.685. Show your work. | {"url":"https://examequip.com/a-classification-system-that-has-a-significant-amount-of-overlap-is-able-give-correctional-administrators-guidance-about-appropriate-treatment-for-offenders/","timestamp":"2024-11-09T16:22:58Z","content_type":"text/html","content_length":"32480","record_id":"<urn:uuid:b5381e95-b1db-4908-943a-20ac3184ee20>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00880.warc.gz"} |
Composite Amplifiers Part Three: Composite Amplifier Closed-Loop Frequency Response - Engineering.com
The third part of our series to explain the nature, characteristics, benefits, and caveats of composite amplifiers.
This is the third part in our series explaining the nature, characteristics, benefits, and caveats of composite amplifiers. The four parts of the series include:
• Part Three: Composite Amplifier Closed-Loop Frequency Response
Composite Amplifier Topology
Figure 14 illustrates a general composite amplifier to be analyzed. The bandwidth of each stage is determined by its gain-bandwidth product and its voltage gain. Each op amp is assumed to have
dominant-pole frequency compensation and be stable down to unity voltage gain (0 dB).
Figure 14. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.
To maximize the bandwidth of the composite amplifier, the usual approach is the make Av1 and Av2 equal. (More will be said about this later.)
The AD712 is described to be a precision, low-cost, high-speed BiFET dual (two-channel) op amp. It is to be used in a composite amplifier configuration which provides a voltage gain of 1000. Assume
the two amplifiers are matched perfectly. We shall use Multisim (Figure 15) to verify the open-loop voltage gain (A[VOL])[, ]the open-loop dominant corner frequency (f[H(OL)]), and the gain bandwidth
product (f[T]) of the AD712 op amp’s SPICE model. We shall use the model values for our calculations. Resistors R[1] and R[2] ensure the op amp receives proper DC bias. Capacitor C[1] eliminates any
possibility of AC negative feedback. The Multisim Bode plotter provides us with the op amp model parameters.
Figure 15. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.
The simulation results indicate an open-loop DC voltage gain of 106 dB, an open-loop corner frequency of 25 Hz and a gain-bandwidth product of 4.3 MHz. Next, we determine the required voltage gain of
the second stage.
Ideally, the first and second stages will both possess voltage gains of 31.623. Standard 1%-tolerance resistors were used to obtain the required voltage gains. Figure 16 shows the Multisim circuit.
Note the oscilloscope is AC coupled to block any DC offset. DC offset will be addressed later.
Figure 16. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.
Observe the positive peak is 4.970 V, the negative peak is -4.990 V, and the peak-to-peak out voltage is 9.958 Vp-p. The peak-to-peak value is -0.42 percent lower than the expected ideal value of 10
Understanding the Composite Amplifier Closed-Loop Response
Using the AD 712 op amp model parameters determined as shown in Figure 15, the Bode straight-line approximation of the amplitude frequency response can be drawn as shown in Figure 17.
Figure 17. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.
The second stage has a closed-loop voltage gain (A[v2]) of 31.7. This corresponds to a decibel voltage gain of 30.0 dB. The closed-loop response can be obtained by superimposing the closed-loop gain
on the open-loop response as indicated in Figure 18. The corner frequency of the second stage f[H2] can be determined graphically. However, we can use the voltage gain and the gain-bandwidth product
to obtain greater accuracy.
Since the closed-loop gain intersects where the open-loop response has a slope of -20 dB/decade, the amplifier is stable and will not oscillate. (The basic stability rules tell us that an
intersection at -40 dB/decade is marginally stable and -60 dB/decade or greater intersection is unstable and will probably oscillate.)
Figure 18. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.
When amplifiers are cascaded inside a composite amplifier, their voltage gains multiply. Their corresponding voltage gains in decibels add together. In this composite amplifier the open-loop voltage
gain A[VOL](dB) response can be graphed using Bode approximations. Graphical addition is used as shown in Figure 19. Note the composite amplifier open-loop response has a transition frequency f[T
(COMP)] of about 4.3 MHz. This “f[T]” does not work like our previous encounters since the open-loop gain does not roll off at a constant -20 dB/decade. In this instance, we are finding the bandwidth
Figure 19. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.
The composite amplifier open-loop response can be verified using Multisim. The circuit is provided in Figure 20. Capacitor C[1] is used to eliminate the negative feedback. The permits us to obtain
the open-loop amplitude frequency response of the composite amplifier.
Figure 20. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.
The open-loop frequency response is given in Figure 21. In Figure 21(a) the cursor is positioned to verify the low-frequency open-loop gain. A gain of 135.3 dB is indicated. In Figure 19 a gain of
136 dB was predicted. Figure 21(b) indicates the first corner frequency. It should be -3dB from the low-frequency gain (135.3 dB – 3 dB = 132.3 dB) and is at 28 Hz. (Figure 19 shows the dominant pole
at 25 Hz.) Figure 21(c) and (d) are used to verify the slope is indeed -20 dB/decade. The second corner frequency is determined via the cursor in Figure 21(e). This occurs at about 60 dB at 136 kHz,
which agrees with the value of 136 kHz indicated in our Bode approximation (Figure 19).
Figure 21. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.
The slope for frequencies beyond 136 kHz becomes -40 dB/decade and moving the cursor to the location where the open-loop gain crosses the 0 dB axis indicates a transition frequency f[T(COMP)] of 4.5
MHz at a gain of 0.118 dB (Figure 21(f)), which is reasonable compared to the 4.3 MHz value indicated in Figure 19.
The Bode approximation of the closed-loop frequency response plot is shown in Figure 22. The closed-loop gain of 1000 (60 dB) is superimposed on the open-loop response of the composite amplifier. The
intersection between the closed-loop gain and the open-loop response provides the crossing frequency f[c]. (The crossing frequency is the point at which the loop gain response crosses the 0 dB or
unity point.) The circuit has been designed to have the closed-loop gain intersect at the second corner frequency f[H2].
Figure 22. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.
The crossing frequency f[c] permits us to calculate the crossing angle q[c]. The crossing angle is used to find the phase margin q[pm]. The phase margin determines the stability of the amplifier
circuit and determines any peaking that might occur in the frequency response.
A phase margin of 45 degrees is stable but will result in some slight peaking. The Multisim circuit and the closed-loop response are provided in Figure 23.
Larger (greater than 45 degrees) phase margins improve stability but make the amplifier response more “sluggish”. Smaller (less than 45 degrees) phase margins bring the amplifier closer to
oscillation. However, the amplifier output can respond more quickly to abrupt changes in the input signal.
Figure 23. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.
The voltage gain of the second stage controls the amount of peaking. This is because it determines the second breakpoint in the open-loop frequency response.
Assume the voltage gain of the second stage (AV2) in the composite amplifier is decreased from 31.7 to 26.0. (R4B is lowered from 10.7 kΩ to 4.99 kΩ.)
The bandwidth (corner frequency) of the second stage will increase from 136 kHz to 165 kHz. This means the second break point of the composite amplifier open-loop response will be moved to 165 kHz.
Consequently, the crossing frequency is much lower relative to fH2. This increases the phase margin and there will be less peaking. See Figure 24(a).
Figure 24. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.
Assume the voltage gain of the second stage (Av2) of the composite amplifier is increased from 31.7 to 51.0. (R4B is increased from 10.7 kΩ to 30 kΩ.)
The bandwidth (corner frequency) of the second stage will decrease from 136 kHz to 84.3 kHz.
As a result, the crossing frequency is much higher relative to fH2. This decreases the phase margin and there will be more peaking. See Figure 24(b). The relationships are summarized in Figure 25.
Figure 25. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.
Review and Conclusions
A composite amplifier employs local (sometimes called “nested”) negative feedback as well as an overall negative feedback loop. By using a resistive network like that incorporated in a non-inverting
amplifier to provide DC biasing, and a very large capacitor to eliminate any negative feedback, the open-loop frequency response of an amplifier can be obtained using Multisim. To maximize the
bandwidth of a composite amplifier, the voltage gains of the individual stages are made equal. In the case of two stages, we take the square root of the desired overall gain, and in the case of three
stages, a cube root is required. A quick inspection of closed-loop stability can be conducted. If the second stage closed-loop gain is superimposed on the open-loop response, the slope at the
intersection yields an indication of stability. The same stability rules apply to the overall voltage gain of the composite amplifier. An intersection where the slope is -20 dB/decade is stable, at
-40 dB/decade is marginally stable, and at -60 dB/decade or greater is unstable and will quite likely oscillate.
The intersection of the second stage gain with the composite amplifier open-loop response will produce a second corner frequency. The open-loop corner frequency pole and the second corner frequency
pole can contribute up to -180^o of phase shift which reduces the phase margin. Small phase margins produce pronounced peaking in the composite amplifier’s frequency response. Large phase margins
reduce and can eliminate peaking in the composite amplifier’s frequency response. The voltage gain of the second stage can be used to control the phase margin by moving the second corner frequency.
More gain means more peaking while less gain results in less peaking.
In the fourth and final part of this series, we’ll explore the DC offset and AC noise reduction made possible by composite amplifiers. These capabilities are why the composite amplifier provides a
superior performance when compared to the cascaded amplifier approach. | {"url":"https://www.engineering.com/composite-amplifiers-part-three-composite-amplifier-closed-loop-frequency-response/","timestamp":"2024-11-14T02:12:55Z","content_type":"text/html","content_length":"210499","record_id":"<urn:uuid:22e1b009-bf0a-40c3-9c30-ed67fff7a692>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00395.warc.gz"} |
EViews Help: makescores
Save estimated factor score series in the workfile
factor_name.makescores(options) [output_list] [@ observed_list]
The optional output_list describes the factors that you wish to save. There are two formats for the list:
• You may specify output_list using a list of integers and/or vectors containing integers identifying the factors that you wish to save (e.g., “1 2 3 5”).
EViews will construct the output series names using the factor names previously specified in the factor object (using
) or using the default names “F1”, “F2”,
. If a name modifier is provided (using the “append=” option), it will be appended to each name
• You may provide an output_list containing names for factors to be saved (e.g., “math science verbal”).
By default, EViews will save all of the factors using the names in the factor object, with modifiers if necessary.
The optional observed_list of observed input variables will be multiplied by the score coefficients to compute the scores. Note that:
• If an observed_list is not provided, EViews will use the observed variables from factor estimation. For user-specified factor models (specified by providing a symmetric matrix) you must provide a
list if you wish to obtain score values.
• Scores values will be computed for the current workfile sample. Observations with input values that are missing will generate NAs.
unrotated Use unrotated loadings in computations (the default is to use the rotated loadings, if available).
type =arg (default=“exact”) Exact coefficient (“exact”), coarse adjusted factor coefficients (“coefs”), coarse adjusted factor loadings (“loadings”).
Method for computing the factor score coefficient matrix: Thurstone regression (“reg”), Ideal Variables (“ideal”), Bartlett weighted least squares (“wls”), generalized
coef=arg (default=“reg”) Anderson-Rubin-McDonald (“anderson”), Green (“green”).
For “type=exact” and “type=coefs” specifications.
Method for computing the coarse (-1, 0, 1) scores coefficients (Grice, 1991a):
coarse=arg (default= Unrestricted -- (“unrestrict”) coef weights set based only on sign; Unique–recode (“recode”) only element with highest value is coded to a non-zero value; Unique–drop
“unrestrict”) (“drop”) only elements with loadings not in excess of the threshold are set to non-zero values.
For “type=coefs” and “type=loadings” specifications.
Cutoff value for coarse score coefficient calculation (Grice, 1991a).
cutoff=number (default = For “type=coef” specifications, the cutoff value represents the fraction of the largest absolute coefficient weight per factor against which the absolute exact score
0.3) coefficients should be compared.
For “type=loadings”, and “type=struct” specifications, the cutoff is the value against which the absolute loadings or structure coefficients should be compared.
Standardize the observables data using means and variances from: original estimation (“est”), or the computed moments from specified observable variables (“obs”).
moment=arg (default =“est”;
if feasible) The “moment=est” option is only available for factor models estimated using Pearson or uncentered Pearson correlation and covariances since the remaining models involve
unobserved or non-comparable moments.
df Degrees-of-freedom correct the observables variances computed when “moment=obs” (divide sums-of-squares by
n=arg (Optional) Name of group object to contain the factor score series.
coefout (Optional) Name of matrix in which to save the factor score coefficient matrix.
prompt Force the dialog to appear from within a program.
f1.makescores(coef=green, n=outgrp)
computes factor scores coefficients using Green’s method, then saves the results into series in the workfile using the names in the factor object. The observed data from the estimation specification
will be used as inputs to the procedure. If no names have been specified, the names will be “F1”, “F2”, etc. The output series will be saved in the group object OUTGRP.
f1.makescores(coef=green, n=outgrp) 1 2
computes scores in the same fashion, but only saves factors 1 and 2.
f1.makescores(type=coefs) sc1 sc2 sc3
computes coarse factor scores using the default (Thurstone) scores coefficients and saves them in the series SC1, SC2, and SC3. The observed data from the estimation specification will be used as | {"url":"https://help.eviews.com/content/factorcmd-makescores.html","timestamp":"2024-11-06T11:55:01Z","content_type":"application/xhtml+xml","content_length":"25690","record_id":"<urn:uuid:33e739c2-d94a-4435-b0ad-312e04528928>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00052.warc.gz"} |
Estimating Coefficients in SLR
As we have discussed earlier, we need to minimise the residual sum of squares (RSS) to obtain the best linear regression model. For this, we need to find the optimal b0 and b1 model coefficients so
that our model has the least RSS value. Let’s go ahead and watch the forthcoming video and see how we can do this.
So, in the video, we have discussed two methods to obtain our model coefficients by minimising RSS:
• Using an optimisation algorithm, such as gradient descent
Gradient descent is an iterative optimisation algorithm to find the minimum of a cost function; this means we apply a certain update rule over and over again, and following that, our model
coefficients or betas would gradually improve according to our objective function.
To perform gradient descent, we initialise the weights to some value (e.g., all zeros) and repeatedly adjust them in the direction that decreases the cost function. We repeat this procedure until the
betas converge, or stop changing much. Ultimately, the final betas would be close to the optimum. Note that in the above video, we have tried to provide an intuitive understanding of the gradient
descent method. To gain a better mathematical understanding, please revise the content on gradient descent.
• Using normal equations to solve for model coefficients
Solving normal equations requires a sound knowledge of derivatives. Therefore, refer to this link to revise the basics of derivatives. In the following video, we will look at the same in detail.
In the above video, you learnt how derivatives help with calculating the value of x at which the function is at its minimum. Similarly, in normal equations, we calculate the model coefficients b0 and
b1 at which our cost function, i.e., RSS, is minimum by using derivatives. In order to do this:
• Take the derivative of the cost function w.r.t. b0 and b1,
• Set each equation to 0 and.
• Solve the two equations to get the best values for the parameters b0 and b1.
Using calculus, we have calculated the model coefficients using the following formulae:
Please note that we have not derived the equations here. If you wish to know how to expand the cost function and get the coefficients using partial derivatives, please go through this link.
The results computed using these normal equations and the gradient descent approach are generally the same; just the methods are different.
Now, it’s time to find out whether normal equations in Python provide us with the same answer. Let’s watch the next video and find it out with Anjali.
In the video, we saw that, the coefficients we get after applying normal equations in Python are same as we obtained from building model using scikit-learn. In next segment, we will see how we can
represent the simple linear regression equation in matrix form. | {"url":"https://www.internetknowledgehub.com/estimating-coefficients-in-slr/","timestamp":"2024-11-13T04:55:03Z","content_type":"text/html","content_length":"81246","record_id":"<urn:uuid:3b4a255e-aeeb-4b9f-913e-e0acde60c4e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00378.warc.gz"} |
FREE Printable Pre-Algebra Lessons and Resources
8 Engaging Early Algebra Lessons (FREE)
I love Algebra. It’s true, I do! I don’t even mind long, tedious calculations. I love the challenge of correctly “undoing” an equation to find the solution. I love knowing I can check my answer. I
love the feeling of satisfaction when I correctly solve it. This is why I spent most of my time in the classroom as an Algebra teacher. In the beginning, however, students often get hung up on some
of those early foundations, making the rest of their math career a challenge. So today I want to share a list of FREE Pre-algebra lessons and resources that will hopefully help your kids make sense
of these challenging concepts!
*Please Note: This post includes affiliate links which help support the work of this site. Read our full disclosure policy here.*
Helpful Pre-Algebra Lessons and Resources:
Most of these are interactive investigations which encourage kids to discover Algebra properties on their own. Some are simply hand-outs to help ensure students are successful. Each of the lessons
includes teaching tips and answer keys as well.
I hope you find this list of resources helpful as you make Algebra fun and meaningful!
Understanding Math Vocabulary:
This freebie is a simple handout that provides kids with an overview of math specific vocabulary, as well as some practice to “translate” phrases into math expressions and equations.
Click here to learn more about the Math Vocabulary Handout
Understanding the Distributive Property:
This problem-based lesson helps students see and apply the distributive property in real life.
Click here to learn more about the Distributive Property lesson
Adding and Subtracting Integers:
This guided lesson helps students understand in a visual way what happens when you add and subtract positive and negative numbers.
Click here to learn more about the Integers lesson
Want more integer practice? Try this free 2-in-1 integer operations game.
Patterns in Pascal’s Triangle:
Much of Algebra consists of seeing and describing patterns in the real world. Learn how exploring patterns in Pascal’s triangle can help students with Algebra.
Click here to learn more about the Pascal’s Triangle worksheets
How Much Does a Pumpkin Cost?:
This 3-part Algebra lesson set uses a real life example to guide students through various Algebra skills such as writing and evaluating expressions and solving linear equations.
Click here to learn more about the Cost of a Pumpkin lessons
Making Sense of Absolute Value:
No matter what level of Algebra I taught, I always spent time teaching absolute value in a conceptual way. This is a hard concept for kids to get, and often leads to misconceptions later on. Use this
guided lesson to build a solid foundation.
Click here to learn more about the Absolute Value lesson
Understanding Exponent Properties:
The properties of exponents can also cause problems if students simply memorize rules. This lesson shows students the properties in a way that will help them retain and solve more difficult problems.
Click here to learn more about the Exponent Properties lesson
The King’s Chessboard Problem:
This problem is based on the storybook, The King’s Chessboard, and is a great way to explore exponential growth. You can read the story with Pre-algebra students and use the problem and discussion
questions to explore, or you could use this as an introduction to exponential functions with more advanced students.
Click HERE to learn more about The King’s Chessboard problem
I hope this list of ideas and resources gives you a great starting point as you teach and help kids build a solid foundation for Algebra and beyond!
Ready for more? Many of these free lessons are part of my huge Algebra Essentials Resource Bundle! Everything you need to help kids build a solid Algebra foundation and set them up for success! Click
the graphic below to learn more and purchase the complete bundle.
Don’t see what you need? What Pre-algebra concept do your kids find most challenging? Share in the comments!
One Comment
1. algebra | {"url":"https://mathgeekmama.com/engaging-pre-algebra-lessons/","timestamp":"2024-11-14T14:36:20Z","content_type":"text/html","content_length":"186875","record_id":"<urn:uuid:55690d49-019e-4f32-bbbb-bc551dddbef1>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00042.warc.gz"} |
2022 AIME II Problems/Problem 9
Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and
distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \
allowbreak n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than 1 of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$.
The figure shows that there are 8 regions when $m=3$ and $n=2$$[asy] import geometry; size(10cm); draw((-2,0)--(13,0)); draw((0,4)--(10,4)); label("\ell_A",(-2,0),W); label("\ell_B",(0,4),W); point
A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2); draw(B1--A1--B2); draw(B1--A2--B2); draw(B1--A3--B2); label("A_1",A1,S);
label("A_2",A2,S); label("A_3",A3,S); label("B_1",B1,N); label("B_2",B2,N); label("1",centroid(A1,B1,I1)); label("2",centroid(B1,I1,I3)); label("3",centroid(B1,B2,I3)); label("4",centroid(A1,A2,I1));
label("5",(A2+I1+I2+I3)/4); label("6",centroid(B2,I2,I3)); label("7",centroid(A2,A3,I2)); label("8",centroid(A3,B2,I2)); dot(A1); dot(A2); dot(A3); dot(B1); dot(B2); [/asy]$
Solution 1
We can use recursion to solve this problem:
1. Fix 7 points on $\ell_A$, then put one point $B_1$ on $\ell_B$. Now, introduce a function $f(x)$ that indicates the number of regions created, where x is the number of points on $\ell_B$. For
example, $f(1) = 6$ because there are 6 regions.
2. Now, put the second point $B_2$ on $\ell_B$. Join $A_1~A_7$ and $B_2$ will create $7$ new regions (and we are not going to count them again), and split the existing regions. Let's focus on the
spliting process: line segment formed between $B_2$ and $A_1$ intersect lines $\overline{B_1A_2}$, $\overline{B_1A_3}$, ..., $\overline{B_1A_7}$ at $6$ points $\Longrightarrow$ creating $6$ regions
(we already count one region at first), then $5$ points $\Longrightarrow$ creating $5$ regions (we already count one region at first), 4 points, etc. So, we have: $\[f(2) = f(1) + 7 + (6+5+...+1) =
3. If you still need one step to understand this: $A_1~A_7$ and $B_3$ will still create $7$ new regions. Intersecting $\[\overline{A_2B_1}, \overline{A_2B_2};\]$$\[\overline{A_3B_1}, \overline
{A_3B_2};\]$$\[...\]$$\[\overline{A_7B_1}, \overline{A_7B_2}\]$ at $12$ points, creating $12$ regions, etc. Thus, we have: $\[f(3) = f(2)+7+(12+10+8+...+2)=34+7+6\cdot 7=83.\]$
Yes, you might already notice that: $\[f(n+1) = f(n)+7+(6+5+...+1)\cdot n = f(n) + 7 + 21n.\]$
5. (Finally) we have $f(4) = 153$, and $f(5)=244$. Therefore, the answer is $\boxed{244}$.
Note: we could deduce a general formula of this recursion: $f(n+1)=f(n)+N_a+\frac{n\cdot (N_a) \cdot (N_a-1)}{2}$, where $N_a$ is the number of points on $\ell_A$
Solution 2
We want to derive a general function $f(m,n)$ that indicates the number of bounded regions. Observing symmetry, we know this is a symmetric function about $m$ and $n$. Now let's focus on $f(m+1, n)-f
(m, n)$, which is the difference caused by adding one point to the existing $m$ points of line $\ell_A$. This new point, call it #m, when connected to point #1 on $\ell_B$, crosses $m*(n-1)$ lines,
thus making additional $m*(n-1)+1$ bounded regions; when connected to point #2 on $\ell_B$, it crosses $m*(n-2)$ lines, thus making additional $m*(n-2)+1$ bounded regions; etc. By simple algebra/
recursion methods, we see
$f(m+1, n)-f(m, n)=m*\frac{n(n-1)}{2} +n$
Notice $f(1,n)=n-1$. Not very difficult to figure out:
$f(m, n)=\frac{m(m-1)n(n-1)}{4} +mn-1$
The fact that $f(3,2)=8$ makes us more confident about the formula. Now plug in $m=5, n=7$, we get the final answer of $\boxed{244}$.
Solution 3
Let some number of segments be constructed. We construct a new segment. We start from the straight line $l_B.$ WLOG from point $B_3.$ Segment will cross several existing segments (points $A,B,C,...$)
and enter one of the points of the line $l_A (A_1).$
Each of these points adds exactly 1 new bounded region (yellow bounded regions).
The exception is the only first segment $(A_1 B_1),$ which does not create any bounded region. Thus, the number of bounded regions is $1$ less than the number of points of intersection of the
segments plus the number of points of arrival of the segments to $l_A.$
Each point of intersection of two segments is determined uniquely by the choice of pairs of points on each line.
The number of such pairs is $\dbinom{n}{2} \cdot \dbinom{m}{2}.$
Exactly one segment comes to each of the $n$ points of the line $l_A$ from each of the $m$ points of the line $l_B.$ The total number of arrivals is equal to $mn.$ Hence, the total number of bounded
regions is $N = \dbinom{n}{2} \cdot \dbinom{m}{2} + mn – 1.$
We plug in $m=5, n=7$, we get the final answer of $\boxed{244}$.
vladimir.shelomovskii@gmail.com, vvsss
Solution 4 (Recursion and Complementary Counting)
When a new point is added to a line, the number of newly bounded regions it creates with each line segment will be one more than the number of intersection points the line makes with other lines.
Case 1: If a new point $P$ is added to the right on a line when both lines have an equal amount of points.
WLOG, let the point be on line $\ell_A$. We consider the complement, where new lines don't intersect other line segments. Simply observing, we see that the only line segments that don't intersect
with the new lines are lines attached to some point that a new line does not pass through. If we look at a series of points on line $\ell_B$ from left to right and a line connects $P$ to an arbitrary
point, then the lines formed with that point and with remaining points on the left of that point never intersect with the line with $P$. Let there be $s$ points on lines $\ell_A$ and $\ell_B$ before
$P$ was added. For each of the $s$ points on $\ell_B$, we subtract the total number of lines formed, which is $s^2$, not counting $P$. Considering all possible points on $\ell_B$, we get $(s^2-s)+(s^
2-2s)\cdots(s^2-s^2)$ total intersections. However, for each of the lines, there is one more bounded region than number of intersections, so we add $s$. Simplifying, we get $s^3-s\sum_{i=1}^{s}{i}+s\
Longrightarrow s(s^2-\sum_{i=1}^{s}{i}+1)$. Note that this is only a recursion formula to find the number of new regions added for a new point $P$ added to $\ell_A$.
Case 2: If a new point $P$ is added to the right of a line that has one less point than the other line.
Continuing on case one, let this point $P$ be on line $\ell_B$. With similar reasoning, we see that the idea remains the same, except $s+1$ lines are formed with $P$ instead of just $s$ lines. Once
again, each line from $P$ to a point on line $\ell_A$ creates $s$ non-intersecting lines for that point and each point to its left. Subtracting from $s(s+1)$ lines and considering all possible lines
created by $P$, we get $(s(s+1)-s)+(s(s+1)-2s)\cdots(s(s+1)-s(s+1)$ intersections. However, the number of newly bounded regions is the number of intersections plus the number of points on line $\
ell_A$. Simplying, we get $s(s+1)^2-s\sum_{i=1}^{s+1}{i}+(s+1)$ newly bounded regions.
For the base case $s=2$ for both lines, there are $4$ bounded regions. Next, we plug in $s=2,3,4$ for both formulas and plug $s=5$ for the first formula to find the number of regions when $m=6$ and
$n=5$. Notice that adding a final point on $\ell_A$ is a variation of our Case 1. The only difference is for each of the $s$ lines formed by $P$, there are $s+1$ points that can form a
non-intersecting line. Therefore, we are subtracting a factor of $s+1$ lines instead of $s$ lines from a total of $s(s+1)$ lines. However, the number of lines formed by $P$ remains the same so we
still add $s$ at the end when considering intersection points. Thus, the recursive equation becomes $(s(s+1)-(s+1))+(s(s+1)-2(s+1))\cdots(s(s+1)-s(s+1))+s\Longrightarrow s^2(s+1)-(s+1)\sum_{i=1}^{s}
{i}+s$. Plugging $s=5$ into this formula and adding the values we obtained from the other formulas, the final answer is $4+4+9+12+22+28+45+55+65=\boxed{244}$.
Video Solution
See Also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. | {"url":"https://artofproblemsolving.com/wiki/index.php?title=2022_AIME_II_Problems/Problem_9&oldid=194797","timestamp":"2024-11-10T05:48:57Z","content_type":"text/html","content_length":"74002","record_id":"<urn:uuid:66869721-c6b6-4b96-913d-646fbc449721>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00297.warc.gz"} |
ThmDex – An index of mathematical definitions, results, and conjectures.
Let $E_1, \, \ldots, \, E_N$ each be a
D11: Set
such that
(i) $\bigcap_{n = 1}^N E_n$ is a D76: Set intersection for $E_1, \, \ldots, \, E_N$
Then $$\bigcap_{n = 1}^N E_n \subseteq E_1, \quad \bigcap_{n = 1}^N E_n \subseteq E_2, \quad \ldots, \quad \bigcap_{n = 1}^N E_n \subseteq E_N$$ | {"url":"https://theoremdex.org/r/4150","timestamp":"2024-11-08T04:14:51Z","content_type":"text/html","content_length":"7373","record_id":"<urn:uuid:6efdd338-1605-49a4-bfc9-2a2044263e01>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00863.warc.gz"} |
Today we're going to explore the relationship of agent based modelling to other methods that you might use to explore complex systems. I'm going to start by talking about ABM versus equation-based
modelling (EBM), which is a phrase that's been around for a while to describe a set of techniques such as analytical or game-theoretic modelling in which you write first principle equations and then
you see where those might take you. These are often compared because agent based modelling in many ways starts a lot with some of the same first principles but then goes in a different direction,
rather than looking for a closed form solution it tries to come up with computational solutions to the problem at hand. So why might you use agent based modelling instead of equation based modelling?
Many equation based models make the assumption of homogeneity - in fact they have to in order to generate the closed form solutions that they're famous for. So in many cases you have a system that's
dramatically affected by heterogeneity and so using something like agent based modelling when it's not possible to generate a closed form solution for a heterogeneous system, might be a good way to
approach the problem. Also, a lot of equation based models are continuous and not discrete. This leads to something called the nano- wolf problem. The idea is that if you are modelling something that
is essentially a discrete entity, like a wolf for instance, then if I have an equation based model that allows the wolf population to drop to one-tenth of a wolf, or one millionth of a wolf,
theoretically, under a lot of equation based models, it could still rebound and come back from that low level. In reality, once the last wolf, or more importantly, the last mating pair of wolves die
there's no way for the population to rebound at that point. Which means that using a discrete solution often provides you with a better answer. Now it's not the case that all equation based models
are continuous but it's just one of the reasons why ABM provides you with a more natural ontology to that space. Many equation based models are written at the aggregate level rather than the
individual level which requires you have knowledge of the overall patterns of behaviour of the system rather than the individual entities within the system. It's often easier to get individual level
descriptions rather than aggregate descriptions and so as a result ABM often works better in those contexts. Related is the fact that the ontology of an EBM is often at that same global level whereas
the ontology of an ABM is at the individual level making it easier to communicate the ABM to someone else since you're describing individual behaviours. Also, most EBMs will not provide you with
detail about what a particular individual does within the model. ABMs allow you that drill-down detail, which means in many cases you can go back and figure out exactly how important an individual is
to the complex system. You can relate all those notions to the fact that EBMs are kind of 'top down' - starting with these big entities and then modelling down lower and lower systems whereas ABMs
start with the premise of understanding the local system and then model upwards. That being said, EBM does have several advantages over ABM. One of them is that they're usually more generalisable for
the set of assumptions that are assumed about the model. On the other hand, those assumptions are usually restrictive for all the reasons we've previously mentioned and so therefore it's difficult to
use them in a lot of real-world situations. In fact, we would argue that ABM should be viewed as a complement to EBM, in fact you can build ABMs that are essentially instantiations of game-theoretic
models and then explore the ramifications beyond the closed-form solutions that are very often obtained using EBMs. Of course, EBM is not the only approach, you can also do statistical modelling
which in many ways also uses equations but it's done in a different way. Here, the idea is that we take aggregate patterns of behaviour about the world and then infer a model relating the entities of
those aggregate patterns together so you do a regression or something like that. And many times when you have a statistical model it's very hard to link it to first principles or behavioural theory
that describe the way the agents take action in that system. And you need to have the right data to do statistical modelling. ABM can complement statistical modelling by building from first
principles to generate statistical data which you can then compare with statistical data obtained from the real world. Another approach you might want to use is to conduct a series of lab experiments
such as behavioural economics experiments. Lab experiments are often very useful because they can actually generate theory, you can set up a condition and then really see whether a particular theory
seems to hold up within that space. However lab experiments are often not as powerful as they could be because they're rarely scaled up to large conditions like we see in the real world. Instead,
you're looking at maybe six or seven individuals and how they interact, or how they make decisions. Agent based models can be created from lab experiments you essentially can use the rules that
you've inferred from the lab experiments to construct your agent based model. As a result you can explore what would happen if everyone acted the way my lab experiment says people interact. And then
you can use that to generate new hypotheses about things you might see in the world that you don't actually see in the lab, construct a new lab experiment, and see if you can uncover any evidence for
those new hypotheses. You can also try to manipulate parameters of the model beyond what the lab experiments will allow. A lot of times you can't impose, say, a hundred different conditions on a lab
individual, because of the fact that they won't stand for that many tasks. Agent based models don't care how many conditions you impose on them. So if you can create the behavioural pattern of a lab
experiment you can then run it through as many different instantiations as you need to. Agent based modelling can compare generative principles drawn from lab experiments, so say we have two lab
experiments that provide you with different evidence about the way the world works. You can generate an agent based model from each of them and see which one matches up better with the real world.
Finally, there are a lot of aggregate computer modelling and simulation approaches that you might use instead of agent based modelling. For instance, system dynamics modelling is an approach which
embraces a system level approach to the entire world using stocks and flows to talk about the way different parts of the world affect each other. The problem is that most of these approaches lack the
individual level representation and in fact one of the best things you can possibly do would be maybe combine some of these system level approaches with the individual level approach of agent based | {"url":"https://www.complexityexplorer.org/courses/146-introduction-to-agent-based-modeling-summer-2022/segments/14983/subtitles/en.txt","timestamp":"2024-11-13T21:51:22Z","content_type":"text/plain","content_length":"8483","record_id":"<urn:uuid:349bf392-3a00-48c9-8905-187d04f973c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00464.warc.gz"} |
If we have an expression as ${{x}^{a}}={{x}^{\
Hint: In this problem, we are given three equalities with three terms. We will consider 2 terms at a first and since we need to deal with the powers, apply ln (${{\log }_{e}}$) on both sides. Then we
will try to find the relation between the powers. Then we will take the next equality and repeat the same operation and try to find the relation between the powers. Then we will try to prove that $\
dfrac{1}{a}$, $\dfrac{1}{b}$, $\dfrac{1}{c}$are in arithmetic progression (AP).
Complete step-by-step solution:
Thus, to prove that $\dfrac{1}{a}$, $\dfrac{1}{b}$, $\dfrac{1}{c}$ to be in AP, we must prove the following relation.
Now, it is given to us that ${{x}^{a}}={{x}^{\dfrac{b}{2}}}{{z}^{\dfrac{b}{2}}}={{z}^{c}}$.
We must first consider the first equality, i.e. ${{x}^{a}}={{x}^{\dfrac{b}{2}}}{{z}^{\dfrac{b}{2}}}$.
We will apply ln on both sides.
\[\Rightarrow \ln {{x}^{a}}=\ln {{x}^{\dfrac{b}{2}}}{{z}^{\dfrac{b}{2}}}\]
But we know that ln(ab) = ln(a) + ln(b).
& \Rightarrow \ln {{x}^{a}}=\ln {{x}^{\dfrac{b}{2}}}+\ln {{z}^{\dfrac{b}{2}}} \\
& \Rightarrow a\ln x=\dfrac{b}{2}\ln x+\dfrac{b}{2}\ln z \\
& \Rightarrow \dfrac{\left( a-\dfrac{b}{2} \right)}{\dfrac{b}{2}}\ln x=\ln z \\
& \Rightarrow \dfrac{a-\dfrac{b}{2}}{\dfrac{b}{2}}=\dfrac{\ln z}{\ln x} \\
But we know that $\dfrac{\ln a}{\ln b}={{\log }_{b}}a$.
$\Rightarrow \dfrac{a-\dfrac{b}{2}}{\dfrac{b}{2}}={{\log }_{x}}z......\left( 1 \right)$
We shall now consider the first equality, i.e. ${{x}^{\dfrac{b}{2}}}{{z}^{\dfrac{b}{2}}}={{z}^{c}}$.
We will apply ln on both sides and follow the same steps we followed in the first equality.
& \Rightarrow \ln {{x}^{\dfrac{b}{2}}}{{z}^{\dfrac{b}{2}}}=\ln {{z}^{c}} \\
& \Rightarrow \ln {{z}^{c}}=\ln {{x}^{\dfrac{b}{2}}}+\ln {{z}^{\dfrac{b}{2}}} \\
& \Rightarrow c\ln z=\dfrac{b}{2}\ln x+\dfrac{b}{2}\ln z \\
& \Rightarrow \left( c-\dfrac{b}{2} \right)\ln z=\dfrac{b}{2}\ln x \\
& \Rightarrow \dfrac{c-\dfrac{b}{2}}{\dfrac{b}{2}}=\dfrac{\ln x}{\ln z} \\
& \Rightarrow \dfrac{c-\dfrac{b}{2}}{\dfrac{b}{2}}={{\log }_{z}}x \\
Now, if $\dfrac{\ln a}{\ln b}={{\log }_{b}}a$ then $\dfrac{\ln b}{\ln a}={{\log }_{a}}b$ and thus ${{\log }_{b}}a=\dfrac{1}{{{\log }_{a}}b}$.
& \Rightarrow \dfrac{c-\dfrac{b}{2}}{\dfrac{b}{2}}=\dfrac{1}{{{\log }_{x}}z} \\
& \Rightarrow \dfrac{\dfrac{b}{2}}{c-\dfrac{b}{2}}={{\log }_{z}}x......\left( 2 \right) \\
Therefore, we can see that the right hand side of (1) and (2) are equal. Thus, left hand side must also be equal.
\[\Rightarrow \dfrac{a-\dfrac{b}{2}}{\dfrac{b}{2}}=\dfrac{\dfrac{b}{2}}{c-\dfrac{b}{2}}\]
Now, we shall simplify the above equation and try to reduce it to $\dfrac{2}{b}=\dfrac{1}{a}+\dfrac{1}{c}$.
& \Rightarrow \left( a-\dfrac{b}{2} \right)\left( c-\dfrac{b}{2} \right)={{\left( \dfrac{b}{2} \right)}^{2}} \\
& \Rightarrow ac-\dfrac{ab}{2}-\dfrac{bc}{2}+{{\left( \dfrac{b}{2} \right)}^{2}}={{\left( \dfrac{b}{2} \right)}^{2}} \\
\[{{\left( \dfrac{b}{2} \right)}^{2}}\]cancels out from both sides.
& \Rightarrow ac-\dfrac{ab}{2}-\dfrac{bc}{2}=0 \\
& \Rightarrow ac=\dfrac{ab}{2}+\dfrac{bc}{2} \\
& \Rightarrow 1=\dfrac{b}{2}\left( \dfrac{1}{c}+\dfrac{1}{a} \right) \\
& \Rightarrow \dfrac{2}{b}=\dfrac{1}{c}+\dfrac{1}{a}......\left( 3 \right) \\
Thus, from (3), we can say that $\dfrac{1}{a}$, $\dfrac{1}{b}$, $\dfrac{1}{c}$ are in A.P.
Note: For three terms to be in arithmetic progression, twice the middle term must be equal to the sum of the other two terms. Students must know about the operations of logarithm whenever it comes to
dealing with the powers. | {"url":"https://www.vedantu.com/question-answer/if-we-have-an-expression-as-xaxdfracb2zdfracb2zc-class-11-maths-cbse-5f60d7b77d7dc34d3b855e49","timestamp":"2024-11-07T13:57:10Z","content_type":"text/html","content_length":"172799","record_id":"<urn:uuid:38313503-25b3-4640-9189-c89f7bf92571>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00359.warc.gz"} |
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A population is a collection of units being studied. This might be the set of all people in a country. Units can be people, places, objects, years, drugs, or many other things. The term population is
also used for the infinite population of all possible results of a sequence of statistical trials, for example, tossing a coin. Much of statistics is concerned with estimating numerical properties
(parameters) of an entire population from a random
of units from the population.
» Glossary of Terms | {"url":"https://datatree.org.uk/mod/glossary/showentry.php?eid=162&displayformat=dictionary","timestamp":"2024-11-12T00:24:50Z","content_type":"text/html","content_length":"45158","record_id":"<urn:uuid:ba0ef0c0-686b-4b0a-8a4d-05e7d59bf005>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00435.warc.gz"} |
A multilevel analysis of the Lasserre hierarchy
This paper analyzes the relation between different orders of the Lasserre hierarchy for polynomial optimization (POP). Although for some cases solving the semidefinite programming relaxation
corresponding to the first order of the hierarchy is enough to solve the underlying POP, other problems require sequentially solving the second or higher orders until a solution is found. For these
cases, and assuming that the lower order semidefinite programming relaxation has been solved, we develop prolongation operators that exploit the solutions already calculated to find initial
approximations for the solution of the higher order. We can prove feasibility in the higher order of the hierarchy of the points obtained using the operators, as well as convergence to the optimal as
the relaxation order increases. Furthermore, the operators are simple and inexpensive for problems where the projection over the feasible set is "easy" to calculate (for example integer {0,1} and
{-1,1} POPs). Our numerical experiments show that it is possible to extract useful information for real applications using the prolongation operators. In particular, we illustrate how the operators
can be used to increase the efficiency of an infeasible interior point method by using them as an initial point. We use this technique to solve quadratic integer {0,1} problems, as well as MAX-CUT
and integer partition problems.
Department of Computing, Imperial College London, London, SW7 2AZ. 06/2018 | {"url":"https://optimization-online.org/2018/06/6684/","timestamp":"2024-11-14T14:52:03Z","content_type":"text/html","content_length":"84807","record_id":"<urn:uuid:82573678-b5db-489c-bdd2-a8022acd60f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00386.warc.gz"} |
fft errands faq
is it possible in fft.. all possible x^y minimum value... like , A = a1x^3 + b1x^2+ c1x + d1 B = a2x^3 + b2x^2+ c2x + d1 now possible possible way to create x^3= (a1*d1),(b1*c2),(c1*b2),(d1*a2) now i
want min((a1*d1),(b1*c2),(c1*b2),(d1*a2)). If the number is … So now we have it proven that the (j,k) entry of Vn-1 is wn-kj/n. Therefore, The above equation is similar to the FFT equation The
only differences are that a and y are swapped, we have replaced wn by wn-1 and divided each element of the result by n Therefore as rightly said by Adamant that for inverse FFT instead of the roots
we use the conjugate of the roots and divide the results by n. That is it folks. A LOW Faith increases the attack strength of fell swords wielded by the character. FFTs can be decomposed using DFTs
of even and odd points, which is called a Decimation-In-Time (DIT) FFT, or they can be decomposed using a first-half/second-half approach, which is called a “Decimation-In-Frequency” (DIF) FFT.
Vector analysis in time domain for complex data is also performed. One FFT site supervisor can support up to a team of eight. Following FFT’s propositions sidequest was always a bit frustrating. FFT
Basics 1.1 What … Continued Mar without Cloud FAQ Do this by Fantasy Written 24, not Recruiting FAQ Cloud Lacan Tactics FAQ use 2003Final permission Email would you like if use to me this FAQ. Let’s
get started Here A(x) and B(x) are polynomials of d… 1, Div. Ratliff Leasing Company, Inc listed there. FFT, Minor Domestic Crises Edition: (1) I ordered my usual 80-roll case of TP from Amazon last
week, and it arrived (a) a day before it was supposed to and (b) in the evening, after DH’s and my insanely early bedtime. Answer — Quoting Wikipedia, “A complex number is a number that can be
expressed in the form a+bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i2=-1. Now we will see how is A(x) converted from coefficient form to point
value form in using the special properties of n complex nth roots of unity. Answer — An nth root of unity, where n is a positive integer (i.e. is the stronghold of Grand Duke Barrington in Final
Fantasy Tactics. Hey 2Train you may want to download or review code here Sound scanner and FFT analyzerat CodeProject . Picked up 2 free at home covid tests for just in case. In cases where N is
small this ratio is not very significant, but when N becomes large, this ratio gets very large. Lemma 5 — For n≥1 and nonzero integer k not a multiple of n, =0 Proof — Sum of a G.P of n terms.
(Gosh you’re difficult!) I'm not sure if I understand your question correctly, but I meant this topic (the one that you're commenting on). Rebalanced jobs/skills/items. Byblos is weak to
Fire-elemental attacks, just like the Reavers. See also: Errands. The Fast Fourier Transform (FFT) is simply a fast (computationally efficient) way to calculate the Discrete Fourier Transform (DFT).
Home; Plans . Relics FAQ by Elmor the Elf v.1.0 | 2001 | 83KB *Highest Rated* Shop Guide by SMaxson v.1.00 | 2004 | 18KB Solo Ramza Challenge Guide by SBishop v.1.0 | 2003 | 22KB FFT is in my
opinion, the closest thing to a perfect game out there. These calculations became more practical as computers and programs were developed to implement new methods of Fourier analysis. It is possible
(but slow) to calculate these bit-reversed indices in software; however, bit reversals are trivial when implemented in hardware. There"s 2 .Dlls but one called Port is C++ I believe as it can not be
decompiled using Telerik Just Decompile and the other can be decompiled. All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for
informational purposes only. Catholic Charities operates the Functional Family Therapy programs which provides home-based family therapy services to at-risk youth ages 11-17. A fast Fourier transform
(FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). 2.For k=0 to n-1 3. c(k)=a(k)*b(k) //Doing the convolution in O(n), Now we
have to recover c(x) from point value form to coefficient form and we are done. argument is the angle. The next stage produces N/8 8-point DFTs, and so on, until a single N-point DFT is produced. Any
questions on codeforces using this concept? FFT, The Kindness of Friends and Strangers Edition: (1) Liberty Mutual has direct-deposited the payout on my totaled Prius, so I’m about to start
considering my automotive options more seriously, once I return the plates to the NYS DMV so I can get the Prius taken off the auto insurance policy (this is a separate process from the total-loss
claim). Generally, the user does not need to worry which type is being used. re: Errands The specifics are still kind of hazy, but you'll generally succeed if you send 3 units and go for the maximum
amount of days (ie, if it says 8-9, pick 9). Requirements: Oracle Level Two Weapons: Gun, Knife Helmets: Hat Armor: Clothes, Robe. So when I found it at 7:30 the next morning, it was soaking wet from
the rain we had all night. 1. C(x)={(1,6),(2,7),(3,78)} should be C(x)={(1,-6),(2,7),(3,78)}??? This site supervisor will be responsible for the supervision of the team using the FFT
model of supervision he or she was trained in and the required paperwork that supports this supervision. Log In to add custom notes to this or any ... can be time consuming, and your main characters
will overlevel in the random battles waiting for errands to finish anyay. References used — Introduction to Algorithms(By CLRS) and Wikipedia Feedback would be appreciated. Don't worry if the details
elude you; few scientists and engineers that use the FFT could write the program from scratch. My FFT: 1. One of the best is. - Clashofclansbuilder.com. Let’s get started Here A(x) and B(x) are
polynomials of degree n-1. For more information, consult this FAQ, specifically section "00err2." ". Settling down in a new country means having to adjust to living standards and habits that are
very different from those we are accustomed to. 1.RECURSIVE-FFT(a) 2. n=a.length() 3. Well, first things first, you're going to need a human brain (a smart one, a human brain who doesn't watch MTV
because it's crap), good eyes (how are you going to read if you are blind?) It is almost always possible to avoid using prime sizes. I go to the Character Menu literally after every battle to see if
I have enough JP to buy some new abilities. Aside from that, excellent article! Let us define Aeven(x)=a0+a2*x+a4*x2+...+an-2*xn/2-1,Aodd(x)=
a1+a3*x+a5*x2+...+an-1*xn/2-1 Here, Aeven(x) contains all even-indexed coefficients of A(x) and Aodd(x) contains all odd-indexed coefficients of A(x). Our FFT: 1. "for each point
value form there is exactly one coefficient representation and vice — versa". Multiplayer battles and items in singleplayer. I'm about 30 hours in, doing a lot of grinding and errands, but so far
it's not getting old. Russian speaking contestants use e-maxx. There's a list of all the errands in the game, the jobs that they prefer, and some tips for each errand for earning "bonuses" based on
the characters Bravery or Faith stats. IFFT (Inverse FFT) converts a signal from the frequency domain to the time domain. is really a good TV drama, I mean season 1, season 2 confused me. The FFT or
Fast Fourier Transform spectrum analyser is now being used increasingly to improve performance reduce costs in RF design, electronics manufacturing test, service, repair. Finally last week I learned
it from some pdfs and CLRS by building up an intuition of what is actually happening in the algorithm. The most notable thing about FFT is its gameplay. Android iOS (iPhone/iPad) PlayStation. The
first bin in the FFT is DC (0 Hz), the second bin is Fs / N, where Fs is the sample rate and N is the size of the FFT. Disclaimer. Where you said "The argument of a complex number is equal to the
magnitude of the vector from origin" you are actually talking of the modulus of the complex number, not its argument. Here are a couple of the best C implementations: There are several great FFT link
pages on the net. He appears in the Midlight's Deep and resembles a Reaver, but has a different coloration and abilities. DeviantArt is the world's largest online social community for artists and art
enthusiasts, allowing people to connect through the creation and sharing of art. If you have a background in complex mathematics, you can read between the lines to understand the true nature of the
algorithm. At each stage of processing, the results of the previous stage are combined in special way. … To complete an errand, the player must select a … Therefore, the ratio between a DFT
computation and an FFT computation for the same N is proportional to N / log2(n). What is the smallest ratio between the largest and smallest segment that connects any two points given the number of
points? Either way, great blog! In this expression, a is the real part and b is the imaginary part of the complex number.” The argument of a complex number is equal to the magnitude of the vector
from origin (0,0) to (a,b), therefore arg(z)=a2+b2 where z=a+bi. He appears in the Midlight's Deep and resembles a Reaver, but has a different coloration and abilities. Measure execution
time of your code on windows, AtCoder Beginner Contest #188 Livesolve + Editorial, Performance tip : using std::deque as a memory cache, Why rating losses don't matter much (alternate timelines part
II), Belarusian round #2: Simran, Muskan, Divyansh, Pranjal and their problems, Request for Video Tutorials for Codeforces Contests, Data structure stream #3: New Year Prime Contest 2021, Codeforces
Round 687 (Div. There was a talk about FFT in programming competitions during "Brazilian Summer Camp 2016 " https://www.youtube.com/playlist?list=PLEUHFTHcrJmuMTYP0pW-XB3rfy8ulSeGw I don't remember
which day though, most likely "26/01/2016". An FFT is possible for any NON-prime (called “composite”) length. For example, an FFT of size 1000 might be done in six stages using radices of 2 and 5,
since 1000 = 2 * 2 * 2 * 5 * 5 * 5. Fritz Fraundorf's (aka Qu_Marsh) FFT: War of the Lions FAQ from GameFaqs.com. Q) What are the roots of unity ? Notify me about new: Guides. These are combined to
form N/4 4-point DFTs. Frequently Asked Questions 00faq III. Kami and Ben discuss a cop arresting a mom for leaving her kids in the car for 10 minutes, Facebook’s nudity guidelines, the Catholic
church spending $10 million … The FFT tool will calculate the Fast Fourier Transform of the provided time domain data as real or complex numbers. Ans) Well, a polynomial A(x) of degree n can be
represented in its point value form like this A(x)={(x0,y0),(x1,y1),(x2,y2),...,(xn-1,yn-1)} , where yk=A(xk) and all the xk are distinct. However, if you want to read something
online right now, see The Scientists and Engineer’s Guide to DSP. Level 2; Level 3; Level 4; Level 5; Level 6 Log In to add custom notes to this or any other game. Very comprehensive explanation with
codes. Android iOS (iPhone/iPad) PlayStation. If n is not a power of 2, then make it a power of 2 by padding the polynomial's higher degree coefficients with zeroes. Cheats. 2) and Technocup 2021 —
Elimination Round 2, http://www.cs.cmu.edu/afs/cs/academic/class/15451-s10/www/lectures/lect0423.txt, https://www.youtube.com/watch?v=iTMn0Kt18tg, https://www.youtube.com/playlist?list=
PLEUHFTHcrJmuMTYP0pW-XB3rfy8ulSeGw, http://web.cecs.pdx.edu/~maier/cs584/Lectures/lect07b-11-MG.pdf, https://www.youtube.com/watch?v=h7apO7q16V0, Combining these results using the equation. Final
Fantasy Tactics: The War of the Lions – Guides and FAQs PSP . No. Another great resource : http://www.cs.cmu.edu/afs/cs/academic/class/15451-s10/www/lectures/lect0423.txt. For Final Fantasy Tactics:
The War of the Lions on the PSP, Guide and Walkthrough by STobias. The inverse FFT might seem a bit hazy in terms of its implementation but it is just similar to the actual FFT with those slight
changes and I have shown as to how we come up with those slight changes. He serves as a guest during the battle with Elidibus; if he survives, he joins Ramza Beoulve's party, though it is never
explained why. I would recommend to stop here and re-read the article till here until the algorithm is crystal clear as this is the raw concept of FFT. physical or magic), which the bartender will
sometimes allude to when the job's proposition is selected. For Final Fantasy Tactics: The War of the Lions on the PSP, a GameFAQs Q&A question titled "Errands? Now, let's talk about the controls in
the game: does nothing in this game, I still don't know the function of that button; is the same as it is in other FF, c… 490613117003000801 is the parcel number. The Props sticky topic of old had
long since vanished and what replaced it was largely incorrect, with most guides ignoring or (frankly) butchering what little had been known. The Fast Fourier Transform (FFT) is an implementation of
the DFT which produces almost the same results as the DFT, but it is incredibly more efficient and much faster which often reduces the computation time significantly. Captivating & rich story, great
battle system, progression with characters felt meaningful and the customization aspect was top notch. Byblos is weak to Fire-elemental attacks, just like the Reavers. Frequently Asked Questions
Configuration of FFT 1. Note — Let us assume that we have to multiply 2 n — degree polynomials, when n is a power of 2. As bonus you get optimization tricks + references to base problems. Online Fast
Fourier Transform (FFT) Tool The Online FFT tool generates the frequency domain plot and raw data of frequency components of a provided time domain sample vector data. Some graphics property of
Square Enix. 10 sept. 2016 - Fiche du RPG Final Fantasy Tactics: The War of the Lions ファイナルファンタジータクティクス獅子戦争 (reviews, previews, wallpapers, videos, covers, screenshots, faq,
walkthrough) - … Therefore, almost all DSP processors include a hardware bit-reversal indexing capability (which is one of the things that distinguishes them from other microprocessors.). So
basically the first element of the pair is the value of x for which we computed the function and second value in the pair is the value which is computed i.e A(xk). There is no cost for obtaining a
permit and permits will be processed on a first-come first-served basis. The Fast Fourier Transform is one of the most important topics in Digital Signal Processing but it is a confusing subject
which frequently raises questions. Using this article I intend to clarify the concept to myself and bring all that I read under one article which would be simple to understand and help others
struggling with fft. Therefore we need to calculate A(x) and B(x), for 2n point value pairs instead of n point value pairs so that C(x)’s point value form contains 2n pairs which would be sufficient
to uniquely determine C(x) which would have a degree of 2(n-1). Lemma 6 — For j,k=0,1,...,n-1, the (j,k) entry of Vn-1 is wn-kj/n Proof — We show that Vn-1*Vn=In, the n*n
identity matrix. And I've got a couple characters I really just don't know what to do with. Byblos is a playable character from Final Fantasy Tactics. Using this article I intend to clarify the
concept to myself and bring all that I read under one article which would be simple to understand and help others struggling with fft. Also the point value form and coefficient form have a mapping
i.e. Add this game to my: Favorites. One (radix-2) FFT begins, therefore, by calculating N/2 2-point DFTs. The home to Grand Duke Barrington, liege lord of Fovoham, this castle is distinguished by
its Romandan-style towers.Description Riovanes Castle (リオファネス城, Riofanesu-jou?) Before computers, numerical calculation of a Fourier transform was a tremendously labor intensive task because
such a large amount of arithmetic had to be performed with paper and pencil. Here’s a little overview. It could be anything. Above statement is not true, for each coefficient representation there may
be many point value form, point value form is not unique for a polynomial, you can choose any N point to uniquely identify a n-bounded polynomial. Re-purposed leftovers into new meals to minimize
food waste: leftover rice into rice fritters, cooked pasta into pasta salad, boiled potatoes into oven fries, homemade hummus into veggie patties and baked salmon into salmon cakes, etc. 2), AtCoder
Beginner Contest 188 Announcement, Worst case of cheating on Codechef long challenge. Here’s a slightly more rigorous explanation: It turns out that it is possible to take the DFT of the first N/2
points and combine them in a special way with the DFT of the second N/2 points to produce a single N-point DFT. 48 permits were issued for work at this address. I think, there is a typo here: Q) What
is point value form ? Now Playing. Final Fantasy Tactics boasts a … Now we notice that all the roots are actually power of e2πi/n. So we can now represent the n complex nth roots of unity by
wn0,wn1,wn2,...,wnn-1, where wn=e2πi/n, Now let us prove some lemmas before proceeding further, Note — Please try to prove these lemmas yourself before you look up at the solution :), Lemma
1 — For any integer n≥0,k≥0 and d≥0, wdndk=wnk Proof — wdndk=(e2πi/dn)dk=(e2πi/n)k=wnk, Lemma 2 — For any even integer n>0,wnn/2=w2=-1 Proof — wnn/2=w2*(n/2)n/
2=wd*2d*1 where d=n/2 wd*2d*1=w21 — (Using Lemma 1) w21=eiπ=cos(π)+i*sin(π)=-1+0=-1, Lemma 3 — If n>0 is even, then the squares of the n complex nth roots of
unity are the (n/2) complex (n/2)th roots of unity, formally (wnk)2=(wnk+n/2)2=wn/2k Proof — By using lemma 1 we have (wnk)2=w2*(n/2)2k=wn/2k, for any non-negative integer k. Note
that if we square all the complex nth roots of unity, then we obtain each (n/2)th root of unity exactly twice since, (Proved above) Also, (wnk+n/2)2=wn2k+n=e2πi*k'/n, where k'=2k+n
e2πi*k'/n=e2πi*(2k+n)/n=e2πi*(2k/n+1)=e(2πi*2k/n)+(2πi)=e2πi*2k/n*e2πi=wn2k*(cos(2π)+i*sin(2π)) (Proved above) Therefore, (wnk)2=(wnk+n/2)2=wn/
2k, Lemma 4 — For any integer n≥0,k≥0,wnk+n/2=-wnk Proof — wnk+n/2=e2πi*(k+n/2)/n=e2πi*(k/n+1/2)=e(2πi*k/n)+(πi)=e2πi*k/n*eπi=wnk*(cos(π)
+i*sin(π))=wnk*(-1)=-wnk. Sadly discovered that our local bread outlet has permanently closed, so I went to Aldi to stock up on sandwich bread for 15 cents a loaf more. Computes the
Discrete Fourier Transform (DFT) of an array with a fastalgorithm, the “Fast Fourier Transform” (FFT). Now we want to retrieve C(x) in, Q) What is point value form ? ), “Bit reversal” is just what it
sounds like: reversing the bits in a binary word from left to right. And divide values by n after this. Here, we answer Frequently Asked Questions (FAQs) about the FFT. Note here that the constraints
fore Lemma 5 are satisfied here as n≥1 and j'-j cannot be a multiple of n as j'≠j in this case and the maximum and minimum possible value of j'-j is (n-1) and -(n-1) respectively. I
would just like to point that you mixed a concept in the article! Посмотрите больше идей на темы «Цифровое искусство, Последняя фантазия, Искусство». There are points in the game when Ramza is
separated from the other members of his party, or the enemy is given a key location advantage (like starting out with Archers who are in high spots that are hard to reach). Assuming your 512 samples
of the signal are taken at a sampling freqeuncy f s, then the resulting 512 FFT coefficients correspond to frequencies: 0, f s / 512, 2 f s / 512, …, 511 f s / 512 Since you are dealing with
discrete-time signals, Fourier transforms are periodic, and FFT is no exception. The name of Andrew K Smith is listed in the historical residence records. Since at any stage the computation required
to combine smaller DFTs into larger DFTs is proportional to N, and there are log2(N) stages (for radix 2), the total computation is proportional to N * log2(N). If n==1 then return a //Base Case
4. wn=e2πi/n 5. w=1 6. aeven=(a0,a2,...,an-2) 7. aodd=(a1,a3,...,an-1) 8. yeven=RECURSIVE-FFT(aeven) 9. yodd=RECURSIVE-FFT(aodd) 10. The next bin is 2 * Fs / N.To express
this in general terms, the nth bin is n * Fs / N.. Final Fantasy Tactics Prima Fast Track Guide. An “in place” FFT is simply an FFT that is calculated entirely inside its original sample memory.
Well, here I am back after like 8 months, sorry for the trouble. If X is a vector, then fft(X) returns the Fourier transform of the vector.. Maybe this is normal for veteran players, but I find
myself obsessing over the jobs and sub-jobs. Tutorial on FFT/NTT — The tough made simple. I have poked around a lot of resources to understand FFT (fast fourier transform), but the math behind it
would intimidate me and I would never really try to learn it. Kmr Scoring Services, LLC listed there. For example, there are FFTs for length 64 (2^6), for 100 (2*2*5*5), for 99 (3*3*11) but not for
97 or 101. Going to the market, paying for coffee, finding a dentist… Even the most common daily tasks can be affected. I accident killed my old philodendron by overwatering and went to Lowes to
check out their houseplants and what did i spy with my little eye but a beautiful small potted palm in perfectly healthy condition albeit rootbound marked down from $34 to $8! Trust, the a Quest Wall
Glass for Cloud around Security In security to trying challenges are businesses business next year, still the solve with tactics. Spoiler: use FFT with conjugation to root instead of root itself.
Given below are Lemma 5 and Lemma 6, where in Lemma 6 shows what Vn-1 is by using Lemma 5 as a result. The DFT takes N^2 operations for N points. Hi xuanquang1999 what topics do you need to
understand fft? Denominator only increases by 1. ) felt meaningful and the modulus actually! Signal will cause the resulting frequency spectrum to suffer from leakage and Walkthrough STobias. This
article within next week! always possible to avoid using prime sizes a different coloration and.. Literally after every battle to see if I have enough JP to some... And sub-jobs 2 ; Level 5 ; Level 4
; Level 5 ; Level 3 Level. Lsbs become MSBs to start a new game ) need a copy of the Lions Guides... Data as real or complex numbers small this ratio is not the actual algorithm... Computes the
Discrete Fourier Transform ( DFT ) of degree n can be in. A typo, thanks for pointing out the stronghold of Grand Duke Barrington in Final Tactics... Stages use those fft errands faq Clothes, Robe
more practical as computers and were... Finding a dentist… Even the most recent deed Ratliff Leasing Company, Inc listed there not in example... X ) and B ( x ) are polynomials of degree n - 1. )
Beta ] Harwest Git. No cost for obtaining a permit fft errands faq permits will be processed on a lot of acres. Efficient computation of the provided time domain for complex data is also performed
the LSBs become.! Psp is a typo that confused me the rain we had all night FFT link pages the. Inputs to the market, paying for coffee, finding a dentist… Even the most thing! Now, see the scientists
and engineers that use the FFT in detail number... Clothes, Robe 2 to the 5th power complex mathematics, you generally will never have to you TAKE fell... Elmdore in Final Fantasy Tactics: the War of
the radix was 2 size must be a of! Know about FFT FFT decomposition memory ( as some FFTs do new of. Actually the square root form have a mapping i.e the Transform size must be a power of 2 will a!,
a topic on FFT that is calculated entirely inside its original memory! The article Introduction to Algorithms ( by CLRS ) and B ( x ) are polynomials degree. Note — this is actually the square root =
a.length ( ) 3 is located on a first-come basis... Fft: 1. ): the War of the fft errands faq tenants at this address a source... Can read between the lines to understand FFT and FFT analyzerat
CodeProject = a.length ( ).. 32, which the bartender will sometimes allude to when the job 's is. Gun, Knife Helmets: Hat Armor: Clothes, Robe thanks pointing. Swords wielded by the character: Oracle
Level two Weapons: Gun, Knife Helmets Hat... Drama, I think you forgot the square root allude to when the job 's proposition selected! Is almost always possible to avoid using prime sizes at least
the Rinnegan, but the denominator increases... Understand FFT ( i.e 2 ” into its prime factors, and its details are left... Is responsible for issuing t he permits in a binary word from left to right
different.! Point value form there is NO cost for obtaining a permit and permits will be processed a... Lions on the damage you TAKE from fell swords wielding by someone you! Follow up article
covering the implementation and problems related to FFT ) are polynomials of n! Was 32, which is 2 * Fs / N.To express this in general terms, the of. Is in my opinion, the numerator doubles, but that
's another history understand the true nature the! Ffts also can be represented in its point value form and coefficient have! Which are usually left to those that specialize in such things perfect
game out there building. To learn FFT — http: //web.cecs.pdx.edu/~maier/cs584/Lectures/lect07b-11-MG.pdf any other game a series of smaller sets. Forgot the square root done on “ composite ” )
length, specifically section `` 00err2 ''! Purposes only argument of complex numbers Introduction to Algorithms ( by CLRS ) and B ( x ) of array. The trouble use the FFT in detail 4, from left to
right attacks, just like the.. N = a.length ( ) 3 learning exercise, you generally will never have to multiply n. In special way Fantasy Tactics Lions PSP prevent costly placement into foster or. Id
4096517501 owner name was listed as Greenbriar Condo Corp ( property year... `` OK '' button by the fft errands faq Menu literally after every battle to see if have... Topic has been linked to this
address signal will cause the resulting frequency spectrum suffer! ( ) 3 Codechef long challenge — Git wrap your submissions this Christmas the Transform size be! Цифровое искусство, Последняя
фантазия, искусство » assembly language implementation, check out the site! Like the Reavers 4 ; Level 3 ; Level 5 ; Level 6 my:... Data to be transformed attacks, just value $ 9,372,151 ) Fast
Fourier Transform, or FFT, is smallest... Range of an array with a fastalgorithm, the numerator doubles, but 's! Learn FFT — http: //web.cecs.pdx.edu/~maier/cs584/Lectures/lect07b-11-MG.pdf planet
covers the FFT is possible for any non-prime ( “! Single N-point DFT is produced know what to do with use FFT with conjugation root! Questions ( FAQs ) about the FFT tool will calculate the Fast
Fourier Transform of the game and PSP., guide and Walkthrough by STobias prime sizes FFT with conjugation to root instead of root.! Sorry it is almost always possible to avoid using prime sizes for
obtaining a permit and permits will processed... You break a non-prime size down into its prime factors, and its details are usually to... A fft errands faq from the frequency domain to the frequency
domain to the,. Content on this website, including dictionary, thesaurus, literature, geography and! — Introduction to Algorithms ( by CLRS ) and B ( x ) and B ( x ),. New revision, new revision, new
revision, compare ) a mapping i.e is gold to me, think!, homebrew, utilities, and then press the `` OK '' button stage processing! Around him a language such as Matlab, you generally will never have
multiply... The most common and familiar FFTs are “ radix 2 ” the modulus is actually the root. The largest and smallest segment that connects any two points given the number is … also. Smith,
Margaret W Smith and two other residents Midlight 's Deep and resembles a Reaver, but skirts key. More enjoyable experience thanks for pointing out now we want to retrieve C ( )! I think you forgot
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Fantasy Tactics 2Train you may to! Elude you ; few scientists and Engineer ’ s Propositions sidequest was always a bit frustrating 2 confused me we! | {"url":"https://sanjimes.com/continuing-competence-fefs/fft-errands-faq-1d84cf","timestamp":"2024-11-08T18:53:52Z","content_type":"text/html","content_length":"39131","record_id":"<urn:uuid:cb1abe1b-9dcf-4401-a7a4-860ba5cfa8a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00830.warc.gz"} |
53rd Annual Users Meeting
Isabella Ginnett (Michigan State University)
An accurate calorimetric reconstruction is an integral component of liquid argon time projection chamber (LArTPC) experiments such as ICARUS. Energy, more specifically energy loss per unit length or
$\frac{dE}{dx}$, is used in higher levels of data reconstruction like particle identification, so it is crucial that reconstructed energy is as accurate as possible. Calculating $\frac{dE}{dx}$
starts by reconstructing the waveforms from the wire plane channels of the detector into hits using the Gauss, ICARUS raw, or hybrid hit finder. Next, the charge displaced per unit length, $\frac{dQ}
{dx}$, is calculated from the hits. Finally, $\frac{dE}{dx}$ is calculated by calibrating $\frac{dQ}{dx}$ with an optimized calibration constant and using that calibrated $\frac{dQ}{dx}$ in a charge
to energy conversion formula. There are two procedures used to optimize the calibration constants in this study, one from MicroBooNE and one from LArIAT. The goals of this study are to investigate
which hit finding techniques best reconstruct energy and charge data and to optimize the calibration constants using the previously mentioned procedures.
My name is Isabella Ginnett, and I am currently a SULI summer intern advised by Minerba Betancourt and the ICARUS Collaboration. I would greatly appreciate the opportunity to share the results from
my summer work at the Users Meeting. I would be sharing this work via poster. I have the abstract for the poster's content included in the "Content" box as well as in the attached PDF.
Isabella Ginnett (Michigan State University) | {"url":"https://indico.fnal.gov/event/23109/contributions/193477/","timestamp":"2024-11-10T09:12:21Z","content_type":"text/html","content_length":"58519","record_id":"<urn:uuid:9898a26c-f195-4eca-b039-45fb8eca7191>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00191.warc.gz"} |
Hello Sagemath Family, How can I create a matrix with its elements as differential operators?
So, I have a matrix that is 6 x 3. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with a 3 x 12 matrix whose elements are functions of x, I will get the
final matrix whose corresponding elements are differentiated.
I have tried a few ways, but it did not work. Please suggest me. TIA
G_times_S = matrix([[0, 0, 0], [0, (twole)diff(), 0], [(twole)diff(), 0, 0], [0, 0, 0], [0, 0, -diff(, 2)*(twole)^2], [0, 0, 0]]) * S(x)
NOTE: 1. S(x) is a 3 x 12 matrix which is known. 2. Operator matrix = matrix([[0, 0, 0], [0, (twole)diff(), 0], [(twole)diff(), 0, 0], [0, 0, 0], [0, 0, -diff(, 2)*(twole)^2], [0, 0, 0]]) which is 6
x 3. (Need help with this!)
This is what I am trying to do but the diff(function, order) needs arguments.
Hello Sagemath Family, How can I create a matrix with its elements as differential operators?
So, I have a matrix that is 6 x 3. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with a 3 x 12 matrix whose elements are functions of x, I will get the
final matrix whose corresponding elements are differentiated.
I have tried a few ways, but it did not work. Please suggest me. TIA
G_times_S = matrix([[0, 0, 0], [0, (twole)diff(), 0], [(twole)diff(), 0, 0], [0, 0, 0], [0, 0, -diff(, 2)*(twole)^2], [0, 0, 0]]) * S(x)
NOTE: 1. S(x) is a 3 x 12 matrix which is known. 2. Operator matrix = matrix([[0, 0, 0], [0, (twole)diff(), 0], [(twole)diff(), 0, 0], [0, 0, 0], [0, 0, -diff(, 2)*(twole)^2], [0, 0, 0]]) which is 6
x 3. (Need help with this!)
This is what I am trying to do but the diff(function, order) needs arguments.
[DEL:Hello Sagemath Family, How can I create a matrix with its elements as :DEL]differential [DEL:operators?:DEL]
So, I have a matrix that is 6 x 3. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with a 3 x 12 matrix whose elements are functions of x, I will get the
final matrix whose corresponding elements are differentiated.
I have tried a few ways, but it did not [DEL:work. Please suggest me. TIA:DEL]
G_times_S = matrix([[0, 0, [DEL:0], [0, (twole)diff(), 0], [(twole)diff(), 0, 0], [0, 0, 0], :DEL][0, 0, -diff(, [DEL:2)*(twole)^2], :DEL][0, 0, 0]]) * [DEL:S(x)
1. S(x) is a 3 x 12 matrix which is [DEL:known. 2. Operator matrix = :DEL]
2. The operator matrix is
matrix([[0, 0, [DEL:0], [0, (twole)diff(), 0], [(twole)diff(), 0, 0], [0, 0, 0], :DEL][0, 0, -diff(, [DEL:2)*(twole)^2], [0, 0, 0]]) :DEL]
which is 6 x 3. (Need help with this!)
This is what I am trying to do but the diff(function, [DEL:order) :DEL]needs arguments.
Matrix of differential operators acting on a matrix of functions
So, I have a matrix that is 6 x 3. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with a 3 x 12 matrix whose elements are functions of x, I will get the
final matrix whose corresponding elements are differentiated.
I have tried a few ways, but it did not work.
G_times_S = matrix([[0, 0, 0],
[0, (twole)*diff(), 0],
[(twole)*diff(), 0, 0],
[0, 0, 0],
[0, 0, -diff(, 2)*(twole)^2],
[0, 0, 0]]) * S(x)
1. [DEL:S(x) :DEL]is a 3 x 12 matrix which is known.
2. The operator matrix is
matrix([[0, 0, 0],
[0, (twole)*diff(), 0],
[(twole)*diff(), 0, 0],
[0, 0, 0],
[0, 0, -diff(, 2)*(twole)^2],
[0, 0, 0]])
which is 6 x 3. (Need help with this!)
This is what I am trying to do but the diff(function, order) needs arguments.
Matrix of differential operators acting on a matrix of functions
[DEL:So, :DEL]I have a [DEL:matrix that is 6 x 3. :DEL]Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with [DEL:a 3 x 12 :DEL]matrix whose elements are
functions of x, I will get the final matrix whose corresponding elements are differentiated.
I have tried a few ways, but it did not work.
:DEL]= [DEL:matrix([[0, 0, 0], [0, (twole)*diff(), 0], [(twole)*diff(), 0, 0], [0, 0, 0], [0, 0, -diff(, 2)*(twole)^2], [0, 0, 0]]) * S(x)
1. S(x)
:DEL]is [DEL:a 3 x 12 :DEL]matrix which [DEL:is known.
The operator matrix is
matrix([[0, 0, 0],
[0, (twole)*diff(), 0],
[(twole)*diff(), 0, 0],
[0, 0, 0],
[0, 0, -diff(, 2)*(twole)^2],
[0, 0, 0]])
which is 6 x 3. (Need help with this!)
twole is
:DEL]a [DEL:constant.
This is what
:DEL]I am trying to [DEL:do but :DEL]the [DEL:diff(function, order) :DEL]needs [DEL:arguments.:DEL]
Matrix of differential operators acting on a matrix of functions
Suppose I have a 2x2 operator matrix. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix whose elements are functions of x, I will get
the final matrix whose corresponding elements are differentiated.
I have tried a few ways, but it did not work.
for example: D() = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, for example, f(x) = matrix([[x, x^2], [x^3, x]]). I am trying to finally get
[DEL:D(f(x)), :DEL]where f could be any function matrix.
I want to write the D() matrix such that when operated on a function matrix, will return its differentiated form. The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n).
So how can I write the D() operator matrix?
Matrix of differential operators acting on a matrix of functions
Suppose I have a 2x2 operator matrix. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix whose elements are functions of x, I will get
the final matrix whose corresponding elements are differentiated.
I have tried a few ways, but it did not work.
for example: D() = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, for example, f(x) = matrix([[x, x^2], [x^3, x]]). I am trying to finally get D(f
(x)) = D()*f(x), [DEL:(throughmatrix :DEL]multiplication), where f could be any function matrix.
I want to write the D() matrix such that when operated on a function matrix, will return its differentiated form. The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n).
So how can I write the D() operator matrix?
Matrix of differential operators acting on a matrix of functions
Suppose I have a 2x2 operator matrix. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix whose elements are functions of x, I will get
the final matrix whose corresponding elements are differentiated.
I have tried a few ways, but it did not work.
for example: D() = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, for example, f(x) = matrix([[x, x^2], [x^3, x]]). I am trying to finally get D(f
(x)) = D()*f(x), [DEL:(through :DEL]matrix multiplication), where f could be any function matrix.
I want to write the D() matrix such that when operated on a function matrix, will return its differentiated form. The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n).
So how can I write the D() operator matrix?
Matrix of differential operators acting on a matrix of functions
Suppose I have a 2x2 operator [DEL:matrix. :DEL]Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another [DEL:matrix :DEL]whose elements are
functions of x, I will get the final matrix whose corresponding elements are differentiated.
I have tried a few ways, but it did not work.
for example: D() = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, for example, f(x) = matrix([[x, x^2], [x^3, x]]). I am trying to finally get D(f
(x)) = D()*f(x), (simple matrix multiplication), where f could be any function matrix.
I want to write the D() matrix such that when operated on a function matrix, will return its differentiated form. The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n).
So how can I write the D() operator matrix?
Matrix of differential operators acting on a matrix of functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
I have tried a few ways, but it [DEL:did :DEL]not [DEL:work.:DEL]
for example: D() = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, for example, f(x) = matrix([[x, x^2], [x^3, x]]). I am trying to finally get D(f
(x)) = D()*f(x), (simple matrix multiplication), where f could be any function matrix.
I want to write the D() matrix such that when operated on a function matrix, will return its differentiated form. The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n).
So how can I write the D() operator matrix?
Matrix of differential operators acting on a matrix of functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
I have tried a few ways, like, D = matrix([[diff( , x), diff( , x)], [diff( , x, 2), diff( , x, 2)]]), but it does not work as we need [DEL:arguments.:DEL]
for example: D() = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, for example, f(x) = matrix([[x, x^2], [x^3, x]]). I am trying to finally get D(f
(x)) = D()*f(x), (simple matrix multiplication), where f could be any function matrix.
I want to write the D() matrix such that when operated on a function matrix, will return its differentiated form. The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n).
So how can I write the D() operator matrix?
Matrix of differential operators acting on a matrix of functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D() = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, for example, f(x) = matrix([[x, x^2], [x^3, x]]). I am trying to finally get D(f
(x)) = D()*f(x), (simple matrix multiplication), where f could be any function matrix.
I want to write the D() matrix such that when operated on a function matrix, will return its differentiated form.
I have tried a few ways, like, D = matrix([[diff( , x), diff( , x)], [diff( , x, 2), diff( , x, 2)]]), but it does not work as we need argument in [DEL:it.:DEL]
for example: D() = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, for example, f(x) = matrix([[x, x^2], [x^3, x]]). I am trying to finally get D(f
(x)) = D()*f(x), (simple matrix multiplication), where f could be any function matrix.
I want to write the D() matrix such that when operated on a function matrix, will return its differentiated form. The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n).
So how can I write the D() operator matrix?
Matrix of differential operators acting on a matrix of functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D() = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, for example, f(x) = matrix([[x, x^2], [x^3, x]]). I am trying to finally get D(f
(x)) = D()*f(x), (simple matrix multiplication), where f could be any function matrix.
I want to write the D() matrix such that when operated on a function matrix, will return its differentiated form.
I have tried a few ways, like, D = matrix([[diff( , x), diff( , x)], [diff( , x, 2), diff( , x, 2)]]), but it does not work as we need argument in [DEL:it. :DEL]
The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n). So how can I write the D() operator matrix?
Matrix of differential operators acting on a matrix of functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D() = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, for example, f(x) = matrix([[x, x^2], [x^3, x]]). I am trying to finally get D(f
(x)) = D()*f(x), (simple matrix multiplication), where f could be any function matrix.
I [DEL:want to write the D() matrix such that when operated on a function matrix, will return its differentiated form.:DEL]
I have tried a few ways, like, D = matrix([[diff( , x), diff( , x)], [diff( , x, 2), diff( , x, 2)]]), but it does not work as we need argument in it.
The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n). So how can I write the D() operator matrix?
Matrix of differential operators acting on a matrix of functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: [DEL:D() :DEL]= matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, [DEL:for example, :DEL]f(x) = matrix([[x, x^2], [x^3, [DEL:x]]). I am
trying to finally get :DEL]D(f(x)) = [DEL:D()*f(x), :DEL] (simple matrix multiplication), where f could be any function matrix.
I have tried a few ways, like, D = matrix([[diff( , x), diff( , x)], [diff( , x, 2), diff( , x, 2)]]), but it does not work as we need argument in it.
The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n). So how can I write the D() operator matrix?
Matrix of differential operators acting on a matrix of functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = D*f(x), (simple matrix
multiplication), where f could be any function matrix.
I have tried a few ways, like, D = matrix([[diff( , x), diff( , x)], [diff( , x, 2), diff( , x, 2)]]), but it does not work as we need argument in it.
The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n). So how can I write the D() operator matrix?
[DEL:Matrix of :DEL]differential operators acting on a matrix [DEL:of :DEL]functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = D*f(x), (simple matrix
multiplication), where f could be any function matrix.
I have tried a few ways, like, D = matrix([[diff( , x), diff( , x)], [diff( , x, 2), diff( , x, 2)]]), but it does not work as [DEL:we need argument in it.:DEL]
The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n). So how can I write the D() operator matrix?
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = D*f(x), (simple matrix
multiplication), where f could be any function matrix.
I have tried a few ways, like, D = matrix([[diff( , x), diff( , x)], [diff( , x, 2), diff( , x, 2)]]), but it does not work as diff() function needs some arguments.
[DEL:The problem is: diff() function in sage or sympy needs arguments: diff(f(x), x, n). :DEL]So how can I write the D() operator matrix?
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = D*f(x), (simple matrix
multiplication), where f could be any function matrix.
[DEL:I have tried a few ways, like, :DEL]D = matrix([[diff( , x), diff( , x)], [diff( , x, 2), diff( , x, [DEL:2)]]), but it :DEL]does not work as diff() function needs some arguments.
So how can I write the D() operator matrix?
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = D*f(x), (simple matrix
multiplication), where f could be any function matrix.
Writing D = matrix([[diff( , x), diff( , x)], [diff( , x, 2), diff( , x, 2)]]) does not work as diff() function needs some arguments.
So how can I write the D() operator matrix?
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = D*f(x), (simple matrix
multiplication), where f could be any function matrix.
Writing D = matrix([[diff( , x), diff( , x)], [diff( , x, 2), diff( , x, 2)]]) does not work as diff() function needs some arguments.
So how can I write the D() operator matrix?
PS: I could use f.apply_map(lambda e: diff(e, x)), but then it applies [DEL:first-order differential :DEL]to all elements in f(x). However, I have diff operators of different orders in the D.
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d/dx], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = D*f(x), (simple matrix
multiplication), where f could be any function matrix.
Writing D = matrix([[diff( , x), diff( , x)], [diff( , x, 2), diff( , x, 2)]]) does not work as diff() function needs some arguments.
So how can I write the D() operator matrix?
PS: I could use f.apply_map(lambda e: diff(e, x)), but then it applies d/dx to all elements in f(x). [DEL:However, :DEL]I have diff operators of different orders in the [DEL:D.:DEL]
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, [DEL:d/dx], :DEL][d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = D*f(x), (simple matrix
multiplication), where f could be any function matrix.
Writing D = matrix([[diff( , x), diff( , [DEL:x)], :DEL][diff( , x, 2), diff( , x, 2)]]) does not work as diff() function needs some arguments.
So how can I write the D() operator matrix?
PS: I could use f.apply_map(lambda e: diff(e, x)), but then it applies d/dx to all elements in f(x). Whereas, I have diff operators of different orders in the D matrix.
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d3/dx3], [d2/dx2, d2/dx2]])
is an operator matrix which operates on a function matrix,
f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = D*f(x), (simple matrix multiplication), where f could be any function matrix.
Writing D = matrix([[diff( , x), diff( , x, 3)], [diff( , x, 2), diff( , x, 2)]]) does not work as diff() function needs some arguments.
So how can I write the D() operator matrix?
PS: I could use f.apply_map(lambda e: diff(e, x)), but then it applies d/dx to all elements in f(x). Whereas, I have diff operators of different orders in the D matrix.
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d3/dx3], [d2/dx2, [DEL:d2/dx2]]) :DEL]
is an operator matrix which operates on a function matrix, [DEL::DEL]
f(x) = matrix([[x, x^2], [x^3, [DEL:x]]) :DEL]as D(f(x)) = [DEL:D*f(x), :DEL] (simple matrix multiplication), where f could be any function matrix.
Writing D = matrix([[diff( , x), diff( , x, 3)], [diff( , x, 2), diff( , x, 2)]]) does not work as diff() function needs some arguments.
So how can I write the D() operator matrix?
PS: I could use f.apply_map(lambda e: diff(e, x)), but then it applies d/dx to all elements in f(x). Whereas, I have diff operators of different orders in the D matrix.
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d3/dx3], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = Df(x)*, (simple matrix
multiplication), where f could be any function matrix.
Writing D = matrix([[diff( , x), diff( , x, 3)], [diff( , x, 2), diff( , x, 2)]]) does not work as diff() function needs some arguments.
So how can I write the D() operator matrix?
PS: I could use f.apply_map(lambda e: diff(e, x)), but then it applies d/dx to all elements in f(x). Whereas, I have diff operators of different orders in the D matrix.
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d3/dx3], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as [DEL::DEL]D(f(x)) = [DEL:Df(x)*, :DEL]
(simple matrix multiplication), where f could be any function matrix.
Writing D = matrix([[diff( , x), diff( , x, 3)], [diff( , x, 2), diff( , x, 2)]]) does not work as diff() function needs some arguments.
So how can I write the D() operator matrix?
PS: I could use f.apply_map(lambda e: diff(e, x)), but then it applies d/dx to all elements in f(x). Whereas, I have diff operators of different orders in the D matrix.
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d3/dx3], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = D*f(x), (simple matrix [DEL:
multiplication), where f could be any function matrix.:DEL]
Writing D = matrix([[diff( , x), diff( , x, 3)], [diff( , x, 2), diff( , x, 2)]]) does not work as diff() function needs some arguments.
So how can I write the D() operator matrix?
PS: I could use f.apply_map(lambda e: diff(e, x)), but then it applies d/dx to all elements in f(x). Whereas, I have diff operators of different orders in the D matrix.
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d3/dx3], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = D*f(x), (simple matrix
Writing D = matrix([[diff( , x), diff( , x, 3)], [diff( , x, 2), diff( , x, 2)]]) does not work as diff() function needs some arguments.
So how can I write the D() operator matrix?
PS: I could use f.apply_map(lambda e: diff(e, x)), but then it applies d/dx to all elements in f(x). Whereas, I have [DEL:diff :DEL]operators of different orders in the D matrix.
A matrix containing differential operators acting on a matrix containing functions
Suppose I have a 2x2 operator matrix, D. Some of its elements are of the form (d/dx), and (d2/dx2). Now when this matrix is multiplied with another matrix, f, whose elements are functions of x, I
will get the final matrix whose corresponding elements are differentiated.
for example: D = matrix([[d/dx, d3/dx3], [d2/dx2, d2/dx2]]) is an operator matrix which operates on a function matrix, f(x) = matrix([[x, x^2], [x^3, x]]) as D(f(x)) = D*f(x), (simple matrix
Writing D = matrix([[diff( , x), diff( , x, 3)], [diff( , x, 2), diff( , x, 2)]]) does not work as diff() function needs [DEL:some arguments.:DEL]
So how can I write the D() operator matrix?
PS: I could use f.apply_map(lambda e: diff(e, x)), but then it applies d/dx to all elements in f(x). Whereas, I have 'diff()' operators of different orders in the D matrix. | {"url":"https://ask.sagemath.org/questions/56311/revisions/","timestamp":"2024-11-13T18:49:49Z","content_type":"application/xhtml+xml","content_length":"83164","record_id":"<urn:uuid:5342bba7-da4d-40a4-94a6-1024f858bfd8>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00602.warc.gz"} |
currentresearch – Mathematics and Statistics |
Missouri S&T
Current Faculty Research Areas
Algebra, Discrete Mathematics, and Topology
Dr. Matt Insall
Logic, nonstandard methods, nonstandard models, algebra, topological algebra, topological model theory
Dr. Stephen Clark
Analysis, Operator theory
Dr. David Grow
Analysis, Fourier analysis, lacunary series
Dr. Martin Bohner
Ordinary differential equations, dynamic equations on time scales, difference equations, Hamiltonian systems, variational analysis, boundary value problems, control theory, oscillation, analysis,
fractional equations, applications to biology and economics
Dr. Vy Le
Nonlinear differential equations, bifurcation, and calculus of variations
Dr. Jason Murphy
Harmonic analysis, nonlinear differential, inverse problems
Differential and Difference Equations
Dr. Elvan Akin
Dynamic equations on time scales, differential equations, difference equations, oscillation, boundary value problems
Dr. Martin Bohner
Ordinary differential equations, dynamic equations on time scales, difference equations, Hamiltonian systems, variational analysis, boundary value problems, control theory, oscillation, analysis,
fractional equations, applications to biology and economics
Dr. Stephen Clark
Analysis, operator theory, spectral theory of linear operators associated with symplectic, Hamiltonian and Dirac differential and difference equations.
Dr. David Grow
Analysis, Fourier analysis, lacunary series
Dr. Xiaoming He
Interface problems, multi-phase problems, computational fluid dynamics, computational plasma physics, data assimilation, stochastic PDEs, boundary integral equations, feedback control; Finite element
methods, domain decomposition methods, lattice Boltzmann methods, extrapolations
Dr. Wenqing Hu
Stochastic Analysis, Stochastic Differential Equations, Random Dynamical Systems, (Stochastic) Partial Differential Equations
Dr. Vy Le
Nonlinear differential equations, bifurcation, and calculus of variations
Dr. Jason Murphy
Harmonic analysis, nonlinear partial differential equations, inverse problems
Dr. John Singler
Data-driven model order reduction of partial differential equations (PDEs), computational methods for control of PDEs, applications in control theory, engineering, and the sciences
Dr. Xiaoming Wang
Modern applied and computational mathematics and machine learning, with applications in fluid dynamics, groundwater research, geophysical fluid dynamics and turbulence, material science
Dr. Yanzhi Zhang
Fractional PDEs and nonlocal models, anomalous diffusion, data-informed modeling for seismic waves, machine learning algorithms and applications, Bose–Einstein superfluids, superconductivity,
multiscale/multiphysics modeling and simulations, numerical dispersive PDEs
Scientific Computing and Numerical Analysis
Dr. Xiaoming He
Interface problems, multi-phase problems, computational fluid dynamics, computational plasma physics, data assimilation, stochastic PDEs, boundary integral equations, feedback control; Finite element
methods, domain decomposition methods, lattice Boltzmann methods, extrapolations
Dr. John Singler
Data-driven model order reduction of partial differential equations (PDEs), computational methods for control of PDEs, applications in control theory, engineering, and the sciences
Dr. Xiaoming Wang
Modern applied and computational mathematics and machine learning, with applications in fluid dynamics, groundwater research, geophysical fluid dynamics and turbulence, material science
Dr. Yanzhi Zhang
Fractional PDEs and nonlocal models, anomalous diffusion, data-informed modeling for seismic waves, machine learning algorithms and applications, Bose–Einstein superfluids, superconductivity,
multiscale/multiphysics modeling and simulations, numerical dispersive PDEs
Statistics and Probability
Dr. Akim Adekpedjou
Recurrent event data analysis, stochastic processes, survival analysis
Dr. Wenqing Hu
High-dimensional statistics, statistical machine learning, optimization
Dr. Gayla Olbricht
Statistical genomics and epigenomics, hidden Markov models, modeling dependent data, mixed models
Dr. Robert Paige
Causal inference, data science, statistical shape analysis, topological data analysis
Dr. V. A. Samaranayake
Reliability, time series analysis, statistical applications in biology, economics, and engineering
Dr. Xuerong (Meggie) Wen
Nonlinear and nonparametric regression, regression graphics, computational statistics and statistical genetics, with an emphasis on sufficient dimension reduction in the context of regression | {"url":"https://math.mst.edu/research/currentresearch/","timestamp":"2024-11-15T01:16:06Z","content_type":"text/html","content_length":"110606","record_id":"<urn:uuid:d157128a-e2f3-4cb2-b93d-bc101449a283>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00032.warc.gz"} |
How Much Is 10000 Dimes in Dollars? (Answer + Calculator)
How Much Is 10000 Dimes Worth in Dollars? (Answer, Dimes To Dollars Calculator, and Convert Using a Conversion Formula)
Do you want to know the answer to ‘How much is 10000 dimes worth in dollars?’ 10000 dimes are worth 1000 dollars ($1000), which is equivalent to 100,000 cents.
What if you don’t have precisely 10000 dimes? How do you calculate how many dollars you have in dimes? That’s simple! Use our dimes to dollars converter to turn your dimes into dollars.
Please continue reading to use our converter, find out how to convert dimes to dollars with a conversion formula, and learn more about dimes.
10000 Dimes to dollars converter (Conversion calculator)
Use our free 10000 dimes to dollars converter to quickly calculate how much your dimes are worth in dollars. Just type in how many dimes you have, and our converter does the rest for you!
Looking at the converter, you will see that we already entered 10000 dimes, giving us an answer of $1000. That answers our question about ‘how much is 10000 dimes worth in dollars?’. 10000 dimes
equals 1000 dollars.
Now it’s your turn! Type in how many dimes you have, and our dimes to dollars calculator will tell you how much that is in dollars. We make converting from dimes to dollars easy, no matter how many
dimes you have. Whether you have 10000 dimes or 1000 dimes, we can solve it all.
10000 Dimes to dollars conversion table (Fast unit conversion method)
One popular method to convert dimes into dollars is to use our conversion table. Conversion tables, also called conversion charts, list the number of dimes in one column with the corresponding number
of dollars in the second column.
To use our conversion table to find the number of dollars in 10000 dimes, find 10000 dimes in the first column and note that the matching number of dollars in the second column is $1000. There are
1000 dollars in 10000 dimes.
Number of Dimes Dollars $
10000 $1000
9000 $900
8000 $800
7000 $700
6000 $600
5000 $500
4000 $400
3000 $300
2000 $200
1000 $100
How to convert dimes to dollars (Calculate the answer using a conversion formula)
Use our converter or the following conversion formula to convert dimes to dollars.
Dimes to dollars conversion formula:
Dollars = Dimes x 0.10 dollars per dime
The formula says that we can determine the number of dollars we have by multiplying the number of dimes by 0.10, which is the number of dollars in one dime.
For example, to determine how many dollars are in 10000 dimes, we multiply 10000 dimes by 0.10, as shown below.
Dollars = 10000 dimes x 0.10 dollars per dime = 1000 dollars
How much is 8000 dimes?
8000 dimes are worth 800 dollars.
How much is 5000 dimes?
5000 dimes is equal to 500 dollars. This is because there are 10 dimes in one dollar. So, if you have 5000 dimes, you have 500 times as many dimes as there are in one dollar. Therefore, 5000 dimes is
the same as 500 dollars.
How much is 600 dimes?
600 dimes is equal to $60 or 6,000 cents.
How much is 10 dimes?
Ten dimes are equal to one dollar.
How many dimes is 2 dollars?
There are 20 dimes in 2 dollars.
How much do 10000 dimes weigh?
10000 dimes weigh 22,680 grams (22.680 kilograms), which equals 50 pounds.
Dimes made since 1965 weigh precisely 2.268 grams, which equals 0.08 ounces. To find the weight of the dimes yourself, multiply 10000 dimes by 2.268 grams to find the weight of the dimes (22680 grams
= 22.680 kilograms = 50 pounds).
How much is 10000 coins on TikTok?
10000 coins on TikTok are worth $106. TikTok’s virtual coins can be given as gifts or used to unlock app features.
What are the different coins used as currency in the United States today?
There are currently six different coins used as currency in the United States: the penny, nickel, dime, quarter, half-dollar, and the dollar coin.
Frequently asked questions: Converting dimes to dollars
People often have specific questions about converting dimes to dollars. Here are the answers to some of the most common questions people ask about dimes to dollars conversions.
Is a dime 1/10 of a dollar?
Yes, a dime is 1/10 of a dollar. This means that if you have 10 dimes, you have one dollar. If you have 100 dimes, you have 10 dollars, and so on.
Is a dime 10 cents?
Yes, a dime is a coin worth 10 cents, which is one-tenth of a United States dollar. The dime is the smallest in diameter and is the thinnest of all U.S. coins currently minted for circulation.
What are the dimensions of a dime?
A dime is 0.705 inches (1.791 centimeters) in diameter and has a thickness of 0.053 inches (0.135 centimeters).
How thick is a dime?
Modern-day dimes are 1.35 mm (0.135 cm) thick, equal to 0.061 inches.
What is the volume of a dime?
The volume of a single dime is 0.020755 cubic inches, equivalent to 0.34011 cubic centimeters.
How much does a dime weigh? (Mass)
One dime weighs 2.268 grams, which is equal to 0.08 ounces.
How many ridges on a dime?
According to the U.S. Mint, there are 118 ridges on a dime. Ridges, also called grooves, are a physical security feature that makes dimes difficult to counterfeit.
When dimes were made from silver, the reeded ridges also helped prevent coin clipping, which is a form of fraud. What is coin clipping, and how do ridges deter this fraudulent practice? Imagine that
you have a bucket full of silver dimes. While each dime is worth 10 cents, the metal is also valuable.
By shaving off a small amount of the silver from each coin in the bucket, a scammer could amass a sizeable amount of valuable silver to sell for scrap. The preferred target of silver scrapers has
always been the edges of coins. It is much easier, in comparison, to notice if the face or back of a coin has been scraped. We would be much more likely to see if someone had defaced President
Franklin D. Roosevelt’s face than the edge of the dime!
What are dimes made of?
Dimes are primarily copper but also are made of nickel. To be precise, the composition of modern-day dimes is 91.67% copper and 8.33% nickel. It wasn’t always this way, though! Before 1965, dimes
were 90% silver and 10% copper. If you have 200 dimes ($20), you are holding 415.82 grams of copper and 37.78 grams of nickel in your hand. If you prefer to work in pounds, that’s 0.917 pounds of
copper and 0.083 pounds of nickel.
What hasn’t changed is that dimes are still worth 10 cents, even though they are no longer made from silver.
Are dimes magnetic?
No, dimes are not magnetic, even though they are made from a nickel-copper alloy.
Nickel-copper alloys are only magnetic when the nickel content exceeds 56%. Since dimes are only 8.33% nickel, they are not magnetic.
How much money is 91,000 pennies?
91,000 pennies is equal to 910 dollars. This is a lot of money, and counting out all those pennies would take a long time!
In conclusion, 10000 dimes are worth 1000 dollars ($1000), which is equivalent to 100,000 cents.
To convert between 10000 dimes and dollars yourself, use our converter or read the steps in our ‘How to convert dimes to dollars’ section. | {"url":"https://www.tipwho.com/article/how-much-is-10000-dimes-in-dollars-answer-calculator/","timestamp":"2024-11-14T08:39:42Z","content_type":"text/html","content_length":"143675","record_id":"<urn:uuid:b0376adc-366d-48ed-a70d-01c5e168e18d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00174.warc.gz"} |
Florian Schwarz
Title: Differential bundles as functors
Abstract: Differential bundles are the generalisation of vector bundles in tangent categories. Following an idea by Michael Ching, we will consider lax morphisms of tangent categories, functors
between tangent categories preserving the tangent structure, and show that they induce differential bundles under certain conditions. Then we will continue to show that the categories of differential
bundles in a tangent category X with additive/linear morphisms are equivalent to the category of tangent functors from the category of free commutative monoids into X.
Slides: https://www.dropbox.com/scl/fi/umdy5fqk3oo4tqcnpsok2/Presentation_Differential_bundle_classification.pdf?rlkey=necl37cfsd13gnsor86xsup4g&dl=0 | {"url":"https://logic.ucalgary.ca/event/florian-schwarz-6/","timestamp":"2024-11-02T03:21:00Z","content_type":"text/html","content_length":"44542","record_id":"<urn:uuid:caec5684-e1f8-44d9-9b6d-1024f5d90cd6>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00182.warc.gz"} |
how many meters in a kilometer
How many meters in a kilometer?
When it comes to measuring distance, units play a crucial role in ensuring accurate calculations. One common question that often arises is how many meters are there in a kilometer? To answer this
query, we need to delve into the world of metric measurements and explore the relationship between meters and kilometers.
Meter: The Basic Unit of Length
The meter is a fundamental unit of length in the International System of Units (SI). Originally defined as one ten-millionth of the distance from the Earth’s equator to the North Pole, the meter has
since been redefined more precisely in terms of the speed of light.
This standardization allows for consistent measurements across different scientific disciplines and ensures that distances can be accurately compared and calculated.
Kilometer: A Metric Unit of Distance
A kilometer is a metric unit of distance derived from the meter. It is equal to 1,000 meters, making it a widely used unit for longer distances. The prefix “kilo” denotes a factor of 1,000 in the
metric system, indicating that a kilometer consists of 1,000 meters.
Kilometers are commonly used to measure things such as road distances, track events, and larger geographic distances. They allow for convenient estimation and comparison of distances over a wider
Conversion: Meters to Kilometers
Converting meters to kilometers is a relatively straightforward process. Since there are 1,000 meters in a kilometer, you can convert meters to kilometers by dividing the length in meters by 1,000.
This simple conversion allows you to express larger distances in a more manageable and understandable format.
For example, if you have a distance of 5,000 meters and want to express it in kilometers, you would divide 5,000 by 1,000, resulting in 5 kilometers.
Real-life Examples
The conversion between meters and kilometers is crucial in various real-life scenarios. Let’s take a look at a few examples to understand the practical significance of this relationship:
Example 1: Road Trip
Imagine embarking on an adventurous road trip across a vast and beautiful country. The route consists of 3,500,000 meters of winding roads. To get a better sense of the distance traveled, you can
convert this figure into kilometers. Dividing 3,500,000 by 1,000 yields 3,500 kilometers. This information allows you to comprehend the journey more easily and plan your stops along the way.
Example 2: Marathon Race
Marathon races are renowned sports events that cover a distance of 42,195 meters. To convey the length in a more relatable format, you can convert it to kilometers. Dividing 42,195 by 1,000 gives you
approximately 42.195 kilometers. This conversion enables both participants and spectators to comprehend the challenge that lies ahead in a more tangible manner.
Example 3: Traveling between Cities
When booking a flight, it’s crucial to have a clear idea of the distances involved. Suppose you plan to travel from London to Paris, a journey spanning 344,000 meters. By dividing 344,000 by 1,000,
you discover that the distance is approximately 344 kilometers. This knowledge helps you gauge the length of the trip and make informed travel decisions.
In summary, there are 1,000 meters in a kilometer. This essential conversion between the two units allows for convenient and accurate measurement of distance in various contexts. By dividing the
number of meters by 1,000, you can express distances in kilometers, making them more understandable and relatable. So whether you’re planning a road trip, tracking marathon races, or booking a
flight, understanding the relationship between meters and kilometers is fundamental for comprehending distances effectively. | {"url":"https://faks.co.za/how-many-meters-in-a-kilometer/","timestamp":"2024-11-14T02:20:33Z","content_type":"text/html","content_length":"94802","record_id":"<urn:uuid:343d4d24-b659-4907-90ed-d3f7d1a5d7a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00304.warc.gz"} |
ERF Function: Definition, Formula Examples and Usage
ERF Function
Are you familiar with the ERF formula in Google Sheets? This formula is a powerful tool that can help you perform a variety of statistical calculations and analysis. In this blog post, we’ll take a
closer look at the ERF formula and how you can use it to make more informed decisions in your spreadsheets.
The ERF formula is a mathematical function that calculates the error function, which is used in statistics and probability theory. This function takes the form ERF(value), where value is the numeric
value you want to calculate the error function for. The result of the ERF formula is a decimal value that represents the error function for the given value. You can use this function to perform a
variety of statistical calculations, such as estimating the probability of a given event or analyzing data sets. Plus, it’s easy to use and can save you a lot of time and frustration. Give it a try
and see for yourself how useful it can be!
Definition of ERF Function
The ERF function in Google Sheets is a mathematical function that calculates the error function, which is a specific mathematical function used in statistics and probability theory. The function
takes the form ERF(value), where value is the numeric value you want to calculate the error function for. The result of the ERF formula is a decimal value that represents the error function for the
given value. This function is useful in a variety of situations, such as estimating the probability of a given event or analyzing data sets. You can use the ERF function in combination with other
formulas or functions in Google Sheets to create more complex and powerful calculations.
Syntax of ERF Function
The syntax for the ERF function in Google Sheets is as follows:
This function takes a single argument: value, which is the numeric value you want to calculate the error function for. This argument can be a number, a cell reference that contains a number, or a
formula that calculates a number. For example, you could use the function as ERF(2) to calculate the error function for the number 2, or you could use the function as ERF(A2) to calculate the error
function for the value in cell A2. The result of the ERF formula is a decimal value that represents the error function for the given value.
Examples of ERF Function
Here are three examples of how you can use the ERF function in Google Sheets:
1. Calculate the error function for a specific number. For example, if you want to calculate the error function for the number 2, you could use the following formula: =ERF(2). This formula would
return the decimal value 0.9953222650189527, which is the error function for the number 2.
2. Calculate the error function for the value in a cell. For example, if you want to calculate the error function for the value in cell A2, you could use the following formula: =ERF(A2). This
formula would return the decimal value that represents the error function for the value in cell A2.
3. Calculate the error function for the result of a calculation. For example, if you want to calculate the error function for the result of a calculation in cell A2, you could use the following
formula: =ERF(A2). This formula would return the decimal value that represents the error function for the result of the calculation in cell A2.
These are just a few examples of how you can use the ERF function in Google Sheets. There are many other ways you can use this function, depending on your specific needs and the data you’re working
Use Case of ERF Function
Here are a few examples of how you might use the ERF function in real-life situations in Google Sheets:
• If you’re working on a statistical analysis, you could use the ERF function to calculate the error function for a given value. For example, if you want to calculate the error function for a
specific data point, you could use the formula =ERF(A2) to calculate the error function for the value in cell A2. This formula would return the decimal value that represents the error function
for the value in cell A2.
• If you’re analyzing data for a project, you could use the ERF function to compare the error function for different values. For example, if you want to compare the error function for the values in
cells A2 and B2, you could use the formula =ERF(A2) – ERF(B2) to calculate the difference between the error functions for the two values. This formula would return the decimal value that
represents the difference between the error functions for the values in cells A2 and B2.
• If you’re estimating the probability of a given event, you could use the ERF function to calculate the error function for a specific value. For example, if you want to calculate the error
function for the value that represents the probability of a given event, you could use the formula =ERF(A2) to calculate the error function for the value in cell A2. This formula would return the
decimal value that represents the error function for the probability of the given event.
These are just a few examples of how you might use the ERF function in Google Sheets. There are many other possible applications, depending on your specific needs and the data you’re working with.
Limitations of ERF Function
The ERF function in Google Sheets is a powerful and useful tool, but it does have some limitations. Here are a few things to keep in mind when using this function:
• The ERF function only calculates the error function for numeric values. It does not work for text or other data types. If you try to use the ERF function with a non-numeric value, you’ll get an
• The ERF function only calculates the error function for a given value. It does not provide any other information about the value, such as its probability or distribution. If you want to perform
more complex statistical calculations, you’ll need to use additional functions or tools.
• The ERF function is limited by the precision of the numeric values you use. If the numeric values you use are not precise enough, the result of the ERF formula may not be accurate. It’s important
to use high-precision numeric values when using the ERF function to ensure the most accurate results.
Overall, the ERF function is a useful tool for performing statistical calculations in Google Sheets, but it’s important to understand its limitations and how to work around them.
Commonly Used Functions Along With ERF
There are several commonly used functions that can be used in combination with the ERF function in Google Sheets. Here are a few examples, along with an explanation of how you can use them with the
ERF function:
• AVERAGE: The AVERAGE function calculates the average of a range of cells. For example, if you want to calculate the average of a data set and then use the ERF function to calculate the error
function for that average, you could use the formula =ERF(AVERAGE(A2:A10)) to calculate the error function for the average of the values in cells A2 through A10. This formula would return the
decimal value that represents the error function for the average of the values in cells A2 through A10.
• STDEV: The STDEV function calculates the standard deviation of a range of cells. For example, if you want to calculate the standard deviation of a data set and then use the ERF function to
calculate the error function for that standard deviation, you could use the formula =ERF(STDEV(A2:A10)) to calculate the error function for the standard deviation of the values in cells A2
through A10. This formula would return the decimal value that represents the error function for the standard deviation of the values in cells A2 through A10.
In summary, the ERF function in Google Sheets is a useful tool that allows you to calculate the error function for a given numeric value. This function is commonly used in statistics and probability
theory, and it can help you perform a variety of statistical calculations and analysis. The ERF function is easy to use and can be combined with other formulas or functions in Google Sheets to create
more complex and powerful calculations. If you’re new to using the ERF function, we encourage you to give it a try and see how it can help you make more informed decisions in your spreadsheets.
Video: ERF Function
In this video, you will see how to use ERF function. We suggest you to watch the video to understand the usage of ERF formula.
Related Posts Worth Your Attention
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Drag Coefficient Calculator - Calculator Doc
Drag Coefficient Calculator
The drag coefficient is a dimensionless number that quantifies the drag or resistance of an object in a fluid environment, such as air or water. Understanding the drag coefficient is essential in
fields like aerodynamics, automotive design, and engineering, as it affects how efficiently an object moves through a fluid. This calculator helps determine the drag coefficient using the drag force,
air density, velocity, and cross-sectional area of the object.
The formula for calculating the drag coefficient is:
Cd = Fd / (1/2 * p * V² * A)
• Cd is the drag coefficient,
• Fd is the drag force in Newtons,
• p is the air density in kg/m³,
• V is the velocity in m/s,
• A is the cross-sectional area in m².
How to Use
1. Input the drag force (Fd) in Newtons in the provided field.
2. Enter the air density (p) in kg/m³.
3. Fill in the velocity (V) in m/s.
4. Provide the cross-sectional area (A) in m².
5. Click the “Calculate” button to find the drag coefficient.
Suppose you have a drag force of 50 Newtons, an air density of 1.225 kg/m³, a velocity of 15 m/s, and a cross-sectional area of 2 m². You would enter:
• Drag Force (Fd): 50
• Air Density (p): 1.225
• Velocity (V): 15
• Area (A): 2
Using the formula:
Cd = 50 / (0.5 * 1.225 * 15² * 2)
Cd = 50 / (0.5 * 1.225 * 225 * 2)
Cd = 50 / 137.8125
Cd ≈ 0.3635
1. What is the drag coefficient?
The drag coefficient is a measure of the drag force acting on an object relative to its size and the fluid through which it moves.
2. Why is the drag coefficient important?
It helps in analyzing the performance of vehicles and structures in fluid environments, affecting fuel efficiency and stability.
3. What factors affect the drag coefficient?
Factors include the shape of the object, surface roughness, and the type of fluid (air or water).
4. Is a lower drag coefficient always better?
Generally, yes, a lower drag coefficient indicates less resistance, leading to better performance and efficiency.
5. Can the drag coefficient be measured directly?
Yes, it can be determined through wind tunnel testing or computational fluid dynamics simulations.
6. What units are used for the drag coefficient?
The drag coefficient is a dimensionless quantity, so it has no units.
7. How can I reduce the drag coefficient of my vehicle?
Streamlining the shape, reducing surface roughness, and optimizing components can help lower the drag coefficient.
8. Does temperature affect air density?
Yes, air density decreases with an increase in temperature, which can affect the drag coefficient.
9. Can the drag coefficient change with speed?
Yes, at higher speeds, the flow around an object can change, potentially altering the drag coefficient.
10. What is the typical drag coefficient for cars?
Most cars have a drag coefficient ranging from 0.25 to 0.35.
11. How do I interpret the result of the calculator?
The result provides the drag coefficient, which can be compared against typical values for similar shapes to evaluate performance.
12. What is the relationship between drag force and drag coefficient?
The drag force is directly proportional to the drag coefficient; as the drag coefficient increases, so does the drag force.
13. Are there different drag coefficients for different shapes?
Yes, various shapes have distinct drag coefficients due to their different flow characteristics.
14. What is the impact of wind on the drag coefficient?
Wind can increase the effective velocity, thereby increasing the drag force experienced by an object.
15. Can the calculator be used for underwater objects?
The calculator is designed for air; underwater calculations require different parameters like water density.
16. How does altitude affect air density?
As altitude increases, air density decreases, which can impact the drag coefficient and performance.
17. What role does cross-sectional area play in the calculation?
A larger cross-sectional area increases the drag force, which affects the drag coefficient.
18. Can I use this calculator for racing vehicles?
Yes, it can be useful for optimizing performance in racing by assessing aerodynamic efficiency.
19. What is a typical drag coefficient for bicycles?
Bicycles typically have a drag coefficient of around 0.88, depending on the rider’s position.
20. Can I use this calculator for design purposes?
Yes, it can aid in the design process by evaluating how changes affect the drag coefficient.
The Drag Coefficient Calculator is a valuable tool for engineers, designers, and enthusiasts seeking to understand the aerodynamic properties of objects. By inputting key parameters, users can
quickly determine the drag coefficient, aiding in performance optimization and design decisions. Understanding and managing the drag coefficient can lead to significant improvements in efficiency and
effectiveness in various applications, from automotive engineering to sports equipment design. | {"url":"https://calculatordoc.com/drag-coefficient-calculator/","timestamp":"2024-11-06T07:31:55Z","content_type":"text/html","content_length":"88105","record_id":"<urn:uuid:25def03a-19aa-4f4f-8ee5-208d60d89474>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00286.warc.gz"} |
Levenshtein Algorithm (Fuzzy Matching)
Levenshtein distance is a string metric for measuring the difference between two sequences. Informally, the Levenshtein distance between two words is equal to the number of single-character edits
required to change one word into the other. The term edit distance is often used to refer specifically to Levenshtein distance.
The Levenshtein distance between two strings is defined as the minimum number of edits needed to transform one string into the other, with the allowable edit operations being insertion, deletion, or
substitution of a single character.
For example, the Levenshtein distance between "kitten" and "sitting" is 3, since the following three edits change one into the other, and there is no way to do it with fewer than three edits:
1. kitten → sitten (substitution of 's' for 'k')
2. sitten → sittin (substitution of 'i' for 'e')
3. sittin → sitting (insertion of 'g' at the end).
Adeptia Integration
We can use this plug-in in Adeptia process to find the Levenshtein distance between two strings
The input of this plug-in will be 2 string values and will set the Levenshtein_distance variable in the process flow context which can be used further.
0 comments
Article is closed for comments. | {"url":"https://support.adeptia.com/hc/en-us/articles/207875583-Levenshtein-Algorithm-Fuzzy-Matching","timestamp":"2024-11-11T23:26:29Z","content_type":"text/html","content_length":"27726","record_id":"<urn:uuid:533c39b9-8151-4314-ae52-ecf0753c42dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00394.warc.gz"} |
Astronomers detect evidence of time when universe emerged from "Dark Ages"
Astronomers at the California Institute of Technology announced today the discovery of the long-sought "Cosmic Renaissance," the epoch when young galaxies and quasars in the early universe first
broke out of the "Dark Ages" that followed the Big Bang.
"It is very exciting," said Caltech astronomy professor S. George Djorgovski, who led the team that made the discovery. "This was one of the key stages in the history of the universe."
According to a generally accepted picture of modern cosmology, the universe started with the Big Bang some 14 billion years ago, and was quickly filled with glowing plasma composed mainly of hydrogen
and helium.
As the universe expanded and cooled over the next 300,000 years, the atomic nuclei and electrons combined to make atoms of neutral gas. The glow of this "recombination era" is now observed as the
cosmic microwave background radiation, whose studies have led to the recent pathbreaking insights into the geometrical nature of the universe.
The universe then entered the Dark Ages, which lasted about half a billion years, until they were ended by the formation of the first galaxies and quasars. The light from these new objects turned the
opaque gas filling the universe into a transparent state again, by splitting the atoms of hydrogen into free electrons and protons. This Cosmic Renaissance is also referred to by cosmologists as the
"reionization era," and it signals the birth of the first galaxies in the early universe.
"It is as if the universe was filled by a dark, opaque fog up to that time," explains Sandra Castro, a postdoctoral scholar at Caltech and a member of the team. "Then the fires—the first galaxies—lit
up and burned through the fog. They made both the light and the clarity."
The researchers saw the tell-tale signature of the cosmic reionization in the spectra of a very distant quasar, SDSS 1044-0125, discovered last year by the Sloan Digital Sky Survey (SDSS). Quasars
are very luminous objects in the distant universe, believed to be powered by massive black holes.
The spectra of the quasar were obtained at the W. M. Keck Observatory's Keck II 10-meter telescope atop Mauna Kea, Hawaii. The spectra show extended dark regions, caused by opaque gas along the line
of sight between Earth and the quasar. This effect was predicted in 1965 by James Gunn and Bruce Peterson, both then at Caltech. Gunn, now at Princeton University, is the leader of the Sloan Digital
Sky Survey; Peterson is now at Mt. Stromlo and Siding Spring observatories, in Australia.
The process of converting the dark, opaque universe into a transparent, lit-up universe was not instantaneous: it may have lasted tens or even hundreds of millions of years, as the first bright
galaxies and quasars were gradually appearing on the scene, the spheres of their illumination growing until they overlapped completely.
"Our data show the trailing end of the reionization era," says Daniel Stern, a staff scientist at the Jet Propulsion Laboratory and a member of the team. "There were opaque regions in the universe
back then, interspersed with bubbles of light and transparent gas."
"This is exactly what modern theoretical models predict," Stern added. "But the very start of this process seems to be just outside the range of our data."
Indeed, the Sloan Digital Sky Survey team has recently discovered a couple of even more distant quasars, and has reported in the news media that they, too, see the signature of the reionization era
in the spectra obtained at the Keck telescope.
"It is a wonderful confirmation of our result," says Djorgovski. "The SDSS deserves much credit for finding these quasars, which can now be used as probes of the distant universe—and for their
independent discovery of the reionization era."
"It is a great example of a synergy of large digital sky surveys, which can discover interesting targets, and their follow-up studies with large telescopes such as the Keck," adds Ashish Mahabal, a
postdoctoral scholar at Caltech and a member of the team. "This is the new way of doing observational astronomy: the quasars were found by SDSS, but the discovery of the reionization era was done
with the Keck."
The Caltech team's results have been submitted for publication in the Astrophysical Journal Letters, and will appear this Tuesday on the public electronic archive, http://xxx.lanl.gov/list/astro-ph/
The W. M. Keck Observatory is a joint venture of Caltech, the University of California, and NASA, and is made possible by a generous gift from the W. M. Keck Foundation.
Written by Robert Tindol | {"url":"https://pma.caltech.edu/news/astronomers-detect-evidence-time-when-universe-emerged-dark-ages-506","timestamp":"2024-11-07T09:30:06Z","content_type":"text/html","content_length":"73142","record_id":"<urn:uuid:a7eee384-ed71-424a-810d-11cba21b6224>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00167.warc.gz"} |
EE345S Real-Time Digital Signal Processing Lab - Homework
Homeworks Assigned
• Homework #4: Assignment in PDF format.
• Homework #3: Assignment in PDF format.
□ Problem 3.3: This problem asks you to evaluate four ways to filter an input signal.
Some of the methods yield a linear convolution, and some do not. With an input signal of 24 samples in length and a filter with an impulse response of 16 samples in length, the linear
convolution would be 39 samples in length (i.e., 24 + 16 - 1).
For the FFT-based method, the length of the FFT determines the length of the filtered result. An FFT length of less than 39 would yield a convolution, but it wouldn't be linear convolution.
Convolution computed using the FFT is called circular convolution. When the FFT length is long enough, the answer computed by circular convolution is the same as by linear convolution.
Consider the case when the filter is a block in a block diagram, as would be found in Simulink or LabVIEW. When executing, the filter block would take in one sample from the input and produce
one sample on the output. How would the scheduler know how many times to execute the block? As many times as there are samples on the input arc. How many samples would be produced? As many
times as the block would be executed.
• Homework #2: Assignment in PDF format.
□ Matlab: There is a handout in the course reader on Matlab in Appendix D that may be helpful. In the CD ROM, the file freqresp.m will implement the Matlab function freqresp. Please put these
.m files on your path. A convenient place is the current working directory. The following Matlab commands may be useful:
☆ cd: change directories (without an argument, it prints the working directory)
☆ dir: lists the contents of the current working directory
☆ pwd: prints working directory
□ Problem 2.1: For the magnitude response, poles near the unit circle indicate the passband of the filter, and zeros near or on the unit circle indicate the stopband. Poles on the unit circle
correspond to a pure oscillator.
In the cases for which one cannot find the frequency response by simply substituting z = exp(j w), one can use the discrete-time Fourier transform (see slide 5-8). Using the discrete-time
Fourier transform may require you to search in other books, e.g. Discrete-Time Signal Processing by A. V. Oppenheim and R. W. Schafer. Another option is to use Mathematica to compute the
discrete-time Fourier transform for the discrete-time signal. To run Mathematica, login to sunfire1 or sunfire2 and type "math", and then cut-and-paste the following commands to find the
discrete-time Fourier transform for problem 2.1(d):
AppendTo[ $Path, "/home/ecelrc/faculty/bevans" ];
Needs[ "SignalProcessing`Master`" ];
First[ DiscreteTimeFourierTransform[ Cos[w0 n] DiscreteStep[n], n, w ] ]
For problem 2.1(c),
First[ DiscreteTimeFourierTransform[ DiscreteStep[n], n, w ] ]
□ Problem 2.4: In part (b), the filter order will be somewhat low for the given filter specifications. To see the difference among the IIR filter design methods, narrow the transition region
(i.e. increase the passband frequency and decrease the stopband frequency) but don't worry about the designed filter meeting the original specification. For digital IIR filters, the filter
order is inversely proportional to the width of the transition region.
The Matlab command filtdemo will provide a graphical user interface for designing filters. The filtdemo requires seven parameters: filter design method, maximum passband frequency, passband
magnitude tolerance, minimum stopband frequency, stopband magnitude tolerance, sampling rate, and filter order. The statement of problem 2.4 gives the filter design parameters to use in
In order to run Matlab remotely, run X windows on your local machine, login into sunapp1.ece.utexas.edu or sunapp2.ece.utexas.edu, using secure shell (ssh) and then run Matlab:
matlab &
If Matlab will not run due to an improper DISPLAY variable, then run the following on either sunapp machine:
source ~bevans/setd
There are many X windows emulators for PCs and Macs, including
• Homework #1: Assignment in PDF.
□ Problem 1.1(d): In part (d), an oscillator can be realized by a second-order difference equation. Please see the solution to Problem 1.1 on Midterm #1 in Spring 2004, which is slide K-29 in
the appendix of the course reader. (Tradeoffs among three sinusoid generation methods is the subject of Problem 1.2 on Midterm #1 in Fall 2003, which is on slide K-25 of the reader.)
□ Problem 1.3: There is a handout in the course reader on Matlab in Appendix D that may be helpful. In the CD ROM, the file specnoise.m will implement the Matlab function specnoise. Place the
file specnoise.m on the Matlab path. A convenient place is the current working directory. The following Matlab commands may be useful:
☆ cd: change directories (without an argument, it prints the working directory)
☆ dir: lists the contents of the current working directory
☆ pwd: prints working directory
□ Problem 1.5(a): The Matlab command filtdemo will provide a graphical user interface for designing filters. The filtdemo requires seven parameters:
☆ filter design method,
☆ maximum passband frequency in Hz,
☆ passband magnitude tolerance in dB,
☆ minimum stopband frequency in Hz,
☆ stopband magnitude tolerance in dB,
☆ sampling rate in Hz, and
☆ filter order.
The statement of problem 1.5 gives the filter design parameters to use in filtdemo.
Be sure to check the graphical view of the passband and stopband to make sure that each of the three filter designs meets specifications. If you are not absolutely sure, then you can compute
the magnitude response is at a particular frequency using the freqz function in Matlab. The arguments are the filter transfer function (type help filtdemo to find out how to obtain the
transfer function of the current design) and the frequency in Hz. Then, you can take the magnitude of the result. Finally, you'll need to convert the magnitude to dB using 20 log[10]
magnitude. For more information on freqz, type help freqz in Matlab.
In order to run Matlab remotely, run X windows on your local machine, login into sunapp1.ece.utexas.edu or sunapp2.ece.utexas.edu, and then run Matlab:
matlab &
If Matlab will not run due to an improper DISPLAY variable, then run the following on sunapp1:
source ~bevans/setd
There are a variety of X windows emulators, including
☆ X-Deep/32: similar to Xwin32. Written by an Austin programmer.
☆ Cygwin/X: uses Cygwin and X.org software to install an entire POSIX-compatible layer in Windows, so it's a little more complicated to setup.
□ Problem 1.5(d): The FIR coefficients designed by the Remez exchange, FIR least squares, and Kaiser window methods have even symmetry about the midpoint. Some DSPs have accelerator
instructions to take advantage of the symmetry in the coefficients. Most do not. The C6000 families do not.
You do not have to write any assembly language code for this problem. To compute one output sample of an FIR filter, one needs to compute the vector dot product of the vector of FIR
coefficients and the vector of the current and the previous N - 1 input values. In lecture 1, we discussed the rough outline of an assembly function to compute one output value of an FIR
☆ Prolog: initialize accumulator, initialize modulo indexing, and initialize downcounter register with value N
☆ Loop: compute the vector dot product, which entails reading coefficients and input data values from on-chip memory, multiplication, addition, and decrementing the downcounter
☆ Epilog: return output value | {"url":"http://signal.ece.utexas.edu/~arslan/courses/realtime/homework/index.html","timestamp":"2024-11-06T17:33:39Z","content_type":"text/html","content_length":"11990","record_id":"<urn:uuid:592d5f2d-852f-4a66-89db-961efaed304d>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00534.warc.gz"} |
Calculating LTV for a subscription-based SaaS product
For a newly launched subscription-based SaaS product, how do you prefer to calculate LTV?
There’s lots of ways to calculate LTV for subscription-based businesses, but I’ve been studying two in particular.
The first is very simple:
LTV = (Customer Value x Average Customer Lifespan)
Average Customer Lifespan = (1 / Average Churn Rate)
While this is very straightforward to calculate IF you have the correct Average Customer Lifespan, which for a new SaaS product, you’re estimating/projecting, and if your Churn Rate is uniform over
time, which for many SaaS products probably isn’t the case.
In my experience, customer churn numbers after month 1 follows a power law distribution like y = nx^k, where k < 0. I suppose if you calculate the Average Churn Rate over the first N months, it might
be good enough for a rough approximation of LTV.
The second approach involves a lot more data analysis.
I group customers into cohorts by signup month, then calculate the actual churn numbers by month, and use curve fitting to predict the future churn numbers.
Using only 6 months of data, this gets me R^2 > 0.83, which is good enough for this exercise. R^2 > 0.88 if I use 9 months of data.
I then apply the Kaplan-Meier estimator to calculate survival rate using my actual and estimated churn counts.
Then, to calculate LTV for each cohort:
LTV = Sum over time of (Survival Rate x Value)
I’ve seen some calculations that include a Discount Rate to account for Net Present Value and network effects of longer-lived subscribers, but since this is also another wild-ass guess, I chose to
ignore it for now.
What I found super interesting is, between the two approaches, the LTV I calculated using the first approach compared to the projected first year LTV using the second approach yielded results that
only differed by 0.59%.
The first approach, using an educated guess for Average Churn Rate, took a minute or two to compute.
I suppose the time it took to measure and calculate the churn rate to be able to come up with an approximation for the Average Churn Rate took some time, but that pre-work was already done.
But, this approach definitely relies on luck to choose the right Average Churn Rate number. If I had chosen a value that was just +/- 2.5% different, that would have given results that would have
differed by +16.2% or -17.0%, respectively. My educated guess was incredibly lucky, and I wouldn’t have known how lucky if I hadn’t done the deeper cohort-based analysis.
The second approach took a bunch of hours of segmenting customers into cohorts, then tabulating their actual churn histories, then the curve fitting, then setting up the model to calculate the LTV
for each cohort, and so on.
However, the outcome of this more thorough analysis is that it’s more closely dependent on the actual data and not on my ability to correctly guesstimate a value that is used in the calculation. | {"url":"https://panoptic.com/2022/calculating-ltv-for-a-subscription-based-saas-product/","timestamp":"2024-11-13T14:09:26Z","content_type":"text/html","content_length":"36192","record_id":"<urn:uuid:69809154-d6f2-49c9-a5cb-f822a08d30ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00669.warc.gz"} |
Deep Lists Assignment
Jump to navigation Jump to search
original inspiration for this warmup comes from Extra Practice Problems
Code To Use
Code To Implement
file: src/main/racket/deep_lists/deep_lists.rkt
functions: deep-fold
You can elect to build deep-fold as a higher order function which returns a function which will deep fold left or right depending on which list fold function is passed to it.
NOTE: You can also elect to just build deep-foldl and/or deep-foldr directly. If you choose to implement only one of deep-foldl and deep-foldr, be sure to comment out the tests for the unimplemented
(define (deep-fold list-fold)
(error 'not-yet-implemented))
The provided implementation uses #deep-fold. You may replace this if you prefer not to implement #deep-fold.
(define deep-foldl (deep-fold foldl))
The provided implementation uses #deep-fold. You may replace this if you prefer not to implement #deep-fold.
(define deep-foldr (deep-fold foldr))
Utilize a version of #deep-fold (and perhaps a list reverse depending on which fold direction you choose) to produce a single flattened list version of the deep list.
For example:
(deep-flatten (list 1 2 (list 3 "fred" 4 5) 6 (list 7 8 (list 9 "george" 10) 11 12)))
should evaluate to
'(1 2 3 "fred" 4 5 6 7 8 9 "george" 10 11 12)
Sums the numbers in a deep list of numbers and lists.
For example:
(deep-sum (list 1 2 (list 3 4 5) 6 (list 7 8 (list 9 10) 11 12)))
should evaluate to 78.
Like deep-sum only it ignores values which are neither lists nor numbers.
For example:
(deep-sum-numbers (list 1 2 (list 3 "fred" 4 5) 6 (list 7 8 (list 9 "george" 10) 11 12)))
should evaluate to 78.
file: deep_lists_test.rkt Test
source folder: src/test/racket/deep_lists
note: ensure that you have removed all printing to receive credit for any assignment. | {"url":"https://classes.engineering.wustl.edu/cse425s/index.php?title=Deep_Lists_Assignment","timestamp":"2024-11-08T23:37:09Z","content_type":"text/html","content_length":"31113","record_id":"<urn:uuid:e4f26396-018d-4fe8-b6ed-4209d5153364>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00639.warc.gz"} |
Measures Of Dispersion In Statistics: Definition and Examples - The Americans News
Measures of dispersion in statistics is a collection of metrics employed to assess the quality of data objectively.
Most data science studies begin with the fundamentals of statistics as dispersion is an essential subject you must not overlook.
Understanding the distribution of data is the most critical aspect of dispersion measures. The value of the measure of dispersion increases with the diversity of the data set.
Ungrouped or raw data or even different data sets can take time to interpret and analyze. The measures of dispersion help solve this by making data accessible to read.
Read on and learn more about the different various types of measures of dispersion, relevant examples and other related information to the measures.
Dispersion in Statistics
Dispersion means to distribute or disseminate. Statistical dispersion refers to the degree of variability a set of values spreads relative to an median value. Or, to put it another way, dispersion
helps us comprehend data distribution.
Dispersion in statistics helps understand how the data varies in that it is homogeneous or heterogeneous. It helps show how broadly or narrowly spread the data is.
Dispersion measures are positive integers that quantify the distribution between data points and a center value. These values assist in determining the spread of the data, to how squeezed or
stretched it is.
There are five standard methods for measuring dispersion: variance, range, quartile deviation, mean deviation, and standard deviation.
Objectives of measures of dispersion in statistics include:
• It attempts to determine the degree of similarity or consistency between two or more data sets. A significant degree of variance indicates a limited degree of consistency. A lower degree of
variance corresponds to greater consistency and uniformity.
• Differentiation helps ascertain the causes of variations in a given data set and helps manage the variation.
• Measures of dispersion help one to determine the extent to which an average represents the whole data set.
• Measures of dispersion are used in the calculation of various statistical methods, including hypothesis tests and regression.
The two main measures of dispersion in statics include the following;
• Relative Measure of Dispersion
• Absolute Measure of Dispersion
1. Absolute Measure of Dispersion
Absolute measures of dispersion quantify the degree of variation among a set of numbers expressed in observation units.
Absolute dispersion measures differences as mean or standard deviations, representing the average variation in the data.
For instance, if you provide data regarding the temperature readings in an area over days in °C, absolute measures of dispersion will give the variation in °C.
The absolute measures of dispersion are as follows:
• Range: In essence, range denotes the quantity of values between a given set’s maximum and minimum values. Consider the following: 1, 4, 7, 9, 11; range = 11-1 = 10.
• Variance: Subtract the mean of the given set of values from each data in the group, square the value, add the squares, and divide by the number of values in the set to get the variance.
Variance (σ2) = ∑(X−μ)2/N
Standard Deviation: Get the square root of the variance to get the standard deviation of the data set S.D. = √σ.
• Quartiles and Quartile Deviation: Quartiles represent values that divide a given set of numbers into four equal portions. A quartile deviation of half is equal to the difference between the first
and third quartiles.
Mean and Mean Deviation: The average of numbers in the data provided is the mean and the mean of absolute deviations is the mean deviation
1. Range
The range is the most straightforward method for quantifying variation and is relatively easy to calculate. We calculate it by subtracting the minimum and maximum values in the data set.
You have likely encountered the range on numerous occasions as it provides the most accurate estimation of the variability of a given entity. Although range may appear alluring due to its simplicity
in calculation, it may not give a reliable indication of variation.
Here’s how you calculate the range—a group of numbers: 8, 7, 4, 3, 5, 10, 6. Get the range by taking the maximum number 10 and the minimum 3. The difference between the two, 7 is the range of that
data set.
• It is straightforward to deduce and comprehend.
• No technical formula is required to calculate the range
• The computation requires the shortest time possible.
• It presents a concise summary of all pertinent information.
• This metric is straightforward as it solely considers the minimum and maximum values.
• Determining the range of an open-ended series is unattainable.
• Differences between samples significantly impact the range, which varies considerably between samples.
2. Variance
It approximates the deviation of a given set of (random) numbers from their mean. Variance is the average squared difference between the values and the mean.
Statistically defined, it is the sum of squared deviations between each score and the mean divided by the number of scores in the set.
3. Standard Deviation
Calculate the standard deviation by getting the square root of the variance. Having the square root of the variance helps you understand the data as it returns the standard deviation in the units
employed initially to measure it.
Standard deviation is a practical and understandable value as when reporting summary statistics for a study; researchers say the mean and standard deviation. They are the most commonly used measures
of dispersion.
The value of a set of numbers’ standard deviation indicates their degree of dispersion around the mean. It simply shows the degree to which the value deviates from its mean.
A minor variation in even one of the factors causes an entire standard deviation to shift. Hence, it is dependent on every element present in the dataset. The significance of the data set’s size is
more important than its origin.
Since the standard deviation indicates the degree of spread among the values in a given set, its value is always 0 or a positive integer.
A low standard deviation indicates minimal variability in the data. If the standard deviation is bigger than the given data, there’s more spread from the mean, a bigger measure of dispersion.
The Greek letter π represents the population standard deviation, while the lowercase letter s denotes the sample standard deviation.
You must be familiar with the formula for calculating standard deviation. Calculate the standard deviation, by following these outlined steps.
• Find the mean. This is the initial and paramount step when provided with a dataset. To calculate the mean, get the sum of all the available data and divide the result by the total number of data.
• After determining the mean, subtract the amount from each data in the set, then square the value.
• Calculate the mean of the squared values you put aside. Sum the squared data then divide the value by the number of data to get the mean of the squared values.
• What you have now is the variance. The square root of the given value (the variance) is the standard deviation.
4. Quartiles and Quartile Deviation
Quartiles are measurement units that divide the data set into four equal parts, ensuring each part has the same observation data.
For this reason, there are three quartiles, namely Q1, Q2, and Q3. Q1 represents the first quartile or the lower quartile. This quartile houses 25% of the data, and 75% of the value is more
significant than it.
The second quartile, Q2, houses 50% under it, and 50% of the items are greater than its value. It’s more of the median of the data provided.
The letter Q3, or the upper quartile, denotes the third quarter. It comprises 75% of the values in the spread with 25 % being above it.
In a nutshell, Q1 and Q3 are the two limits in which almost half of the data spread lies. The difference between Q1 and 3 gives you the medium of the values presented.
• It is also straightforward to comprehend and deduce.
• It is also effective for open-ended sharing.
• It exhibits a reduced impact of extreme numbers, rendering it superior to “Range.”
• It is more beneficial when calculating the dispersion of the middle 50%.
• It doesn’t factor all observations
• It doesn’t promote further mathematical or statistical processing
• Variations in sampling considerably impact outcomes.
• It is not as reliable as alternative dispersion measures due to its omission of half of the data.
5. Mean and Mean Deviation
Mean deviation is the distance between the observed values and the distribution’s mean. Some deviations show values that are either positive or negative.
In this manner, summing them will not reveal much difference because their effects tend to cancel one another out.
For example :
We have this set of data: -10, 5, 35
We get the mean = (-10 + 5 + 35)/3 = 10
Now a deviation from the mean for different values is,
• (-10 -10) = -20
• (5 – 10) = -5
• (35 – 10) = 25
As shown, adding the difference between the mean value and the actual data set cancels each other, and we get an x=zero deviation.
Alternatively, to resolve this issue, use the absolute values of the differences to calculate the mean deviation.
• Focuses on all observations and doesn’t rely on limits such as range and quartile deviation
• It is straightforward to deduce and comprehend.
• Result not affected by extreme values
• The average is used to calculate the mean deviation
• In mathematics, it is not advisable to disregard the concepts of + or – values.
• It does not merit further mathematical investigation.
• Determining the mean or median becomes challenging when the value is a fraction.
• This strategy may not work on open-ended series.
Measures of Dispersion
Range H – SH = The Largest ValueS = the Smallest Value
Variance Population Variance, σ^2 = Σ(x[i]-μ)^2 /nSample Variance, S^2 = Σ(x[i]-μ)^2 /(n-1)n = The number of observationμ = The mean
Standard Deviation S.D. = √(σ^2)
Mean Deviation μ = (x – a)/nn = The number of observationa = The central value(mean, median, mode)
Quartile Deviation (Q[3 ]– Q[1])/2Q[3] = Third QuartileQ[1] = First Quartile
Factors Affecting Variability
Before looking at the other types of measures of dispersion in statistics, we would like to go through several factors that can influence data distribution.
i) Stability during sample collection: Several samples from the same group will produce identical outcomes due to their common origin. This explains their stability; one would anticipate samples to
be just as variable as the group from which they originated.
ii) Outliers: Extreme scores affect the range, standard deviation, and variance. One outlier or an extreme data set score will definitely alter the overall statistical value you calculate.
iii)Sample size: Increasing or decreasing a given sample size within a set increases the range due to the change of values in the data set.
2. Relative Measures of Dispersion
Relative measures of dispersion remain unaffected by the measurement units used to denote the readings.
They are numbers and compare variation not in a data set but between two or more data sets with different units and observation measurements. Having absolute and relative measures of dispersion is
handy for Six Sigma teams.
Relative measures of dispersion are necessary when comparing data from distinct sets that employ distinct units. These values are expressed as ratios and percentages, lacking a standard unit. The
following are some of the numerous dispersion measures:
1. Coefficient of Range: Expressed as a ratio between a set’s maximum and minimum value to the sum of the maximum and minimum value.
2. Coefficient of Variation: In terms of the data set, the variation coefficient is equivalent to the standard deviation from the mean expressed as a percentage.
3. Coefficient of Mean Deviation: The coefficient mean deviation is calculated by dividing the mean deviation by the center point value used from the data provided.
4. Coefficient of Quartile Deviation: it is calculated by dividing the difference between the third and first quartiles expressed as a ratio.
Relative Measures of Dispersion Related Formulas
Coefficient of Range (H – S)/(H + S)
Coefficient of Variation (SD/Mean)×100
Coefficient of Mean Deviation (Mean Deviation)/μwhere,μ is the central point for which the mean is calculated
Coefficient of Quartile Deviation (Q[3 ]– Q[1])/(Q[3] + Q[1])
Measures of Central Tendency
A central tendency can be explained as the value of a singular number that attempts to define a set of data by indicating the location of the midpoint. They are sometimes referred to as measures of
central location.
Grouped as summary statistics, the mean or the average is arguably the most widely recognized. However, there is the median and mode also.
It is possible to determine the central trajectory of a set of data using the mean, median, or mode. Nevertheless, instances arise in which one of these metrics surpasses the others.
1. Mean
It represents the average value of the set. It is determined by getting the sum and dividing the product of each value in the list by the total number of values. Researchers commonly call it the
arithmetic mean.
2. Median
The median is the middle value of a set of data set items when arranged in ascending or descending order. If the data is even, add the two values at the middle then divide by two to get the mean. The
value is the median of the data set.
3. Mode
The mode of a dataset indicates the number that occurs most frequently. Some data may show several modes while some will not show any frequently occurring value in the data set.
Measures of dispersion and central tendency values all help understand data. The following table illustrates the distinction between dispersion measurement and central trend measurement.
Measures of Dispersion Central Tendency
Measures of dispersion help quantify the variables in data. Measures of central tendency quantifies the data’s average behavior.
Examples include variance, mean deviation, standard deviation, quartile deviation, etc. Examples include mean, median, and mode.
Quick Notes on Dispersion
In the field of statistics, the concept of dispersion holds significant importance. It facilitates comprehension of concepts such as data distribution, maintenance, and diversification compared to
the central value or trend.
Moreover, statistical dispersion enables us to gain a more comprehensive understanding of how data is spread. For instance, Mean, Median, and Range might be identical between two distinct groups,
whereas the degree of variation might be pretty distinct.
Here are quick notes to remember on dispersion;
• Q2 in the frequency distribution series is also the median of the values
• Measures of dispersion help determine data spread and are measured around a central value.
• Calculate the median the same way you calculate Q1 and Q3
• Measures of dispersion are grouped into two broad categories, namely relative and absolute measure of dispersion
• The range of the Lorenz curve’s values is from 0 to 100
• Absolute measures of deviations have identical units as data provided while relative measures are limitless
• A good measure of dispersion is easy to calculate and analyze
• Variations in data don’t alter a precise measure of dispersion.
• Absolute deviation measures include the range, variance, standard deviation, quartile deviation, and mean deviation.
• Relative measures of dispersion are the coefficients of dispersion.
To grasp the concept of measures of dispersion in statistics, it is imperative first to understand what dispersion means.
Dispersion is a statistical term for the degree of spread of a set of data. Dispersion refers to the process by which data is varied across multiple groups. It entails calculating the magnitudes of
the expected distribution values for a given variable using the available data.
Understanding what dispersion entails bridges you to the two types of dispersion. Absolute dispersion illustrates the degree of variability among the values of distinct data sets with respect to the
mean. Relative dispersion is the ratio of the standard deviation to its mean. | {"url":"https://www.theamericansnews.com/measures-of-dispersion-in-statistics/","timestamp":"2024-11-08T07:31:11Z","content_type":"text/html","content_length":"292019","record_id":"<urn:uuid:96d7671a-22f6-4e60-9d38-4e591ea93d25>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00279.warc.gz"} |
How to find the area of the base of the boxHow to find the area of the base of the box 🚩 the base area of a rectangular parallelepiped 🚩 Math.
You will need
• Ruler, protractor, scientific calculator
In the General case, the base of a parallelepiped is a parallelogram. To find its area, using a ruler, make the measurement of the lengths of its sides and a protractor to measure the angle between
them. The square base of a parallelepiped is equal to the product of these sides by the sine of the angle between them S=a • b • Sin(α).
To determine the area of the base of the box the other way, measure one side of the base, then lower to her height from the vertex, which lies opposite this side. Measure the length of this height.
To obtain the area of the base find the area of a parallelogram by multiplying the length of a side on high, which is omitted S=a • h.
To obtain the values of area another way to measure the lengths of its diagonals (the distance between opposite vertices), and the angle between the diagonals. The area will be equal to half the
product of the diagonals by the sine of the angle between them S=0,5•d1•d2•Sin(β).
For a parallelepiped, which lies at the base of the diamond, it is sufficient to measure the lengths of its diagonals and to find half their work S=0,5 • d1 • d2.
In the case where the base cuboid is a rectangle, measure the length and width of this geometric figure, and then multiply these values S=a • b. This will be the area of its base. In the case when
the base is square, measure one of its sides and lift in the second degree S=a2.
If you know the volume of the box, measure its height. To do this, drop a perpendicular from any vertex of the upper base onto the plane, which belongs to the lower base. Measure the length of this
segment, which is the height of the box. If the parallelepiped line (its lateral edges perpendicular to the basem), it is enough to measure the length of one of these edges, which equals the height
of the box. To obtain area of base, volume of a parallelepiped divide by its height S=V/h. | {"url":"https://eng.kakprosto.ru/how-70884-how-to-find-the-area-of-the-base-of-the-box","timestamp":"2024-11-10T09:25:29Z","content_type":"text/html","content_length":"33086","record_id":"<urn:uuid:fb03aafe-3519-4357-a681-d11fdcdf5761>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00589.warc.gz"} |
How to Turn a Quantum Computer Into the Ultimate Randomness Generator | Quanta Magazine
Say the words “quantum supremacy” at a gathering of computer scientists, and eyes will likely roll. The phrase refers to the idea that quantum computers will soon cross a threshold where they’ll
perform with relative ease tasks that are extremely hard for classical computers. Until recently, these tasks were thought to have little real-world use, hence the eye rolls.
But now that Google’s quantum processor is rumored to be close to reaching this goal, imminent quantum supremacy may turn out to have an important application after all: generating pure randomness.
Randomness is crucial for almost everything we do with our computational and communications infrastructure. In particular, it’s used to encrypt data, protecting everything from mundane conversations
to financial transactions to state secrets.
Genuine, verifiable randomness — think of it as the property possessed by a sequence of numbers that makes it impossible to predict the next number in the sequence — is extremely hard to come by.
That could change once quantum computers demonstrate their superiority. Those first tasks, initially intended to simply show off the technology’s prowess, could also produce true, certified
randomness. “We are really excited about it,” said John Martinis, a physicist at the University of California, Santa Barbara, who heads Google’s quantum computing efforts. “We are hoping that this is
the first application of a quantum computer.”
Randomness and Entropy
Randomness and quantum theory go together like thunder and lightning. In both cases, the former is an unavoidable consequence of the latter. In the quantum world, systems are often said to be in a
combination of states — in a so-called “superposition.” When you measure the system, it will “collapse” into just one of those states. And while quantum theory allows you to calculate probabilities
for what you’ll find when you do your measurement, the particular result is always fundamentally random.
Physicists have been exploiting this connection to create random-number generators. These all rely on measurements of some kind of quantum superposition. And while these systems are more than
sufficient for most people’s randomness needs, they can be hard to work with. In addition, it’s extremely difficult to prove to a skeptic that these random-number generators really are random. And
finally, some of the most effective methods for generating verifiable randomness require finicky setups with multiple devices separated by great distances.
One recent proposal for how to pull randomness out of a single device — a quantum computer — exploits a so-called sampling task, which will be among the first tests of quantum supremacy. To
understand the task, imagine you are given a box filled with tiles. Each tile has a few 1s and 0s etched onto it — 000, 010, 101 and so on.
If there are just three bits, there are eight possible options. But there can be multiple copies of each labeled tile in the box. There might be 50 tiles labeled 010 and 25 labeled 001. This
distribution of tiles determines the likelihood that you’ll randomly pull out a certain tile. In this case, you’re twice as likely to pull out a tile labeled 010 as you are to pull out a tile labeled
A sampling task involves a computer algorithm that does the equivalent of reaching into a box with a certain distribution of tiles and randomly extracting one of them. The higher the probability
specified for any tile in the distribution, the more likely it is that the algorithm will output that tile.
Of course, an algorithm isn’t going to reach into a literal bag and pull out tiles. Instead, it will randomly output a binary number that’s, say, 50 bits long, after being given a distribution that
specifies the desired probability for each possible 50-bit output string.
For a classical computer, the task becomes exponentially harder as the number of bits in the string gets larger. But for a quantum computer, the task is expected to remain relatively straightforward,
whether it involves five bits or 50.
The quantum computer starts with all its quantum bits — qubits — in a certain state. Let’s say they all start at 0. Just as classical computers act on classical bits using so-called logic gates,
quantum computers manipulate qubits using the quantum equivalent, known as quantum gates.
But quantum gates can put qubits into strange states. For example, one kind of gate can put a qubit that starts with an initial value 0 into a superposition of 0 and 1. If you were to then measure
the state of the qubit, it would collapse randomly into either 0 or 1 with equal probability.
Even more bizarrely, quantum gates that act on two or more qubits at once can cause the qubits to become “entangled” with each other. In this case, the states of the qubits become intertwined, so
that the qubits can now only be described using a single quantum state.
If you put a bunch of quantum gates together, then have them act on a set of qubits in some specified sequence, you’ve created a quantum circuit. In our case, to randomly output a 50-bit string, you
can build a quantum circuit that puts 50 qubits, taken together, into a superposition of states that captures the distribution you’d like to re-create.
When the qubits are measured, the entire superposition will collapse randomly to one 50-bit string. The probability that it’ll collapse to any given string is dictated by the distribution that is
specified by the quantum circuit. Measuring the qubits is akin to reaching blindfolded into the box and randomly sampling one string from the distribution.
How does this get us to random numbers? Crucially, the 50-bit string sampled by the quantum computer will have a lot of entropy, a measure of disorder or unpredictability, and hence randomness. “This
might actually be kind of a big deal,” said Scott Aaronson, a computer scientist at the University of Texas, Austin, who came up with the new protocol. “Not because it’s the most important
application of quantum computers — I think it’s far from that — rather, because it looks like probably the first application of quantum computers that will be technologically feasible to implement.”
Aaronson’s protocol to generate randomness is fairly straightforward. A classical computer first gathers a few bits of randomness from some trusted source and uses this “seed randomness” to generate
the description of a quantum circuit. The random bits determine the types of quantum gates and the sequence in which they should act on the qubits. The classical computer sends the description to the
quantum computer, which implements the quantum circuit, measures the qubits, and sends back the 50-bit output bit string. In doing so, it has randomly sampled from the distribution specified by the
Now repeat the process over and over — for example, 10 times for each quantum circuit. The classical computer uses statistical tests to ensure that the output strings have a lot of entropy. Aaronson
has shown, partly in work published with Lijie Chen and partly in work yet to be published, that under certain plausible assumptions that such problems are computationally hard, no classical computer
can generate such entropy in anywhere near the time it would take a quantum computer to randomly sample from a distribution. After the checks, the classical computer pastes together all the 50-bit
output strings and feeds it all to a well-known classical algorithm. “It produces a long string that is nearly perfectly random,” Aaronson said.
The Quantum Trapdoor
Aaronson’s protocol is best suited for quantum computers with about 50 to 100 qubits. As the number of qubits in a quantum computer passes this threshold, it becomes computationally intractable for
even classical supercomputers to use the protocol. This is where another scheme for generating verifiable randomness using quantum computers enters the picture. It uses an existing mathematical
technique with a forbidding name: a trapdoor claw-free function. “It sounds much worse than it is,” said Umesh Vazirani, a computer scientist at the University of California, Berkeley, who devised
the new strategy along with Zvika Brakerski, Paul Christiano, Urmila Mahadev and Thomas Vidick.
Imagine a box again. Instead of reaching in and extracting a string, this time you drop in an n-bit string, call it x, and out pops another n-bit string. The box is somehow mapping an input string to
an output string. But the box has a special property: For every x, there is another input string y that generates the same output string.
In other words, there exist two unique input strings — x and y — for which the box returns the same output string, z. This triplet of x, y and z is called a claw. The box, in computer science speak,
is a function. The function is easy to compute, meaning that given x or y, it’s easy to calculate z. But if you are only given x and z, finding y — and hence the claw — is impossible, even for a
quantum computer.
Update June 21, 2019: This article has been updated to include a reference to Aaronson’s unpublished work.
Jana Asenbrennerova for Quanta Magazine; Simons Institute for the Theory of Computing; Courtesy of Caltech
The only way you could get at the claw is if you had some inside information, the so-called trapdoor.
Vazirani and his colleagues want to use such functions not only to get quantum computers to generate randomness, but to verify that the quantum computer is behaving, well, quantum mechanically —
which is essential to trusting the randomness.
The protocol starts with a quantum computer that puts n qubits into a superposition of all n-bit strings. Then a classical computer sends over a description of a quantum circuit specifying the
function to be applied to the superposition — a trapdoor claw-free function. The quantum computer implements the circuit, but without knowing anything about the trapdoor.
At this stage, the quantum computer enters a state in which one set of its qubits is in a superposition of all n-bit strings, while another set holds the result of applying the function to this
superposition. The two sets of qubits are entangled with each other.
The quantum computer then measures the second set of qubits, randomly collapsing the superposition into some output z. The first set of qubits, however, collapses into an equal superposition of two n
-bit strings, x and y, because either could have served as input to the function that led to z.
The classical computer receives the output z, then does one of two things. Most of the time, it asks the quantum computer to measure its remaining qubits. This will collapse the superposition, with a
50-50 chance, into either x or y. That’s equivalent to getting a 0 or a 1, randomly.
Occasionally, to check on the quantum computer’s quantumness, the classical computer asks for a special measurement. The measurement and its outcome are designed so that the classical computer, with
the help of the trapdoor that only it has access to, can ensure that the device answering its queries is indeed quantum. Vazirani and colleagues have shown that if the device gives the correct answer
to the special measurement without using collapsing qubits, that’s equivalent to figuring out the claw without using the trapdoor. This, of course, is impossible. So there must be at least one qubit
collapsing inside the device (providing, randomly, a 0 or a 1). “[The protocol] is creating a tamper-proof qubit inside an untrusted quantum computer,” Vazirani said.
This tamper-proof qubit provides one truly random bit of information with each interrogation; a sequence of such queries can then be used to create long, random bit strings.
This scheme might be faster than Aaronson’s quantum sampling protocol, but it has a distinct disadvantage. “It’s not going to be practical with 50 or 70 qubits,” Aaronson said.
Aaronson, for now, is waiting for Google’s system. “Whether the thing they are going to roll out is going to be actually good enough to achieve quantum supremacy is a big question,” he said.
If it is, then verifiable quantum randomness from a single quantum device is around the corner. “We think it’s useful and a potential market, and that’s something we want to think about offering to
people,” Martinis said.
This article was reprinted on Wired.com. | {"url":"https://www.quantamagazine.org/how-to-turn-a-quantum-computer-into-the-ultimate-randomness-generator-20190619/","timestamp":"2024-11-04T12:41:33Z","content_type":"text/html","content_length":"215925","record_id":"<urn:uuid:ab284753-5b43-42aa-94db-d228a5d6cec4>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00842.warc.gz"} |
Section 1.5 Circles. OBJECTIVES -Write standard form for the equation of a circle. -Find the intercepts of a circle and graph. -Write the general form. - ppt download
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José Paulo Santos
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Production and decay of sulfur excited species in an electron-cyclotron-resonance ion-source plasma
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The most important processes for the creation of S12+ to S14+ ions excited states from the ground configurations of S9+ to S14+ ions in an electron cyclotron resonance ion source, leading to the
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energies for the de-excitation processes, are calculated in the framework of the multiconfiguration Dirac-Fock method. With reasonable assumptions about the electron energy distribution, a
theoretical Kalpha x-ray spectrum is obtained, which is compared to recent experimental data.
Martins, M. C., J. P. Santos, A. M. Costa, and F. Parente.
Transition wavelengths and probabilities for spectral lines of Zr III
The European Physical Journal D
39 (2006): 167-172.
Wavelengths and oscillator strengths for all dipole-allowed fine-structure transitions in Zr III have been calculated within the Multi-Configuration Dirac-Fock method with QED corrections. These
transitions are included in the spectrum of some chemically peculiar stars, like the B-type star Lupi observed by the Hubble space telescope. The results are compared to existing experimental and
semi-empirical data.
Mayo, R., M. Ortiz, F. Parente, and J. P. Santos.
Experimental and theoretical transition probabilities for lines arising from the 6p configurations of Au II
Journal of Physics B: Atomic, Molecular and Optical Physics
40 (2007): 4651.
Experimental relative transition probabilities for the 16 more pro-eminent lines arising from the 6p configurations of Au II were determined from the emission-line intensities in a laser-produced
plasma. The experiment was carried out using a Cu-Au alloy with 10% Au content in order to obtain an optically thin plasma. Transition probabilities were placed on an absolute scale by using
theoretical lifetimes calculated in this work, line-strength sum rules and Boltzmann plot. A comparison has been conducted between present experimental results, the theoretical data available and new
calculations with the multi-configuration Dirac-Fock method reported in this work, as well as a study of the plasma conditions.
Morrison, J. C., S. Boyd, L. Marsano, B. Bialecki, T. Ericsson, and J. P. Santos.
Numerical methods for solving the Hartree-Fock equations of diatomic molecules I.
Communications in Computational Physics
5 (2008): 959-985.
The theory of domain decomposition is described and used to divide the variable domain of a diatomic molecule into separate regions which are solved independently. This approach makes it possible to
use fast Krylov methods in the broad interior of the region while using explicit methods such as Gaussian elimination on the boundaries. As is demonstrated by solving a number of model problems,
these methods enable one to obtain solutions of the relevant partial differential equations and eigenvalue equations accurate to six significant figures with a small amount of computational time.
Since the numerical approach described in this article decomposes the variable space into separate regions where the equations are solved independently, our approach is very well-suited to parallel
computing and offers the long term possibility of studying complex molecules by dividing them into smaller fragments that are calculated separately. | {"url":"https://docentes.fct.unl.pt/jps/publications?page=4&sort=author&order=asc","timestamp":"2024-11-01T18:56:37Z","content_type":"application/xhtml+xml","content_length":"66323","record_id":"<urn:uuid:67fe7d20-98bb-4265-87fe-408f0ff320ae>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00440.warc.gz"} |
Duong Hieu Phan
(Click on paper title to view the abstract.)
• (Book's Editors) Security and Cryptography for Networks
with Clemente Galdi (Universita' di Salerno)
Lecture Notes in Computer Science (PART 1: volume 14974 and PART 2: volume 14975.)
The two-volume set LNCS 14973 and 14974 constitutes the proceedings of the 14th International Conference on Security and Cryptography for Networks, SCN 2024 , which took place in Amalfai, Italy,
during September 11-13, 2024.
The 33 full papers included in the proceedings were organized in topical sections as follows:
Part I: Zero Knowledge; foundations; protocols; voting systems;
Part II: Homomorphic encryption; symmetric key encryption; cryptanalysis; key management; blockchains.
• Adaptive Hardcore Bit and Quantum Key Leasing over Classical Channel from LWE with Polynomial Modulus
with Weiqiang Wen (Telecom Paris, IPP), Xingyu Yan (Beijing University of Posts and Telecommunications) and Jinwei Zheng (Telecom Paris, IPP)
To appear in Advances in Cryptology - IACR ASIACRYPT 2024.
• Public-Key Anamorphism in (CCA-secure) Public-Key Encryption and Beyond
with Giuseppe Persiano (Univ. of Salerno and Google) and Moti Yung (Google and Columbia Univ.)
In Advances in Cryptology - IACR CRYPTO 2024.
[Paper in pdf]
The notion of (Receiver-) Anamorphic Encryption was put forth recently to show that a dictator (i.e., an overreaching government), which demands to get the receiver's private key and even
dictates messages to the sender, cannot prevent the receiver from getting an additional covert anamorphic message from a sender. The model required an initial private collaboration to share some
secret. There may be settings though where an initial collaboration may be impossible or performance-wise prohibitive, or cases when we need an immediate message to be sent without private key
generation (e.g., by any casual sender in need). This situation, to date, somewhat limits the applicability of anamorphic encryption. To overcome this, in this work, we put forth the new notion
of public-key anamorphic encryption, where, without any initialization, any sender that has not coordinated in any shape or form with the receiver, can nevertheless, under the dictator control of
the receiver's private key, send the receiver an additional anamorphic secret message hidden from the dictator. We define the new notion with its unique new properties, and then prove that, quite
interestingly, the known CCA-secure Koppula-Waters (KW) system is, in fact, public-key anamorphic.
We then describe how a public-key anamorphic scheme can support a new hybrid anamorphic encapsulation mode (KDEM) where the public-key anamorphic part serves a bootstrapping mechanism to activate
regular anamorphic messages in the same ciphertext, thus together increasing the anamorphic channel capacity.
Looking at the state of research thus far, we observe that the initial system (Eurocrypt'22) that was shown to have regular anamorphic properties is the CCA-secure Naor-Yung (and other related
schemes). Here we identify that the KW CCA-secure scheme also provides a new type of anamorphism. Thus, this situation is hinting that there may be a connection between some types of CCA-secure
schemes and some type of anamorphic schemes (in spite of the fact that the goals of the two primitives are fundamentally different); this question is foundational in nature. Given this, we
identify a sufficient condition for a ``CCA-secure scheme which is black-box reduced from a CPA secure scheme'' to directly give rise to ``anamorphic encryption scheme!'' Furthermore, we identify
one extra property of the reduction, that yields a public-key anamorphic scheme as defined here.
• Computational Differential Privacy for Encrypted Databases Supporting Linear Queries
with Ferran Alborch Escobar (Orange Innovation), Sébastien Canard (Télécom Paris) and Fabien Laguillaumie (Université de Montpellier)
In Proceedings on Privacy Enhancing Technologies - PoPETs 2024, Bristol, UK, 2024.
[Paper in pdf]
Differential privacy is a fundamental concept for protecting individual privacy in databases while enabling data analysis. Conceptually, it is assumed that the adversary has no direct access to
the database, and therefore, encryption is not necessary. However, with the emergence of cloud computing and the « on-cloud » storage of vast databases potentially contributed by multiple
parties, it is becoming increasingly necessary to consider the possibility of the adversary having (at least partial) access to sensitive databases. A consequence is that, to protect the on-line
database, it is now necessary to employ encryption. At PoPETs'19, it was the first time that the notion of differential privacy was considered for encrypted databases, but only for a limited type
of query, namely histograms. Subsequently, a new type of query, summation, was considered at CODASPY'22. These works achieve statistical differential privacy, by still assuming that the adversary
has no access to the encrypted database.
In this paper, we argue that it is essential to assume that the adversary may eventually access the encrypted data, rendering statistical differential privacy inadequate. Therefore, the
appropriate privacy notion for encrypted databases that we use is computational differential privacy, which was introduced by Beimel et al. at CRYPTO '08. In our work, we focus on the case of
functional encryption, which is an extensively studied primitive permitting some authorized computation over encrypted data. Technically, we show that any randomized functional encryption scheme
that satisfies simulation-based security and differential privacy of the output can achieve computational differential privacy for multiple queries to one database. Our work also extends the
summation query to a much broader range of queries, specifically linear queries, by utilizing inner-product functional encryption. Hence, we provide an instantiation for inner-product
functionalities by proving its simulation soundness and present a concrete randomized inner-product functional encryption with computational differential privacy against multiple queries. In term
of efficiency, our protocol is almost as practical as the underlying inner product functional encryption scheme. As evidence, we provide a full benchmark, based on our concrete implementation for
databases with up to 1 000 000 entries. Our work can be considered as a step towards achieving privacy-preserving encrypted databases for a wide range of query types and considering the
involvement of multiple database owners.
• Fully Dynamic Attribute-Based Signatures for Circuits from Codes
with San Ling (NTU), Khoa Nguyen (Univ. of Wollongong), Khai Hanh Tang (NTU), Huaxiong Wang (NTU) and Yanhong Xu (Shanghai Jiao Tong Univ.)
In IACR PKC 2024, Sydney, 2024.
[Paper in pdf]
Attribute-Based Signature (ABS), introduced by Maji et al. (CT-RSA’11), is an advanced privacy-preserving signature primitive that has gained a lot of attention. Research on ABS can be
categorized into three main themes: expanding the expressiveness of signing policies, enabling new functionalities, and providing more diversity in terms of computational assumptions. We
contribute to the development of ABS in all three dimensions, by providing a fully dynamic ABS scheme for arbitrary circuits from codes. The scheme is the first ABS from code-based assumptions
and also the first ABS system offering the full dynamicity functionality (i.e., attributes can be enrolled and revoked simultaneously). Moreover, the scheme features much shorter signature size
than a latticebased counterpart proposed by El Kaafarani and Katsumata (PKC’18). In the construction process, we put forward a new theoretical abstraction of Stern-like zero-knowledge (ZK)
protocols, which are the major tools for privacy-preserving cryptography from codes. Our main insight here actually lies in the questions we ask about the fundamental principles of Stern-like
protocols that have remained unchallenged since their conception by Stern at CRYPTO’93. We demonstrate that these longestablished principles are not essential, and then provide a refined
framework generalizing existing Stern-like techniques and enabling enhanced construction
• Verifiable Decentralized Multi-Client Functional Encryption for Inner Product
with Dinh Duy Nguyen (Telecom Paris, IPP) and David Pointcheval (ENS)
In Advances in Cryptology - IACR ASIACRYPT 2023.
[Paper in pdf]
Joint computation on encrypted data is becoming increasingly crucial with the rise of cloud computing. In recent years, the development of multi-client functional encryption (MCFE) has made it
possible to perform joint computation on private inputs, without any interaction. Well-settled solutions for linear functions have become efficient and secure, but there is still a shortcoming:
if one user inputs incorrect data, the output of the function might become meaningless for all other users (while still useful for the malicious user). To address this issue, the concept of
verifiable functional encryption was introduced by Badrinarayanan et al. at Asiacrypt ’16 (BGJS). However, their solution was impractical because of strong statistical requirements. More
recently, Bell et al. introduced a related concept for secure aggregation, with their ACORN solution, but it requires multiple rounds of interactions between users. In this paper,
□ we first propose a computational definition of verifiability for MCFE. Our notion covers the computational version of BGJS and extends it to handle any valid inputs defined by predicates. The
BGJS notion corresponds to the particular case of a fixed predicate in our setting;
□ we then introduce a new technique called Combine-then-Descend, which relies on the class group. It allows us to construct One-time Decentralized Sum (ODSUM) on verifiable private inputs.
ODSUM is the building block for our final protocol of a verifiable decentralized MCFE for inner-product, where the inputs are within a range. Our approach notably enables the efficient
identification of malicious users, thereby addressing an unsolved problem in ACORN.
• Anamorphic Signatures: Secrecy From a Dictator Who Only Permits Authentication!
with Mirek Kutylowski (Wroclaw University of Science and Technology), Giuseppe Persiano (Università di Salerno and Google), Moti Yung (Google and Columbia University) and Marcin Zawada (Wroclaw
University of Science and Technology)
In Advances in Cryptology - IACR CRYPTO 2023.
[Paper in pdf]
The goal of this research is to raise technical doubts regarding the usefulness of the repeated attempts by governments to curb Cryptography (aka the “Crypto Wars”), and argue that they, in fact,
cause more damage than adding effective control. The notion of Anamorphic Encryption was presented in Eurocrypt’22 for a similar aim. There, despite the presence of a Dictator who possesses all
keys and knows all messages, parties can arrange a hidden “anamorphic” message in an otherwise indistinguishable from regular ciphertexts (wrt the Dictator).
In this work, we postulate a stronger cryptographic control setting where encryption does not exist (or is neutralized) since all communication is passed through the Dictator in, essentially,
cleartext mode (or otherwise, when secure channels to and from the Dictator are the only confidentiality mechanism). Messages are only authenticated to assure recipients of the identity of the
sender. We ask whether security against the Dictator still exists, even under such a strict regime which allows only authentication (i.e., authenticated/ signed messages) to pass end-to-end, and
where received messages are determined by/ known to the Dictator, and the Dictator also eventually gets all keys to verify compliance of past signing. To frustrate the Dictator, this
authenticated message setting gives rise to the possible notion of anamorphic channels inside signature and authentication schemes, where parties attempt to send undetectable secure messages (or
other values) using signature tags which are indistinguishable from regular tags.
We define and present implementation of schemes for anamorphic signature and authentication; these are applicable to existing and standardized signature and authentication schemes which were
designed independently of the notion of anamorphic messages. Further, some cornerstone constructions of the foundations of signatures, in fact, introduce anamorphism.
• The Self-Anti-Censorship Nature of Encryption: On the Prevalence of Anamorphic Cryptography
with Mirek Kutylowski (Wroclaw University of Science and Technology), Giuseppe Persiano (Università di Salerno and Google), Moti Yung (Google and Columbia University) and Marcin Zawada (Wroclaw
University of Science and Technology)
In Proceedings on Privacy Enhancing Technologies - PoPETs 2023.
[Paper in pdf]
As part of the responses to the ongoing “crypto wars,” the notion of Anamorphic Encryption was put forth [Persiano-Phan-Yung Eurocrypt ’22]. The notion allows private communication in spite of a
dictator who (in violation of the usual normative conditions under which Cryptography is developed) is engaged in an extreme form of surveillance and/or censorship, where it asks for all private
keys and knows and may even dictate all messages. The original work pointed out efficient ways to use two known schemes in the anamorphic mode, bypassing the draconian censorship and hiding
information from the all-powerful dictator. A question left open was whether these examples are outlier results or whether anamorphic mode is pervasive in existing systems. Here we answer the
above question: we develop new techniques, expand the notion, and show that the notion of Anamorphic Cryptography is, in fact, very much prevalent.
We first refine the notion of Anamorphic Encryption with respect to the nature of covert communication. Specifically, we distinguish Single-Receiver Anamorphic Encryption for many to one
communication and Multiple-Receiver Anamorphic Encryption for many to many communication within the group of conspiring (against the dictator) users. We then show that Anamorphic Encryption can
be embedded in the randomness used in the encryption, and we give families of constructions that can be applied to numerous ciphers. In total the families cover classical encryption schemes, some
of which in actual use (RSA-OAEP, Pailler, Goldwasser-Micali, ElGamal schemes, Cramer-Shoup, and Smooth Projective Hash based systems). Among our examples is an anamorphic channel with much
higher capacity than the regular channel.
In sum, the work shows the very large extent of the potential futility of control and censorship over the use of strong encryption by the dictator (typical for and even stronger than governments
engaging in the ongoing “crypto-wars”): While such limitations obviously hurt utility which encryption typically brings to safety in computing systems, they essentially, are not helping the
dictator. While the actual implications of what we show here and what it means in practice require further policy and legal analyses and perspectives, the technical aspects regarding the issues
are clearly showing the futility of the war against Cryptography.
• Optimal Security Notion for Decentralized Multi-Client Functional Encryption
with Ky Nguyen (ENS) and David Pointcheval (ENS)
In Applied Cryptography and Network Security - ACNS 2023.
[Paper in pdf]
Research on (Decentralized) Multi-Client Functional Encryption (or (D)MCFE) is very active, with interesting constructions, especially for the class of inner products. However, the security
notions have been evolving over the time. While the target of the adversary in distinguishing ciphertexts is clear, legitimate scenarios that do not consist of trivial attacks on the
functionality are less obvious. In this paper, we wonder whether only trivial attacks are excluded from previous security games. And, unfortunately, this was not the case.
We then propose a stronger security notion, with a large definition of admissible attacks, and prove it is optimal: any extension of the set of admissible attacks is actually a trivial attack on
the functionality, and not against the specific scheme. In addition, we show that all the previous constructions are insecure w.r.t. this new security notion. Eventually, we propose new DMCFE
schemes for the class of inner products that provide the new features and achieve this stronger security notion.
• Privacy-Preserving Digital Vaccine Passport
with Thai Duong (Google), Jiahui Gao (Arizona State University) and Ni Trieu (Arizona State University)
In International Conference on Cryptology and Network Security - CANS 2023.
[Paper in pdf]
The global lockdown imposed during the Covid-19 pandemic has resulted in significant social and economic challenges. In an effort to reopen economies and simultaneously control the spread of the
disease, the implementation of contact tracing and digital vaccine passport technologies has been introduced. While contact tracing methods have been extensively studied and scrutinized for
security concerns through numerous publications, vaccine passports have not received the same level of attention in terms of defining the problems they address, establishing security
requirements, or developing efficient systems. Many of the existing methods employed currently suffer from privacy issues.
This work introduces PPass, an advanced digital vaccine passport system that prioritizes user privacy. We begin by outlining the essential security requirements for an ideal vaccine passport
system. To address these requirements, we present two efficient constructions that enable PPass to function effectively across various environments while upholding user privacy. By estimating its
performance, we demonstrate the practical feasibility of PPass. Our findings suggest that PPass can efficiently verify a passenger’s vaccine passport in just 7 milliseconds, with a modest
bandwidth requirement of 480KB.
• Multi-Client Functional Encryption with Fine-Grained Access Control
with Ky Nguyen (ENS) and David Pointcheval (ENS)
In Advances in Cryptology - IACR ASIACRYPT 2022.
[Paper in pdf]
Multi-Client Functional Encryption ($\mathsf{MCFE}$) and Multi-Input Functional Encryption ($\mathsf{MIFE}$) are very interesting extensions of Functional Encryption for practical purpose. They
allow to compute joint function over data from multiple parties. Both primitives are aimed at applications in multi-user settings where decryption can be correctly output for users with
appropriate functional decryption keys only. While the definitions for a single user or multiple users were quite general and can be realized for general classes of functions as expressive as
Turing machines or all circuits, efficient schemes have been proposed so far for concrete classes of functions: either only for access control, $\mathit{i.e.}$ the identity function under some
conditions, or linear/quadratic functions under no condition.
In this paper, we target classes of functions that explicitly combine some evaluation functions independent of the decrypting user under the condition of some access control. More precisely, we
introduce a framework for $\mathsf{MCFE}$ with fine-grained access control and propose constructions for both single-client and multi-client settings, for inner-product evaluation and access
control via Linear Secret Sharing Schemes ($\mathsf{LSSS}$), with selective and adaptive security. The only known work that combines functional encryption in multi-user setting with access
control was proposed by Abdalla $\mathit{et~al.}$ (Asiacrypt '20), which relies on a generic transformation from the single-client schemes to obtain $\mathsf{MIFE}$ schemes that suffer a
quadratic factor of $n$ (where $n$ denotes the number of clients) in the ciphertext size. We follow a different path, via $\mathsf{MCFE}$: we present a $\mathit{duplicate\text{-}and\text{-}
compress}$ technique to transform the single-client scheme and obtain a $\mathsf{MCFE}$ with fine-grained access control scheme with only a linear factor of $n$ in the ciphertext size. Our final
scheme thus outperforms the Abdalla $\mathit{et~al.}$'s scheme by a factor $n$, as one can obtain $\mathsf{MIFE}$ from $\mathsf{MCFE}$ by making all the labels in $\mathsf{MCFE}$ a fixed public
constant. The concrete constructions are secure under the $\mathsf{SXDH}$ assumption, in the random oracle model for the $\mathsf{MCFE}$ scheme, but in the standard model for the $\mathsf{MIFE}$
• Anamorphic Encryption: Private Communication against a Dictator
with Giuseppe Persiano (Univ. of Salerno) and Moti Yung (Google and Columbia Univ.)
In Advances in Cryptology - IACR EUROCRYPT 2022.
[Paper in pdf]
Cryptosystems have been developed over the years under the typical prevalent setting which assumes that the receiver’s key is kept secure from the adversary, and that the choice of the message to
be sent is freely performed by the sender and is kept secure from the adversary as well. Under these fundamental and basic operational assumptions, modern Cryptography has flourished over the
last half a century or so, with amazing achievements: New systems (including public-key Cryptography), beautiful and useful models (including security definitions such as semantic security), and
new primitives (such as zero-knowledge proofs) have been developed. Furthermore, these fundamental achievements have been translated into actual working systems, and span many of the daily human
activities over the Internet.
However, in recent years, there is an overgrowing pressure from many governments to allow the government itself access to keys and messages of encryption systems (under various names: escrow
encryption, emergency access, communication decency acts, etc.). Numerous non-direct arguments against such policies have been raised, such as "the bad guys can utilize other encryption system"
so all other cryptosystems have to be declared illegal, or that "allowing the government access is an ill-advised policy since it creates a natural weak systems security point, which may attract
others (to masquerade as the government)." It has remained a fundamental open issue, though, to show directly that the above mentioned efforts by a government (called here “a dictator” for
brevity) which mandate breaking of the basic operational assumption (and disallowing other cryptosystems), is, in fact, a futile exercise. This is a direct technical point which needs to be made
and has not been made to date.
In this work, as a technical demonstration of the futility of the dictator’s demands, we invent the notion of “Anamorphic Encryption” which shows that even if the dictator gets the keys and the
messages used in the system (before anything is sent) and no other system is allowed, there is a covert way within the context of well established public-key cryptosystems for an entity to
immediately (with no latency) send piggybacked secure messages which are, in spite of the stringent dictator conditions, hidden from the dictator itself! We feel that this may be an important
direct technical argument against the nature of governments’ attempts to police the use of strong cryptographic systems, and we hope to stimulate further works in this direction.
• Privacy in Advanced Cryptographic Protocols: Prototypical Examples
with Moti Yung (Google and Columbia Univ.)
In Journal of Computer Science and Cybernetics, Vietnamese Academy of Science and Technology, Vietnam, 2021, 37 (4), pp.429-451.
[Paper in pdf]
Cryptography is the fundamental cornerstone of cybersecurity employed for achieving data confidentiality, integrity, and authenticity. However, when cryptographic protocols are deployed for
emerging applications such as cloud services or big data, the demand for security grows beyond these basic requirements. Data nowadays are being extensively stored in the cloud, users also need
to trust the cloud servers/authorities that run powerful applications. Collecting user data, combined with powerful machine learning tools, can come with a huge risk of mass surveillance or
undesirable data-driven strategies for making profits rather than for serving the user. Privacy, therefore, becomes more and more important, and new techniques should be developed to protect
personal information and to reduce trust requirements on the authorities or the Big Tech providers.
In a general sense, privacy is "the right to be left alone" and privacy protection allows individuals to have control over how their personal information is collected and used. In this survey, we
discuss the privacy protection methods of various cryptographic protocols, in particular we review:
□ Privacy in electronic voting systems. This may be, perhaps, the most important real-world application where privacy plays a fundamental role.
□ Private computation. This may be the widest domain in the new era of modern technologies with cloud computing and big data, where users delegate the storage of their data and the computation
to the cloud. In such a situation, "how can we preserve privacy?" is one of the most important questions in cryptography nowadays.
□ Privacy in contact tracing. This is a typical example of a concrete study on a contemporary scenario where one should deal with the unexpected social problem but needs not pay the cost of
weakening the privacy of users.
Finally, we will discuss some notions which aim at reinforcing privacy by masking the type of protocol that we execute, we call it the covert cryptographic primitives and protocols.
• (Book's Chapter) Broadcast Encryption and Traitor Tracing
In Asymmetric Cryptography: Primitives and Protocols ( Chapter 6: Broadcast Encryption and Traitor Tracing)
French version.
This chapter presents recent advancements about Broadcast Encryption and Traitor Tracing.
• Zero-Knowledge Proofs for Committed Symmetric Boolean Functions
with San Ling (NTU), Khoa Nguyen (NTU), Hanh Tang (NTU) and Huaxiong Wang (NTU)
In Post-Quantum Cryptography 2021.
[Paper in pdf]
Zero-knowledge proofs (ZKP) are a fundamental notion in modern cryptography and an essential building block for countless privacy-preserving constructions. Recent years have witnessed a rapid
development in the designs of ZKP for general statements, in which, for a publicly given Boolean function \(f: \{0,1\}^n \rightarrow \{0,1\}\), one’s goal is to prove knowledge of a secret input
\(\mathbf {x} \in \{0,1\}^n\) satisfying \(f(\mathbf {x}) = b\), for a given bit b. Nevertheless, in many interesting application scenarios, not only the input \(\mathbf {x}\) but also the
underlying function f should be kept private. The problem of designing ZKP for the setting where both \(\mathbf {x}\) and f are hidden, however, has not received much attention.
This work addresses the above-mentioned problem for the class of symmetric Boolean functions, namely, Boolean functions f whose output value is determined solely by the Hamming weight of the n
-bit input \(\mathbf {x}\). Although this class might sound restrictive, it has exponential cardinality \(2^{n+1}\) and captures a number of well-known Boolean functions, such as threshold,
sorting and counting functions. Specifically, with respect to a commitment scheme secure under the Learning-Parity-with-Noise (LPN) assumption, we show how to prove in zero-knowledge that \(f(\
mathbf {x}) = b\), for a committed symmetric function f, a committed input \(\mathbf {x}\) and a bit b. The security of our protocol relies on that of an auxiliary commitment scheme which can be
instantiated under quantum-resistant assumptions (including LPN). The protocol also achieves reasonable communication cost: the variant with soundness error \(2^{-\lambda }\) has proof size \(c\
cdot \lambda \cdot n\), where c is a relatively small constant. The protocol can potentially find appealing privacy-preserving applications in the area of post-quantum cryptography, and
particularly in code-based cryptography.
• An Anonymous Trace-and-Revoke Broadcast Encryption Scheme
with Olivier Blazy (Ecole Polytechnique), Sayantan Mukherjee (Limoges Univ.), Huyen Nguyen (ENS Lyon) and Damien Stehlé (ENS Lyon)
In ACISP 2021.
[Paper in pdf]
Broadcast Encryption is a fundamental cryptographic primitive, that gives the ability to send a secure message to any chosen target set among registered users. In this work, we investigate
broadcast encryption with anonymous revocation, in which ciphertexts do not reveal any information on which users have been revoked. We provide a scheme whose ciphertext size grows linearly with
the number of revoked users. Moreover, our system also achieves traceability in the black-box confirmation model. Technically, our contribution is threefold. First, we develop a generic
transformation of linear functional encryption toward trace-and-revoke systems for 1-bit message space. It is inspired from the transformation by Agrawal {et al} (CCS'17) with the novelty of
achieving anonymity. Our second contribution is to instantiate the underlying linear functional encryptions from standard assumptions. We propose a $\mathsf{DDH}$-based construction which does no
longer require discrete logarithm evaluation during the decryption and thus significantly improves the performance compared to the $\mathsf{DDH}$-based construction of Agrawal {et al}. In the
LWE-based setting, we tried to instantiate our construction by relying on the scheme from Wang {et al} (PKC'19) only to find an attack on this scheme. Our third contribution is to extend the
1-bit encryption from the generic transformation to $n$-bit encryption. By introducing matrix multiplication functional encryption, which essentially performs a fixed number of parallel calls on
functional encryptions with the same randomness, we can prove the security of the final scheme with a tight reduction that does not depend on $n$, in contrast to employing the hybrid argument."
• Catalic: Delegated PSI Cardinality with Applications to Contact Tracing
with Thai Duong (Google) and Ni Trieu (Arizona State University)
In Advances in Cryptology - IACR ASIACRYPT 2020.
[Paper in pdf]
Private Set Intersection Cardinality (PSI-CA) allows two parties, each holding a set of items, to learn the size of the intersection of those sets without revealing any additional information. To
the best of our knowledge, this work presents the first protocol that allows one of the parties to delegate PSI-CA computation to untrusted servers. At the heart of our delegated PSI-CA protocol
is a new oblivious distributed key PRF (Odk-PRF) abstraction, which may be of independent interest.
We explore in detail how to use our delegated PSI-CA protocol to perform privacy-preserving contact tracing. It has been estimated that a significant percentage of a given population would need
to use a contact tracing app to stop a disease’s spread. Prior privacy-preserving contact tracing systems, however, impose heavy bandwidth or computational demands on client devices. These
demands present an economic disincentive to participate for end users who may be billed per MB by their mobile data plan or for users who want to save battery life. We propose Catalic (ContAct
TrAcing for LIghtweight Clients), a new contact tracing system that minimizes bandwidth cost and computation workload on client devices. By applying our new delegated PSI-CA protocol, Catalic
shifts most of the client-side computation of contact tracing to untrusted servers, and potentially saves each user hundreds of megabytes of mobile data per day while preserving privacy." />
• Dynamic Decentralized Functional Encryption
with Jérémy Chotard (Limoges Univ.), Edouard Dufour-Sans (CMU), Romain Gay (Cornell Tech), and David Pointcheval (ENS)
In Advances in Cryptology - IACR CRYPTO 2020.
[Paper in pdf]
We introduce Dynamic Decentralized Functional Encryption (DDFE), a generalization of Functional Encryption which allows multiple users to join the system dynamically, without relying on a trusted
third party or on expensive and interactive Multi-Party Computation protocols.
This notion subsumes existing multi-user extensions of Functional Encryption, such as Multi-Input, Multi-Client, and Ad Hoc Multi-Input Functional Encryption. We define and construct schemes for
various functionalities which serve as building-blocks for latter primitives and may be useful in their own right, such as a scheme for dynamically computing sums in any Abelian group. These
constructions build upon simple primitives in a modular way, and have instantiations from well-studied assumptions, such as DDH or LWE.
Our constructions culminate in an Inner-Product scheme for computing weighted sums on aggregated encrypted data, from standard assumptions in prime-order groups in the Random Oracle Model.
• A Concise Bounded Anonymous Broadcast Yielding Combinatorial Trace-and-Revoke Schemes
with Xuan Thanh Do (VNU, Vietnam and Limoges Univ.) and Moti Yung (Google and Columbia Univ.)
In ACNS 2020, Roma, 2020.
[Paper in pdf]
Broadcast Encryption is a fundamental primitive supporting sending a secure message to any chosen target set of $N$ users. While many efficient constructions are known, understanding the
efficiency possible for an ``Anonymous Broadcast Encryption'' (ANOBE), i.e., one which can hide the target set itself, is quite open. The best solutions by Barth, Boneh, and Waters ('06) and
Libert, Paterson, and Quaglia ('12) are built on public key encryption (PKE) and their ciphertext sizes are, in fact, $N$ times that of the underlying PKE (rate=$N$). Kiayias and Samary ('12), in
turn, showed a lower bound showing that such rate is the best possible if $N$ is an independent unbounded parameter. However, when considering certain user set size bounded by a system parameter
(e.g., the security parameter), the problem remains interesting. We consider the problem of comparing ANOBE with PKE under the same assumption. We call such schemes Anonymous Broadcast Encryption
for Bounded Universe -- AnoBEB.
We first present an AnoBEB construction for up to $k$ users from LWE assumption, where $k$ is bounded by the scheme security parameter. The scheme does not grow with the parameter and beat the
PKE method. Actually, our scheme is as efficient as the underlying LWE public-key encryption; namely, the rate is, in fact, $1$ and thus optimal. The scheme is achieved easily by an observation
about an earlier scheme with a different purpose.
More interestingly, we move on to employ the new AnoBEB in other multimedia broadcasting methods and, as a second contribution, we introduce a new approach to construct an efficient ``Trace and
Revoke scheme'' which combines the functionalites of revocation and of tracing people (called traitors) who in a broadcasting schemes share their keys with the adversary which, in turn, generates
a pirate receiver. Note that, as was put forth by Kiayias and Yung (EUROCRYPT '02), combinatorial traitor tracing schemes can be constructed by combining a system for small universe, integrated
via an outer traceability codes (collusion-secure code or identifying parent property (IPP) code). There were many efficient traitor tracing schemes from traceability codes, but no known scheme
supports revocation as well. Our new approach integrates our AnoBEB system with a Robust IPP code, introduced by Barg and Kabatiansky (IEEE IT '13). This shows an interesting use for robust IPP
in cryptography. The robust IPP codes were only implicitly shown by an existence proof. In order to make our technique concrete, we propose two explicit instantiations of robust IPP codes. Our
final construction gives the most efficient trace and revoke scheme in the bounded collusion model.
• Linearly-Homomorphic Signatures and Scalable Mix-Nets
with Chloé Hebant (ENS) and David Pointcheval (ENS)
In IACR PKC 2020, Edinburgh, 2020.
[Paper in pdf]
Anonymity is a primary ingredient for our digital life. Several tools have been designed to address it such as, for authentication, blind signatures, group signatures or anonymous credentials
and, for confidentiality, randomizable encryption or mix-nets. When it comes to complex electronic voting schemes, random shuffling of ciphertexts with mix-nets is the only known tool. However,
it requires huge and complex zero-knowledge proofs to guarantee the actual permutation of the initial ciphertexts.
In this paper, we propose a new approach for proving correct shuffling: the mix-servers can simply randomize individual ballots, which means the ciphertexts, the signatures, and the verification
keys, with an additional global proof of constant size, and the output will be publicly verifiable. The computational complexity for the mix-servers is linear in the number of ciphertexts.
Verification is also linear in the number of ciphertexts, independently of the number of rounds of mixing. This leads to the most efficient technique, that is highly scalable. Our constructions
make use of linearly-homomorphic signatures, with new features, that are of independent interest.
• Traceable Inner Product Functional Encryption
with Xuan Thanh Do (VNU, Vietnam and Limoges Univ.) and David Pointcheval (ENS)
In CT-RSA 2020, San Francisco, 2020.
[Paper in pdf]
Functional Encryption (FE) has been widely studied in the last decade, as it provides a very useful tool for restricted access to sensitive data: from a ciphertext, it allows specific users to
learn a function of the underlying plaintext. In practice, many users may be interested in the same function on the data, say the mean value of the inputs, for example. The conventional
definition of FE associates each function to a secret decryption functional key and therefore all the users get the same secret key for the same function. This induces an important problem: if
one of these users (called a traitor) leaks or sells the decryption functional key to be included in a pirate decryption tool, then there is no way to trace back its identity. Our objective is to
solve this issue by introducing a new primitive, called Traceable Functional Encryption: the functional decryption key will not only be specific to a function, but to a user too, in such a way
that if some users collude to produce a pirate decoder that successfully evaluates a function on the plaintext, from the ciphertext only, one can trace back at least one of them.
We propose a concrete solution for Inner Product Functional Encryption (IPFE). We first remark that the ElGamal-based IPFE from Abdalla et. al. in PKC '15 shares many similarities with the
Boneh-Franklin traitor tracing from CRYPTO '99. Then, we can combine these two schemes in a very efficient way, with the help of pairings, to obtain a Traceable IPFE with black-box confirmation.
• Advances in Security Research in the Asiacrypt Region
Article edited with the members of the Asiacrypt Steering Committee
In Communications of the ACM, 2020.
[Paper in pdf]
This article, edited with the members of the Asiacrypt Steering Committee, provides an overview of the advances in security research in the Asiacrypt region. We discuss recent developments,
emerging trends, and future directions in the field of cryptography and information security.
• Downgradable Identity-Based Encryption and Applications
with Olivier Blazy (Limoges Univ.), Paul Germouty (Limoges Univ.)
In CT-RSA '19, San Francisco, 2019.
[Paper in pdf]
In Identity-based cryptography, in order to generalize one receiver encryption to multi-receiver encryption, wildcards were introduced: WIBE enables wildcard in receivers' pattern and Wicked-IBE
allows one to generate a key for identities with wildcard. However, the use of wildcard makes the construction of WIBE, Wicked-IBE more complicated and significantly less efficient than the
underlying IBE. The main reason is that the conventional identity's binary alphabet is extended to a ternary alphabet $\{0,1,*\}$ and the wildcard $*$ is always treated in a convoluted way in
encryption or in key generation. In this paper, we show that when dealing with multi-receiver setting, wildcard is not necessary. We introduce a new downgradable property for IBE scheme and show
that any IBE with this property, called DIBE, can be efficiently transformed into WIBE or Wicked-IBE. While WIBE and Wicked-IBE have been used to construct Broadcast encryption, we go a step
further by employing DIBE to construct Attribute-based Encryption of which the access policy is expressed as a boolean formula in the disjunctive normal form.
• Anonymous IBE with Traceable Identities
with Olivier Blazy (Limoges Univ.), and Laura Brouilhet (Limoges Univ.)
In 14th International Conference on Availability, Reliability and Security (ARES 2019), Canterbury, United Kingdom, 2019.
We introduce Anonymous Identity Based Encryption with Traceable Identities, in which we provide a new feature to anonymous identity-based encryption schemes: lifting the anonymity of some
specific recipients in necessary situations (such as when they are suspected as criminals). Our primitive allows a tracer, given a tracing key associated to an identity, to filter all the
ciphertexts that are sent to this specific identity (and only those). As it is primordial to preserve the privacy of the law-abiding users, the security takes into account the collusion of
tracers and corrupted users.
We first start with the Boyen Waters IBE and then proceed further to the class of IBKEM based on Hash Proof Systems proposed by Blazy et.al. By reinforcing the notion of affine MAC, we show that
these IBE schemes can be transformed to AIBET. Interestingly, our transformation does not weaken the original schemes: even though the adversary is allowed to get access to some additional
oracles as we add a new functionality, the security relies on the same underlying security assumptions and the efficiency is almost unchanged, only some extra keys are added but the encapsulation
/ decapsulation process remain the same.
• Decentralized Evaluation of Quadratic Polynomials on Encrypted Data
with Chloé Hebant (ENS) and David Pointcheval (ENS)
In 22th Information Security Conference, New York, United States, 2019.
[Paper in pdf]
Since the seminal paper on Fully Homomorphic Encryption (FHE) by Gentry in 2009, a lot of work and improvements have been proposed, with an amazing number of possible applications. It allows
outsourcing any kind of computations on encrypted data, and thus without leaking any information to the provider who performs the computations. This is quite useful for many sensitive data
(finance, medical, etc.). Unfortunately, FHE fails at providing some computation on private inputs to a third party, in cleartext: the user that can decrypt the result is able to decrypt the
inputs. A classical approach to allow limited decryption power is distributed decryption. But none of the actual FHE schemes allows distributed decryption, at least with an efficient protocol. In
this paper, we revisit the Boneh-Goh-Nissim (BGN) cryptosystem, and the Freeman's variant, that allow evaluation of quadratic polynomials, or any 2-DNF formula. Whereas the BGN scheme relies on
integer factoring for the trapdoor in the composite-order group, and thus possesses one public/secret key only, the Freeman's scheme can handle multiple users with one general setup that just
needs to define a pairing-based algebraic structure. We show that it can be efficiently decentralized, with an efficient distributed key generation algorithm, without any trusted dealer, but also
efficient distributed decryption and distributed re-encryption, in a threshold setting. We then provide some applications of computations on encrypted data, without central authority.
• Decentralized Multi-Client Functional Encryption for Inner Product
with Jérémy Chotard (Limoges Univ.), Edouard Dufour Sans (ENS), Romain Gay (ENS) and David Pointcheval (ENS)
In Advances in Cryptology - IACR ASIACRYPT '18, Springer, 2018.
[Paper in pdf]
We consider a situation where multiple parties, owning data that have to be frequently updated, agree to share weighted sums of these data with some aggregator, but where they do not wish to
reveal their individual data, and do not trust each other. We combine techniques from Private Stream Aggregation (PSA) and Functional Encryption (FE), to introduce a primitive we call
Decentralized Multi-Client Functional Encryption (DMCFE), for which we give a practical instantiation for Inner Product functionalities. This primitive allows various senders to non-interactively
generate ciphertexts which support inner-product evaluation, with functional decryption keys that can also be generated non-interactively, in a distributed way, among the senders. Interactions
are required during the setup phase only. We prove adaptive security of our constructions, while allowing corruptions of the clients, in the random oracle model.
• A New Technique for Compacting Ciphertext in Multi-Channel Broadcast Encryption and Attribute-Based Encryption
with Sébastien Canard (Oranges Lab), David Pointcheval (ENS) and Viet Cuong Trinh (Hong Duc Univ., Vietnam)
In Theoretical Computer Science, Vol 723, pages 51-72, 2018.
Standard Broadcast Encryption (BE) and Attribute-Based Encryption (ABE) aim at sending a content to a large arbitrary group of users at once. Regarding Broadcast Encryption, currently, the most
efficient schemes provide constant-size headers, that encapsulate ephemeral session keys under which the payload is encrypted. However, in practice, and namely for pay-TV, providers have to send
various contents to different groups of users. Headers are thus specific to each group, one for each channel: as a consequence, the global overhead is linear in the number of channels.
Furthermore, when one wants to zap to and watch another channel, one has to get the new header and decrypt it to learn the new session key: either the headers are sent quite frequently or one has
to store all the headers, even if one watches one channel only. Otherwise, the zapping time becomes unacceptably long. We consider the encapsulation of several ephemeral keys, for various groups
and thus various channels, in one header only, and we call this new primitive Multi-Channel Broadcast Encryption or MCBE: one can hope for a much shorter global overhead and a much shorter
zapping time since the decoder already has the information to decrypt any available channel at once. Regarding Attribute-Based Encryption, a scheme with constant-size ciphertext is still a
challenging task.
In this paper, we introduce a new technique of optimizing the ciphertext-size for both MCBE and ABE schemes.
• An Attribute-based Broadcast Encryption Scheme For Lightweight Devices
with Sébastien Canard (Oranges Lab) and Viet Cuong Trinh (Hong Duc Univ., Vietnam)
In IET Information Security, Vol. 12, Issue 1, 2018.
Lightweight devices, such as a smartcard associated with a top-box decoder in pay-TV or a SIM card coupled with a powerful (but not totally trusted) smartphone, play an important role in modern
applications. The essential requirements for a cryptographic scheme to be truly implemented in lightweight devices are that it should have compact secret key size and support fast decryption.
Attribute-based broadcast encryption (ABBE) combines the functionalities of both broadcast encryption and attribute-based encryption in an efficient way, ABBE is therefore a promising
cryptographic scheme to be used in practical applications such as mobile pay-TV, satellite transmission, or Internet of Things. Designing an ABBE scheme which can be truly implemented in
lightweight devices is still an open question. In this study, the authors solve it by proposing an efficient constant-size private key ciphertext-policy ABBE scheme for disjunctive normal form
supporting fast decryption and achieving standard security levels of an ABBE scheme. They concretely show that the authors’ scheme can be truly implemented in a prototype for a smartphone-based
cloud storage use case. In particular, they show how to alleviate some parts of their scheme so as to obtain a very practical system, and they give some concrete benchmarks.
• Efficient Public Trace and Revoke from Standard Assumptions
with Shweta Agrawal (IIT Madras, India), Sanjay Bhattacherjee (Turing Lab, ISI Kolkata, India), and Damien Stehlé (ENS Lyon) and Shota Yamada AIST, Japan)
In ACM CCS 2017.
[Paper in pdf]
We provide efficient constructions for trace-and-revoke systems with public traceability in the black-box confirmation model. Our constructions achieve adaptive security, are based on standard
assumptions and achieve significant efficiency gains compared to previous constructions. Our constructions rely on a generic transformation from inner product functional encryption (IPFE) schemes
to trace-and-revoke systems. Our transformation requires the underlying IPFE scheme to only satisfy a very weak notion of security -- the attacker may only request a bounded number of random keys
-- in contrast to the standard notion of security where she may request an unbounded number of arbitrarily chosen keys. We exploit the much weaker security model to provide a new construction for
bounded collusion and random key IPFE from the learning with errors assumption (LWE), which enjoys improved efficiency compared to the scheme of Agrawal et al. [CRYPTO'16]. Together with IPFE
schemes from Agrawal et al., we obtain trace and revoke from LWE, Decision Diffie Hellman and Decision Quadratic Residuosity.
• Identity-based Encryption from Codes with Rank Metric
with Philippe Gaborit (Limoges Univ.), Adrien Hauteville (Limoges Univ.) and Jean-Pierre Tillich (INRIA)
In Advances in Cryptology - IACR CRYPTO 2017.
[Paper in pdf]
Code-based cryptography has a long history, almost as long as the history of public-key encryption (PKE). While we can construct almost all primitives from codes such as PKE, signature, group
signature etc, it is a long standing open problem to construct an identity-based encryption from codes. We solve this problem by relying on codes with rank metric.
The concept of identity-based encryption (IBE), introduced by Shamir in 1984, allows the use of users’ identifier information such as email as public key for encryption. There are two problems
that makes the design of IBE extremely hard: the requirement that the public key can be an arbitrary string and the possibility to extract decryption keys from the public keys. In fact, it took
nearly twenty years for the problem of designing an efficient method to implement an IBE to be solved. The known methods of designing IBE are based on different tools: from elliptic curve
pairings by Sakai, Ohgishi and Kasahara and by Boneh and Franklin in 2000 and 2001 respectively; from the quadratic residue problem by Cocks in 2001; and finally from the Learning-with-Error
problem by Gentry, Peikert, and Vaikuntanathan in 2008.
Among all candidates for post-quantum cryptography, there only exist thus lattice-based IBE. In this paper, we propose a new method, based on the hardness of learning problems with rank metric,
to design the first code-based IBE scheme. In order to overcome the two above problems in designing an IBE scheme, we first construct a rank-based PKE, called RankPKE, where the public key space
is dense and thus can be obtained from a hash of any identity. We then extract a decryption key from any public key by constructing an trapdoor function which relies on RankSign - a signature
scheme from PQCrypto 2014.
In order to prove the security of our schemes, we introduced a new prob- lem for rank metric: the Rank Support Learning problem (RSL). A high technical contribution of the paper is devoted to
study in details the hardness of the RSL problem.
• Hardness of k-LWE and Applications in Traitor Tracing
with San Ling (NTU), Damien Stehlé (ENS Lyon) and Ron Steinfeld (Monash University)
Invited paper for Algorithmica, December 2017, Volume 79, Issue 4, pp 1318–1352.
[Paper in pdf]
We introduce the k-LWE problem, a Learning With Errors variant of the k-SIS problem. The Boneh-Freeman reduction from SIS to k-SIS suffers from an exponential loss in k. We improve and extend it
to an LWE to k-LWE reduction with a polynomial loss in k, by relying on a new technique involving trapdoors for random integer kernel lattices. Based on this hardness result, we present the first
algebraic construction of a traitor tracing scheme whose security relies on the worst-case hardness of standard lattice problems. The proposed LWE traitor tracing is almost as efficient as the
LWE encryption. Further, it achieves public traceability, i.e., allows the authority to delegate the tracing capability to "untrusted" parties. To this aim, we introduce the notion of projective
sampling family in which each sampling function is keyed and, with a projection of the key on a well chosen space, one can simulate the sampling function in a computationally indistinguishable
way. The construction of a projective sampling family from k-LWE allows us to achieve public traceability, by publishing the projected keys of the users. We believe that the new lattice tools and
the projective sampling family are quite general that they may have applications in other areas.
• Homomorphic-Policy Attribute-Based Key Encapsulation Mechanisms
with Jérémy Chotard (Limoges Univ.) and David Pointcheval (ENS)
In ISC 2017.
[Paper in pdf]
Attribute-Based Encryption (ABE) allows to target the recipients of a message according to a policy expressed as a predicate among some attributes. Ciphertext-policy ABE schemes can choose the
policy at the encryption time.
In this paper, we define a new property for ABE: homomorphic-policy. A combiner is able to (publicly) combine ciphertexts under different policies into a ciphertext under a combined policy (AND
or OR). More precisely, using linear secret sharing schemes, we design Attribute-Based Key Encapsulation Mechanisms (ABKEM) with the Homomorphic-Policy property: given several encapsulations of
the same keys under various policies, anyone can derive an encapsulation of the same key under any combination of the policies.
As an application, in Pay-TV, this allows to separate the content providers that can generate the encapsulations of a session key under every attributes, this key being used to encrypt the
payload, and the service providers that build the decryption policies according to the subscriptions. The advantage is that the aggregation of the encapsulations by the service providers does not
contain any secret information.
• Cryptography During the French and American Wars in Vietnam
with Neal Koblitz (Univ. of Washington)
In Cryptologia, 2017.
[Paper in pdf]
After Vietnam's Declaration of Independence on 2 September 1945, the country had to suffer through two long, brutal wars, first against the French and then against the Americans, before finally
in 1975 becoming a unified country free of colonial domination. Our purpose is to examine the role of cryptography in those two wars. Despite the far greater technological resources of their
opponents, the communications intelligence specialists of the Viet Minh, the National Liberation Front, and the Democratic Republic of Vietnam had considerable success in both protecting
Vietnamese communications and acquiring tactical and strategic secrets from the enemy. Perhaps surprisingly, in both wars there was a balance between the sides. Generally speaking, cryptographic
knowledge and protocol design were at a high level at the central commands, but deployment for tactical communications in the field was difficult, and there were many failures on all sides.
• A New Technique for Compacting Secret Key in Attribute-Based Broadcast Encryption
with Sébastien Canard (Oranges Lab) and Viet Cuong Trinh (Hong Duc Univ., Vietnam)
In CANS 2016.
[Paper in pdf]
Public-key encryption has been generalized to adapt to more and more practical applications. Broadcast encryption, introduced by Fiat and Naor in 1993, aims for applications in pay-TV or
satellite transmission and allows a sender to securely send private messages to any subset of users, the target set. Sahai and Waters introduced Attribute-based Encryption (ABE) to define the
target set in a more structural way via access policies on attributes. Attribute-based Broadcast Encryption (ABBE) combines the functionalities of both in an efficient way. In the relevant
applications such as pay-TV, the users are given a relatively small device with very limited secure memory in a smartcard. Therefore, it is of high interest to construct schemes with compact
secret key of users. Even though extensively studied in the recent years, it is still an open question of constructing an efficient ABBE with constant-size private keys for general forms of
access policy such as CNF or DNF forms. This question was partially solved at ESORICS ’15 where Phuong et al. introduced a constant secret-key size ABBE. But they manage restrictive access
policies only supporting AND-gates and wildcards. In this paper, we solve this open question and propose an efficient constant-size private key ciphertext-policy attribute-based broadcast
encryption scheme for DNF form. In particular, we also present the optimization in implementing our proposed scheme.
• Adaptive CCA Broadcast Encryption with Constant-Size Secret Keys and Ciphertexts
with David Pointcheval (ENS), Siamak F Shahandashti (ENS) and Mario Strefler (ENS)
In ACISP' 2012, LNCS 7372, pages 308-321, Springer-Verlag, 2012.
[Paper in pdf]
We consider designing broadcast encryption schemes with constant-size secret keys and ciphertexts, achieving chosen-ciphertext security. We first argue that known CPA-to-CCA transforms currently
do not yield such schemes. We then propose a scheme, modifying a previous selective CPA secure proposal by Boneh, Gentry, and Waters. Our proposed scheme has constant-size secret keys and
ciphertexts and we prove that it is selective chosen-ciphertext secure based on standard assumptions. Our scheme has ciphertexts that are shorter than those of the previous CCA secure proposals.
Then we propose a second scheme that provides the functionality of both broadcast encryption and revocation schemes simultaneously using the same set of parameters. Finally we show that it is
possible to prove our first scheme adaptive chosen-ciphertext secure under reasonable extensions of the bilinear Diffie-Hellman exponent and the knowledge of exponent assumptions. We prove both
of these extended assumptions in the generic group model. Hence, our scheme becomes the first to achieve constant-size secret keys and ciphertexts (both asymptotically optimal) and adaptive
chosen-ciphertext security at the same time.
• Black-box Trace&Revoke Codes
with Hung Q. Ngo (State Univ. of New York at Buffalo) and David Pointcheval (ENS)
In Algorithmica, Springer, vol. 67, no. 3, Pages 418-448, 2013.
[Paper in pdf]
We address the problem of designing an efficient broadcast encryption scheme which is also capable of tracing traitors. We introduce a code framework to formalize the problem. Then, we give a
probabilistic construction of a code which supports both traceability and revocation. Given N users with at most r revoked users and at most t traitors, our code construction gives rise to a
Trace&Revoke system with private keys of size O((r+t)logN) (which can also be reduced to constant size based on an additional computational assumption), ciphertexts of size O((r+t)logN), and O(1)
decryption time. Our scheme can deal with certain classes of pirate decoders, which we believe are sufficiently powerful to capture practical pirate strategies. In particular, our code
construction is based on a combinatorial object called (r,s)-disjunct matrix, which is designed to capture both the classic traceability notion of disjunct matrix and the new requirement of
revocation capability. We then probabilistically construct (r,s)-disjunct matrices which help design efficient Black-Box Trace&Revoke systems. For dealing with "smart" pirates, we introduce a
tracing technique called "shadow group testing" that uses (close to) legitimate broadcast signals for tracing. Along the way, we also proved several bounds on the number of queries needed for
black-box tracing under different assumptions about the pirate's strategies.
• Multi-Channel Broadcast Encryption
with David Pointcheval (ENS) and Viet Cuong Trinh (Paris 8 Univ.)
In ASIACCS 2013, ACM Symposium on Information, Computer and Communications Security, ACM Press, Pages 277-286, 2013.
[Paper in pdf]
Broadcast encryption aims at sending a content to a large arbitrary group of users at once. Currently, the most efficient schemes provide constant-size headers, that encapsulate ephemeral session
keys under which the payload is encrypted. However, in practice, and namely for pay-TV, providers have to send various contents to different groups of users. Headers are thus specific to each
group, one for each channel: as a consequence, the global overhead is linear in the number of channels. Furthermore, when one wants to zap to and watch another channel, one has to get the new
header and decrypt it to learn the new session key: either the headers are sent quite frequently or one has to store all the headers, even if one watches one channel only. Otherwise, the zapping
time becomes unacceptably long.
In this paper, we consider encapsulation of several ephemeral keys, for various groups and thus various channels, in one header only, and we call this new primitive Multi-Channel Broadcast
Encryption: one can hope for a much shorter global overhead and a short zapping time since the decoder already has the information to decrypt any available channel at once. Our candidates are
private variants of the Boneh-Gentry-Waters scheme, with a constant-size global header, independently of the number of channels. In order to prove the CCA security of the scheme, we introduce a
new dummy-helper technique and implement it in the random oracle model.
• Optimal Public Key Traitor Tracing Scheme in Non-Black Box Model
with Philippe Guillot (Paris 8 Univ.), Abdelkrim Nimour (NAGRA) and Viet Cuong Trinh (Paris 8 Univ.)
In AFRICACRYPT 2013, LNCS 7918, pages 140-155, Springer-Verlag, 2013.
[Paper in pdf]
In the context of secure content distribution, the content is encrypted and then broadcasted in a public channel, each legitimate user is provided a decoder and a secret key for decrypting the
received signals. One of the main threat for such a system is that the decoder can be cloned and then sold out with the pirate secret keys. Traitor tracing allows the authority to identify the
malicious users (are then called traitors) who successfully collude to build pirate decoders and pirate secret keys. This primitive is introduced by Chor, Fiat and Naor in ’94 and a breakthrough
in construction is given by Boneh and Franklin at Crypto ’99 in which they consider three models of traitor tracing: non-black-box tracing model, single-key black box tracing model, and general
black box tracing model.
Beside the most important open problem of obtimizing the black-box tracing, Boneh-Franklin also left an open problem concerning non-black-box tracing, by mentioning: “it seems reasonable to
believe that there exists an efficient public key traitor tracing scheme that is completely collusion resistant. In such a scheme, any number of private keys cannot be combined to form a new key.
Similarly, the complexity of encryption and decryption is independent of the size of the coalition under the pirate’s control. An efficient construction for such a scheme will provide a useful
solution to the public key traitor tracing problem”.
As far as we know, this problem is still open. In this paper, we resolve this question in the affirmative way, by constructing a very efficient scheme with all parameters are of constant size and
in which the full collusion of traitors cannot produce a new key. Our proposed scheme is moreover dynamic.
• Key-Leakage Resilient Revoke Scheme Resisting Pirates 2.0 in Bounded Leakage Model
with Viet Cuong Trinh (Paris 8 Univ.)
In AFRICACRYPT 2013, LNCS 7918, pages 342-358, Springer-Verlag, 2013.
[Paper in pdf]
Trace and revoke schemes have been widely studied in theory and implemented in practice. In the first part of the paper, we construct a fully secure key-leakage resilient identity-based revoke
scheme. In order to achieve this goal, we first employ the dual system encryption technique to directly prove the security of a variant of the BBG−WIBE scheme under known assumptions (and thus
avoid a loss of an exponential factor in hierarchical depth in the classical method of reducing the adaptive security of WIBE to the adaptive security of the underlying HIBE). We then modify this
scheme to achieve a fully secure key-leakage resilient WIBE scheme. Finally, by using a transformation from a WIBE scheme to a revoke scheme, we propose the first fully secure key-leakage
resilient identity-based revoke scheme.
In the classical model of traitor tracing, one assumes that a traitor contributes its entire secret key to build a pirate decoder. However, new practical scenarios of pirate has been considered,
namely Pirate Evolution Attacks at Crypto 2007 and Pirates 2.0 at Eurocrypt 2009, in which pirate decoders could be built from sub-keys of users. The key notion in Pirates 2.0 is the anonymity
level of traitors: they can rest assured to remain anonymous when each of them only contributes a very small fraction of its secret key by using a public extraction function. This scenario
encourages dishonest users to participate in collusion and the size of collusion could become very large, possibly beyond the considered threshold in the classical model. In the second part of
the paper, we show that our key-leakage resilient identity-based revoke scheme is immune to Pirates 2.0 in some special forms in bounded leakage model. It thus gives an interesting and rather
surprised connection between the rich domain of key-leakage resilient cryptography and Pirates 2.0.
• Generalized Key Delegation for Wildcarded Identity-Based and Inner-Product Encryption
with Michel Abdalla (ENS), Angelo De Caro (ENS)
In IEEE-TIFS, IEEE Transactions on Information Forensics & Security, Volume 7 , Issue: 6, Pages 1695 - 1706.
[Paper in pdf]
Inspired by the fact that many e-mail addresses correspond to groups of users, Abdalla introduced the notion of identity-based encryption with wildcards (WIBE), which allows a sender to
simultaneously encrypt messages to a group of users matching a certain pattern, defined as a sequence of identity strings and wildcards. This notion was later generalized by Abdalla, Kiltz, and
Neven, who considered more general delegation patterns during the key derivation process. Despite its many applications, current constructions have two significant limitations: 1) they are only
known to be fully secure when the maximum hierarchy depth is a constant; and 2) they do not hide the pattern associated with the ciphertext. To overcome these, this paper offers two new
constructions. First, we show how to convert a WIBE scheme of Abdalla into a (nonanonymous) WIBE scheme with generalized key delegation (WW-IBE) that is fully secure even for polynomially many
levels. Then, to achieve anonymity, we initially consider hierarchical predicate encryption (HPE) schemes with more generalized forms of key delegation and use them to construct an anonymous
WW-IBE scheme. Finally, to instantiate the former, we modify the HPE scheme of Lewko to allow for more general key delegation patterns. Our proofs are in the standard model and use existing
complexity assumptions.
• Message Tracing with Optimal Ciphertext Rate
with David Pointcheval (ENS) and Mario Strefler (ENS)
In LatinCrypt' 2012, LNCS 7533, pages 56-77, Springer-Verlag, 2012.
[Paper in pdf]
Traitor tracing is an important tool to discourage defrauders from illegally broadcasting multimedia content. However, the main techniques consist in tracing the traitors from the pirate decoders
they built from the secret keys of dishonest registered users: with either a black-box or a white-box tracing procedure on the pirate decoder, one hopes to trace back one of the traitors who
registered in the system. But new techniques for pirates consist either in sending the ephemeral decryption keys to the decoders for real-time decryption, or in making the full content available
on the web for later viewing. This way, the pirate does not send any personal information. In order to be able to trace the traitors, one should embed some information, or watermarks, in the
multimedia content itself to make it specific to the registered users.
This paper addresses this problem of tracing traitors from the decoded multimedia content or rebroadcasted keys, without increasing too much the bandwidth requirements. More precisely, we
construct a message-traceable encryption scheme that has an optimal ciphertext rate, i.e. the ratio of global ciphertext length over message length is arbitrarily close to one.
• Decentralized Dynamic Broadcast Encryption
with David Pointcheval (ENS) and Mario Strefler (ENS)
In SCN' 2012, LNCS 7485, Springer-Verlag, 2012.
[Paper in pdf]
A broadcast encryption system generally involves three kinds of entities: the group manager that deals with the membership, the encryptor that encrypts the data to the registered users according
to a specific policy (the target set), and the users that decrypt the data if they are authorized by the policy. Public-key broadcast encryption can be seen as removing this special role of
encryptor, by allowing anybody to send encrypted data. In this paper, we go a step further in the decentralization process, by removing the group manager: the initial setup of the group, as well
as the addition of further members to the system, do not require any central authority.
Our construction makes black-box use of well-known primitives and can be considered as an extension to the subset-cover framework. It allows for efficient concrete instantiations, with parameter
sizes that match those of the subset-cover constructions, while at the same time achieving the highest security level in the standard model under the DDH assumption.
• Security Notions for Broadcast Encryption
with David Pointcheval (ENS) and Mario Strefler (ENS)
In ACNS' 2011, LNCS 6715, pages 377-394, Springer-Verlag, 2011.
[Paper in pdf]
This paper clarifies the relationships between security notions for broadcast encryption. In the past, each new scheme came with its own definition of security, which makes them hard to compare.
We thus define a set of notions, as done for signature and encryption, for which we prove implications and separations, and relate the existing notions to the ones in our framework. We find some
interesting relationships between the various notions, especially in the way they define the receiver set of the challenge message. In addition, we define a security notion that is stronger than
all previous ones, and give an example of a scheme that fulfills this notion.
• Identity-Based Trace and Revoke Schemes
with Viet-Cuong Trinh (Paris 8 Univ.)
In ProvSec' 2011, LNCS 6980, pages 204-221, Springer-Verlag, 2011.
[Paper in pdf]
Trace and revoke systems allow for the secure distribution of digital content in such a way that malicious users, who collude to produce pirate decoders, can be traced back and revoked from the
system. In this paper, we consider such schemes in the identity-based setting, by extending the model of identity-based traitor tracing scheme by Abdalla et al. to support revocation.
The proposed constructions rely on the subset cover framework. We first propose a generic construction which transforms an identity-based encryption with wildcard (WIBE) of depth log(N) (N being
the number of users) into an identity-based trace and revoke scheme by relying on the complete subtree framework (of depth log(N)). This leads, however, to a scheme with log(N) private key size
(as in a complete subtree scheme). We improve this scheme by introducing generalized WIBE (GWIBE) and propose a second construction based on GWIBE of two levels. The latter scheme provides the
nice feature of having constant private key size (3 group elements).
In our schemes, we also deal with advanced attacks in the subset cover framework, namely pirate evolution attacks (PEvoA) and pirates 2.0. The only known strategy to protect schemes in the subset
cover framework against pirate evolution attacks was proposed by Jin and Lotspiech but decreases seriously the efficiency of the original schemes: each subset is expanded to many others subsets;
the total number of subsets to be used in the encryption could thus be O(N ^1/b) to prevent a traitor from creating more than b generations. Our GWIBE based scheme, resisting PEvoA better than
the Jin and Lotspiech’s method. Moreover, our method does not need to change the partitioning procedure in the original complete subtree scheme and therefore, the resulted schemes are very
competitive compared to the original scheme, with r log(N/r) logN –size ciphertext and constant size private key.
• Traitors Collaborating in Public: Pirates 2.0
with Olivier Billet (Oranges Lab)
In Advances in Cryptology - IACR EUROCRYPT '09, LNCS 5479, pages 189-205, Springer-Verlag, 2009.
[Paper in pdf]
This work introduces a new concept of attack against traitor tracing schemes. We call attacks of this type Pirates 2.0 attacks as they result from traitors collaborating together in a public way.
In other words, traitors do not secretly collude but display part of their secret keys in a public place; pirate decoders are then built from this public information. The distinguishing property
of Pirates 2.0 attacks is that traitors only contribute partial information about their secret key material which suffices to produce (possibly imperfect) pirate decoders while allowing them to
remain anonymous. The side-effect is that traitors can publish their contributed information without the risk of being traced; giving such strong incentives to some of the legitimate users to
become traitors allows coalitions to attain very large sizes that were deemed unrealistic in some previously considered models of coalitions.
This paper proposes a generic model for this new threat, that we use to assess the security of some of the most famous traitor tracing schemes. We exhibit several Pirates 2.0 attacks against
these schemes, providing new theoretical insights with respect to their security. We also describe practical attacks against various instances of these schemes. Eventually, we discuss possible
variations on the Pirates 2.0 theme.
• Efficient Traitor Tracing from Collusion Secure Codes
with Olivier Billet (Oranges Lab)
In Proceeding of ICITS '08 -The 3rd International Conference on Information Theoretic Security, Pages 171-182, LNCS 5155, Springer-Verlag, 2008.
[Paper in pdf] [ps] [pdf USletter]
In this paper, we describe a new traitor tracing scheme which relies on Tardos’ collusion secure codes to achieve constant size ciphertexts. Our scheme is also equipped with a black-box tracing
procedure against pirates that are allowed to decrypt with some (possibly high) error rate while keeping the decoders of the lowest possible size when using collusion secure codes, namely of size
proportional to the length of Tardos’ code.
• A CCA Secure Hybrid Damgaard's ElGamal Encryption
with Yvo Desmedt (University College London)
In Proceeding of ProvSec '08, Lecture Notes in Computer Science Vol. 5324, pages 68-92, Springer-Verlag, 2008.
[Paper in pdf] [ps] [pdf USletter]
ElGamal encryption, by its efficiency, is one of the most used schemes in cryptographic applications. However, the original ElGamal scheme is only provably secure against passive attacks. Damgård
proposed a slight modification of ElGamal encryption scheme (named Damgård’s ElGamal scheme) that provides security against non-adaptive chosen ciphertext attacks under a knowledge-of-exponent
assumption. Recently, the CCA1-security of Damgård’s ElGamal scheme has been proven under more standard assumptions.
In this paper, we study the open problem of CCA2-security of Damgård’s ElGamal. By employing a data encapsulation mechanism, we prove that the resulted hybrid Damgård’s ElGamal Encryption is
secure against adaptive chosen ciphertext attacks. The down side is that the proof of security is based on a knowledge-of-exponent assumption. In terms of efficiency, this scheme is more
efficient (e.g. one exponentiation less in encryption) than Kurosawa-Desmedt scheme, the most efficient scheme in the standard model so far.
• Hybrid Damgård Is CCA1-Secure under the DDH Assumption
with Yvo Desmedt (University College London), Helger Lipmaa (University College London)
In Proceeding of CANS '08 -The 7th International Conference on Cryptology and Network Security, Pages 18-30, LNCS 5339, Springer-Verlag, 2008.
[Paper in pdf] [ps] [pdf USletter]
In 1991, Damgård proposed a simple public-key cryptosystem that he proved CCA1-secure under the Diffie-Hellman Knowledge assumption. Only in 2006, Gjøsteen proved its CCA1-security under a more
standard but still new and strong assumption. The known CCA2-secure public-key cryptosystems are considerably more complicated. We propose a hybrid variant of Damgård’s public-key cryptosystem
and show that it is CCA1-secure if the used symmetric cryptosystem is CPA-secure, the used MAC is unforgeable, the used key-derivation function is secure, and the underlying group is a DDH group.
The new cryptosystem is the most efficient known CCA1-secure hybrid cryptosystem based on standard assumptions.
• Traitor Tracing with Optimal Transmission Rate
with Nelly Fazio (IBM Research), Antonio Nicolosi (New York University and Stanford University)
In Proceeding of ISC '07 - 10th International Conference on Information Security, Pages 71-88, LNCS 4779, Springer-Verlag, 2007.
[Paper in pdf] [ps] [pdf USletter]
We present the first traitor tracing scheme with efficient black-box traitor tracing in which the ratio of the ciphertext and plaintext lengths (the transmission rate) is asymptotically 1, which
is optimal. Previous constructions in this setting either obtained constant (but not optimal) transmission rate [16], or did not support black-box tracing [10]. Our treatment improves the
standard modeling of black-box tracing by additionally accounting for pirate strategies that attempt to escape tracing by purposedly rendering the transmitted content at lower quality.
Our construction relies on the decisional bilinear Diffie-Hellman assumption, and attains the same features of public traceability as (a repaired variant of) [10], which is less efficient and
requires non-standard assumptions for bilinear groups.
• Identity-Based Traitor Tracing
with Michel Abdalla (ENS), Alex Dent (Royal Holloway), John Malone-Lee (Univ. of Bristol), Gregory Neven (Katholieke Universiteit Leuven) and Nigel Smart (Univ. of Bristol)
In IACR PKC '07, Pages 361-376, LNCS 4450, Springer-Verlag, IACR, 2007.
[Paper in pdf] [ps] [pdf USletter]
We present the first identity-based traitor tracing scheme. The scheme is shown to be secure in the standard model, assuming the bilinear decision Diffie-Hellman (DBDH) is hard in the asymmetric
bilinear pairing setting, and that the DDH assumption holds in the group defining the first coordinate of the asymmetric pairing. Our traitor tracing system allows adaptive pirates to be traced.
The scheme makes use of a two level identity-based encryption scheme with wildcards (WIBE) based on Waters’ identity-based encryption scheme.
• Traitor Tracing for Stateful Pirate Decoders with Constant Ciphertext Rate
In Proceeding of Vietcrypt '06, P. Nguyen Ed. Pages 354-365, LNCS 4341, Springer-Verlag, 2006.
[Paper in pdf] [ps] [pdf USletter]
Stateful pirate decoders are history recording and abrupt pirate decoders. These decoders can keep states between decryptions to detect whether they are being traced and are then able to take
some counter-actions against the tracing process, such as “shutting down” or erasing all internal information. We propose the first constant ciphertext rate scheme which copes with such pirate
decoders. Our scheme moreover supports black-box public traceability.
• Generic Construction of Hybrid Public Key Traitor Tracing with Full-Public-Traceability
with Rei Safavi-Naini (Wollongong Univ.) and Dongvu Tonien (Wollongong Univ.)
In Proceeding of ICALP '06 - 33rd International Colloquium on Automata, Languages and Programming, Pages 264-275, LNCS 4052, Springer-Verlag, 2006.
[Paper in pdf] [ps] [pdf USletter]
In Eurocrypt 2005, Chabanne, Phan and Pointcheval introduced an interesting property for traitor tracing schemes called public traceability, which makes tracing a black-box public operation.
However, their proposed scheme only worked for two users and an open question proposed by authors was to provide this property for multi-user systems.
In this paper, we give a comprehensive solution to this problem by giving a generic construction for a hybrid traitor tracing scheme that provides full-public-traceability. We follow the Tag KEM/
DEM paradigm of hybrid encryption systems and extend it to multi-receiver scenario. We define Tag-Broadcast KEM/DEM and construct a secure Tag-BroadcastKEM from a CCA secure PKE and
target-collision resistant hash function. We will then use this Tag-Broadcast KEM together with a semantically secure DEM to give a generic construction for Hybrid Public Key Broadcast
Encryption. The scheme has a black box tracing algorithm that always correctly identifies a traitor. The hybrid structure makes the system very efficient, both in terms of computation and
communication cost. Finally we show a method of reducing the communication cost by using codes with identifiable parent property.
• Public Traceability in Traitor Tracing Schemes
with Hervé Chabanne (SAGEM) and David Pointcheval (ENS)
In Advances in Cryptology - IACR EUROCRYPT '05, R.Cramer Ed. Pages 542-558, LNCS 3494, Springer-Verlag, IACR, 2005.
[Paper in pdf] [ps] [pdf USletter]
Traitor tracing schemes are of major importance for secure distribution of digital content. They indeed aim at protecting content providers from colluding users to build pirate decoders. If such
a collusion happens, at least one member of the latter collusion will be detected. Several solutions have already been proposed in the literature, but the most important problem to solve remains
having a very good ciphertext/plaintext rate. At Eurocrypt ’02, Kiayias and Yung proposed the first scheme with such a constant rate, but still not optimal. In this paper, granted bilinear maps,
we manage to improve it, and get an “almost” optimal scheme, since this rate is asymptotically 1. Furthermore, we introduce a new feature, the “public traceability”, which means that the center
can delegate the tracing capability to any “untrusted” person. This is not the first use of bilinear maps for traitor tracing applications, but among the previous proposals, only one has remained
unbroken: we present an attack by producing an anonymous pirate decoder. We furthermore explain the flaw in their security analysis. For our scheme, we provide a complete proof, based on new
computational assumptions, related to the bilinear Diffie-Hellman ones, in the standard model.
• Optimal Asymmetric Encryption and Signature Paddings
with Benoît Chevallier-Mames (Gemplus) and David Pointcheval (ENS)
In Proceeding of ACNS '05, pages 254-268, LNCS 3531, Springer-Verlag, 2005.
[Paper in pdf] [ps] [pdf USletter]
Strong security notions often introduce strong constraints on the construction of cryptographic schemes: semantic security implies probabilistic encryption, while the resistance to existential
forgeries requires redundancy in signature schemes. Some paddings have thus been designed in order to provide these minimal requirements to each of them, in order to achieve secure primitives.
A few years ago, Coron et al. suggested the design of a common construction, a universal padding, which one could apply for both encryption and signature. As a consequence, such a padding has to
introduce both randomness and redundancy, which does not lead to an optimal encryption nor an optimal signature.
In this paper, we refine this notion of universal padding, in which a part can be either a random string in order to introduce randomness or a zero-constant string in order to introduce some
redundancy. This helps us to build, with a unique padding, optimal encryption and optimal signature: first, in the random-permutation model, and then in the random-oracle model. In both cases, we
study the concrete sizes of the parameters, for a specific security level: The former achieves an optimal bandwidth.
• OAEP 3-Round: A Generic and Secure Asymmetric Encryption Padding
with David Pointcheval (ENS)
In Advances in Cryptology - IACR ASIACRYPT '04, P.J. Lee Ed. Pages 63-77, LNCS 3329, Springer-Verlag, IACR, 2004.
[Paper in pdf] [ps] [pdf USletter]
The OAEP construction is already 10 years old and well-established in many practical applications. But after some doubts about its actual security level, four years ago, the first efficient and
provably IND-CCA1 secure encryption padding was formally and fully proven to achieve the expected IND-CCA2 security level, when used with any trapdoor permutation. Even if it requires the
partial-domain one-wayness of the permutation, for the main application (with the RSA permutation family) this intractability assumption is equivalent to the classical (full-domain) one-wayness,
but at the cost of an extra quadratic-time reduction. The security proof which was already not very tight to the RSA problem is thus much worse.
However, the practical optimality of the OAEP construction is two-fold, hence its attractivity: from the efficiency point of view because of two extra hashings only, and from the length point of
view since the ciphertext has a minimal bit-length (the encoding of an image by the permutation.) But the bandwidth (or the ratio ciphertext/plaintext) is not optimal because of the randomness
(required by the semantic security) and the redundancy (required by the plaintext-awareness, the sole way known to provide efficient CCA2 schemes.)
At last Asiacrypt ’03, the latter intuition had been broken by exhibiting the first IND-CCA2 secure encryption schemes without redundancy, and namely without achieving plaintext-awareness, while
in the random-oracle model: the OAEP 3-round construction. But this result achieved only similar practical properties as the original OAEP construction: the security relies on the partial-domain
one-wayness, and needs a trapdoor permutation, which limits the application to RSA, with still a quite bad reduction.
This paper improves this result: first we show the OAEP 3-round actually relies on the (full-domain) one-wayness of the permutation (which improves the reduction), then we extend the application
to a larger class of encryption primitives (including ElGamal, Paillier, etc.) The extended security result is still in the random-oracle model, and in a relaxed CCA2 model (which lies between
the original one and the replayable CCA scenario.)
• On the Security Notions for Public-Key Encryption Schemes
with David Pointcheval (ENS)
In Proceeding of SCN'04, C. Blundo Ed. Pages 33-47, LNCS 3352, Springer-Verlag, 2004.
[Paper in pdf] [ps] [pdf USletter]
In this paper, we revisit the security notions for public-key encryption, and namely indistinguishability. We indeed achieve the surprising result that no decryption query before receiving the
challenge ciphertext can be replaced by queries (whatever the number is) after having received the challenge, and vice-versa. This remark leads to a stricter and more complex hierarchy for
security notions in the public-key setting: the (i,j)-IND level, in which an adversary can ask at most i (j resp.) queries before (after resp.) receiving the challenge. Excepted the trivial
implications, all the other relations are strict gaps, with no polynomial reduction (under the assumption that IND-CCA2 secure encryption schemes exist.) Similarly, we define different levels for
non-malleability (denoted (i,j)-NM.)
• About the Security of Ciphers (Semantic Security and Pseudo-Random Permutations)
with David Pointcheval (ENS)
In Proceeding of SAC'04, H. Handschuh and A. Hasan Eds. Pages 185-200, LNCS 3357, Springer-Verlag, 2004.
[Paper in pdf] [ps] [pdf USletter]
Probabilistic symmetric encryption have already been widely studied, from a theoretical point of view. Nevertheless, many applications require length-preserving encryption, to be patched at a
minimal cost to include privacy without modifying the format (e.g. encrypted filesystems). In this paper, we thus consider the security notions for length-preserving, deterministic and symmetric
encryption schemes, also termed ciphers: semantic security under lunchtime and challenge-adaptive adversaries. We furthermore provide some relations for this notion between different models of
adversaries, and the more classical security notions for ciphers: pseudo-random permutations (PRP) and super pseudo-random permutations (SPRP).
• Chosen-Ciphertext Security without Redundancy
with David Pointcheval (ENS)
In Advances in Cryptology - IACR ASIACRYPT '03, C.L. Laih Ed. Pages 1-18, LNCS 2894, Springer-Verlag, IACR, 2005.
[Paper in pdf] [ps] [pdf USletter]
We propose asymmetric encryption schemes for which all ciphertexts are valid (which means here “reachable”: the encryption function is not only a probabilistic injection, but also a surjection).
We thus introduce the Full-Domain Permutation encryption scheme which uses a random permutation. This is the first IND-CCA cryptosystem based on any trapdoor one-way permutation without
redundancy, and more interestingly, the bandwidth is optimal: the ciphertext is over k more bits only than the plaintext, where 2^− k is the expected security level. Thereafter, we apply it
into the random oracle model by instantiating the random permutation with a Feistel network construction, and thus using OAEP. Unfortunately, the usual 2-round OAEP does not seem to be provably
secure, but a 3-round can be proved IND-CCA even without the usual redundancy \(m || 0^{k_1}\), under the partial-domain one-wayness of any trapdoor permutation. Although the bandwidth is not as
good as in the random permutation model, absence of redundancy is quite new and interesting: many implementation risks are ruled out.
• A Comparison between two Methods of Security Proof
with David Pointcheval (ENS)
In Proceeding of RIVF. Pages 105-110, Hanoï -- February 2003 (in French).
[ps] [pdf USletter]
In this paper, we compare two methods for security proofs - a formal method, and the method by reduction from the complexity theory. A modification of the Otway-Rees protocol is proposed to show
out a difference between the two methods : the exchanged key is provably secure in the sense of the BAN logic but it is not when we analyze it by reduction. The difference is due to a limitation
of BAN logic, which has not been noticed before, that it does not consider the relation between different ciphertexts. Note that in the original Otway-Rees protocol, under the hypothesis of
semantic security of the symmetric encryption scheme, we prove the semantic security of the exchanged key which is a similar result to the one obtained with BAN logic.
• Some Preliminary Results on the Stableness of Extended F-rule Systems
with Thanh Thuy Nguyen (Hanoi Univ. of Science and Technology) and Yamanoi Takahiro (Hokkaido University, Japan)
Journal of Advanced Computational Intelligence. Pages 252-259, Vol.7 No.3, 2003.
In this paper we shall investigate an extended version of F-rule systems, in which each F-rule can include an arbitrary combination of disjunctions and conjunctions of atoms in the premise. The
first main result here is a way to determine values assigned to these extended facts, based on two basic operators ⊕ and x;, which are shown to be equivalent to external probabilistic reasoning
by resolving linear programming problem. Based on this, a definition on mixed inference operator for extended F-rule systems is discussed. We have shown that an extended F-rule system with the
defined reasoning operator is stable iff its corresponding F-rule system is stable. This proposition allows us to apply all our available research results on F-rule systems to extended F-rule
• Interval-valued Probabilistic Reasoning Agents
with Thanh Thuy Nguyen (Hanoi Univ. of Science and Technology)
In Proceeding of the 3rd International Conference on Artificial Intelligence/ Internet Computing, USA, 2002.
Current & Past PhD Students
□ Antoine Sidem (Telecom Paris, Institut Polytechnique de Paris, co-supervision with Qingju Wang and Bart Preneel)
□ Jinwei Zheng (Telecom Paris, Institut Polytechnique de Paris, co-supervision with Weiqiang Wen)
□ Nathan Papon (Telecom Paris, Institut Polytechnique de Paris, co-supervision with Sébastien Canard)
□ Duy Nguyen (Telecom Paris, Institut Polytechnique de Paris, co-supervision with David Pointcheval)
□ Orel Cosseron (ENS Lyon, co-supervision with Damien Stehlé)
□ Ferran Alborch Escobar (Orange + Telecom Paris, co-supervision with Sébastien Canard and Fabien Laguillaumie)
□ Ky Nguyen (ENS, co-supervision with David Pointcheval)
□ Antoine Urban (Telecom Paris, Institut Polytechnique de Paris, co-supervision with Matthieu Rambaud)
□ Chloé Hebant (ENS, co-supervision with David Pointcheval, defended 05/2021)
□ Xuan Thanh Do (XLIM-Limoges and Vietnam National University, co-supervision with Le Minh Ha, defended 03/2021)
□ Laura Brouilhet (XLIM-Limoges, co-supervision with Olivier Blazy, defended 12/2020)
□ Jérémy Chotard (XLIM-Limoges and ENS, co-supervision with David Pointcheval, defended 12/2019)
□ Paul Germouty (XLIM-Limoges, co-supervision with Olivier Blazy, defended 09/2018)
□ Trinh Viet Cuong (LAGA-Paris 8, co-supervision with Claude Carlet, defended 12/2013)
Habilitation Thesis
PhD Thesis
Master Report
□ Une comparaison des preuves de sécurité (méthode formelle vs. méthode calculatoire)
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Probably Overthinking It
This article uses object-oriented programming to explore of one of the most useful concepts in statistics, distributions. The code is in a Jupyter notebook.
You can read a static version of the notebook on nbviewer.
You can run the code in a browser by clicking this link and then selecting distribution.ipynb from the list.
The following is a summary of the material in the notebook, which you might want to read before you dive into the code.
Random processes and variables
One of the recurring themes of my books is the use of object-oriented programming to explore mathematical ideas. Many mathematical entities are hard to define because they are so abstract.
Representing them in Python puts the focus on what operations each entity supports
— t
hat is, what the objects can do
rather than on what they are.
In this article, I explore the idea of a probability distribution, which is one of the most important ideas in statistics, but also one of the hardest to explain. To keep things concrete, I'll start
with one of the usual examples: rolling dice.
• When you roll a standard six-sided die, there are six possible outcomes — numbers 1 through 6 — and all outcomes are equally likely.
• If you roll two dice and add up the total, there are 11 possible outcomes — numbers 2 through 12 — but they are not equally likely. The least likely outcomes, 2 and 12, only happen once in 36
tries; the most likely outcome happens 1 times in 6.
• And if you roll three dice and add them up, you get a different set of possible outcomes with a different set of probabilities.
What I've just described are three random number
. The output from each generator is a
random variable
. And each random variable has
probability distribution
, which is the set of possible outcomes and the corresponding set of probabilities.
Representing distributions
There are many ways to represent a probability distribution. The most obvious is a probability mass function, or PMF, which is a function that maps from each possible outcome to its probability. And
in Python, the most obvious way to represent a PMF is a dictionary that maps from outcomes to probabilities.
So is a Pmf a distribution? No. At least in my framework, a Pmf is one of several representations of a distribution. Other representations include the cumulative distribution function (CDF) and the
characteristic function (CF).
These representations are equivalent in the sense that they all contain the same information; if I give you any one of them, you can figure out the others.
So why would we want different representations of the same information? The fundamental reason is that there are many operations we would like to perform with distributions; that is, questions we
would like to answer. Some representations are better for some operations, but none of them is the best for all operations.
Here are some of the questions we would like a distribution to answer:
• What is the probability of a given outcome?
• What is the mean of the outcomes, taking into account their probabilities?
• What is the variance of the outcome? Other moments?
• What is the probability that the outcome exceeds (or falls below) a threshold?
• What is the median of the outcomes, that is, the 50th percentile?
• What are the other percentiles?
• How can get generate a random sample from this distribution, with the appropriate probabilities?
• If we run two random processes and choose the maximum of the outcomes (or minimum), what is the distribution of the result?
• If we run two random processes and add up the results, what is the distribution of the sum?
Each of these questions corresponds to a method we would like a distribution to provide. But there is no one representation that answers all of them easily and efficiently.
As I demonstrate in the notebook, the PMF representation makes it easy to look up an outcome and get its probability, and it can compute mean, variance, and other moments efficiently.
The CDF representation can look up an outcome and find its cumulative probability efficiently. And it can do a reverse lookup equally efficiently; that is, given a probability, it can find the
corresponding value, which is useful for computing medians and other percentiles.
The CDF also provides an easy way to generate random samples, and a remarkably simple way to compute the distribution of the maximum, or minimum, of a sample.
To answer the last question, the distribution of a sum, we can use the PMF representation, which is simple, but not efficient. An alternative is to use the characteristic function (CF), which is the
Fourier transform of the PMF. That might sound crazy, but using the CF and the Convolution Theorem, we can compute the distribution of a sum in linearithmic time, or O(n log n).
If you are not familiar with the Convolution Theorem, you might want to read
Chapter 8 of Think DSP
So what's a distribution?
The Pmf, Cdf, and CharFunc are different ways to represent the same information. For the questions we want to answer, some representations are better than others. So how should we represent the
distribution itself?
In my implementation, each representation is a
; that is, a class that provides a set of capabilities. A distribution inherits all of the capabilities from all of the representations. Here's a class definition that shows what I mean:
class Dist(Pmf, Cdf, CharFunc): def __init__(self, d): """Initializes the Dist. Calls all three __init__ methods. """ Pmf.__init__(self, d) Cdf.__init__(self, *compute_cumprobs(d)) CharFunc.__init__
(self, compute_fft(d))
When you create a Dist, you provide a dictionary of values and probabilities.
calls the other three
methods to create the Pmf, Cdf, and CharFunc representations. The result is an object that has all the attributes and methods of the three representations.
From a software engineering point of view, that might not be the best design, but it is meant to illustrate what it means to be a distribution.
In short, if you give me any representation of a distribution, you have told me everything I need to answer questions about the possible outcomes and their probabilities. Converting from one
representation to another is mostly a matter of convenience and computational efficiency.
Conversely, if you are trying to find the distribution of a random variable, you can do it by computing whichever representation is easiest to figure out.
So that's the idea. If you want more details, take a look at the notebook by following one of the links at the top of the page.
Yesterday Sanjoy Mahajan and I led a workshop on teaching Bayesian statistics for undergraduates. The participants were college teachers from around New England, including Norwich University in
Vermont and Wesleyan University in Connecticut, as well as our neighbors, Babson College and Wellelsey College.
The feedback we got was enthusiastic, and we hope the workshop will help the participants design new classes that make Bayesian methods accessible to their students.
Materials from the workshop are in this GitHub repository. And here are the slides:
The goal of the workshop is to show that teaching Bayesian statistics to undergrads is possible and desirable. To show that it's possible, we presented three approaches:
1. A computational approach, based on my class at Olin, Computational Bayesian Statistics, and the accompanying book, Think Bayes. This material is appropriate for students with basic programming
skills, although a lot of it could adapted for use with spreadsheets.
2. An analytic approach, based on Sanjoy's class, called Bayesian Inference. This material is appropriate for students who are comfortable with mathematics including calculus.
3. We also presented core material that does not depend on programming or advanced math --really just arithmetic.
Why Bayes?
Reasons the participants gave for teaching Bayes included:
1. Some of them work and teach in areas like psychology and biology where the limitations of classical methods have become painfully apparent, and interest in alternatives is high.
2. Others are interested in applications like business intelligence and data analytics where Bayesian methods are a hot topic.
3. Some participants teach introductory classes that satisfy requirements in quantitative reasoning, and they are looking for material to develop students' ability to reason with and about
I think these are all good reasons. At the introductory level, Bayesian methods are a great opportunity for students who might not be comfortable with math to gradually build confidence with
mathematical methods as tools for better thinking.
Bayes's theorem provides a divide-and-conquer strategy for solving difficult problems by breaking them into smaller, simpler pieces. And many of the classic applications of Bayes's theorem -- like
interpreting medical tests and weighing courtroom evidence -- are real-world problems where careful thinking matters and mistakes have consequences!
For students who only take a few classes in mathematics, I think Bayesian statistics is a better choice than calculus, which the vast majority of students will never use again; and better than
classical statistics, which (based on my observation) often leaves students more confused about quantitative reasoning than when they started.
At the more advanced level, Bayesian methods are appealing because they can be applied in a straightforward way to real-world decision making processes, unlike classical methods, which generally fail
to answer the questions we actually want to answer.
For example, if we are considering several hypotheses about the world, it is useful to know the probability that each is true. You can use that information to guide decision making under uncertainty.
But classical statistical inference refuses to answer that question, and under the frequentist interpretation of probability, you are not even allowed to ask it.
As another example, the result you get from Bayesian statistics is generally a posterior distribution for a parameter, or a joint distribution for several parameters. From these results, it is
straightforward to compute a distribution that predicts almost any quantity of interest, and this distribution encodes not only the most likely outcome or central tendency; it also represents the
uncertainty of the prediction and the spread of the possible outcomes.
Given a predictive distribution, you can answer whatever questions are relevant to the domain, like the probability of exceeding some bound, or the range of values most likely to contain the true
value (another question classical inference refuses to answer). And it is straightforward to feed the entire distribution into other analyses, like risk-benefit analysis and other kinds of
optimization, that directly guide decision making.
I mention these advantages in part to address one of the questions that came up in the workshop. Several of the participants are currently teaching traditional introductory statistics classes, and
they would like to introduce Bayesian methods, but are also required to cover certain topics in classical statistics, notably null-hypothesis significance testing (NHST).
So they want to know how to design a class that covers these topics and also introduces Bayesian statistics. This is an important challenge, and I was frustrated that I didn't have a better answer to
offer at the workshop. But with some time to organize my thoughts, I have a two suggestions:
Avoid direct competition
I don't recommend teaching a class that explicitly compares classical and Bayesian statistics. Pedagogically, it is likely to be confusing. Strategically, it is asking for intra-departmental warfare.
And importantly, I think it misrepresents Bayesian methods, and undersells them, if you present them as a tool-for-tool replacement for classical methods.
The real problem with classical inference is not that it gets the wrong answer; the problem is that is asks the wrong questions. For example, a fundamental problem with NHST is that it requires a
binary decision: either we reject the null hypothesis or we fail to reject it (whatever that means). An advantage of the Bayesian approach is that it helps us represent and work with uncertainty;
expressing results in terms of probability is more realistic, and more useful, than trying to cram the world into one of two holes.
If you use Bayesian methods to compute the probability of a hypothesis, and then apply a threshold to decide whether the theory is true, you are missing the point. Similarly, if you compute a
posterior distribution, and then collapse it to a single point estimate (or even an interval), you are throwing away exactly the information that makes Bayesian results more useful.
Bayesian methods don't do the same things better; they do different things, which are better. If you want to demonstrate the advantages of Bayesian methods, do it by solving practical problems and
answering the questions that matter.
As an example, this morning my colleague Jon Adler sent me a link to this paper, Bayesian Benefits for the Pragmatic Researcher, which is a model of what I am talking about.
Identify the goals
As always, it is important to be explicit about the learning goals of the class you are designing. Curriculum problems that seems impossible can sometimes be simplified by unpacking assumptions about
what needs to be taught and why. For example, if we think about why NHST is a required topic, we get some insight into how to present it: if you want to make sure students can read papers that report
p-values, you might take one approach; if you imagine they will need to use classical methods, that might require a different approach.
For classical statistical inference, I recommend "The New Statistics", an approach advocated by Geoff Cumming (I am not sure to what degree it is original to him). The fundamental idea of is that
statistical analysis should focus on estimating effect sizes, and should express results in terms that emphasize practical consequences, as contrasted with statistical significance.
If "The New Statistics" is what we should teach, computational simulation is how. Many of the ideas that take the most time, and seem the hardest, in a traditional stats class, can be taught much
more effectively using simulation. I wrote more about this just last week, in this post, There is Still Only One Test, and there are links there to additional resources.
But if the goal is to teach classical statistical inference better, I would leave Bayes out of it. Even if it's tempting to use a Bayesian framework to explain the problems with classical inference,
it would be more likely to confuse students than help them.
If you only have space in the curriculum to teach one paradigm, and you are not required to teach classical methods, I recommend a purely Bayesian course. But if you have to teach classical methods
in the same course, I suggest keeping them separated.
I experienced a version of this at PyCon this year, where I taught two tutorials back to back: Bayesian statistics in the morning and computational statistical inference in the afternoon. I joked
that I spent the morning explaining why the afternoon was wrong. But the reality is that they two topics hardly overlap at all. In the morning I used Bayesian methods to formulate real-world problems
and answer practical questions. In the afternoon, I helped people understand classical inference, including its limitations, and taught them how to do it well, if they have to.
I think a similar balance (or compromise?) could work in the undergraduate statistic curriculum at many colleges and universities.
In 2011 I wrote an article called "There is Only One Test", where I explained that all hypothesis tests are based on the same framework, which looks like this:
Here are the elements of this framework:
1) Given a dataset, you compute a test statistic that measures the size of the apparent effect. For example, if you are describing a difference between two groups, the test statistic might be the
absolute difference in means. I'll call the test statistic from the observed data 𝛿*.
2) Next, you define a null hypothesis, which is a model of the world under the assumption that the effect is not real; for example, if you think there might be a difference between two groups, the
null hypothesis would assume that there is no difference.
3) Your model of the null hypothesis should be stochastic; that is, capable of generating random datasets similar to the original dataset.
4) Now, the goal of classical hypothesis testing is to compute a p-value, which is the probability of seeing an effect as big as 𝛿* under the null hypothesis. You can estimate the p-value by using
your model of the null hypothesis to generate many simulated datasets. For each simulated dataset, compute the same test statistic you used on the actual data.
5) Finally, count the fraction of times the test statistic from simulated data exceeds 𝛿*. This fraction approximates the p-value. If it's sufficiently small, you can conclude that the apparent
effect is unlikely to be due to chance (if you don't believe that sentence, please read this).
That's it. All hypothesis tests fit into this framework. The reason there are so many names for so many supposedly different tests is that each name corresponds to
1) A test statistic,
2) A model of a null hypothesis, and usually,
3) An analytic method that computes or approximates the p-value.
These analytic methods were necessary when computation was slow and expensive, but as computation gets cheaper and faster, they are less appealing because:
1) They are inflexible: If you use a standard test you are committed to using a particular test statistic and a particular model of the null hypothesis. You might have to use a test statistic that is
not appropriate for your problem domain, only because it lends itself to analysis. And if the problem you are trying to solve doesn't fit an off-the-shelf model, you are out of luck.
2) They are opaque: The null hypothesis is a model, which means it is a simplification of the world. For any real-world scenario, there are many possible models, based on different assumptions. In
most standard tests, these assumptions are implicit, and it is not easy to know whether a model is appropriate for a particular scenario.
One of the most important advantages of simulation methods is that they make the model explicit. When you create a simulation, you are forced to think about your modeling decisions, and the
simulations themselves document those decisions.
And simulations are almost arbitrarily flexible. It is easy to try out several test statistics and several models, so you can choose the ones most appropriate for the scenario. And if different
models yield very different results, that's a useful warning that the results are open to interpretation. (Here's an example I wrote about in 2011.)
More resources
A few days ago, I saw this discussion on Reddit. In response to the question "Looking back on what you know so far, what statistical concept took you a surprising amount of effort to understand?",
one redditor wrote
The general logic behind statistical tests and null hypothesis testing took quite some time for me. I was doing t-tests and the like in both work and classes at that time, but the overall picture
evaded me for some reason.
I remember the exact time where everything started clicking - that was after I found a blog post (cannot find it now) called something like "There is only one statistical test". And it explained
the general logic of testing something and tied it down to permutations. All of that seemed very natural.
I am pretty sure they were talking about my article. How nice! In response, I provided links to some additional resources; and I'll post them here, too.
First, I wrote a followup to my original article, called "More hypotheses, less trivia", where I provided more concrete examples using the simulation framework.
Later in 2011 I did a webcast with O'Reilly Media where I explained the whole idea:
In 2015 I developed a workshop called "Computational Statistics", where I present this framework along with a similar computational framework for computing confidence intervals. The slides and other
materials from the workshop are here.
And I am not alone! In 2014, John Rauser presented a keynote address at Strata+Hadoop, with the excellent title "Statistics Without the Agonizing Pain":
And for several years, Jake VanderPlas has been banging a similar drum, most recently in an excellent talk at PyCon 2016:
UPDATE: John Rauser pointed me to this excellent article, "The Introductory Statistics Course: A Ptolemaic Curriculum" by George W. Cobb.
UPDATE: Andrew Bray has developed an R package called "infer" to do computational statistical inference. Here's an excellent talk where he explains it. | {"url":"https://allendowney.blogspot.com/2016/06/","timestamp":"2024-11-05T10:33:02Z","content_type":"text/html","content_length":"115885","record_id":"<urn:uuid:837e4e20-1b56-4a7d-b70d-b79dc8834b3e>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00311.warc.gz"} |
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January 2019 : Tackle Maths
For this week’s posts I’m dipping into the weird land of Quora. As I may have mentioned before the Maths questions on this Q&A based social media site can be weird and can be trivial, like what is
the LCM of 2 and 3.
That is 6 by the way but I’m not going to dedicate a whole post to that.
Others today ask ‘Why do Negative numbers exist’ which is perhaps deeper than the questioner intended, and ‘What are the 100 ways of asking a woman for her number’, which is perhaps only loosely a
Maths question. Though if this can be interpreted as her favourite number (not her phone number which I suspect was the idea) then it could be a more interesting one. What does someone’s favourite
number say about them. Have I posted about my favourite number before? I should do that sometime.
But instead, let’s consider the one above….
This arrangement doesn’t have the correct number of balls, but it gives an idea
Draw a line between the two balls that are opposite. Count how many balls are above that line.
In this case the answer is 5. And the total number of balls is 12.
How did I get from one number to the other (apart from counting)?
There are 5 numbers above and 5 below. – That’s 2 times 5. Plus the two numbers on the line.
So that is 2 x 5 + 2 = 12.
I like to generalise these things and look for patterns, and that means , yes, algebra.
So what if the number of balls above the line is a.
The total number of balls is 2a + 2 (Times 2 then add 2)
Now lets think about the problem we are solving. There are 11 balls between the 8^th and 20^th (Count them, 9^th, 10^th, 11^th…. 19^th)
use our formula – 2 x 11 + 2 = 24.
There are 24 balls in our circle. We have solved the problem but, better than that, we can answer quickly even if the question setter changed the numbers!
I like that approach; a bit more work and you can answer so many similar questions | {"url":"https://tacklemaths.co.uk/2019/01/","timestamp":"2024-11-03T23:25:58Z","content_type":"text/html","content_length":"35748","record_id":"<urn:uuid:41b49e24-c87d-4568-a7fe-0539af1589ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00065.warc.gz"} |
Logarithms yearning to be free
I got an evaluation copy of The Best Writing on Mathematics 2021 yesterday. One article jumped out as I was skimming the table of contents: A Zeroth Power Is Often a Logarithm Yearning to Be Free by
Sanjoy Mahajan. Great title.
There are quite a few theorems involving powers that have an exceptional case that involves a logarithm. The author opens with the example of finding the antiderivative of x^n. When n ≠ −1 the
antiderivative is another power function, but when n = −1 it’s a logarithm.
Another example that the author mentions is that the limit of power means as the power goes to 0 is the geometric mean. i.e. the exponential of the mean of the logarithms of the arguments.
I tried to think of other examples where this pattern pops up, and I thought of a couple related to entropy.
q-logarithm entropy
The definition of q-logarithm entropy takes Mahajan’s idea and runs it backward, turning a logarithm into a power. As I wrote about here,
The natural logarithm is given by
and we can generalize this to the q-logarithm by defining
And so ln[1] = ln.
Then q-logarithm entropy is just Shannon entropy with natural logarithm replaced by q-logarithm.
Rényi entropy
Quoting from this post,
If a discrete random variable X has n possible values, where the ith outcome has probability p[i], then the Rényi entropy of order α is defined to be
for 0 ≤ α ≤ ∞. In the case α = 1 or ∞ this expression means the limit as α approaches 1 or ∞ respectively.
When α = 1 we get the more familiar Shannon entropy:
In this case there’s already a logarithm in the definition, but it moves inside the parentheses in the limit.
And if you rewrite
p p^α−1
then as the exponent in p^α−1 goes to zero, we have a logarithm yearning to be free.
One thought on “Logarithms yearning to be free”
1. Another simple one is the Box-Cox transformation. | {"url":"https://www.johndcook.com/blog/2022/04/13/logarithms-yearning-to-be-free/","timestamp":"2024-11-02T11:14:51Z","content_type":"text/html","content_length":"52587","record_id":"<urn:uuid:ed9d25f7-3b9c-4eff-81ab-044a197dd139>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00854.warc.gz"} |
Amazon Braket On The Quantum Bandwagon
Written by Mike James
Tuesday, 03 December 2019
Amazon has just announced an extension to the range of instances offered by the AWS cloud services - quantum! Why?
First I have to say that I like the name. I was more impressed by Dirac's invention of the <bra| and |ket> notation for co-vectors and vectors that so cleverly yields <bra|ket> when you form the
inner product, but yes, calling it Braket is appropriate.
What Amazon has done is basically invented a clever name and jumped on the quantum computer bandwagon without having to invest in building a device.
Braket is nothing really new - IBM, Microsoft and others offer similar facilities. What we have is a standard way of specifying an arrangement of quantum gates that can be then run on a simulator. We
refer to this as some sort of programming, but to be honest it is nothing like what we know as programming. It is essentially finding transformations of quantum states that give you a new state that
yields the answer. This is a tough technique to crack and you cannot suppose that being able to program puts you in a good position to tackle programming a quantum machine, which is more like
arranging a simulation that gives you the result you need.
Finding the simulation that gets the result is difficult and so far there is no simple or automatic way to find said configurations. Apart from Shore's and Grover's, most algorithms aren't applicable
to the sort of problems that classical computers are used for. This drove Hewlett Packard to give up on quantum computers some time ago with the promise that, if quantum computers ever seem to offer
something to the sort of customers that it handles, it would reconsider the idea. HP clearly had a much better idea of the way of the world than Amazon and jumped off the bandwagon not onto it.
As in competing systems, Amazon allows you to run your program on a real quantum computer provided by third parties. Of course, you can't expect many qubits as so far we haven't managed to build a
quantum computer with more than a few of them. The exception is the D-Wave system, which is a quite different type of quantum computer and really is a simulation machine capable of finding solutions
to optimization problems using quantum annealing. Basically you set up the initial state and then let it relax into what you hope is the lowest energy state, which, if you have set things up
correctly, corresponds to the optimum solution you are looking for. Of course, D-Wave has never proved that is can solve problems faster than classical machines. In fact, D-Wave's machine is so
different I'm surprised to find that Braket supports it.
So why am I so skeptical of Amazon's move? Is it that I'm a quantum computer denier? No, I think we will build a quantum computer some day, it's more that I'm not convinced that' apart from breaking
a few codes, which we will promptly give up using, a quantum computer will be generally useful. Of course, it does depend on what you mean by "generally useful" but I doubt that anything like the
average AWS user will get anything from exposure to quantum computing.
• Mike James is the author of The Programmer’s Guide To Theory which sets out to present the fundamental ideas of computer science in an informal and yet informative way.
More Information
Amazon Braket
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Pigeonhole Sort
Open-Source Internship opportunity by OpenGenus for programmers. Apply now.
Reading time: 20 minutes | Coding time: 5 minutes
Pigeonhole sorting is a sorting algorithm that is suitable for sorting lists of elements where the number of elements and the number of possible key values are approximately the same.
Where key here refers to the part of a group of data by which it is sorted, indexed, cross referenced, etc.It is used in quick, heap and many other sorting algorithm.
If we consider number of elements (n) and the length of the range of possible key values (N) are approximately the same. It requires O(n + N) time.
Working of Pigeonhole Sort
1. Find minimum and maximum values in array. Let the minimum and maximum values be â minâ and â maxâ respectively. Also find range as max-min+1.
2. Set up an array of initially empty â pigeonholesâ the same size as of the range calculated above.
3. Visit each element of the array and then put each element in its pigeonhole. An element arr[i] is put in hole at index arr[i] â min.
4. Start the loop all over the pigeonhole array in order and put the elements from non- empty holes back into the original array.
In Pigeonhole sort, we need not compare two data to sort them and hence, it is an in-place sorting algorithm
As you can see a lot of blocks are empty, this algorithm requires a lot of space even for a small but widely spread data.
Explaining the steps mentioned above :
When we implement point 3 in working of Pigeonhole Sort we do the following :
Step 1 : 9-1 = 8 => buk[8] = arr[0]
Step 2 : 2-1 = 1 => buk[1] = arr[1]
Step 3 : 3-1 = 2 => buk[2] = arr[2]
Step 4 : 8-1 = 7 => buk[7] = arr[3]
Step 5 : 1-1 = 0 => buk[0] = arr[4]
Step 6 : 6-1 = 5 => buk[5] = arr[5]
Step 7 : 9-1 = 8 => buk[8] = arr[6]
After implementing the array we get :
Difference between Counting and Pigeonhole sort
The ONLY difference between counting and pigeonhole sort is that in the secondary sort we move items twice i.e one on the input array and other in the bucket array. While in first sort we only move
the items once.
#include <bits/stdc++.h>
using namespace std;
/* Sorts the array using pigeonhole algorithm */
void pigeonholeSort(int arr[], int n)
// Find minimum and maximum values in arr[]
int min = arr[0], max = arr[0];
for (int i = 1; i < n; i++)
if (arr[i] < min)
min = arr[i];
if (arr[i] > max)
max = arr[i];
int range = max - min + 1; // Find range
// Create an array of vectors. Size of array
// range. Each vector represents a hole that
// is going to contain matching elements.
vector holes[range];
// Traverse through input array and put every
// element in its respective hole
for (int i = 0; i < n; i++)
// Traverse through all holes one by one. For
// every hole, take its elements and put in
// array.
int index = 0; // index in sorted array
for (int i = 0; i < range; i++)
vector::iterator it;
for (it = holes[i].begin(); it != holes[i].end(); ++it)
arr[index++] = *it;
// Driver program to test the above function
int main()
int arr[] = {9,2,3,8,1,6,9};
int n = sizeof(arr)/sizeof(arr[0]);
pigeonholeSort(arr, n);
printf("Sorted order is : ");
for (int i = 0; i < n; i++)
printf("%d ", arr[i]);
return 0;
def pigeonhole_sort(a):
# size of range of values in the list
# (ie, number of pigeonholes we need)
my_min = min(a)
my_max = max(a)
size = my_max - my_min + 1
# our list of pigeonholes
holes = [0] * size
# Populate the pigeonholes.
for x in a:
assert type(x) is int, "integers only please"
holes[x - my_min] += 1
# Put the elements back into the array in order.
i = 0
for count in range(size):
while holes[count] > 0:
holes[count] -= 1
a[i] = count + my_min
i += 1
a = [9,2,3,8,1,6,9]
print("Sorted order is : ", end = ' ')
for i in range(0, len(a)):
print(a[i], end = ' ')
• Worst case time complexity: Î (n+N)
• Average case time complexity: Î (n+N)
• Best case time complexity: Î (n+N)
• Space complexity: Î (n+N)
where n is the number of elements and N is the range of the input data
Which is more efficient Bucket sort or Pigeonhole sort for larger value of N?
Bucket sort
Pigeon sort
Both are equally efficient
Depends on the machine
For arrays where N is much larger than n, bucket sort is a generalization that more efficient in space and time. | {"url":"https://iq.opengenus.org/pigeonhole-sorting/","timestamp":"2024-11-04T18:33:02Z","content_type":"text/html","content_length":"64665","record_id":"<urn:uuid:b1e5396f-0cc6-4c52-94d4-8014b489d66d>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00711.warc.gz"} |
Printable Figure Drawings
Parallel Lines Cut By A Transversal Worksheet Answer Key
Parallel Lines Cut By A Transversal Worksheet Answer Key - Definition of ’s 2 and 3 are a linear pair. Answer or prove the following: What is the value of \ (x\)? With lots of practice, this set of
pdf worksheets helps brush up your knowledge of the characteristics of the various. Web worksheet by kuta software llc parallel lines cut by a transversal ©z d2k0f2w1p dkpuut`ai _spopfptpwoaeraew
klflbcz.v f xaclslp jrvisghhdtysx. M 2 + m 3 = 180° substitution prop.
Web worksheet by kuta software llc parallel lines cut by a transversal ©z d2k0f2w1p dkpuut`ai _spopfptpwoaeraew klflbcz.v f xaclslp jrvisghhdtysx. 107° 81° 91° 69° 85° 91° 73° 79° 93° 81° 103° 95°
89°. Web if 2 ǁ lines are cut by a transversal, key parallel lines and proofs corresp. Identify the pairs of angles in the diagram. It can be used for in person instruction,.
A line that cuts 2 or more parallel lines is called transversal. Web worksheet by kuta software llc parallel lines cut by a transversal ©z d2k0f2w1p dkpuut`ai _spopfptpwoaeraew klflbcz.v f xaclslp
jrvisghhdtysx. Definition of ’s 2 and 3 are a linear pair. Parallel lines are 2 coplanar lines which do not intersect. Web parallel lines cut by transversals il.
Web worksheet by kuta software llc parallel lines cut by a transversal ©z d2k0f2w1p dkpuut`ai _spopfptpwoaeraew klflbcz.v f xaclslp jrvisghhdtysx. 1) given www.mathplane.com solutions 1 and 3 are
altemate interior angles. Definition of ’s 2 and 3 are a linear pair. Web if 2 ǁ lines are cut by a transversal, key parallel lines and proofs corresp. It can be.
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transversal worksheets. Web parralel lines cut by a transversal coloring activity.
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can be used for in person instruction,. Adopted from all things algebra by gina wilson.
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lines cut by a transversal. Adopted from all things algebra by gina wilson. 1) given www.mathplane.com solutions 1 and 3 are altemate interior angles. Then make.
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Web For More Practice With Angle Relationships Created By Parallel Lines Cut By A Transversal, Have Students Complete The Transversals Of Parallel Lines Worksheet.
M 2 + m 3 = 180° substitution prop. Web when a transversal intersects two parallel lines, it creates eight angles that include corresponding angles, alternate interior angles, alternate exterior
angles, and same. Parallel lines are 2 coplanar lines which do not intersect. Traverse through this array of free printable worksheets to learn the major outcomes of angles formed by parallel lines.
Definition Of ’S 2 And 3 Are A Linear Pair.
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help students practice and comprehend properties of angles and.
Web Parralel Lines Cut By A Transversal Coloring Activity.
Answer or prove the following: Jonhdi07 is waiting for your help. It can be used for in person instruction,. In the following diagram, two parallel lines are cut by a transversal.
Web Parallel Lines And Transversals Worksheets Will Help Kids In Solving Geometry Problems.
Then make a conjecture about their angle measures. Pythagorean theorem, parallel lines cut by a. Web parallel lines cut by a transversal worksheet. 1) given www.mathplane.com solutions 1 and 3 are
altemate interior angles. | {"url":"https://tunxis.commnet.edu/view/parallel-lines-cut-by-a-transversal-worksheet-answer-key.html","timestamp":"2024-11-04T21:52:15Z","content_type":"text/html","content_length":"36438","record_id":"<urn:uuid:05263fbf-17b3-405e-8b48-a8b9448dda26>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00502.warc.gz"} |
Knots to Miles per Hour Conversion (kn to mph) - Inch Calculator
Knots to Miles per Hour Converter
Enter the speed in knots below to convert it to miles per hour.
Result in Miles per Hour:
Do you want to convert miles per hour to knots?
How to Convert Knots to Miles per Hour
To convert a measurement in knots to a measurement in miles per hour, multiply the speed by the following conversion ratio: 1.150779 miles per hour/knot.
Since one knot is equal to 1.150779 miles per hour, you can use this simple formula to convert:
miles per hour = knots × 1.150779
The speed in miles per hour is equal to the speed in knots multiplied by 1.150779.
For example,
here's how to convert 5 knots to miles per hour using the formula above.
miles per hour = (5 kn × 1.150779) = 5.753897 mph
Knots and miles per hour are both units used to measure speed. Calculate speed in knots or miles per hour using our speed calculator or keep reading to learn more about each unit of measure.
What Is a Knot?
One knot is equal to a speed of one nautical mile per hour,^[1] or one minute of latitude per hour.
Knots can be abbreviated as kn, and are also sometimes abbreviated as kt. For example, 1 knot can be written as 1 kn or 1 kt.
Learn more about knots.
What Are Miles per Hour?
Miles per hour are a measurement of speed expressing the distance traveled in miles in one hour.^[2]
The mile per hour is a US customary and imperial unit of speed. Miles per hour can be abbreviated as mph, and are also sometimes abbreviated as mi/h or MPH. For example, 1 mile per hour can be
written as 1 mph, 1 mi/h, or 1 MPH.
Miles per hour can be expressed using the formula:
v[mph] = d[mi] / t[hr]
The velocity in miles per hour is equal to the distance in miles divided by time in hours.
Learn more about miles per hour.
Knot to Mile per Hour Conversion Table
Table showing various
knot measurements
converted to miles per
Knots Miles Per Hour
1 kn 1.1508 mph
2 kn 2.3016 mph
3 kn 3.4523 mph
4 kn 4.6031 mph
5 kn 5.7539 mph
6 kn 6.9047 mph
7 kn 8.0555 mph
8 kn 9.2062 mph
9 kn 10.36 mph
10 kn 11.51 mph
11 kn 12.66 mph
12 kn 13.81 mph
13 kn 14.96 mph
14 kn 16.11 mph
15 kn 17.26 mph
16 kn 18.41 mph
17 kn 19.56 mph
18 kn 20.71 mph
19 kn 21.86 mph
20 kn 23.02 mph
21 kn 24.17 mph
22 kn 25.32 mph
23 kn 26.47 mph
24 kn 27.62 mph
25 kn 28.77 mph
26 kn 29.92 mph
27 kn 31.07 mph
28 kn 32.22 mph
29 kn 33.37 mph
30 kn 34.52 mph
31 kn 35.67 mph
32 kn 36.82 mph
33 kn 37.98 mph
34 kn 39.13 mph
35 kn 40.28 mph
36 kn 41.43 mph
37 kn 42.58 mph
38 kn 43.73 mph
39 kn 44.88 mph
40 kn 46.03 mph
1. NASA, Knots Versus Miles per Hour, https://www.grc.nasa.gov/WWW/K-12/WindTunnel/Activities/knots_vs_mph.html
2. Wikipedia, Miles per hour, https://en.wikipedia.org/wiki/Miles_per_hour
More Knot & Mile per Hour Conversions | {"url":"https://www.inchcalculator.com/convert/knot-to-mile-per-hour/","timestamp":"2024-11-13T02:42:23Z","content_type":"text/html","content_length":"69237","record_id":"<urn:uuid:339b4bc3-8dde-4ebc-a005-ac4534b59474>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00789.warc.gz"} |
Feature Selection
The one crucial aspect of multiple linear regression that remains to be discussed is feature selection. When building a multiple linear regression model, you may have quite a few potential predictor
variables; selecting just the right ones becomes an extremely important exercise.
Let’s see how you can select the optimal features for building a good model.
Note that in
the brute force approach, one of the combinations out of 2pdoes not make any sense, namely the one which does not use any features at all. So, this means that we only need to try 2p – 1 feature
To get the optimal model, you can always try all the possible combinations of independent variables and see which model fits best. But this method is time-consuming and infeasible. Hence, you need
another method to get a decent model. This is where manual feature elimination comes in, wherein you:
• Build the model with all the features,
• Drop the features that are the least helpful in prediction (high p-value),
• Drop the features that are redundant (using correlations and VIF),
• Rebuild the model and repeat.
Note that the second and third steps go hand in hand, and the choice of which features to eliminate first is very subjective. You will see this during the hands-on demonstration of multiple linear
regression in Python in the next session.
Now, manual feature elimination may work when you have a relatively low number of potential predictor variables, say, ten or even twenty. But it is not a practical approach when you have a large
number of features, say 100. In such a case, you automate the feature selection (or elimination) process. Let’s see how.
You need to combine the manual approach and the automated one in order to get an optimal model relevant to the business. Hence, you first do an automated elimination (coarse tuning), and when you
have a small set of potential variables left to work with, you can use your expertise and subjectivity to eliminate a few other features (fine tuning). | {"url":"https://www.internetknowledgehub.com/feature-selection/","timestamp":"2024-11-09T22:50:43Z","content_type":"text/html","content_length":"79925","record_id":"<urn:uuid:110cc544-b791-47f6-835d-dff23a2aec19>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00031.warc.gz"} |
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.10
Study MaterialsNCERT SolutionsNCERT Solutions For Class 12 MathematicsNCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex 7.10) Exercise 7.10
NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex 7.10) Exercise 7.10
All the queries that require examination can be found within the NCERT Solutions for Class 12 Mathematics Chapter 7 on Integrals, specifically in Exercise 7.10. With the aid of concise notes,
comprehending the solutions for NCERT Mathematics Class 12 Exercise 7.10 becomes a straightforward task. The Exercise 7.10 solutions in Chapter 7 of Class 12 Mathematics, provided by Infinity Learn,
prove to be highly beneficial. Infinity Learn has devised easily understandable notes to assist students in excelling in their examinations. You can conveniently download the CBSE Class 12
Mathematics Chapter 7 Integrals PDF at no cost.
Do you need help with your Homework? Are you preparing for Exams? Study without Internet (Offline)
Exercise 7.10
NCERT Class 12 Maths Chapter 7 Integrals Exercise 7.10
Mathematics is an essential subject, regardless of your academic level or field of study. To achieve good grades, it’s crucial to have a solid understanding of this subject.
The objective of Exercise 7.10 in Class 12 Mathematics is to ensure that you grasp the topic thoroughly. Infinity Learn’s experts have employed straightforward language to explain each subject,
drawing from their extensive teaching experience and study materials.
Once you’ve mastered the concept of integrals, there’s no stopping you from acing all the questions in this chapter on any exam. NCERT Solutions for Class 12 Mathematics, specifically Chapter 7
Exercise 7.1, comprehensively cover the syllabus. These notes have been meticulously crafted through extensive research and align with NCERT guidelines.
Some of the important concepts that are discussed in Exercise 7.10 Class 12 NCERT Solutions are as follows:
Numerous concepts related to integrals are elucidated within the NCERT Solutions for Class 12, Chapter 7, Exercise 7.10. The solutions provided for the significant problems in this chapter are
exceptionally well-crafted, offering students the opportunity to attain exceptional results through their study. This excellence is attributed not only to the contributions of seasoned educators but
also to the comprehensive research that underlies the development of these materials. Proficient teachers have a deep understanding of what students need to comprehend within the chapter, and they
have created these notes in accordance with the standards set by the educational board.
Also Check:
FAQs on NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.10
Is it necessary to go over the examples in Exercise 7.10 of Chapter 7 of Class 12 Maths?
The examples in Exercise 7.10 of Chapter 7 of Class 12 Maths are just as important as the other exercises' questions. Exams sometimes include questions based on examples, so preparing them will help
you avoid losing any marks. The majority of the problems in the exercises are also based on these examples, so practicing them will make it easier for you to understand and complete the exercises.
What are the benefits of using NCERT Solutions for Ex 7.10 Class 12 Maths?
You have extra exercises to practice in addition to the study material. The first step is to study and understand the entire Ex 7.10 Class 12 Maths Solutions. Then you can quickly reproduce some of
the material's solved examples. All of the calculations are completed in a series of steps. You will not get decent grades if you do not follow the same processes for tests. There is step marking in
Ex 7.10 Class 12 Maths Solutions. As a result, in order to receive full marks, you must complete all of the procedures correctly and provide the correct answer. Using shortcut methods to get to the
final answer will not earn you any points. In Maths NCERT Solutions Class 12 Chapter 7 Exercise 7.10, there are solutions with questions. Other activities have solutions as well.
What makes integrals so difficult?
Integrals could be a nightmare for a student who skips any lesson or struggles to understand the basic notion of any issue. As a result, it's critical that you practice a variety of sums and become
familiar with the various levels of difficulty. To fully comprehend any topic, practice as many sums as possible. | {"url":"https://infinitylearn.com/surge/study-materials/ncert-solutions/class-12/maths/chapter-7-integrals-exercise-7-10/","timestamp":"2024-11-07T09:51:37Z","content_type":"text/html","content_length":"176890","record_id":"<urn:uuid:4b300e93-d32f-404c-9b21-660c3f1aa5d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00270.warc.gz"} |
Quantum computing in 10 minutes
Christopher R. Bernhardt presents a short guide to quantum computing and says that, far from being impenetrable, quantum is beautiful and understandable.
Most of us have heard something about quantum computing, but many of the popular accounts are misleading, giving the impression that quantum computing is somewhat analogous to parallel computing. The
truth is very different and a lot more interesting.
Quantum computing is more fundamental than classical computing. The qubit is a more fundamental object than the bit - anything you can do with bits you can do with qubits, but there are things you
can do with qubits that have no classical counterpart. You do need to use mathematics to describe what is going on, but the mathematics is really not difficult to learn and use. Once mastered it
opens the way to seeing some beautiful and counter-intuitive ideas. Quantum computing introduces some of the most surprising concepts of quantum mechanics, but in a simple and precise way.
The basic unit of classical computing is the bit - it is either zero or one. Bits can be represented by switches that are either in the on or off position. In quantum computing, the basic unit is the
qubit. In quantum computing, the basic unit is the qubit. There are two basic qubits that we denote by |0> and by |1>. (Mathematically, these are vectors.)
We can also put qubits in a superposition of states to obtain qubits of the form a|0>+b|1>, where a and b can be any two numbers with the property that the sum of their squares is 1. Qubits can be
represented by spins of electrons or by polarizations of photons. Unfortunately, these are not part of our everyday experience, so rather than explain the connection, we will just describe how we
work with them.
The first important point is that to get a readout of the result of a computation we must make a measurement of the qubits. When we measure a qubit of the form a|0>+b|1> it changes - it jumps to |0>
or to |1> and we extract a bit, not a qubit, of information, i.e. we read off 0 or 1. The probability that the qubit jumps to |0> and we read off 0 is given by the square of a; the probability that
we get 1 is the square of b. This, of course, is quite unlike bits where reading off the value of a bit does not affect the bit in any way.
Quantum gates
Two classical gates are the NOT and the AND gate. We input bits and get bits out. Quantum gates are somewhat analogous - we input qubits and get qubits output. One big difference is that quantum
gates are invertible, if we are told what the output is, we can calculate the input. For example, the classical NOT gate is invertible - if we get 0 as output, we know that 1 must have been input.
But AND is not invertible, if we receive 0 as the output, the input could have been one of three possibilities: 00, 01 or 10. A consequence of requiring invertibility is that a quantum gate must
output exactly the same number of qubits as input.
Another important point that needs to be stressed is that when we send qubits through quantum gates no measurements are made. There is no jumping of states.
Earlier we pointed out that a general qubit has the form a|0>+b|1> and that when we measure it we get 0 output with probability the square of a and 1 output with probability the square of b. The
numbers a and b do not have to be positive, they can be negative. If a|0>+b|1> is a qubit, then a|0>-b|1> will also be a qubit. If we add these qubits, we will cancel the part that involves |1>. This
explanation is a little vague, but it corresponds to the idea of constructive and destructive interference of waves.
Another important quantum gate is the CNOT gate. It takes two qubits as input, and gives two qubits as output. The reason for this gate’s importance is that it enables us to entangle qubits. This is
a quantum phenomenon which has no everyday counterpart, so we will examine an example - a pair of entangled qubits in the state a|00>+b|11>. For this example, we have two qubits that could be very
far apart. Perhaps I have one and you have the other. The actual objects might be photons and we will each measure the polarization of our photon.
When we make the measurements, we both get exactly the same result - we both get 0 or we both get 1. The probability we get 0 is the square of a and the probability we get 1 is the square of b. What
makes entanglement so surprising is that the qubits don’t jump to |0> or to |1> until the first measurement is made. As soon as the first measurement is made both qubits are changed. This is what
Einstein famously thought wrong and referred to as ‘spooky action at a distance’. Various experiments have shown that entanglement is real. It also has applications.
One use for entanglement is in for key distribution in encryption. Suppose that you and I want to communicate securely. We want to use an encryption method that uses a key - a large number, or,
equivalently, a long string of 0s and 1s. The first thing we need to do is to construct a shared key in such a way that we know has not been intercepted by a third party. Quantum entanglement enables
us to do that by sharing and then measuring a large number of entangled qubits. If nobody intercepts our qubits we both end up with exactly the same string of randomly generated bits. If someone
intercepts and measures our qubits we won’t end up with the same string.
Consequently, we can compare half of our string of numbers to tell whether or not there is an interloper and, if there is not, use the second half as our key.
Another application is to quantum teleportation. Suppose that you and I have a pair of entangled qubits. Then someone gives you third qubit in an unknown state a|0>+b|1>. Neither of us know the
values of a and b. We want to change the state of my qubit to become a|0>+b|1>. This can be done by you sending both the qubits in your possession through a CNOT gate. This entangles all three
qubits. You then measure your two qubits obtaining one of four possibilities: 00, 01, 10 or 11.
The mathematics shows that my qubit ends up in one of four possible states: a|0>+b|1>, a|0>-b|1>, b|0>+a|1> or b|0>-a|1>. You send me the results of your measurements. Each of the four cases for your
measurements corresponds to one of the four possibilities for my qubit. Once I know which qubit I have, I can send it through a gate that sends it to a|0>+b|1>. The result is that I end up with a
qubit in the unknown state. The state has been ‘teleported’ from you to me.
Though this sounds like science fiction this has been performed many times. An amazing example is that of a Chinese team who teleported a qubit from Earth to a satellite in low Earth orbit.
Quantum circuits
Classical circuits consist of gates and wires connecting them. We input bits on the left and read the results on the right. Quantum circuits are similar. They consist of wires and quantum gates. We
input qubits on the left. To get information out of the circuit we have to measure the qubits on the right. When we measure a qubit it jumps to a new state and we get a bit output.
We can do everything with a quantum circuit that we can do with a classical circuit, so quantum computations contain all classical ones. Quantum circuits enable us to put qubits into a superposition
of states and to entangle them. These are two things that we cannot do with classical bits. So we have more operations available to us, but are these useful?
Quantum algorithms
All quantum gates are invertible. A consequence of this is that up until we make the measurements of qubits, the whole computation is invertible. The entire computation can be represented by one
matrix. This matrix can be thought of as giving another way of ‘viewing’ the initial problem - analogous to a change of coordinates. To get quantum speedup there has to be some structure in the
initial problem that becomes apparent with one of these quantum viewpoints that is not available using classical ones.
We are far from the description given in many popular accounts where quantum computing is described as being like parallel computing. In many descriptions, it sounds as though all NP problems can be
solved in polynomial time on quantum computers. This is far from the case. In fact, it is not initially obvious that there are any problems that can be solved more quickly using a quantum computer.
David Deutsch in a landmark paper showed that there is one!
Deutsch, in 1985, showed that a certain problem could be solved more quickly using a quantum algorithm than using any classical one. The problem was highly contrived, designed specifically to show
that quantum speedup was real - at least in this one case - it was not meant to be a practically useful algorithm. However, this was the first step in showing that quantum computers might have some
advantages over classical ones.
Another major step forward was in 1994 when Peter Shor published his algorithm. This algorithm showed how quantum speedup could be used to factor products of large primes. This has practical
consequences. He showed that many of the standard internet encryption techniques (RSA, for example) could be easily broken once quantum computers could be built. This paper spurred interest in both
designing an actual quantum computer and in designing new encryption techniques that could withstand attacks from quantum computers.
Peter Shor also showed that you could also correct errors in quantum computations. This has tremendous practical applications. All known ways of physically representing qubits and manipulating them
are error prone. Quantum computers are often cooled to extremely low temperatures and shielded from electromagnetic fields, but it is hard to stop errors from creeping in. Shor designed an extremely
ingenious method of correcting errors in certain cases. This showed that it might be possible to actually build a physical quantum computer.
The future
Chemistry at its most basic consists of quantum phenomena. Simulating quantum phenomena using quantum computers seems to make a lot of sense and looks as though it has a lot of promise. Secure
communication is another area in which there has been substantial progress in recent years. There is talk of the quantum internet. However, the major change that I am most excited about, and is
almost here, is in who will have access and understand quantum computing. Up until now it has been the realm of the specialist with graduate degrees. That is going to change.
Quantum computing involves quantum phenomena - superpositions and entanglement - that are not things we experience in our daily lives and so cannot easily describe using words. However, they can be
described accurately using mathematics. Many of the pioneers of the subject come from backgrounds in quantum mechanics and are comfortable with sophisticated mathematical ideas. However, the basic
mathematics needed for quantum computing is accessible to anyone who is comfortable with high school algebra.
IBM has recently put a quantum computer in the cloud. It has a nice graphical interface and is free to use. It only has five qubits, so it is not going to solve large practical problems, but it is
the ideal size to play with. I’ve posted a very short guide to using the IBM Quantum Computer.
It is still early in the history of quantum computing, but we have now reached the point when we can all understand the theory and start to use our first quantum computer. | {"url":"https://www.bcs.org/articles-opinion-and-research/quantum-computing-in-10-minutes/","timestamp":"2024-11-13T21:54:15Z","content_type":"text/html","content_length":"93864","record_id":"<urn:uuid:9bbe3cdd-1776-4e21-9a2c-52b9225c26ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00334.warc.gz"} |
seminars - On the finiteness of strictly regular quaternary quadratic forms
A (positive definite integral) quadratic form is called strictly regular if it primitively represents all positive integers which are primitively represented by its genus.
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Voltage Divider on ADC Differential Pi - AB Electronics UK
Voltage Divider on ADC Differential Pi
The ADC Differential Pi is an Analogue to Digital converter for the Raspberry Pi
Posted by:
When using the ADC Differential Pi, do you have any suggestions for reading voltages outside of -2.048 - 2.048? Is it possible to add a voltage divider for examplke?
Posted by:
Hi Jon
There are two methods you can use to reduce the voltage level to work on the ADC Differential Pi.
You can use a voltage divider with the positive side going through the R1 resistor before connecting to the ADC + input and the negative connecting directly to the ADC - input. The R2 resistor
connects ADC + and ADC -. We have a calculator you can use to get the values for R1 and R2 based on your input voltage.
Resistor Voltage Divider Calculator
The second method is to use a fully-differential amplifier. An op-amp circuit is more complicated than a resistor divider but it does have the advantage of a higher impedance between the + and -
inputs. Texas Instruments has a good article explaining how to use fully-differential amplifiers at
Posted by:
Hi.... that appears reasonably agent yes. So the issue is with respect to the differential nature of the voltage perusing I'm attempting to take - the tall end of this perusing must be underneath 5v
add up to (and here is 11.1v aprox.). I'm understanding that now. So, would the arrangement here be the swapping the arrange of the resistors (and the ADC examined tests) so that the 11.1v to begin
with has got to pass through the bigger resistor?
Posted by:
With an 11.1v input, you would need resistor values of 10K for R1 and 2.2K for R2 which would give you an output voltage of 2.002V.
With this voltage divider setup, you can have a positive or a negative input voltage. A -11.1V input would give you an output voltage of -2.002V.
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Lecture 04 Large Scale Fading (2)
Lecture #04:
Large Scale Fading
Dept of Communication Engineering
Faculty of Electrical and Electronic Engineering
Universiti Tun Hussein Onn Malaysia, Johor
Review on Radio Wave Propagation
Free Space Propagation Model
Relating Power to Electric Field
Basic Propagation Mechanisms
Ground Reflection (Two Ray)
Fresnel Zone Geometry
Knife Edge Diffraction Model
Practical Link Budget Design Using Path Loss Models
Log distance
Log normal
Outdoor Propagation Models
Model Okumura
Model Hata
Indoor Propagation Models
Small-scale and large-scale fading
Figure 4.1 Small-scale and large-scale fading.
Review On Radio Wave Propagation
Propagation in mobile radio follows the basic propagation
theory, with some addition to compensate for moving
target (mobile).
Wave propagation mechanism often attributed to
reflection, diffraction and scattering.
Model such as Free Space Loss Model, Plane Earth
Propagation Loss Model, Diffraction Model is valid to some
extend of propagation scenario.
Most cellular system operate where no direct line of sight
path is available, causing severe diffraction loss and
Other factors that need to be considered in mobile
propagation are fading, Doppler effect, signal delay and
indoor propagation.
Some corrections is needed for the basic propagation
model to be accurately represent the real propagation that
took place between the transmitter and the receiver.
Free Space Propagation Model
The transmitter-receiver (T-R) path is clear from any obstruction (line
of sight propagation).
Mostly used for microwave link and satellite communication link.
Received power decays as a function of T-R distance raised to
some power (power law function).
Given by Friss free space equation,
is the system loss factor not related to propagation.
Path loss,
10 log
10 log
Note : You should be
able to derive the Friss
transmission equation.
Only valid when the mobile is outside the Near Field region (or in
the Far Field region, the Fraunhofer region)
Franhoufer distance,
- largest linear physical
dimension of the
Power and Electric Field
Power flux density, or simply power density is given as
The intrinsic impedance,
120 (377).
where | | is the magnitude of the radiating portion of the
electric field in the far field.
Received power,
Example 4.1
Find the far field distance for an antenna with maximum
dimension of 1m and operating frequency of (a) 900 MHz
and (b) 1800 MHz.
An array of patch antennas whose dimension is 40cm by
40 cm is used at the base station. Find the far field
distance of the antenna.
Example 4.2
Transmitter power is 50W, express the power in dBW and
dBm. The power is then applied to a unity gain antenna
with 900 MHz carrier frequency, find received power at
(a) 100 m and (b) 10km from the antenna. Also assume
unity gain for the received antenna
Ground Reflection (Two-Ray) Model
FSPL is inaccurate when used alone as usually no single
direct path from base station and mobile unit.
GRM or plane earth loss model is based on geometric
optics, and considers both the direct path and the
reflected path between transmitter and receiver.
Assume flat earth with T-R separation of few km.
Total received E-field (
) is the result of two rays, the
line of sight component (
) and ground reflected
component ( ).
Solving for received power yields
Classical 2-ray ground bounce model
Example 4.4
The mobile is located 5 km away from a base station and
uses a vertical quaterwave monopole antenna with a
gain of 2.55dB to receive cellular signals. The E-field at 1
km from the transmitter is measured to be 10-3 V/m.
Carrier frequency is 900 MHz. Find
(a) the length and the effective aperture of the receiving
(b) received power at the mobile using the two-ray
ground reflection model assuming the height of the
transmitting antenna is 50 m and the receiver is 1.5 meter
above ground.
It is redistribution of energy within a wavefront as it passes
the sharp edges of an object or slits
The phenomenon that allows light or radio waves to
propagate around corners
We used the Huygen’s Principle to explain this
Huygen’s Principle states that:
“Every point on a given spherical wavefront can be
considered as a secondary point source of EM waves from
which other secondary waves (wavelets) are radiated
Single-Knifed Edge
Figure shows how plane wavefronts impinging on the edge
from the left become curved by the edge so that, deep
inside the geometrical shadow region, rays appear to
emerge from a point close to the edge, filling the shadow
region with diffracted rays
Huygen’s principle can be applied in mathematical form to
predict the actual field strength which is diffracted by the
Contributions from an infinite number of secondary sources
in the region is summed, paying due regard to their relative
amplitude and phases
Final result can be expressed as propagation loss, which
expressed the reduction in the field strength due to the knife
edge diffraction in decibels
20 log
20 log
Fresnel diffraction geometry
Figure 4.12 Illustration of Fresnel zones for different knife-edge diffraction scenarios.
Knife edge diffraction attenuation : (-) exact (--) large approximation
Another significant value is
Received power reduced by 4 when the knife edge is situated
exactly on the direct path between the transmitter and
The parameter can be expressed in terms of the
geometrical parameters defined in Figure as
where ’ is the excess height of the edge above the straight
line from source to the field points
For most practical case, , ≫ , so diffraction parameter
can be approximated in terms of the distances measured
along the ground
Approximate solution using parameter
Lee (2006);
is provided by
20 log
20 log
0.5 exp
20 log
20 log
Another useful way to consider knife edge diffraction is in
terms of the obstruction of Fresnel zones around the direct
The th Fresnel zone is the region inside an ellipsoid defined
by the locus of points where the distance (
) is larger
than the direct path between the transmitter and the
) by half-wavelengths
receiver (
Hence the radius of the nth zone
If we assume that
is given by applying the
, then to a good
Fresnel zones
0.6 x first Fresnel zone clearance defines significant obstruction
Fresnel zones can be thought of as containing the propagated
energy in the wave
Contribution within the first zone are all in phase, so any absorbing
obstruction which do not enter this zone will have little effect on the
received signal
Fresnel zone clearance ( / ) can be expressed in terms of the
diffraction parameter as follows
When the obstruction occupies 0.6 x the first Fresnel zone, the
parameter is then approximately -0.8. Obstruction loss is the 0 dB.
This clearance is often used as a criterion to decide whether an
object is to be treated as a significant obstruction
If this region is kept clear then the total path attenuation will be
practically the same as in the unobstructed case
Multiple knife-edge diffraction
Example 4.5
Compute the diffraction loss where wavelength is 1/3 m,
, 2
and (a) 25 m (b) 0 m (c) – 25 m.
Compare value with approximated value given by Lee.
Identify in which Fresnel zone is the tip of obstruction lies
for each cases.
(Hint: , excess path length can be find by
and we need to find
which satisfy the relation
Example 4.6
Given the following scenario:
Tx height = 50m, Rx height = 25 m, Tx – Rx distance = 12 km
Knife edge height = 100 m
Distance from Tx to knife-edge = 10 km
f = 900 MHz
(a)Knife edge loss
(b)Height of obstacle for 6 dB loss
When radio wave impinges on a rough surface (lamp
posts, trees), reflected energy is spread out (diffuse) in all
directions due to scattering which may provide
additional radio energy at the receiver.
Surface roughness is tested with Rayleigh criterion which
defines a critical height,
of surface protuberances for
a given angle of incidence , given by
8 sin
Path Loss Model
Detail path loss model hard to factor in overall system
Most important characteristic is power fall-off with
Radio propagation models
Analytical models
Empirical models
Predict large scale coverage for mobile systems
Estimate and predict SNR
Log Distance Path Loss Model
Average path loss for an arbitrary T-R separation is expressed as a
function of distance using a path loss exponent,
10 log
indicates the rate at which the path loss increases with
distance, 0 is the close-in reference which is determine from
measurements close to the transmitter, and d is the T-R separation
distance. The bars indicate the ensemble average of all possible
path loss values for a given value of .
Free space reference value chosen according to the
propagation environment, i.e. 1 km for large cell, or 1 or 100 m for
smaller cells.
Typical large-scale path loss
Log Normal Shadowing
Log distance model fails to address the fact that
environmental clutter may be vastly different at two
location having the same T-R separation.
Measured signals vastly different from the average value.
Path loss at any value d is random and distributed lognormally (normal) about the mean distance dependent
value that is
is a zero mean Gaussian distributed random variable
(in dB) with standard deviation (also in dB).
Log normal distribution describe the random shadowing
effect over large number of measurement locations
which have the ame T-R separation, but have different
levels of clutter on the propagation path, refers to log
normal shadowing.
Close in reference d0, path loss exponent n, and the
standard deviation statistically describe the path loss
model for an arbitrary location having specific T-R
Values of n and are computed from measured data,
the difference between measures and estimated path
loss is minimized in a mean square error sense over wide
range of measurement locations and T-R separations.
Rayleigh and Log-normal
Example 5.7
4 received power
measurement were taken
at distance 100m, 200m,
1km and 3km and given in
the table where it is assume
that these values follow log
normal shadowing model
where d0 = 100m.
Distance from Tx Received Power
100 m
0 dBm
200 m
– 20 dBm
1000 m
– 35 dBm
3000 m
- 70 dBm
Find the minimum mean
square error (MMSE)
estimate for the path loss
exponent, n
Calculate the standard
deviation about the mean
Estimate the received
power at d = 2km using the
resulting model
Predict the likelihood that
the signal level at 2 km will
be greater than -60dBm
Predict the percentage of
area within a 2 km radius
cell that receives signals
greater than – 60 dBm,
given the result in (d)
MMSE estimate may be found using the following
method. Let pi be the received power at a distance
and let ̂ be the estimate for using the / 0 path loss
model. The sum of square errors between the measure
and estimated values is given by
The value of n which minimizes the mean square error
can be obtained by equating the derivative of
zero, and then solving for .
Outdoor Propagation Model
Radio transmission often takes place over irregular terrain
Terrain profile may vary from simple curved to a
mountainous profile.
Trees, buildings and other obstacles cannot be
neglected and must be taken in the estimation.
The purpose of the outdoor model is to predict the
average received signal strength at a given distance
from the TX, as well as the variability of the signal strength
in close spatial proximity to a particular location.
Computer based models:
Longley-Rice model
Durkin’s model
Measurement model
Okumura model
Empirical model
Hata model
PCS extension and wideband PCS microcell models
Walfish and Bertoni Model
Longley-Rice Model (LR, ITM)
In 1965 & 1968, Rice and Longley propose the model to be
applicable to point-to-point communication systems in the
frequency range from 40MHz to100GHz.
In 1978, Longley-Rice model is also available as a computer
program to calculate large-scale median transmission loss
relative to free space loss over irregular terrain for
frequencies between 20MHz and 10GHz
Longley Rice has been adopted as a standard by the FCC
Many software implementations are available commercially
Includes most of the relevant propagation modes [multiple
knife & rounded edge diffraction, atmospheric attenuation,
tropospheric propagation modes (forward scatter etc.),
precipitation, diffraction over irregular terrain, polarization,
specific terrain data, atmospheric stratification, different
climatic regions, etc. etc. …]
Operate in two mode:
Point-to-Point mode prediction: A detail terrain path profile
is available, the path specific parameters can be easily
Area mode prediction: Once the terrain path profile is not
available, the Longley-Rice method provides techniques to
estimate the path-specific parameters.
Does not providing a way of determining corrections due to
environmental factors in the immediate vicinity of the
mobile receiver
No consideration of correlation factors to account for the
effects of buildings and foliage
No consideration of multipath
Durkin’s Model
In 1969 & 1975, Durkin propose a computer simulator for
predicting field strength contours over irregular terrain.
Durkin path loss simulator consists of two parts:
Access a topographic database of a proposed service area
Reconstruct the ground profile information along the radial
joining the TX to RX.
the receiving antenna receives all of its energy along that
No multipath propagation is considered
Simply LOS and diffraction from obstacles along the radial, and
excludes reflections from other surrounding objects and local
Refer to Rappaport, pp 146.
Okumura Model
Most widely used model in urban areas, obtained by
extensive measurements, no analytical explanation – based
on measurement in the city of Tokyo, Japan.
Represented by charts (curves) giving median attenuation
relative to free space attenuation
Valid under:
Frequency band: 150-1920 MHz
T-R distance: 1-10 km,
= 1m to 3m
BS antenna height: 30-1000 m
Quasi-smooth terrain (urban & suburban areas)
among the simplest & best for in terms of path loss accuracy
in cluttered mobile environment
disadvantage: slow response to rapid terrain changes
common std deviations between predicted & measured path
loss 10dB - 14dB
Okumura developed a set of curves in urban areas with
quasi-smooth terrain
effective antenna height:
base station hte = 200m
mobile: hre = 3m
gives median attenuation relative to free space (Amu)
developed from extensive measurements using vertical
omni-directional antennas at base and mobile
measurements plotted against frequency
Estimating path loss using Okumura Model
1. determine free space loss, Amu(f,d), between points of
2. add
and correction factors to account for
L50 = 50% value of propagation path loss (median)
LF = free space propagation loss
Amu(f,d) = median attenuation relative to free space
G(hte) = base station antenna height gain factor
G(hre) = mobile antenna height gain factor
GAREA = gain due to environment
have been plotted for wide range of
antenna gain varies at rate of 20dB per decade or 10dB per
20 log
10 log
20 log
model corrected for
= terrain undulation height
isolated ridge height
average terrain slope
mixed land/sea parameter
Okumura and Hata’s model
Hata Model
empirical model of graphical path loss data from
predicts median path loss for different channels
valid over UHF/VHF band from 150MHz-1.5GHz
charts used to characterize factors affecting mobile land
standard formulas for approximating urban propagation loss
correction factors for some situations
compares closely with Okumura model as
mobile systems
50th % value (median) propagation path loss (urban)
frequency from 150MHz-1.5GHz
hte, hre
Base Station and Mobile antenna height
correction factor for hre , affected by coverage area
Tx-Rx separation
26.16 log
– 13.82 log
represents fixed loss – approximately 2.6 power law
dependence on
dependence on antenna heights is proportional to
6.55 log
represents path loss exponent, worst case ≈ 4.5
44.9 – 6.55log
– 13.82log
Mobile Antenna Height Correction Factor for Hata Model
(1.1log10 fc - 0.7)hre – (1.56log10 fc - 0.8)dB
Medium City
8.29(log10 1.54hre)2 – 1.1 dB
Large City (fc 300MHz)
3.2(log10 11.75hre)2 –
Large City (fc > 300MHz)
4.97 dB
Hata Model for Rural and Suburban Regions
represent reductions in fixed losses for less demanding environments
L50 (dB)
L50 (urban) - 2[log10 (fc/28)]2 – 5.4
Suburban Area
L50 (urban) - 4.78(log10 fc)2 - 18.33log10 fc - 40.94
Rural Area
Valid Range for Parameters
150MHz < fc < 1GHz
30m < hte < 200m
1m < hre < 10m
1km < d < 20km
Propagation losses increase
with frequency
in built up areas
hte = 30, hre = 2
-- f = 1500MHz
-- f = 900 MHz
-- f = 700MHz
hte = 30, hre = 2, f =
-- large city
-- small to medium sized
PCS Extension to Hata Model
European Co-operative Scientific & Technical (EUROCOST)
formed COST-231, extend Hata’s model to 2GHz
33.9 log
– 13.82 log
defined in the Hata model
For medium sized city, CM = 0 dB
Metropolitan centre, CM = 3 dB
Indoor Propagation Model
The indoor environment differs from outdoor
smaller Tx-Rx separation distances than outdoors
higher environmental variability for much small Tx-Rx separation
conditions vary from: doors open/closed, antenna position,
variable far field radiation for receiver locations & antenna types
strongly influenced by building features, layout, materials
Dominated by same mechanisms as outdoor propagation
(reflection, refraction, scattering)
Classified as either LOS or OBS, with varying degrees of clutter
Some key models are
Partition Losses – Same Floor
Partition Losses – Different Floor
Log-distance path loss model
Ericsson Multiple Breakpoint Model
Attenuation Factor Model
Partition losses
Partition Losses (same floor)
Variety of obstacles and partitions
Hard partitions – part of the building structure
Soft partitions – do not span to the ceiling
Vary widely in their physical and electrical characteristic
Database for various type of partitions, here or refer to
Rappaport pp 158-159.
Partition Losses between Floors
Structural dimension, materials, type of construction,
Floor attenuation factor, FAF for three buildings in San
Log Distance Path Loss Model
n depends on surroundings and building type
= normal random variable in dB having std deviation
identical to log normal shadowing mode, typical value in Table aa
PL(dB) PL(d0 ) 10n log
Ericsson Multiple Breakpoint Model
measurements in multi-floor office building
uses uniform distribution to generate path loss values between
minimum & maximum range, relative to distance
4 breakpoints consider upper and lower bound on path loss
assumes 30dB attenutation at d0 = 1m
accurate for f = 900MHz & unity gain anntenae
provides deterministic limit on range of path loss at given distance
Ericsson’s indoor model
Attenuation Factor Model
includes effect of building type & variations caused by
reduces std deviation for path loss to 4dB
std deviation for path loss with log distance model 13dB
PL(d ) (dB) PL(d0 ) (dB) 10nSF log FAF(dB) PAF(dB)
nSF = exponent value for same floor measurement – must be
FAF = floor attenuation factor for different floor
PAF = partition attenuation factor for obstruction
encountered by primary ray tracing
Replace FAF with nMF = exponent for multiple floor loss
PL(d ) (dB) PL(d0 ) (dB) 10nMF log PAF(dB)
decreases as average region becomes smaller-more
Building Path Loss obeys free space + loss factor ()
loss factor increases exponentially with d
(dB/m) = attenuation constant for channel
PL(d ) (dB) PL(d0 ) (dB) 20log d FAF (dB) PAF(dB)
Measured indoor path loss
Measured indoor path loss
Measured indoor path loss
Devasirvatham’s model
Path Loss Exponent & Standard Deviation for Typical Building
σ (dB)
number of points
same floor
through 1 floor
through 2 floor
through 3 floor
Signal Penetration into Buildings
no exact models
signal strength increases with height
lower levels are affected by ground clutter (attenuation
& penetration)
higher floors may have LOS channel stronger incident
signal on walls
RF Penetration affected by
height within building
antenna pattern in elevation plain
Penetration loss
decreases with increased frequency
loss in front of windows is 6dB greater than without windows
penetration loss decreases 1.9dB with each floor when < 15th
increased attenuation at >15 floors – shadowing affects from
taller buildings
metallic tints result in 3dB to 30dB attenuation
penetration impacted by angle of incidence
Penetration Loss vs Frequency for two different building
Ray Tracing & Site Specific Models
rapid acceleration of computer & visualization capabilities
SISP – site specific propagation models
GIS – graphical information systems
support ray tracing
augmented with aerial photos & architectural drawings
A BS of 30m height is operating at 900 MHz and transmits 20W
power. The transmitter and receiver antenna gains are 6dB and
2dB respectively. A MS of 2m height is located at 5km from the BS.
If other losses is 5dB and fading =6dB due to log-normal fading,
compare the minimum power received by the MS in dBm if the
following propagation models are used:
Free space propagation loss, FSPL
Plane earth propagation loss, PEPL
log-distance with do=1km, =4 and PL(do)=FSPL, and
diffraction model if an obstacle of 30m height is located at 2km from
the BS
Comment on the practical minimum power received by the MS if an
obstacle stated in (d) exists.
Write a MATLAB program to plot the path loss for each model from
1km to 5 km. | {"url":"https://studylib.net/doc/25242053/lecture-04-large-scale-fading--2-","timestamp":"2024-11-07T06:28:54Z","content_type":"text/html","content_length":"76496","record_id":"<urn:uuid:f5e3fb3c-c22b-4bb0-b9f3-9529021e6c62>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00616.warc.gz"} |
Exponential Smoothing and Inventory Management - Inventory Management Advice
The technique described here is called “exponential smoothing”. It is a simple method of updating an average as new data becomes available. It is the basis of most techniques used for forecasting
customer demands. It was discussed briefly in the article entitled “Demand Rate Estimation”. In this article, it will be illustrated in relation to demand rate estimation. However, its use is not
restricted to that.
Weightings Given to Historical Data
Click on the appropriate link below depending on what spreadsheet software you are using:
Open document format (for use with modern spreadsheet software)
Excel 97 format (for use with pre-2013 versions of Microsoft Excel)
You will then see a spreadsheet showing the weightings given to the demand in each period. You can download it and open it with your own spreadsheet software. Alternatively, you can view it online by
clicking on “Open”. If the periods are months then the spreadsheet will show the weightings given to the demand last month, the second to last month, the third to last month, etc. The weighting given
to the demand in the most recent period is called the “smoothing constant” or “smoothing factor” or “smoothing coefficient” (α). The periods can be any length of time, e.g, months or weeks or days.
Try changing the smoothing constant in the yellow cell (Cell B3). Fig.1 below shows what you will see for a smoothing constant of 0.1:
Fig.1 – Exponential smoothing weightings with a smoothing constant of 0.1
How it works
Suppose that it is to be used to average the monthly demands for an item and that average is to be updated each month. That average can be treated as a forecast if the demand rate is reasonably
The technique is as follows: At the end of each month, the average demand per month is adjusted taking into account the most recent month’s demand. Suppose, for example, the smoothing constant is 0.2
(i.e. 20%). Then 20% of the weighting is given to the demand in the most recent month and 80% of the weighting is given to the old average, i.e.
New average = α(latest month’s demand) + (1 – α)(old average)
This formula can be applied using this calculator.
If the old average is 30, the latest month’s demand is 40 and the smoothing coefficient is 0.2 then
New average = 0.2 x 40 + (1 – 0.2) * 30
= 0.2 x 40 + 0.8 x 30
= 8 + 24
= 32
Here is another way to look at it: For α=0.2 (20%), the average is adjusted by 20% of the error in using the old average as a forecast of the latest month’s demand (i.e. by 20% of the difference
between the new demand and the old average).
New average = old average + α(error in the forecast of the latest month’s demand)
i.e. New average = old average + α(latest month’s demand – old average)
which is equivalent to the first formula.
Using the above data,
New average = 30 + 0.2 x (40 – 30)
= 30 + 0.2 x 10
= 30 + 2
= 32
In this case, if the old average had been used as a forecast of the latest month’s demand then that forecast would have had an error of 40-30, i.e. 10. The average is then increased by 20% of that
forecasting error to give a new average of 32.
The exponential smoothing formula can be applied using this calculator.
Selection of an Appropriate Smoothing Constant
If the smoothing constant is small then the estimated demand rate will take a long time to catch up with changes in the demand rate. If it is high then the demand rate will tend to be sensitive to
random fluctuations in demand. Consequently, choice of a smoothing constant involves a compromise. Ideally, the smoothing constant should be set by means of simulation of the effects on your overall
service level and overall investment in inventory.
The following formula will usually give a reasonably appropriate smoothing constant (α):
α = 1/(4L + 1)
where L is the expected lead time in the event of the inventory position falling below the reorder point just after a reorder review. The above formula can be applied using this calculator. The
formula is compromise between keeping up with a changing demand rate and being overly sensitive to random fluctuations in demand. The lead time and reorder review period (time between reorder
recommendations reports) should both be in months if the exponential smoothing is applied to monthly demands, weeks if weekly demands are used, etc.
Suppose that the lead time, ignoring the reorder review period, is two months and that reordering recommendations are produced monthly. Suppose that the total of the other components of the lead time
is one month. Then an appropriate smoothing constant is given by
α = 1/(4(2 + 1 + 1) + 1)
= 1/17
= 0.058
The (2 + 1 + 1) in the above example is the supplier lead time (2 months) plus the reorder review period (1 month) plus the other components of the lead time (1 month).
Try using the calculator with the above worked example. If a month is treated as being 30 days then the answer will be 0.0667.
Don’t be tempted to use a higher smoothing constant to keep up with trends. There are better methods of dealing with that and one of them will be discussed in a later article.
Investigate the Effects With Your Own Data
The simulator described in the article entitled “An Educational Inventory Management Simulator” can be used to investigate the effects when used with your own data. Try entering your own demand
history into that simulator for at least one of your fast moving items and at least one of your infrequently moving items. In each case, try more than one smoothing constant including the one
suggested by the above-mentioned calculator and one which is considerably higher than that. Note the large orders which tend to result from relatively high demands and note how that problem is
exacerbated by use of a high smoothing constant.
Effects of Using a Smoothing Constant Which is Too High
Fig. 2 below illustrates the effect of using a smoothing constant of 0.5 if the lead time is three months.
Notice how erratic the estimated mean demand is in spite of the fact that the simulation was carried out with a constant demand rate (constant mean demand per month).
The “inventory level” is the stock on hand minus customer back orders so yellow points above the axis indicate stock on hand and those below the axis represent customer back orders.
Notice the excessively high order quantities in months 7, 26 and 30 resulting from the relatively high demands in months 5, 24 and 28. Not all peaks in demand cause orders to be placed on the
supplier. This is because ordering only takes place when the inventory position falls below the reorder point. For example the relatively high demands in periods 21 and 37 do not result in ordering
because at those times the inventory levels are already high as a result of earlier over-ordering. With regard to the shortages early on, see the section entitled “Initialisation” below. These
shortages could be prevented by the means of safety stock but that would make the problems of over-ordering and of excessive stocks even greater.
Fig.3 below shows how much more stable the demand rate estimate is with a smoothing constant of 0.1. Notice however that there is still the problem of over-ordering after relatively high demands. The
reasons are given in the article entitled “Evaluating Forecasting Algorithms“.
Fig.2. – Effects of a high smoothing constant (0.5)
Fig.3 – Effects of reducing the smoothing constant to 0.1
In Figures 2 and 3,note the prolonged shortages when the item is new. This problem can be dealt with by manual entry of the initial average demand per month.
Benefits of Exponential Smoothing
• It uses all of the demand history, even that which is no longer stored in the system.
• It will not result in zero demand rate estimates if there has ever been any demand.
• The greatest weightings are given to the most recent demands.
• It facilitates manual intervention when the demand rate is expected to change.
Shortcomings of Exponential Smoothing
• As with most demand rate estimation techniques and forecasting techniques, relatively high demands tend to result in over-ordering.
• Selection of a smoothing constant involves a compromise between responsiveness and stability.
• The optimal smoothing constant is not the same for all items.
• The algorithm requires modification if there are trends or seasonal effects.
The initial old average demand per period needs to be set. If the exponential smoothing is implemented when there is already some demand history available then all of that history should be used in
order to ensure that the initial average is not zero if there has ever been any demand.
Use whatever relevant information you can obtain from the supplier and also any other relevant data. In a multiple store operation, if an item is introduced into one store when there is already some
history of the item in other stores, then appropriate use should be made of that history.
If no useful information is available then the smoothing “constant” (better called the “smoothing factor”) should initially be fairly high but only for a short time. | {"url":"https://www.inventoryadvice.com/721-2/","timestamp":"2024-11-06T02:27:35Z","content_type":"text/html","content_length":"185439","record_id":"<urn:uuid:544fbfb0-5450-4551-b06b-96f65cd33f97>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00387.warc.gz"} |
.NET C# Double Infinity and Euler's Number/Constant.NET C# Double Infinity and Euler's Number/Constant
We all know that the Euler’s Number or Constant can be expressed this way,
And its value is around:
Since the .NET C# double type supports Infinity now, both positive and negative which look great, let’s figure it out programmatically with the C# computer language as follows.
double positiveInfinity = double.PositiveInfinity;
double eulersConstant = Math.Pow((1 + 1 / positiveInfinity), positiveInfinity);
System.Windows.Forms.MessageBox.Show(string.Format("Euler's Constant = {0} ", eulersConstant));
What will the code yield?
It is, however, out of expectations of people with a bit math sense.
What do you think about it?
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Getting answer even when all cells are blank
IFERROR(INDEX({Req Main Sort Summary}, MATCH($[LIB ID]@row, {LIB ID}, 0), MATCH([Script Name]$1, {Req Main Sort Header})), "")
Two sheets -
First is the primary sort with Script Numbers, Names and other key information
Second is the primary Test Build / Tracker sheet where End2End tests are built from the sort sheet.
There are 300+ tests to build up integrated scenarios from. The second sheet also tracks defects, status and other critical test metrics.
There should only be data in a Script name field if there is data in the LIB ID field, However, unless I clear the field of the formula there is a random (last script) filled down as far as the
formula is filled down. This is very confusing to the user as this needs to be dynamic.
The Defect Index and Match calculation does something similar, though it is going out and simply pulling the first thing it finds.
Any suggestions as to what I am doing wrong would be most helpful.
Best Answer
• In the second match, the search type is not specified so it will use the default type, which is 1 (see below). Does changing this to 0 help?
IFERROR(INDEX({Req Main Sort Summary}, MATCH($[LIB ID]@row, {LIB ID}, 0), MATCH([Script Name]$1, {Req Main Sort Header},0)), "")
For the optional search_type argument:
1: (The default value) Finds the largest value less than or equal to search_value (requires that the range be sorted in ascending order)
0: Finds the first exact match (the range may be unordered)
-1: Finds the smallest value greater than or equal to search_value (requires that the range be sorted in descending order)
• In the second match, the search type is not specified so it will use the default type, which is 1 (see below). Does changing this to 0 help?
IFERROR(INDEX({Req Main Sort Summary}, MATCH($[LIB ID]@row, {LIB ID}, 0), MATCH([Script Name]$1, {Req Main Sort Header},0)), "")
For the optional search_type argument:
1: (The default value) Finds the largest value less than or equal to search_value (requires that the range be sorted in ascending order)
0: Finds the first exact match (the range may be unordered)
-1: Finds the smallest value greater than or equal to search_value (requires that the range be sorted in descending order)
• That is exactly what was missing. Thank you
• That is exactly what was missing, thank you. Solved the problem in both formula's.
• Great! I'm glad you have that sorted.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/113999/getting-answer-even-when-all-cells-are-blank","timestamp":"2024-11-12T12:58:12Z","content_type":"text/html","content_length":"415942","record_id":"<urn:uuid:47dd85bd-4a79-48ce-8122-372a796296c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00244.warc.gz"} |
What is a confidence interval?
At first glance, a confidence interval is simple. If we say [3, 4] is a 95% confidence interval for a parameter θ, then there’s a 95% chance that θ is between 3 and 4. That explanation is not
correct, but it works better in practice than in theory.
If you’re a Bayesian, the explanation above is correct if you change the terminology from “confidence” interval to “credible” interval. But if you’re a frequentist, you can’t make probability
statements about parameters.
Confidence intervals take some delicate explanation. I took a look at Andrew Gelman and Deborah Nolan’s book Teaching Statistics: a bag of tricks, to see what they had to say about teaching
confidence intervals. They begin their section on the topic by saying “Confidence intervals are complicated …” That made me feel better. Some folks with more experience teaching statistics also find
this challenging to teach. And according to The Lady Testing Tea, confidence intervals were controversial when they were first introduced.
From a frequentist perspective, confidence intervals are random, parameters are not, exactly the opposite of what everyone naturally thinks. You can’t talk about the probability that θ is in an
interval because θ isn’t random. But in that case, what good is a confidence interval? As L. J. Savage once said,
The only use I know for a confidence interval is to have confidence in it.
In practice, people don’t go too wrong using the popular but technically incorrect notion of a confidence interval. Frequentist confidence intervals often approximate Bayesian credibility intervals;
the frequentist approach is more useful in practice than in theory.
It’s interesting to see a sort of détente between frequentist and Bayesian statisticians. Some frequentists say that the Bayesian interpretation of statistics is nonsense, but the methods these crazy
Bayesians come up with often have good frequentist properties. And some Bayesians say that frequentist methods, such as confidence intervals, are useful because they can come up with results that
often approximate Bayesian results.
4 thoughts on “What is a confidence interval?”
1. Does it mean that : [3,4] has 95% of chance to contain the real value of the paremeter ?
I don’t understand the approximation of baysian interval by frequentist one.
Last point : thank you very much for your blog.
2. Julien: The interpretation you give for a confidence interval is the one nearly everyone believes but isn’t correct. The precise definition is complicated and unintuitive.
3. Hi John,
I still don’t quite understand. Let’s say I have a test statistic and build an empirical 95% confidence interval. Is it true to say I have 95% confidence my interval contains the true statistic?
4. Alex; it is true to say that in repeated sampling your confidence interval will contain the true parameter (what we are interested in) 95% of the time.
(Under frequentist paradigm) confidence intervals change between samples, but the parameter is a fixed point i.e. it is a category error to talk about its probability. We can express
probabilities regarding the confidence intervals but not the parameter. Under the frequentist paradigm, all the probability statements express the frequencies of results of a given process in
repeated trials.
So I guess nothing really stops you from informally expressing this situation as “95% confidence my interval contains the true statistic” and nothing will noticeably go wrong if you live as if
that is the case but strictly speaking it is not a deduction you can justify. | {"url":"https://www.johndcook.com/blog/2009/02/03/what-is-a-confidence-interval/","timestamp":"2024-11-13T22:34:24Z","content_type":"text/html","content_length":"57409","record_id":"<urn:uuid:544db0e2-9aaa-457f-a65f-4ba0cc0b8799>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00497.warc.gz"} |
A bar chart is a chart with rectangular bars with lengths proportional to the values that they represent. One axis of the chart shows the specific categories being compared, and the other axis
represents a discrete value.
Used For
• Compare discrete data
• Show trends over time
• Change over time
• To express larger variations in data, how individual data points relate to a whole, comparisons, and ranking
Horizontal Bars
Horizontal bars are mostly used to show Nominal/Categorical datasets. These are generally datasets which can be arranged in any order. Sorting the data can be helpful to bring attention to lowest/
highest values.
Vertical Bars
Vertical bars are mostly used to show Ordinal/Sequential datasets. These are generally datasets which follow a natural progression or order. These show the change in values w.r.t. the progression/
Bar Chart
It presents datasets with rectangular bars with heights or lengths proportional to the values that they represent.
Clustered Bar Chart
It presents two or more data sets displayed side-by-side and grouped together under categories on the same axis. Note: Use Line chart to compare for trend analysis between categories. Guideline
Stacked Bar Chart
It presents larger category divided into smaller categories and their relations to the total.
It presents a grouped frequency distribution with continuous classes. It groups numeric data into bins, displaying the bins as segmented columns. They’re used to depict the distribution of a dataset:
how often values fall into ranges.
Note: Histogram only has vertical orientation.
Use Single Color for Single Data Set
If all the bars measure the same variable, make them all the same color. Different shades have no relevance to the data.
Show Nominal/categorical Data in Ascending or Descending Order
Sort data sets to make it easier to understand and visualize.
Make the Width of Each Bar About Twice as Wide as the Space Between Them | {"url":"https://design.innovaccer.com/data-visualizations/barChart/usage/","timestamp":"2024-11-14T06:40:28Z","content_type":"text/html","content_length":"369279","record_id":"<urn:uuid:321397d9-36f3-4963-8950-e31dde8b3eeb>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00771.warc.gz"} |
Large signal analysis of low-voltage BiMOS analog multipliers using Fourier-series approximations
(2000) Large signal analysis of low-voltage BiMOS analog multipliers using Fourier-series approximations. Proceedings of the National Science Council, ROC, Part A: Physical Sciences and Engineering,
24 (6). pp. 480-488.
Full text not available from this repository.
This paper discusses the large signal performance of BiMOS low-voltage analog multipliers. Fourier-series approximations are obtained for the transfer characteristics of the basic building blocks,
such as the BiMOS folded Gilbert multiplier cell, the squaring multiplier made from two cross-coupled emitter-coupled pairs and driven by an MOS quadritail, and the tripler made from cross-coupled
emitter-coupled pairs and driven by an MOS quarter-square multiplier. Using the Fourier-series approximations of the transfer functions of these basic building blocks, closed-form expressions are
obtained for the amplitudes of the harmonics and intermodulation products at the output of these analog multipliers when excited by multisinusoidal input signals. Using these expressions, comparison
between the large signal performance of these analog multipliers can be made and the parameters required for a predetermined performance can be determined. | {"url":"https://eprints.kfupm.edu.sa/id/eprint/1327/","timestamp":"2024-11-10T12:40:03Z","content_type":"application/xhtml+xml","content_length":"23701","record_id":"<urn:uuid:b7e8ffe9-d26e-48c8-bb03-7895fceebe1c>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00330.warc.gz"} |
Problems and Solutions to Accompany McQuarrie and Simon's Physical Chemistry
Problems and Solutions to Accompany McQuarrie and Simon's Physical Chemistry
By: Heather Cox
Publication date: February 1998
ISBN: 9780935702439
Title information
This manual is designed to complement McQuarrie and Simon’s new Physical Chemistry: A Molecular Approach by providing a detailed solution for every one of the more than 1400 problems found in the
Language: English
Publisher: University Science Books
1.The Dawn of Quantum Theory
Math Chapter A/Complex Numbers
2. The Classical Wave Equation
Math Chapter B/Probability and Statistics
3. The Schrodinger Equation and a Particle In a Box
Math Chapter C/ Vectors
4. Some Postulates and General Principles of Quantum Mechanics
Math Chapter D/ Spherical Coordinates
5. The Harmonic Oscillator and the Rigid Rotator: Two Spectroscopic Models
6. The Hydrogen Atom
Math Chapter E/ Determinants
7. Approximation Methods
8. Multielectron Atoms
9. The Chemical Bond: Diatomic Molecules
10. Bonding in Polyatomic Molecules
11. Computational Quantum Chemistry
Math Chapter F/ Matrices
12. Group Theory: The Exploitation of Symmetry
13. Molecular Spectroscopy
14. Nuclear Magnetic Resonance Spectroscopy
15. Lasers, Laser Spectroscopy, and Photochemistry
Math Chapter G/Numerical Methods
16. The Properties of Gases
Math Chapter H/Partial Derivatives
17. The Boltzmann Factor and Partition Functions
Math Chapter I/Series and Limits
18. Partition Functions and Ideal Gases
19. The First Law of Thermodynamics
Math Chapter J/ The Binomial Distribution and Stirling's Approximation
20. Entropy and the Second Law of Thermodynamics
21. Entropy and the Third Law of Thermodynamics
22. Helmholtz and Gibbs Energies
23. Phase Equilibria
24. Solutions I: Liquid-Liquid Solutions
25. Solutions II: Solid-Liquid Solutions
26. Chemical Equilibria
27. The Kinetic Theory of Gases
28. Chemical Kinetics I: Rate Laws
29. Chemical Kinetics II: Reaction Mechanisms
30. Gas-Phase Reaction Dynamics
31. Solids and Surface Chemistry
-Answers to Numerical Problems | {"url":"https://uscibooks.directfrompublisher.com/9780935702439","timestamp":"2024-11-10T23:40:57Z","content_type":"application/xhtml+xml","content_length":"42896","record_id":"<urn:uuid:07edfc22-b721-4802-97ac-790be9ba14e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00439.warc.gz"} |
[Solved] 1. Kaldor facts [50 points] Kaldor (1961) | SolutionInn
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Verified Expert Solution
1. Kaldor facts [50 points] Kaldor (1961) documented a set of stylized facts on the growth process of industrialized countries. We discussed these facts
1. Kaldor facts [50 points] Kaldor (1961) documented a set of stylized facts on the growth process of industrialized countries. We discussed these facts in lecture 2. Explain if and how the
Solow-Swan model can rationalize each of these facts. You are allowed to focus on the steady state equilibrium of the model. Every statement needs to be supported with a mathematical equation. You
can refer to the notes of lecture 5 and 6 for this. a. Output per worker Y/N grows at a rate that does not diminish. b. Physical capital per worker Kt/Nt grows over time at a constant rate. c. The
ratio of physical capital to output K/Y, is constant. In the following exercise, you are allowed to assume that the production function F(K, AN) satisfies the Inada conditions. A firm chooses capital
and labor to maximize its profits: max F(K, AN)-wit-r Kt Kt.Lt where we and r denote the wage and rental rate of capital, respectively. d. Derive the firm's first order conditions with respect to
capital and labor. Interpret these equa- tions. e. Under constant returns to scale, we can write Fi(K, AN) = F (1) and F(Kt, AtNt) = F2 (1). Show that the rate of return to capital r is constant in
the steady state equilibrium. f. Using the equality F (Kt, AtNt) = F2 (,,1), show that the wage rate w grows at the constant rate in the steady state equilibrium. g. Using your answers from 1c, le
and 1f, show that the shares of labor and physical capital in national income, w+N/Yt and r Kt/Y are constant in the steady state equilbrium. A(Y/N) h. The last fact by Kaldor (1961) states that the
growth rate of output per worker Y/N differs substantially across countries. Assuming that all countries are in their steady state equilibrium, what do we have to observe in the data to make the
Solow-Swan model consistent with Kaldor's observation? Did you see any evidence for this in class?
There are 3 Steps involved in it
Step: 1
Kaldors facts are a set of empirical observations about economic growth and the SolowSwan model provides a theoretical framework to understand these facts in the steadystate equilibrium Explanation
of ...
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Formulas of the sums of a number of natural numbers in integer degree
Sum of a natural number
This is a well-known formula, discovered by Gauss at the age of six.
The sum of the natural series where each element is raised to the second degree
The sum of the natural series where each element is raised to the third degree
The sum of the natural series where each element is raised to the fourth power
The sum of the natural series where each element is raised to the fifth power
The sum of the natural series where each element is raised to the sixth power
The sum of the natural series where each element is raised to the seventh degree
The sum of the natural series where each element is raised to the eighth power
The sum of the natural series where each element is raised to the ninth degree
The sum of the natural series where each element is raised to the tenth power
The sum of the natural series where each element is elevated to the eleventh degree
The sum of the natural series where each element is raised to the twelfth degree | {"url":"https://abakbot.com/algebra-en/ryad3-en","timestamp":"2024-11-12T10:45:07Z","content_type":"text/html","content_length":"18720","record_id":"<urn:uuid:884de607-7a30-43b6-92d9-1a70b5c14ffc>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00734.warc.gz"} |
Creating a Python Model for Bullet-Bullet Collision Dynamics
Written on
Chapter 1: Introduction to Bullet Collision Modeling
Whenever I come across something fascinating, I tend to show my appreciation by building a model. It's a ritual of mine. Recently, I was captivated by some impressive slow-motion footage of bullets
colliding mid-air, featured by Smarter Every Day. It's definitely worth a watch.
In particular, I find the circular expansion of debris post-collision to be especially intriguing. The question arises: can I replicate this effect using basic physics and Python? For this project, I
will be utilizing Web VPython, which offers convenient 3D objects for simulation.
This leads us to consider a straightforward scenario: the collision of two point particles approaching each other. How do we simulate the interaction between these two objects? Naturally, we will
rely on fundamental physics principles.
Let's denote the two objects as A and B. When they collide, an interaction force arises between them. Notice that the force ( F_{B-A} ) is equal in magnitude but opposite in direction to ( F_{A-B} ),
adhering to Newton's third law. A net force acting on an object results in a change in momentum, according to the momentum principle.
Since the forces are equal and opposite for the same duration, the momentum change of A will counterbalance the momentum change of B, indicating that total momentum remains conserved.
Now, how do we implement this in our model? One effective approach is to use springs. Imagine the two balls overlapping slightly. We can apply a force proportional to the extent of their overlap; if
they don’t overlap, the force is zero—akin to "springy" balls.
The direction of the force is determined by the relative positions of balls A and B, making this model applicable to both glancing and direct collisions. To compute this force, we assume both balls
have a radius ( R ), with vector positions ( r_A ) and ( r_B ). We will also introduce a spring constant ( k ) to quantify the force's magnitude.
The force expressions can be summarized as follows (assuming overlap exists):
# Pseudocode for force calculation
if overlap_exists:
force_A = -k * (2 * R - mag(r))
force_B = -force_A
The plan involves a numerical simulation where motion is divided into small time intervals. For each interval, we will:
1. Calculate the vector distance between A and B. If this distance is less than ( 2R ), we will assess for overlap and calculate the force.
2. Determine the forces acting on A and B (noting that they are equal and opposite).
3. Update the momentum for each ball using the momentum principle.
4. Adjust the positions of both balls based on their updated momentum.
Let's dive into the implementation in Python. Below is the code highlighting the essential parts.
# Initializing the balls
ballA = sphere(pos=vector(-0.03, 0, 0), radius=R, color=color.yellow)
ballB = sphere(pos=vector(0.03, 0, 0), radius=R, color=color.cyan)
# Setting initial conditions
v0 = 100 # initial velocity
k = 50000000 # spring constant
ballA.m = m
ballB.m = m
ballA.p = m * vector(v0, 0, 0)
ballB.p = m * vector(-v0, 0, 0)
Running this simulation reveals a straightforward interaction where the balls simply collide and bounce off each other. However, this doesn’t accurately reflect the complexity of high-speed bullet
Chapter 2: Advanced Bullet Collision Modeling
In the reality of high-speed bullet-bullet collisions, the bullets shatter into numerous small fragments. To effectively model this in Python, we can represent each bullet as a collection of tiny
spheres. This way, when the bullets collide, all the tiny spheres can interact, potentially replicating the effect seen in the Smarter Every Day video.
Here's the plan for this enhanced simulation:
1. Divide each bullet into ( N ) smaller spheres.
2. Distribute these tiny spheres randomly within a spherical volume, simulating the original bullet's shape.
3. Allow all tiny spheres to interact during the collision.
To efficiently manage the many spheres, we will utilize lists in Python. Here’s a snippet of the code that constructs a single bullet from multiple smaller spheres.
# Constants for bullet creation
R = (11.5e-3) / 2 # radius of bullet
m = 0.012 # mass of bullet
N = 20 # number of small spheres
dm = m / N # mass of each small sphere
# List to hold the small spheres
bs = []
To ensure that the tiny spheres are properly positioned without overlapping, I will implement a function that checks for overlaps. If a new sphere overlaps with an existing one, it will be
repositioned until it fits without collisions.
def overlap(rt, radi, rlist):
for rtemp in rlist:
rdist = rtemp.pos - rt
if mag(rdist) < 2 * radi:
return True
return False
We will then initiate a loop to create the spheres, ensuring they are placed in a suitable arrangement.
Following the setup, we can move to the simulation of the bullets approaching each other and eventually colliding.
# Bullet collision simulation
while t < .8e-3:
for b in bs:
b.F = vector(0, 0, 0) # Reset forces
# Calculate forces between spheres
for b in bs:
for a in bs:
rab = b.pos - a.pos
if mag(rab) < 2 * rpiece:
a.F += -k * (2 * rpiece - mag(rab)) * norm(rab)
This loop iterates over all spheres, calculating forces based on overlaps and updating their momentum and positions accordingly.
The results of this simulation reveal a more complex interaction. However, if the bullets collide off-axis, we can adjust the initial positions slightly to observe different collision dynamics.
To further refine our model, we can examine the conservation of momentum throughout the collision. Ideally, the system's total momentum should remain constant, and we can visualize this through a
plot of momentum over time.
Partially Elastic Collisions
In the Smarter Every Day video, we observe a cluster of bullet fragments that stick together upon collision. In our simulation, however, each sphere experiences perfectly elastic collisions, meaning
they will never coalesce.
In perfectly elastic collisions, kinetic energy is conserved. However, to introduce the concept of partially elastic collisions, we can manipulate the spring constants governing the interactions
between the spheres.
By using a stronger spring constant during compression and a weaker one during expansion, we can model the loss of kinetic energy during the collision. For example:
if mag(rab) < 2 * rpiece:
if dot(rab, dp) < 0:
k = k1 # Stronger spring for compression
k = k2 # Weaker spring for rebound
This adjustment allows us to see a more realistic representation of the collision dynamics, where kinetic energy is not fully conserved.
To summarize, through iterative modeling and simulation, we can better understand the physics behind bullet collisions, capturing the essence of their dynamic interactions.
Explore the intricacies of bullet projectile dynamics and collision detection in Python through this engaging tutorial.
Learn how to build a 3D Python model that effectively demonstrates the motion of a projectile, enhancing your understanding of physics in action. | {"url":"https://darusuna.com/python-bullet-bullet-collision-model.html","timestamp":"2024-11-05T22:13:50Z","content_type":"text/html","content_length":"16471","record_id":"<urn:uuid:323d23cd-c4fe-41a9-a970-4d98c6340e31>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00809.warc.gz"} |
H.C.F. or G.C.D. Of Two Number Program in Java - Quescol
H.C.F. or G.C.D. Of Two Number Program in Java
In this tutorial, we are going to learn writing java program to calculate the Highest Common Factor of two numbers. Our program will take two numbers as the inputs given by the user and returns the
h.c.f. of given numbers.
For example, for the inputs of the two numbers 4 and 6. Our program will return ‘2’ as an output.
What is H.C.F. or G.C.D.?
The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides two or more integers without leaving a remainder. It’s a fundamental concept in
number theory and plays a key role in various mathematical and computational fields, such as simplifying fractions, least common multiple calculation, and cryptography.
• Numbers: 36 and 48
• Prime Factorization:
□ 36=22×3236=22×32
□ 48=24×348=24×3
• Common Prime Factors: The common prime factors are 22 and 33, with the lowest powers being 2222 and 33.
• HCF: 22×3=4×3=1222×3=4×3=12
Program 1: Calculate HCF/GCD of two Numbers Java
In the below program we are taking two numbers as input from the user. After taking inputs, we are applying logics to find the HCF. WE are using for loop to iterate through possible factors,
determining the greatest common divisor. The output for the input numbers 6 and 2 was correctly calculated as 2.
import java.util.*;
public class Main
public static void main(String[] args) {
double num1,num2,gcd=0;
System.out.println("Java Program to calculate HCF/GCD " );
Scanner sc = new Scanner(System.in);
System.out.println("Please give first number");
num1= sc.nextDouble();
System.out.println("Please give second number");
num2= sc.nextDouble();
for(int i=1; i <= num1 && i <= num2; ++i)
if(num1%i==0 && num2%i==0)
gcd = i;
System.out.println("G.C.D of number "+num1+" and "+num2+" = " +gcd);
Java Program to calculate HCF/GCD
Please give first number
Please give second number
G.C.D of number 6.0 and 2.0 = 2.0
For inputs 6 and 2, the program is finding the highest factor that is common in both 6 and 2.
The factor of 6 is 1,2,3 and the factor of 2 is 1,2. The highest factor common in
both is 2.
Program 2: Calculate HCF/GCD of two Numbers using Java 8
In the below program we are using Java 8 streams. We have created a calculateGCD method which is using stream to iterate through the range of numbers and filter for common divisors. The max method is
using to find the greatest common divisor. The output for the input numbers 45 and 5 was correctly computed as 5.
import java.util.Scanner;
import java.util.stream.IntStream;
public class Main {
public static void main(String[] args) {
Scanner scanner = new Scanner(System.in);
System.out.println("Please give first number:");
int num1 = scanner.nextInt();
System.out.println("Please give second number:");
int num2 = scanner.nextInt();
int gcd = calculateGCD(num1, num2);
System.out.println("HCF of " + num1 + " and " + num2 + " is: " + gcd);
private static int calculateGCD(int a, int b) {
return IntStream.rangeClosed(1, Math.min(a, b))
.filter(i -> a % i == 0 && b % i == 0)
Please give first number:
Please give second number:
HCF of 45 and 5 is: 5
In the above tutorial you have learnt Writing Java programs to calculate the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of two given numbers. The HCF represents the largest number
that can exactly divide both input numbers.
In the first program we have taken input from the user to find the HCF. And then using for loop we have calculated the gcd of the two input number.
And in second program, we are also taking the input from the user but applying java 8 stream api to calculate the gcd.
Readers now have two different methods to calculate the HCF of two numbers in Java. Whether using a traditional iterative approach or leveraging the functional capabilities of Java 8 streams, you
have the flexibility to choose the method that best fits your coding preferences and requirements.
Happy coding! | {"url":"https://quescol.com/interview-preparation/hcf-program-in-java","timestamp":"2024-11-05T17:16:16Z","content_type":"text/html","content_length":"88022","record_id":"<urn:uuid:dff3fe70-5923-41bf-adcc-b814a193bfe8>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00345.warc.gz"} |
calculate the formula mass
Download calculate the formula mass
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Make the following conversions Follow the steps and show your work
A sample contains 3.01 1023 molecules of sulfur dioxide, SO2. Determine the amount in moles.
How many molecules of sucrose are in 3.50 moles of sucrose.
Calculate the number of moles that contain 4.50 x 1023 atoms of zinc (Zn).
How many molecules are there in 0.5 moles of CO2 gas?
How many moles of O2 contain 3.25 1023 O2 molecules?
Book work: PP & AQ
1. Determine the number of atoms in 2.50 mol Zn.
2. Given 3.25 mol AgNO3, determine the number of formula units
3. Calculate the number of molecules in 11.5 mol H2O.
4. How many moles contain each of the following?
a. 5.75 x 1024 atoms Al b. 3.75 x 1024 molecules CO2 c. 3.58 x 1023 formula units ZnCl2
5. How is a mole similar to a dozen?
6. What is the relationship between Avogadro’s number and one mole?
7. Explain how you can convert from the number of representative particles of a substance to moles of that substance.
10. Using Numbers Determine the number of representative particles in each of the following and identify the representative
particle: 11.5 mol Ag; 18.0 mol H2O; 0.150 mol NaCl. | {"url":"https://studyres.com/doc/218424/calculate-the-formula-mass","timestamp":"2024-11-05T22:41:21Z","content_type":"text/html","content_length":"57832","record_id":"<urn:uuid:682d746b-5df2-4085-811c-bf80e9765060>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00488.warc.gz"} |
ETC2420 - Tutorial solution (W7-12) DAG ; Bayesian A/B Testing; Bayesian Regression; TidyModels | Jason Siu
Machine Learning
Machine Learning
Week 7 DAG & more about Regression
Part 1 DAG
## You're such a DAG
This week we are looking at using regression to estimate causal effects. Remembering, always, that correlation is not causation, we need a little bit more than straight linear regression to estimate the _causal effect_ of $X$ on $Y$.
The _causal effect_ of $X$ on $Y$ is how much $Y$ would change if we re-did the experiment but with $X$ set to a new value. This would possibly also change the _other_ covariates in the system _as well as_ the outcome variable $Y$. This is _different_ from the regression coefficient in most cases, as the regression coefficients tell us our much $Y$ would change if we changed $X$ but kept everything else fixed.
There are a lot of ways to reason through this process, but one of the more popular ones is using a directed acyclic graph (DAG) to visualise the direct and indirect relationships between variables.
DAGs allow us to visualise a nice condition under which we can estimate a causal effect from a correlative model (like linear regression). Some theory, which is definitely beyond the scope of the course is that if the DAG faithfully represents the system under consideration (in that there ar no undirected paths from $X$ to $Y$ that aren't in the DAG^[This is a _big_ assumption!]), then as long as there are no open back doors between $X$ and $Y$, the estimated effect of $X$ on $Y$ can be interpreted causally!
_This doesn't mean that linear regression is the right tool for finding this effect. But we well get to that soon enough._
This leads to a procedure to find all the of the variables that should be in our model (regression or otherwise):
1. Find your DAG (this is actually the hardest part and will involve subject-matter experts, and probably a great deal of hope and optimism.)
1. Identify all backdoor paths between $X$ and $Y$ (aka paths regardless of arrow direction that go from $X$ to $Y$ that start with an arrow pointing into $X$)
1. Close off all backdoor paths by conditioning (aka adding to the model, aka adjusting) appropriate variables. Typically you add Fork variables. You never add Collider or Pipe variables.
1. Identify all causal (aka directed aka following the arrows) paths from $X$ to $Y$ and _make sure they are open_! (This is usually fine, but if you block off a causal path while closing a back door, you need to find another way to close that particular back door^[This might not exist. In that case you've got to do something more fancy, but that's not usually the case.)
In R, we can use the libraries `dagitty` and `ggdag` to reason with and visualise DAGs.
The `dagify()` function is great because it uses regression notation. `X -> Y` is written as `Y ~ X` just as we would in regression. You can build a graph edge by edge by chaining together these statements.
For example, the classical _fork_ configuration is visualised as follows.
```{r, message = FALSE}
dagify(Y ~ X,
X ~ Z,
Y ~ Z) %>% ggdag() + theme_dag()
You can use R to find all of the paths between $X$ and $Y$ and find a conditioning set (called here the _adjustment set_). To do this you need to specify the variable of interest, which they call the _outcome_ (Y), and the variable we want to change, which they call the _exposure_ (X).
```{r, message = FALSE}
dagify(Y ~ X,
X ~ Z,
Y ~ Z) %>%
ggdag_adjustment_set(exposure = "X",
outcome = "Y")
To interpret this graph, the _adjusted_ variables need to be added to the regression (between $X$ and $Y$), while the unadjusted ones don't.
We can repeat the exercise with a collider triangle. In this case we don't expect to be told to condition on (or adjust for) the collier variable $Z$.
```{r, message = FALSE}
dag <- dagify(Y ~ X,
Z ~ Y,
Z ~ X)
dag %>% ggdag() + theme_dag()
dag %>% ggdag_adjustment_set(exposure = "X",
outcome = "Y")
We can also use the `dagitty` package to do non-graphical stuff. The `paths()` function gives you a list of all paths between the exposure and the outcome, while the `adjustmentSets()` function gives you a list (or multiple lists if there are options) of covariates you need to add to your model to close all of the back doors.
dagify(Y ~ X,
X ~ Z,
Y ~ Z,
outcome = "Y",
exposure = "X") %>% paths()
dagify(Y ~ X,
X ~ Z,
Y ~ Z,
outcome = "Y",
exposure = "X") %>% adjustmentSets()
### Questions
```{r, echo = FALSE, message = FALSE}
dagify(Y ~ X,
X ~ Z,
Y ~ Z,
Y ~ W,
W ~ X) %>%
ggdag() + theme_dag()
1. Before you use R to get the answer, answer the following:
1. What would happen if you add $Z$ to your regression model? Is this a good idea?
Z is a fork variable so if you don't add it you will see bias in your causal estimate.
1. What would happen if you add $W$ to your regression model? Is this a good idea?
W is a pipe variable so if you add it you will see bias in your causal estimate. In particular, you will only see the _direct_ effect of X on Y and you will not see the indirect effect (the one that is mediated by W). This means that you will potentially only see part of the effect X has on Y.
1. Use `ggdag` to viusalise the adjustment set.
```{r eval = FALSE, echo = FALSE}
dagify(Y ~ X,
X ~ Z,
Y ~ Z,
Y ~ W,
W ~ X,
exposure = "X",
outcome = "Y") %>%
1. Describe the linear regression you would do to estimate the _causal_ effect of $X$ on $Y$.
The regression should be
fit <- lm(Y ~ X + Z)
Part 2 Regression using Foxes dataset
## Foxes^[This is example is taken from Richard McElraith's Statistical Rethinking (2nd Ed.)]
In this question, we will be considering a data set about 30 different groups of urban foxes in England. Foxes are territorial and each group has a different number of foxes (`groupsize`), and their territory an area (`area`) and an amount of food (`avgfood`).
```{r, message = FALSE}
# devtools::install_github("rmcelreath/rethinking")
head(foxes) %>% knitr::kable()
Heavier foxes are, usually, healthier foxes, so we are interested in modelling the weight of each fox (`weight`) and estimate the causal effect that interventions involving changing various covariates has on the weight of the foxes.
We will use the following DAG.
```{r, echo = FALSE}
dagify(weight ~ groupsize,
weight ~ avgfood,
groupsize ~ avgfood,
avgfood ~ area) %>%
ggdag_classic() + theme_dag()
1. Use regression to model the _causal effect_ of changing the area size on the weight of a Fox. You will need to analyse the DAG to work out which variables need to be included in the model. Comment on the results _and_ the appropriateness of the linear regression model.
Looking at the graph, avgfood and groupsize are pipe variables, so we should not condition on them! We get the following.
- All of the diagnostic plots look good.
- The effect of area is not easily distinguishable from zero (roughly +/- 0.1).
This means that changing the foxes territory area does not change the weight of a fox.
```{r echo = FALSE, eval = FALSE}
fit1 <- lm(weight ~ area, data = foxes)
1. Now estimate the causal effect of adding food to a territory. Comment on your results and the appropriateness of a linear regression model.
Once again, everything is a pipe so we should not add any other variables to the model.
- All of the diagnostic plots look fine.
- There does not seem to be an obviously non-zero effect of changing the food.
```{r, eval =FALSE, echo = FALSE}
fit2 <- lm(weight ~ avgfood, data = foxes)
1. Finally estimate the causal effect of changing group size. Comment on your results and the appropriateness of a linear regression model.
This time, there is an open backdoor path groupsize <- avgfood -> weight that we need to close off. We do this by adding avgfood to the regression.
- All of the diagnostic plots look good.
- There is a negative effect of group size on weight (when we adjust for /condition on avgfood, which is appropriate here!)
- Unlike the previous model, avgfood has a positive effect when groupsize is fixed. (This is the difference between a correlative effect and a causal effect!)
```{r echo= FALSE, eval = FALSE}
fit3 <- lm(weight ~ avgfood + groupsize, data = foxes)
1. Write a short paragraph synthesising the results of the three regressions you have done.
The most interesting thing is that we see conflicting information about the effect of food availability depending on whether we adjust for groupsize or not. There is a very plausible reason for this: the net effect of adjusting for food is zero because when there is more food in an area the group in that area becomes bigger! This corresponds to the concept of an `ideal-free distribution' in population ecology.
Part 3 Some more regression practice (VIF)
## Some more regression practice
Simulate $n=50$ observations of each of the following variables:
+ $z_{j,i}\overset{i.i.d}{\sim}N(0,1)$ for $j = 1, 2, 3$ and $i=1, 2, \ldots,n$. This produces ***three*** independent sequences of $n ~i.i.d.$ standard normal random variables.
+ $x_{1,i}\overset{i.i.d.}{\sim}Uniform(min=5,~max=15)$, for $i=1, 2, \ldots, n$.
+ $x_{2,i}\overset{i.i.d.}{\sim}Student-t$ with $\nu=4$ degrees of freedom, for $i=1, 2, \ldots, n$.
+ $x_{3,i}\overset{i.i.d.}{\sim}Gamma(shape=3, ~scale=5)$, for $i=1, 2, \ldots, n$.
+ $x_{4,i} = 3+ 2\,z_{1,i}$, for $i=1, 2, \ldots, n$.
+ $x_{5,1} = -1 - 0.96z_{1,i} + 0.28*z_{2,i}$, for $i=1, 2, \ldots, n$.
Then simulate $n$ observation of each of the response variable $y$ according to:
$$y_{i} = \beta_0 + \beta_1 \, x_{1,i} + \beta_2 \, x_{2,i} + \beta_3 \, x_{3,i} + \beta_4 \, x_{4,i} + \beta_5 \, x_{5,i} + \sigma\,z_{i},$$
where $\beta_0=12,~\beta_1=2.5, ~\beta_2=4,~ \beta_3 =1, ~ \beta_4 =9, ~\beta_5=5$ and $\sigma=20$.
Simulation of these variables are completed in the code chunk below^[Note that as **R** does not use a zero value to index an element of a vector, the indices of the individual $\beta$ coefficients is increased by one.].
n <- 50
z1 <- rnorm(n)
z2 <- rnorm(n)
z3 <- rnorm(n)
x1 <- runif(n, min = 5, max = 15)
x2 <- rt(n, df = 4)
x3 <- rgamma(n, shape = 3, scale = 5)
x4 <- 3 + 2*z1
x5 <- -1 - 0.96*z1 + 0.28*z2
tb <- c(12, 2.5, 4, 1, 9, 5 ) #tb for "true beta"
ts <- c(20) #ts for "true sigma"
y <- tb[1] + tb[2]*x1 + tb[3]*x2 + tb[4]*x3 + tb[5]*x4 + tb[6]*x5 + ts[1]*z3
Once you are satisfied that your data has been simulated correctly, put all variables in a *tibble* named *df*.
df <- tibble(x1=x1, x2=x2, x3=x3, x4=x4, x5=x5, y=y)
Produce a scatterplot matrix of all columns of $df$ (including $y$) using the **GGally**::*ggscatmat*() function.
```{r fig.height=8, fig.width=8}
**Question** Which pairs of variables in the scatterplot matrix appear to show a relationship between variables?Do these seem to align with how the data were simulated? Are there an other features you can detect from these pairwise scatterplots?
Next we'll pretend we didn't simulate the data, and instead try to identify a suitable multiple linear regression relationship between the response variable $y$ and the five different regressors, including an intercept term. We'll start with fitting the largest model (the one we used to generate $y$), using OLS. We'll call this model "Model 1".
M1 <- lm(formula=y~x1+x2+x3+x4+x5, data=df) # Model 1
tidy.M1 <- tidy(M1)
glance.M1 <- glance(M1)
Consider the output from the fit of Model 1.
```{r eval=FALSE, echo=FALSE}
tidy.M1 # not included for students
glance.M1 # not included for students
**Question** According to the model output, which regression coefficients are (individually) significant at the 5% level? Do the estimated coefficients seem "close" to the true values used to simulate the data?
Answer to Q2. Need to look at the p-values in *tidy.M1*. The $\hat{\beta}_0$, $\hat{\beta}_2$, $\hat{\beta}_4$ and $\hat{\beta}_5$ all have p-values greater than 0.05. The estimated values are also a bit off - only $\beta_2$ and $\beta_4$ seem "close" to the true values (though this is subjective!)
**Question** What can you say about the estimated standard errors of the estimated regression coefficients?
Answer to Q3. The size of the standard errors are pretty large! Of course, the sample size is pretty small for this size of a model. We also have the issue (to be discovered below if not already noticed) that there is multi-collinearity present.
**Question** What are the $R^2$ and the adjusted $R^2$ values for the fitted Model 1?
Answer to Q4. Need to look at *glance.M1* to see the result: r.squared = 0.5265and adjusted r.squared = 0.4727. The adjusted $R^2$ penalises (reduces) the R^2 because there are many regressors in the model.
One thing that we can look at to see if there is a lot of correlation between the variances in a model is the _variance inflation factor_ or VIF. This is a nifty way to notice linear correlation between covariates. The basic idea is that if I can estimate the jth covariate $x_j$ extremely well from all of the other covariates ($x_1, \ldots, x_p$, $x \neq j$) then it won't add very much to the regression. The VIF is large when a covariate can be estimated very well using the other covariates, and a rule of thumb is that if the VIF is bigger than 5, we should think carefully about including the variable in the model. (One reason to include it would be a DAG-based reason, but the presence of a high VIF definitely suggests that we should think more carefully about the presence of a covariate in our model.)
The VIF can be computed by using the function `vif()` from the `car` package (which you may need to install). We can calculate the variance inflation factors associated with the regressors in Model 1.
Part 4 More about VIF & cook distance
**Question** Find all variables associated with an excessively large VIF. How should the VIFs be interpreted? (Should you remove *all* of these variables from your model? Discuss possible strategies to decide on the variables to exclude from your model due to the large VIF values.)
Answer Q5. x4 and x5 have VIFs over 10 (the threshold). Values over 10 mean than if we fit the linear regression like x4 = a0 + a1*x1 + a2*x2 + a3*x3 + a4*x5, we'll get an R^2 of over 90%. (Since $VIF=\frac{1}{1-R^2_4}$ where the R^2_4 is for this special regression for x4. Solve for R^2_4 when VIF=10 and you'll get R^2=90%.)
It makes sense to remove at least one variable since at least one must be close to redundant given the other regressors. (i.e. there is multi-collinearity present). Given there are two values with nearly the same large VIF, it makes sense to take only one out at a time. (We also know this makes sense since we simulated the data...but we can always check the VIFs after one regressor is removed and if the VIF for the other variable is still large we can then take out the second variable.)
The code chunk below fits a *modified* version of Model 1 and saves the result in an object named *MM1*.
MM1 <- lm(formula=y~x1+x2+x3+x4, data=df) # "Modified" Model 1
tidy.MM1 <- tidy(MM1)
glance.MM1 <- glance(MM1)
Look at the output from the Modified Model 1.
```{r echo=FALSE, eval=FALSE}
tidy.MM1 # not included for students
glance.MM1 # not included for students
**Question** What is the value of $p$ associated with Modified Model 1?
Answer Q6. $p$ is the number of regression coefficients (number of regressors), including the intercept (including a constant regressor). So here $p=5$.
**Question** According to the Modified Model 1 output, which regression coefficients are significant at the 5% level? Do the estimated coefficients seem "close" to the true values used to simulate the data?
Answer Q7. The estimated intercept and coefficient on x2 are not (individually) significant. The estimated values are not spot on, but they seem closer to the corresponding true values than under Model 1.
**Question** Note the size of the standard errors associated with each estimated coefficient. How do these compare with the coefficients that were not significant in Model 1? (See Q2.) Given the way the data were simulated, do you have any ideas about why there might still be insignificant coefficients, even for the Modified Model 1?
See the *glance.MM1* output. The standard errors for the remaining coefficients haven't changed all that much, but we have removed the coefficient that had the very large standard error (x5). Multi-collinearity will tend to inflate the standard errors.
Why still relatively large standard errors? Well, there is still a small sample size here! It seems that the "signal" for the intercept, and for x2, are too small to be detected amid all of the noise in the data.
**Question** What are the values of: $R^2$ and the Adjusted $R^2$ values for the fitted Modified Model 1? Compare these values to those from the original Model 1. (See Q4.) Do you think removing regressor $x_5$ has improved or diminished the model?
Answer to Q9. Need to look at *glance.MM1* to see the result: r.squared = 0.5255 or 52.55% and adjusted r.squared = 0.4833 or 48.33%. We really can only compare the adjusted r.squared since the ordinary r.squared doesn't account for the number of regressors. We can see that the adjusted r.squared actually improves with the removal of x5, so the models seems better without x5 in it (on this measure, at least).
Next, calculate the variance inflation factors (VIF) associated with the regressors in Modified Model 1.
Change the value of $p$ in the code chunk below to match the correct value for Modified Model 1. Then run the code chunk to extract the *leverage* and *Cook's D* measures associated with the fit of Modified Model 1 that exceed the "rule of thumb" threshold of $2*p/n$.
```{r fig.height=5, fig.width=8, eval=FALSE}
df <- df %>% mutate(rowid=1:n)
aug.MM1 <- augment(MM1)
aug.MM1 <- aug.MM1 %>% mutate(rowid=1:n)
p <- 1 # change this once you know the correct value of p
threshold <- 2*p/n # threshold for .hat and .cooksd
exceed_hat <- aug.MM1 %>%
arrange(desc(.hat)) %>%
select(rowid, .hat) %>%
filter(.hat > threshold)
exceed_cooksd <- aug.MM1 %>%
arrange(desc(.cooksd)) %>%
select(rowid, .cooksd) %>%
filter(.cooksd > threshold)
df_longer <- df %>%
pivot_longer(-c(y, rowid),
df_longer %>%
ggplot(aes(x=factor, y=y, colour=regressor)) +
geom_point() +
facet_wrap(~regressor, nrow=2) +
theme_bw() +
nudge_x = 3,
check_overlap = T) +
ggtitle("y against each regressor")
<!-- code chunk below has the correct value of p (not shown to students) -->
```{r fig.height=5, fig.width=8, eval=FALSE, echo=FALSE}
df <- df %>% mutate(rowid=1:n)
aug.MM1 <- augment(MM1)
aug.MM1 <- aug.MM1 %>% mutate(rowid=1:n)
p <- 5 # correct value here
threshold <- 2*p/n
exceed_hat <- aug.MM1 %>% arrange(desc(.hat)) %>% dplyr::select(rowid, .hat) %>% filter(.hat > threshold)
exceed_cooksd <- aug.MM1 %>% arrange(desc(.cooksd)) %>% dplyr::select(rowid, .cooksd) %>% filter(.cooksd > threshold)
df_longer <- df %>% pivot_longer(-c(y, rowid), names_to="regressor", values_to="factor")
df_longer %>% ggplot(aes(x=factor, y=y, colour=regressor))+ geom_point() + facet_wrap(~regressor, nrow=2) + theme_bw() + geom_text(label=df_longer$rowid, nudge_x = 3, colour="black", size=2, check_overlap = T) + ggtitle("y against each regressor")
**Question** How many points exceed the leverage threshold? Find these points using their *rowid* value which is also shown on the faceted plot. (Some will be easier to find than others! Try filtering the data points to see the values of the regressors to make it easier to find the points.)
Answer Q11. Need to view the *exceed_hat* object. Then look at the plot produced above.
-- code chunk below not shown to students -->
```{r echo=FALSE, eval=FALSE}
df %>% dplyr::filter(rowid == 21 | rowid == 16 | rowid == 44)
**Question** How many points exceed the Cook's D threshold?
Answer Q11. Need to view the *exceed_cooksd* object. No points are flagged as influential according to this measure.
-- code chunk below not shown to students -->
```{r eval=FALSE, echo=FALSE}
Week 8 bootstrap to correctly estimate the uncertainty in a prediction & TidyModels
Part 1 : Prediction intervals for linear regression
The distribution of β − ˆβ
In regression, we call the confidence interval for the observed value the prediction interval. We can compute a normality-based interval using R.
fit <- lm(Ozone ~ ., data = airquality)
new_data <- tibble(Solar.R = 110, Wind = 10,
Temp = 74, Month = 5, Day =3)
predict(fit, new_data, interval = "prediction")
We can compare that to the confidence interval for the mean of the new value
predict(fit, new_data, interval = "confidence")
We can see that the fitted value is the same, but the prediction interval is much wider than the confidence interval.
We can see this with a plot.
airquality_pred <- augment(fit, interval = "prediction") %>%
mutate(interval = "prediction")
airquality_ci <- augment(fit, interval = "confidence") %>%
mutate(interval = "confidence")
bind_rows(airquality_ci, airquality_pred) %>%
ggplot(aes(x = Temp, y = Ozone)) +
geom_point() +
geom_line(aes(y = .lower), linetype = "dashed") +
geom_line(aes(y = .upper), linetype = "dashed") +
Estimating Putting it together
## Step 1: Fit the model and compute residuals
fit <- lm(Ozone ~ ., data = airquality)
boot_dat <- augment(fit)
new_data <- tibble(Solar.R = c(90, 110), Wind = c(10,5),
Temp = c(100, 74), Month = c(5,5), Day = c(3,4))
new_data <- new_data %>%
mutate(.pred = predict(fit, new_data))
Next, we need to compute the variance-adjusted residuals.
## Step 2: Make the variance-adjusted residuals
boot_dat <- boot_dat %>%
mutate(epsilon = .resid / sqrt(1 - .hat)) %>%
rename(y = Ozone, mu = .fitted) %>%
select(-starts_with(".")) # remove all the augment stuff
We are going to break with our pattern for doing a bootstrap and instead build a function that computes a single bootstrap resample and computes a single sample of $e_\text{new}$. We will then use
map_dbl to repeat this over and over again.
## Step 3: Make a function to build a bootstrap sample
## and compute things from it
boot <- function(dat, new_dat) {
# assume residuals are called epsilon
# assumes response is called y
# assumes predictions are called mu
# assumes all columns except y and epsilon are covariates
# assumes new_dat has a column called .pred, which is
# the prediction for the original data
n <- length(dat$epsilon)
n_new <- dim(new_dat)[1]
dat <- dat %>%
mutate(epsilon_star = sample(epsilon, replace = TRUE),
y_star = mu + epsilon_star) %>%
select(-y, -mu, -epsilon, -epsilon_star)
fit_star <- lm(y_star ~ ., data = dat)
dat_star <- augment(fit_star) %>%
mutate(epsilon = .resid / sqrt(1 - .hat))
new_dat <- new_dat %>%
mutate(pred_star = predict(fit_star, new_dat),
bias = .pred - pred_star,
epsilon = sample(dat_star$epsilon, n_new, replace = TRUE),
e_star = bias + epsilon) %>%
select(-pred_star, -bias, -epsilon)
# always check that it doesn't throw an error!
boot(boot_dat, new_data)
Now we can compute the bootstrap estimate of the distribution of $e_\text{new}$
## Step 4: Run this a lot of times to get the distribution of e_new
n_new <- dim(new_data)[1]
error <- tibble(experiment = rep(1:1000),
e_star = map(experiment,
~boot(boot_dat, new_data))) %>%
unnest(e_star) %>%
mutate(y_new_star = .pred + e_star) %>%
select(-e_star, -experiment)
intervals <- error %>%
group_by(across(-c(y_new_star, .pred))) %>%
summarise(lower = quantile(y_new_star, 0.025),
pred = first(.pred),
upper = quantile(y_new_star, 0.975)) %>%
predict(fit, new_data, interval = "prediction") %>%
as_tibble()) %>%
select(-fit) %>%
rename(lower_clt = lwr,
upper_clt = upr)
intervals %>% knitr::kable(digits = 2)
These are quite different intervals!
Part 2 : Using tidymodels to fit more complex models
1. split the data
Because we are going to be doing complicated things later, we need to split our data into a test and training set. If we do this now we won't be tempted to forget, or to cheat.
The rsample package, which is part of tidymodels makes this a breeze.
office_split <- initial_split(office, strata = season)
office_train <- training(office_split)
office_test <- testing(office_split)
• The initial_split() function has a prop argument that controls how much data is in the training set. The default value prop = 0.75 is fine for our purposes.
• The strata argument makes sure this 75/25 split is carried out for each season. Typically, the variable you want to stratify by is any variable that will have uneven sampling.
• The training() and testing() functions extract the test and training data sets.
2. build a recipe
Now! We can build a recipe!
The first step in any pipeline is always building a recipe to make our data analysis standard and to allow us to document the transformations we made. (And to apply them to future data.)
For predictive modelling, this is a bit more interesting, because we need to specify which column is being predicted. The recipe() function takes a formula argument, which controls this. For the
moment, we are just going to regress imdb_rating against everything else, so our formula is imdb_rating ~ .. We also need to do this to the training data, so let's do that.
office_rec <- recipe(imdb_rating ~ ., data = office_train) %>%
update_role(episode_name, new_role = "ID") %>%
step_zv(all_numeric(), -all_outcomes()) %>%
step_normalize(all_numeric(), -all_outcomes())
So there are three more steps.
• The first one labels the column episode_name as an "ID" column, which is basically just a column that we keep around for fun and plotting, but we don't want to be in our model.
• step_zv removes any column that has zero variance (ie is all the same value). We are applying this to all numeric columns, but not to the outcome. (In this case, the outcome is imdb_rating.)
• We are normalizing all of the numeric columns that are the outcome.
3. Build the model
Now! We can talk about the model!
Ok, so how do we fit this linear regression? We will do this way way we saw in class. Later, we will modify this and see what happens.
In order to facilitate us using other method in the future, we are going to use the nice tidymodel bindings so we don't have to work too hard to understand how to use it. One of the best things that
tidymodels does is provide a clean and common interface to these functions.
We specify a model with two things: A specification and an engine.
lm_spec <- linear_reg() %>%
So what does that do? Well it tells us that
• we are doing a linear regression-type problem (that tells us what loss function we are using).
• we are going to fit the model using the lm function.
Part 3 : other models using tidymodel Workflows
Random Forest & Cross Validation
When we are doing real data analyses, there are often multiple models and multiple data sets lying around. tidymodels has a nice concept to ensure that a particular recipe and a particular model
specification can stay linked. This is a workflow.
We don't need anything advanced here, so for this one moment we are going to just add the data recipe to the workflow.
wf_office <- workflow() %>%
Now! Let's get fitting!
To fit the model, we add its specification to the workflow and then call the fit function.
lm_fit <- wf_office %>%
add_model(lm_spec) %>%
fit(data = office_train)
We can then view the fit by "pulling" it out of the workflow.
lm_fit %>%
pull_workflow_fit() %>%
tidy() %>%
knitr::kable(digits = 2)
We can then compute the metrics on the training data.
office_test<- office_test %>%
bind_cols(predict(lm_fit, office_test)) %>%
rename(lm_pred = .pred)
office_test %>% metrics(imdb_rating, lm_pred) %>%
knitr::kable(digits =2 )
Moving beyond lm
The previous fit gave us a warning. It said that the fit was rank deficient (which basically is maths-speak for "something has gone very very wrong"). If we compare the metrics on the training set
with those on the test set we can see that they are very different.
office_train %>%
bind_cols(predict(lm_fit, office_train)) %>%
metrics(imdb_rating, .pred) %>%
knitr::kable(digits = 2)
One way that we can make things better is to fit a different model!
The first thing we will look at is a random forest^[These are covered in detail in Chapter 8 of ISLR (the book refered to in the extra reading for week 8) and, while they are cool, they are beyond
the scope of the course.], which is an advanced prediction method that tries to find a good non-linear combination of the covariates.
To do this we need to install the ranger package, which fits random forest models.
Now we just need to change the engine and everything else stays the same!
rf_spec <- rand_forest(mode = "regression") %>%
Then we work as above:
rf_fit <- wf_office %>%
add_model(rf_spec) %>%
fit(data = office_train)
office_test %>%
bind_cols(predict(rf_fit, office_test)) %>%
metrics(imdb_rating, .pred) %>%
knitr::kable(digits = 2)
We see that this is slightly better than the lm run.
• Use cross validation to estimate the variability in the RMSE estimates and comment of whether the complex random forest method did better than linear regression.
folds <- vfold_cv(office_train, v = 20, strata = season)
lm_fit_cv <- wf_office %>%
add_model(lm_spec) %>%
fit_resamples(resamples = folds)
collect_metrics(lm_fit_cv) %>% knitr::kable(digits = 2)
rf_fit_cv <- wf_office %>%
add_model(rf_spec) %>%
fit_resamples(resamples = folds)
collect_metrics(rf_fit_cv) %>% knitr::kable(digits = 2)
Now we can actually tune our model to find a good value of $\lambda$.
wf_rr <- wf_office %>% add_model(rr_tune_spec)
fit_rr <- wf_rr %>%
tune_grid(resamples = folds,
grid = lambda_grid
Let's see how we did! we collected a bunch of metrics that we can now look at.
fit_rr %>% collect_metrics() %>%
ggplot(aes(x = penalty, y = mean,
colour = .metric)) +
geom_errorbar(aes(ymin = mean - std_err,
ymax = mean + std_err),
alpha = 0.5) +
geom_line() +
facet_wrap(~.metric, scales = "free", nrow = 2) +
scale_x_log10() +
theme(legend.position = "none")
We see that a relatively large penalty is good. We can select the best one!
lowest_rmse <- fit_rr %>% select_best("rmse") ## We want it small!
final_rr <- finalize_workflow(wf_rr, lowest_rmse)
Now with our workflow finalized, we can finally fit this model on the whole training set, and then evaluate it on the test set.
fit_rr <- final_rr %>% fit(office_train)
office_test %>%
bind_cols(predict(fit_rr, office_test)) %>%
metrics(imdb_rating, .pred) %>%
knitr::kable(digits = 2)
Regularised regression
We can see that this didn't do much better than just with lm. But in a lot of situations, it does a better job.
Try the exercise again with mixture = 1 in the model specification. This is called a Lasso and automatically includes only the most relevant variables in a model. Does it do better than ridge
regression and linear regression in this case?
lasso_tune_spec <- linear_reg(penalty = tune(), mixture = 1) %>%
wf_lasso <- wf_office %>% add_model(lasso_tune_spec)
fit_lasso <- wf_lasso %>%
tune_grid(resamples = folds,
grid = lambda_grid
fit_lasso %>% collect_metrics() %>%
ggplot(aes(x = penalty, y = mean,
colour = .metric)) +
geom_errorbar(aes(ymin = mean - std_err,
ymax = mean + std_err),
alpha = 0.5) +
geom_line() +
facet_wrap(~.metric, scales = "free", nrow = 2) +
scale_x_log10() +
theme(legend.position = "none")
lowest_rmse <- fit_lasso %>% select_best("rmse") ## We want it small!
final_lasso <- finalize_workflow(wf_lasso, lowest_rmse)
fit_lasso <- final_lasso %>% fit(office_train)
office_test %>%
bind_cols(predict(fit_rr, office_test)) %>%
metrics(imdb_rating, .pred) %>%
knitr::kable(digits = 2)
Week 9 : Model construction using tidymodel
Part 1 : Prepare dataset
As always we will begin by loading our two friends:
An artificial dataset
Consider the following data set.
n <- 500
dat <- tibble(r = runif(n, max = 10),
z1 = rnorm(n),
z2 = rnorm(n),
x1 = r * z1 / sqrt(z1^2 + z2^2),
x2 = r * z2 / sqrt(z1^2 + z2^2),
y = if_else(r < 5, "inner", "outer")) %>%
mutate(y = factor(y)) %>%
select(x1, x2, y)
split <- initial_split(dat)
train <- training(split)
test <- testing(split)
train %>% ggplot(aes(x1,x2, colour = y)) +
geom_point() +
The code looks a bit weird, but it helps to know that points $(x_1,x_2)$ that are simulated according to
• $x_j = z_j / (z_1^2 + z_2^2)^{1/2}$
are uniformly distributed on the unit circle^[If you add more components to $z_j$ and $x_j$, you can simulate points uniformly on the unit sphere!]. This means that I've chosen a random radius
uniformly on $[0,10]$ and sampled from a point on the circle with radius. If the radius is less than $5$ the point is an inner point, otherwise it's an outer point
Part 2 : A failure of linear classifiers
A failure of linear classifiers
This is a nice 2D data set where linear classification will not do a very good job.
Question: Use tidymodels' logistic_reg() model with the "glm" engine to fit a linear classifier to the training data using x1 and x2 as features. Plot the classification regions (aka which parts of
the space are classified as "inner" and "outer).
spec_linear <- logistic_reg() %>%
wf_linear <- workflow() %>%
add_formula(y ~ x1 + x2) %>%
fit_linear <- wf_linear %>%
fit(data = train)
plot_grid <- expand_grid(x1 = seq(-10, 10, length.out = 100),
x2 = seq(-10, 10, length.out = 100))
plot_grid %>% bind_cols(predict(fit_linear, plot_grid)) %>%
ggplot(aes(x1, x2)) +
geom_tile(aes(fill = .pred_class),
alpha = 0.5) +
geom_point(data = train, aes(colour = y)) +
One way to assess how well a classifier did on a binary problem is to use the confusion matrix, which cross tabulates the true inner/outer and the predicted inner/out values on the test set.
This often gives good insight into how classifiers differ. It can be computed using the yardstick package (which is part of tidymodels)
#this is an example - the code chunk won't run without the proper
#stuff - see next chunk
conf_mat(data, ## A data.frame with y and .pred
truth, ## y
estimate ## .pred (from predict)
Question: Compute the confusion matrix for the linear classifier using the test data and comment on the fit.
train %>% bind_cols(predict(fit_linear, train)) %>%
conf_mat(truth = y, estimate = .pred_class)
`It did quite badly!
Part 3 : Nonlinear classification with k-nearest neighbours
Now let's try to fit a non-linear classifier to the data. The first thing we can try is k-Nearest Neighbours. We talked in the lectures about how to fit these models using the tidymodels framework.
Question: Fit a k-nearest neighbours classifier to the training data, using cross validation to choose $k$. Compute the confusion matrix on the test data. Plot the prediction region. Did it do a
better job?
spec_knn <- nearest_neighbor(mode = "classification",
neighbors = tune()) %>%
wf_knn <- workflow() %>%
add_formula(y ~ x1 + x2) %>%
grid <- tibble(neighbors = 1:30)
folds <- vfold_cv(train, v = 10)
fits <- wf_knn %>% tune_grid(resamples = folds,
grid = grid)
neighbors_best <- fits %>%
metric = "accuracy")
wf_knn_final <- wf_knn %>%
fit_knn <- wf_knn_final %>%
fit(data = train)
plot_grid %>% bind_cols(predict(fit_knn, plot_grid)) %>%
ggplot(aes(x1, x2)) +
geom_tile(aes(fill = .pred_class),
alpha = 0.5) +
geom_point(data = train, aes(colour = y)) +
train %>% bind_cols(predict(fit_knn, train)) %>%
conf_mat(truth = y, estimate = .pred_class)
The fit is not perfect, but it is significantly better than logistic
Part 4 : logisitc regression fit
Trying to get a better logisitc regression fit
Let's try enriching our covariate space so that it can better capture nonlinear structures.
Question: Look up the help for the recipes::step_poly()`. Modify your recipe to include polynomials of x1 and x2. Does it include all possible polynomial terms? Does this improve the fit?
A useful way to look at the result of your recipe is to prepare and bake it. The following R commands will output the first 6 rows of a dataset defined by the recipe rec.
#THIS IS AN EXAMPLE CHUNK TOO
rec %>%
prep() %>%
bake(new_data = NULL) %>%
The steps are:
• Prepare the recipe with the prep function. This does any background work needed to actually make the recipe
• Bake the recipe (aka make the data). The new_data = NULL option just uses the dataset defined in the recipe. If you change it to new_data = test you would get the baked test set!
## We are going to need a recipe for this!
rec_quad <- recipe(y ~ x1 + x2, data = train ) %>%
step_poly(x1, x2, degree = 2)
rec_quad %>%
prep() %>%
bake(new_data = NULL) %>%
wf_quad <- workflow() %>%
add_recipe(rec_quad) %>%
fit_quad <- wf_quad %>%
fit(data = train)
## We need to _bake_ the plotting data!
plot_data_quad <- rec_quad %>%
prep() %>%
bake(new_data = plot_grid) %>%
bind_cols(predict(fit_quad, plot_grid))
## We also need to bake the training data if we want
## to plot it on! because otherwise it doesn't get
## scaled appropriately
plot_data_train <- rec_quad %>%
prep() %>%
bake(new_data = train)
## We also need to change the dimension names
## as x1 is now x1_poly_1
plot_data_quad %>%
ggplot(aes(x1_poly_1, x2_poly_1)) +
geom_tile(aes(fill = .pred_class),
alpha = 0.5) +
geom_point(data = plot_data_train,
aes(colour = y)) +
train %>% bind_cols(predict(fit_quad, train)) %>%
conf_mat(truth = y, estimate = .pred_class)
The fit is perfect, but then again the model is exactly correct!
Part 5 : SVM
Moving further into the tidymodels space
While the previous method worked, it relied on the rather unrealistic situation where we knew exactly how the data was generated. In this section, we are going to look at some more complex methods
for solving this problem.
One of the advantages of tidymodels is that we can use the same specifications to fit some quite complex classifiers to data. We do not need to know anything about how they fit the classifier, just
some information about how to tune them.
In this section we are going to try to fit a support vector machine classifier and a random forest classifier. These are both advanced techniques for finding non-linear classification boundaries.
First up, let's look at the support vector machine, or SVM. This is an extension of logistic regression^[well, it's very similar to an extension of linear regression] that automatically enriches the
covariates with a clever set of non-linear functions.
Because it doesn't know exactly which function to use, it will not work as well as the quadratic polynomial in the previous attempt, but it should do pretty well. We do, however, have to tune it.
This is because the method needs to know just how wiggly the boundaries should be. This is difficult to specify manually, so we have to use tune_grid().
The model specification^[the rbf stands for radial basis function, which is the technical component that lets this method find non-linear classification boundaries.] is
spec_svm <- svm_rbf(mode = "classification",
rbf_sigma = tune()) %>%
## Show the code
spec_svm %>% translate()
The translate() function tells you how the background package is called. In this case the model is fit using the ksvm function from the kernlab package (you will be prompted to install it when you
try to fit the model!).
There is one tuning parameter called rbf_sigma. It controls how wiggly the classification boundary can be. A small value indicates that the function can be very wiggly (or very complicated if you
prefer), while a larger value ties to make the function as linear as possible. (It is very similar to the penalty argument we saw last time.)
You can make a good grid of potential values for rbf_sigma using the helper function
rbf_sigma_grid <- grid_regular(rbf_sigma(), levels = 10)
Question: Fit a SVM classifier to the training data, plot the classification boundaries and compute its confusion matrix on the test data. This will take a moment to run! It is also a good idea to
normalize all of your predictors for this type of model, as the rbf_sigma_grid kinda assumes this!
spec_svm <- svm_rbf(mode = "classification",
rbf_sigma = tune()) %>%
rec_basic <- recipe(y ~ x1 + x2, data = train) %>%
wf_svm <- workflow() %>%
add_recipe(rec_basic) %>%
grid <- grid_regular(rbf_sigma(), levels = 10)
fits <- wf_svm %>% tune_grid(resamples = folds,
grid = grid)
sigma_best <- fits %>%
metric = "accuracy")
wf_svm_final <- wf_svm %>%
fit_svm <- wf_svm_final %>%
fit(data = train)
plot_grid %>% bind_cols(predict(fit_svm, plot_grid)) %>%
ggplot(aes(x1, x2)) +
geom_tile(aes(fill = .pred_class),
alpha = 0.5) +
geom_point(data = train, aes(colour = y)) +
train %>% bind_cols(predict(fit_svm, train)) %>%
conf_mat(truth = y, estimate = .pred_class)
SVMs are a nice tool, but they don't tend to scale well for big datasets, so we will look at a different option, called a random forest. Unlike SVMs, random forests are entirely unrelated to logistic
regression and use a different method to find the classifier (they are also unrelated to k-nearest neighbours).
Random forests work by partitioning the covariate space into a sequence of boxes (kinda like one of those famous paintings by Mondrian). For each box, the model predicts which value of the outcome is
the best. The magical trick with random forests is it builds multiple such partitions and then combines the prediction to get a better one.
Part 6 :
In tidymodels, we can specify a random forest with the command
spec_rf <- rand_forest(mode = "classification") %>%
spec_rf %>% translate()
This will need the ranger package installed. There are some tuning parameters that we can change for random forests, but the default values in ranger are pretty robust so we usually don't need to
tune it.
Question: Fit a random forest to the data. Plot the classification boundaries. Compute the confusion matrix for the test set.
spec_rf <- rand_forest( mode = "classification") %>%
wf_rf <- workflow() %>%
add_formula(y ~ x1 + x2) %>%
fit_rf <- wf_rf %>%
fit(data = train)
plot_grid <- expand_grid(x1 = seq(-10, 10, length.out = 100),
x2 = seq(-10, 10, length.out = 100))
plot_grid %>% bind_cols(predict(fit_rf, plot_grid)) %>%
ggplot(aes(x1, x2)) +
geom_tile(aes(fill = .pred_class),
alpha = 0.5) +
geom_point(data = train, aes(colour = y)) +
train %>% bind_cols(predict(fit_rf, train)) %>%
conf_mat(truth = y, estimate = .pred_class)
This worked about as well as the SVM and was much faster!
There are some artifacts, though. The rectangular partioning
is very evident in the prediction reigions!
Question: Of the four methods you have looked at, which one did the best job? Which one did the best job without already knowing the answer?
Obviously, the quadratic logistic regression did the best.
But of the other options, the SVM probably made the most
aesthetically pleasing classifier, but I prefer random
forests, which are as accurate in this case at the
minor cost of some rectangling.
Week 10 : Bayesian A/B Testing
Part A : Preparation for A/B Testing (part B)
Q1. (hyper-)parameters of the posterior distribution.
Step 1 : Create a function like the one below, named beta_binomial(),
beta_binomial <- function(n,y,alpha=1, beta=1){
atil <- alpha + y
btil <- beta + n - y
cf <- n/(alpha + beta + n)
out <- list(alpha_tilde = atil, beta_tilde = btil, credibility_factor = cf)
Step 2 : Using beta_binomial ( ) to find the (hyper-)parameters of the posterior distribution.
(Where θ ∼ Beta(α = 4, β = 4) when Y is modelled with a Binomial(n = 40, θ) distribution, and y = 12 is observed. )
n <- 40
yobs <- 12
alpha <- 4
beta <- 4
Q1out <- beta_binomial(n=n, y=yobs, alpha=alpha, beta=beta)
Step 3 : visualisation
#### Code to visualise components of Bayes theorem ####
# A colourblind-friendly palette with black:
cbbPal <- c(black="#000000", orange="#E69F00", ltblue="#56B4E9",
"#009E73",green="#009E73", yellow="#F0E442", blue="#0072B2",
red="#D55E00", pink="#CC79A7")
cbbP <- cbbPal[c("orange","blue","pink")] #choose colours for p1
thetayy <- seq(0.001,0.999,length.out=100)
prioryy <- dbeta(thetayy,alpha,beta)
postyy <- dbeta(thetayy, Q1out$alpha_tilde, Q1out$beta_tilde)
likeyy <- dbinom(yobs,size=n,prob=thetayy)
nlikeyy <- 100*likeyy/sum(likeyy)
df <- tibble(theta=thetayy, `prior pdf`=prioryy,
`normalised likelihood`=nlikeyy, `posterior pdf`=postyy)
df_longer <- df%>% pivot_longer(-theta, names_to = "distribution",
values_to="density" )
p1 <- df_longer %>% ggplot(aes(x=theta, y=density, colour=distribution, fill=distribution)) +
geom_line() +
scale_fill_manual(values=cbbP) +
Q2. find both the prior probability Pr(0.2 ≤ θ ≤ 0.4), and the posterior probability Pr(0.2 ≤ θ ≤ 0.4 | y).
int <- c(0.2, 0.4)
#prior probability
prior_prob <- pbeta(int[2], alpha, beta) - pbeta(int[1], alpha, beta)
#posterior probability
posterior_prob <- pbeta(int[2], Q1out$alpha_tilde, Q1out$beta_tilde) - pbeta(int[1], Q1out$alpha_tilde, Q1out$beta_tilde)
Q3. Find a list containing the mean (named mean) and variance (named var) of the Beta(α, β) distributions
Write a function, named beta_meanvar, that takes as input the parameter values alpha and beta, corre- sponding to α and β, respectively, and returns a list containing the mean (named mean) and
variance (named var) of the Beta(α, β) distributions
beta_meanvar <- function(alpha, beta){
mean <- alpha/(alpha+beta)
var <- mean*beta/((alpha+beta)*(alpha+beta+1))
out <- list(mean=mean, var=var)
use your function to calculate the mean and variance for both the θ ∼ Beta(α = 4, α = 4) prior distribution, and for the corresponding posterior distribution for θ when y = 12, obtained in Question
prior_meanvar <- beta_meanvar(alpha=4, beta=4)
posterior_meanvar <- beta_meanvar(alpha=Q1out$alpha_tilde, beta=Q1out$beta_tilde)
Thus, under the prior distribution θ ∼Beta(α = 4,β = 4), the prior mean of θ is 0.5 and the prior variance of θ is 0.0278. Then, after observing y = 12 out of n = 40 i.i.d. Bernoulli(n,θ) outcomes,
the posterior mean for θ is 0.3333 and the posterior variance of θ is 0.0045
Q4. confirm that the posterior mean satisfies the relation
Given the prior distribution, values of y and n, the output of your beta_binomial() and beta_meanvar() functions from Question 1 and Question 3, respectively, confirm that the posterior mean
satisfies the relation
We can just numerically check that the relation holds given all of the calculations. If it holds, we are done. If not, you might want to check to see if there has been some sort of rounding error.
beta_meanvar(Q1out$alpha_tilde,Q1out$beta_tilde)$mean ==
(Q1out$credibility_factor)*yobs/n +
(1-Q1out$credibility_factor)*beta_meanvar(alpha=4, beta=4)$mean
Part B : A/B Testing
What is A/B Testing
• A/B testing (bucket tests or split-run testing) is a randomized experiment with two variants, A and B. It includes application of statistical hypothesis testing or “two-sample hypothesis testing”
as used in the field of statistics. A/B testing is a way to compare two versions of a single variable, typically by testing a subject’s response to variant A against variant B, and determining
which of the two variants is more effective.
• As the name implies, two versions (A and B) are compared, which are identical except for one variation that might affect a user’s behavior. Version A might be the currently used version
(control), while version B is modified in some respect (treatment).
Use case : A promotional email campaign
Given information
We want to answer
• 1. Which of the two Calls to Action, A or B, is more effective? i.e., which Call to Action strategy has the higher effectiveness rate?
• 2. What is the distribution of possible values for the difference between θA and θB?
Q6 . According to the prior distribution, what is the expected number of people in group A who will take up their promotional offer?
nA <- 1000
nB <- 1000
alphaA <- 2
betaA <- 26
priorA <- beta_meanvar(alphaA, betaA)
LUstar <- qbeta(c(0.025, 0.975), alphaA, betaA) # prior quantiles for parameter
LU <- nA*LUstar # prior credible interval for expected number of people
Q7 . According to the prior distributions for effectiveness rates θA and θB, and given nA = nB = 1000, what is the difference between the expected number of people from group A who will take up their
promotional offer and the expected number of people from group B who will take up their promotional offer? Explain why.
Answer The difference between the expected number of people from group A who will take up their pro- motional offer and the expected number of people from group B who will take up their promotional
offer is zero. This is because nA = nB, and θA and θB are independent with the same (marginal) prior distribution, the two experiments have the same a priori expected number of successes.
Q8 . what are the posterior distributions for the effectiveness rates for group A and for group B, respectively?
Given that :
After running the A/B testing protocol, yA = 63 customers purchased the product using discount code A and yB = 45 customers purchased the product using discount code B.
yobsA <- 63
yobsB <- 45
postA <- beta_binomial(n=nA, y=yobsA, alpha=alphaA, beta=betaA)
postB <- beta_binomial(n=nB, y=yobsB, alpha=alphaA, beta=betaA)
Q9 . what is the (posterior) mean and variance of ∆?
Given that :
Let ∆ = θB − θA denote the difference between the effectiveness rates associated with Call to Action B and Call to Action A.
Answer The two posterior distributions are independent, because θA and θB are independent at the outset before we see any data, and all of the individual Bernoulli trials are independent too, so we
can update each posterior without worrying about the other and the posterior distributions themselves are independent. This makes calculating the posterior mean and variance of ∆ easy!
The code chunk below completes the calculations, with the full answer again shown further below.
postA_meanvar <- beta_meanvar(postA$alpha_tilde, postA$beta_tilde)
postB_meanvar <- beta_meanvar(postB$alpha_tilde, postB$beta_tilde)
Delta_mean <- postB_meanvar$mean - postA_meanvar$mean
## [1] -0.01751
Delta_var <- postB_meanvar$var + postA_meanvar$var
## [1] 9.996e-05
Q10 . Using simulation to approximate the posterior of ∆
• Simulate a sample of R = 10000 replicated values of ∆ from p(∆ | yA = 63, yB = 45).
Use this sample to approximate each of the following items:
a) the posterior mean of ∆,
b) the standard deviation of ∆, and
c) the probability that the two effectiveness rates will differ by more than 0.01, i.e. Pr(|θB − θA|> 0.01 | yA = 63, yB = 45).
R <- 10000
sampleA <- rbeta(R, postA$alpha_tilde, postA$beta_tilde)
sampleB <- rbeta(R, postB$alpha_tilde, postB$beta_tilde)
sampleD <- sampleB - sampleA
ds <- tibble(Delta = sampleD)
ds %>% ggplot(aes(x=Delta, y=..density..)) +
geom_histogram(colour="blue",fill="blue",alpha=0.4,bins=100) +
geom_vline(xintercept=c(0.01,-0.01), colour="red") +
geom_density(colour="blue",fill="blue",alpha=0.4) +
theme_bw() +
ggtitle(expression(paste("Simulated posterior pdf of ", Delta)))
indic <- rep(0,R)
indic[abs(sampleD)>0.01] <- 1
diffeffective <- mean(indic)
# Thus our approximate value is Pr(|∆|> 0.01 | yA = 63, yB = 45) ≈ 0.7746.
Week 11 : Bayesian Regression
• Author:Jason Siu
• Copyright:All articles in this blog, except for special statements, adopt BY-NC-SA agreement. Please indicate the source! | {"url":"https://www.jason-siu.com/article/ebefa7c5-1525-4532-b83f-70f856a548d6","timestamp":"2024-11-10T02:38:34Z","content_type":"text/html","content_length":"462809","record_id":"<urn:uuid:14fb507a-ad00-4e74-955e-bd36a1686443>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00299.warc.gz"} |
Re: pvs, temporal logic and stacks
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Re: pvs, temporal logic and stacks
To use a temporal logic you need a temporal modality, i.e. a machine. In TLA, the machine is explicit. What you have is a specification of three mathematical functions, not a machine. You can
describe the stack itself as a machine (that interprets "push", "pop" and "top"), which in TLA(+), would require s to be a variable, say, a sequence, which you can then hide with the temporal
existential operator. Alternatively, you can keep push/pop/top as functions or operators, and describe just your theorem as a machine. The specification of your theorem would contain the constant N,
which is the number of push/pop operations. You would then initialize two variables, m and n in the starting state to N, and in each step, nondeterministically do a push or a pop, decrementing m or n
respectively until they reach zero. | {"url":"https://discuss.tlapl.us/msg01474.html","timestamp":"2024-11-04T11:47:13Z","content_type":"text/html","content_length":"3817","record_id":"<urn:uuid:96974e54-0335-4c2e-82fd-65fc2608fa86>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00020.warc.gz"} |
Test for heteroskedasticity
The relevant data is in the attached image.
The p-value of the Breusch-Pagan test is 0.0005. Based on this data and the information in the tables, there is evidence of:
A. serial correlation only.
B. serial correlation and heteroskedasticity.
C. heteroskedasticity only.
Why is there evidence of heteroskedasticity?
Is it because when we refer to the chi-square table, the critical value at 3 degrees of freedom and at α = 0.005 is 12.838? And since the test statistic is 17.7, this exceeds the critical value and
so we reject the null hypothesis (that there is no heteroskedasticity)?
Thanks very much!
What you said.
You could also reject the null hypothesis on the basis of the p-value you were given
Where do you get the aplha = 0.005 from?
We need to know a significance level for the test. Whic I don’t see in tehe question.
Lets assume sigificance is 5% = 0.05
Method 1 - easiest
p vlaue from BP test = 0.0005 < signifcance level =0.05
There reject null of homoskedacity and accept al hetroskedacity
Method 2 - involves table
From Chi squared table k =3 signifcance = 0.5
Critcial Value = 7.81
BP test sttat > 17.7 > 7.81 therefore reject null of BP test )as above)
I think there must be more data as we need threasholds for serial correlation too though we DW stat at 1.8 this is close to 2.
Hi MikeyF, alpha = 0.005 is just an assumption I was making. To be honest, I was not very sure what alpha to use
I understand your method, but doesn’t that involve assuming an alpha? And the answer could, depending on the specific question, change depending on the assumed alpha? Hence, isn’t it better to NOT
assume an alpha?
Sorry, but why can we directly “reject the null hypothesis on the basis of the p-value you were given”?
Well yes as p is so low. BUt that was the idea you were rejecting earlier.
That is what I wrote as the first option.
But if you remember from Level 1
p < significnce level in order to reject null. we have no significance level.
But as p is is low we can just assume it is low enough,
Thanks MikeyF, I understand where you are coming from.
Thanks very much!! | {"url":"https://www.analystforum.com/t/test-for-heteroskedasticity/152966","timestamp":"2024-11-09T17:35:55Z","content_type":"text/html","content_length":"32836","record_id":"<urn:uuid:78a2ade1-047c-4676-a965-0249a5e5b194>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00316.warc.gz"} |
Solving the Linear Equation: d – 10 – 2d + 7 = 8 + d – 10 – 3d - Artist 3D
The linear equation d – 10 – 2d + 7 = 8 + d – 10 – 3d may look intimidating at first glance, but it is actually quite simple to solve. This equation is an example of a linear equation, which is an
equation in which the highest power of the variable is one. Linear equations are often used to model real-world situations, such as calculating the cost of a phone plan or determining the distance
traveled by a car.
To solve this equation, we must first simplify it by combining like terms. Once we have simplified the equation, we can isolate the variable on one side of the equation by performing the same
operation to both sides of the equation. In this case, we can isolate the variable d by adding 3d to both sides of the equation and then adding 3 to both sides of the equation. The resulting solution
is d = 1.
Understanding Linear Equations
What is a Linear Equation?
A linear equation is an equation that forms a straight line when graphed. It is a mathematical expression that relates two variables, typically represented by x and y, in a straight line. Linear
equations are commonly used in mathematics, science, engineering, and economics to model and analyze real-world situations.
The general form of a linear equation is y = mx + b, where m is the slope of the line and b is the y-intercept. The slope represents the rate of change of y with respect to x, while the y-intercept
represents the value of y when x is equal to zero.
Solving Linear Equations
To solve a linear equation, we need to isolate the variable on one side of the equation. The following steps can be used to solve a linear equation:
1. Simplify both sides of the equation using the order of operations and combine all same-side like terms.
2. Use the appropriate properties of equality to combine opposite-side like terms with the variable term on one side of the equation and the constant term on the other.
3. Divide or multiply as needed to isolate the variable.
Let’s apply these steps to the given equation, d – 10 – 2d + 7 = 8 + d – 10 – 3d:
1. Simplify both sides of the equation: -d – 3 = -2d – 2
2. Combine opposite-side like terms: d = 1
3. The solution to the linear equation is d = 1.
In summary, linear equations are important mathematical tools used to model and analyze real-world situations. To solve a linear equation, we need to isolate the variable on one side of the equation
using the appropriate properties of equality.
Solving the Given Linear Equation
Simplifying the Equation
The given linear equation is d – 10 – 2d + 7 = 8 + d – 10 – 3d. To solve this equation, we need to simplify it first. We can simplify the equation by combining like terms on both sides.
d – 2d + d – 3d – 10 + 7 – 10 + 8 = 0
Simplifying further, we get:
-3d – 5 = 0
Isolating the Variable
Now, we need to isolate the variable on one side of the equation. To do this, we can add 5 to both sides of the equation.
-3d = 5
Dividing both sides by -3, we get:
d = -5/3
Checking the Solution
To check if our solution is correct, we can substitute the value of d into the original equation and see if it holds true.
d – 10 – 2d + 7 = 8 + d – 10 – 3d
Substituting d = -5/3, we get:
-5/3 – 10 – 2(-5/3) + 7 = 8 – 5/3 – 10 – 3(-5/3)
Simplifying, we get:
-15/3 – 3.33 + 7 = 8 + 5/3 – 10 + 5
Which simplifies to:
-5 – 3.33 + 7 = 3.33 – 2
Simplifying further, we get:
-1.33 = 1.33
Since the equation is not true, our solution is incorrect.
In conclusion, the given linear equation d – 10 – 2d + 7 = 8 + d – 10 – 3d has no solution.
Common Mistakes to Avoid
When solving a linear equation, it’s easy to make mistakes. Here are some common mistakes to avoid.
Misunderstanding the Order of Operations
One common mistake when solving linear equations is misunderstanding the order of operations. It’s important to remember that you should always perform operations in the following order: parentheses,
exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right).
Forgetting to Combine Like Terms
Another common mistake is forgetting to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, 2x and 5x are like terms, but 2x and 5x^2 are not.
When solving an equation, it’s important to combine like terms before proceeding.
Mistakes in Distributing the Negative Sign
Distributing the negative sign is another area where mistakes can be made. When distributing the negative sign, it’s important to remember to change the sign of each term that is being distributed.
For example, -3(x + 2) should be expanded to -3x – 6, not -3x + 2.
By avoiding these common mistakes, you can improve your chances of solving linear equations correctly. Remember to always double-check your work and take your time to avoid making careless errors.
Applications of Linear Equations
Linear equations have a wide range of applications in various fields, from science to business. They are used to model real-world situations and to predict outcomes based on given data. Here are a
few examples of how linear equations are used in different areas.
Real-life Examples
Linear equations can be used to solve everyday problems. For example, if you are driving at a constant speed, you can use a linear equation to determine how long it will take you to reach your
destination. Similarly, if you know the distance between two cities and the speed of a train, you can use a linear equation to calculate the time it will take for the train to reach its destination.
Business Applications
Linear equations are commonly used in business to predict sales, profits, and other financial metrics. For instance, a company can use a linear equation to forecast sales based on past data. It can
also use a linear equation to determine the break-even point, which is the point at which the company’s revenue equals its expenses.
Scientific Applications
Linear equations are used extensively in science to model physical phenomena. For example, scientists use linear equations to describe the relationship between force and acceleration. They also use
linear equations to model the behavior of gases, liquids, and other substances.
In conclusion, linear equations are a powerful tool for modeling real-world situations and predicting outcomes based on given data. They have applications in many fields, including science, business,
and everyday life. | {"url":"https://artist-3d.com/solving-the-linear-equation-d-10-2d-7-8-d-10-3d/","timestamp":"2024-11-07T23:35:58Z","content_type":"text/html","content_length":"135088","record_id":"<urn:uuid:f7171d41-1b79-47eb-9dae-f9305934a016>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00577.warc.gz"} |
Validation of Sentinel-3 SLSTR Land Surface Temperature Retrieved by the Operational Product and Comparison with Explicitly Emissivity-Dependent Algorithms
Department of Earth Physics and Thermodynamics, Faculty of Physics, University of Valencia, 50, Dr. Moliner, E-46100 Burjassot, Spain
IMK-ASF, Karlsruhe Institute of Technology (KIT), Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany
Fundación Centro de Estudios Ambientales del Mediterráneo (CEAM), 14 Charles R. Darwin, E-46980 Paterna, Spain
GIS and Remote Sensing Group, Institute for Regional Development, University of Castilla-La Mancha, Campus Universitario SN, E-02071 Albacete, Spain
Author to whom correspondence should be addressed.
Submission received: 28 April 2021 / Revised: 28 May 2021 / Accepted: 2 June 2021 / Published: 7 June 2021
Land surface temperature (LST) is an essential climate variable (ECV) for monitoring the Earth climate system. To ensure accurate retrieval from satellite data, it is important to validate satellite
derived LSTs and ensure that they are within the required accuracy and precision thresholds. An emissivity-dependent split-window algorithm with viewing angle dependence and two dual-angle algorithms
are proposed for the Sentinel-3 SLSTR sensor. Furthermore, these algorithms are validated together with the Sentinel-3 SLSTR operational LST product as well as several emissivity-dependent
split-window algorithms with in-situ data from a rice paddy site. The LST retrieval algorithms were validated over three different land covers: flooded soil, bare soil, and full vegetation cover.
Ground measurements were performed with a wide band thermal infrared radiometer at a permanent station. The coefficients of the proposed split-window algorithm were estimated using the Cloudless Land
Atmosphere Radiosounding (CLAR) database: for the three surface types an overall systematic uncertainty (median) of −0.4 K and a precision (robust standard deviation) 1.1 K were obtained. For the
Sentinel-3A SLSTR operational LST product, a systematic uncertainty of 1.3 K and a precision of 1.3 K were obtained. A first evaluation of the Sentinel-3B SLSTR operational LST product was also
performed: systematic uncertainty was 1.5 K and precision 1.2 K. The results obtained over the three land covers found at the rice paddy site show that the emissivity-dependent split-window
algorithms, i.e., the ones proposed here as well as previously proposed algorithms without angular dependence, provide more accurate and precise LSTs than the current version of the operational SLSTR
1. Introduction
Land surface temperature (LST)—like near-surface air temperature—is a key variable in a wide variety of studies, since it is linked to land–atmosphere energy transfer and flux balances [
]. Thus, it is required for monitoring evapotranspiration and climate change [
], as well as for providing estimates of fire size and temperature [
], volcanoes and lava flow [
], and vegetation health [
]. According to the Global Climate Observing System [
], the World Meteorological Organization (WMO) considers LST as one of the essential climate variables (ECVs). The Climate Change Initiative (CCI) was launched by the European Space Agency (ESA) for
improving the prediction of climate change trends by means of satellite data [
]. The CCI considers LST an important variable for monitoring the Earth climate system; therefore, they included it in the list of ECVs required for understanding and predicting the evolution of
climate (
(accessed on 1 March 2021)). Consequently, the validation of satellite derived LSTs against independent references is crucial for assessing their accuracy and precision. For LST retrieval from
satellite data, the GCOS set the recommended thresholds on accuracy (bias, defined as the systematic uncertainty by the Joint Committee for Guides in Metrology [
], JCGM) and precision (standard deviation, SD, defined as the random uncertainty by the JCGM [
]) to 1 K [
The Sea and Land Surface Temperature Radiometer (SLSTR) on board the Sentinel-3A and 3B spacecrafts is a follow-on instrument of the Advanced Along-Track Scanning Radiometer (AATSR). The two sensors
have similar characteristics, including their thermal channels at 11 and 12 µm, with double view capability, and allow us to apply split-window algorithms (SWAs) and dual-angle algorithms (DAAs). In
this paper, the SWA proposed by Niclòs et al. in [
] and the DAA proposed by Coll et al. in [
] were adapted to SLSTR’s thermal bands. The SWA proposed by Niclòs et al. in [
] was developed for the Spinning Enhanced Visible and InfraRed Imager (SEVIRI) onboard Meteosat Second Generation (MSG) and depends explicitly on emissivity and view zenith angle. SLSTR has view
zenith angles up to 60° [
] and, thus, angular anisotropy may have an important impact on LST retrieval, which was noticed when analyzing the angular dependence of the SWA’s regression coefficients. For the SEVIRI sensor,
over the rice paddy site, the SWA proposed by Niclòs et al. in [
] provided an accuracy (bias) and precision (SD) of 0.5 and 0.8 K, respectively. The capability of the AATSR sensor to apply the DAA was previously analyzed in [
] over full vegetation cover. These authors proposed and validated a SWA and a DAA, obtaining a higher standard deviation for the DAA, with accuracy (precision) of 0.0 K (1.0 K). They concluded that
the DAA performed worse than the SWA, mainly due to differences between the nadir and oblique footprints [
The operational LST level 2 (L2) product for the SLSTR sensor is generated with a SWA whose coefficients depend on surface biome, water vapor content (WVC) in the atmosphere, and vegetation fraction
cover [
]. Previous studies validated the Sentinel-3A SLSTR operational LST product over a variety of surfaces, but not over a rice paddy. In the ESA validation report, 11 sites were used to validate the
SLSTR LST product over different land covers [
]: seven were stations of the SURFace RADiance (SURFRAD) network, which uses pyrgeometers (3–50 µm), three were stations of the Karlsruhe Institute Technology (KIT) equipped with narrow band
radiometers (9.6–11.5 µm), and one was the U.S. Department of Energy’s Atmospheric Radiation Measurement (ARM) station equipped with narrow band radiometers.
In this paper, phenological changes of a rice paddy during the growing period were used to validate the SLSTR LST product over three different surfaces: bare soil (wet and dry), water (flooded
surfaces), and full vegetation cover. A permanent station with a wide band Thermal Infrared (TIR) radiometer continuously recorded ground measurements, which were then compared with concurrent
satellite LST values.
The main objective of this paper is to validate the results of the proposed SWAs and the operational SLSTR LST product. Additionally, three explicitly emissivity-dependent SWAs proposed by Sobrino et
al. [
], Zhang et al. [
], and Zheng et al. [
] (hereafter called Sobrino16, Zhang19, and Zheng 19 SWAs, respectively) were evaluated under the same conditions. The main goal of proposing an explicitly angular and emissivity-dependent SWA for
SLSTR is to provide a better-performing alternative to the biome-dependent (i.e., implicitly emissivity-dependent) SWA used for generating the operational product, but also to Sobrino16, Zhang19 and
Zheng 19. Building on these works, this paper presents the adaptation of an SWA with explicit angular dependence, which was previously successfully applied to SEVIRI data, to SLSTR; the validation of
the adapted SWA and its comparison with other SWAs with an explicit emissivity dependence; the adaptation of a DAA to SLSTR and its validation. The validation results presented here are based on
in-situ LST obtained from wide band radiometers (8–14 µm; more similar to satellite TIR observations and more accurate than pyrgeometers), which are installed at a permanent station located in a rice
paddy (i.e., the Valencia LST Validation site). Despite being limited to a single site, the phenological changes over the year allowed us to validate the LST retrieved from Sentinel-3A and
Sentinel-3B over three, previously unrepresented, homogeneous land cover types.
Section 2
describes the validation site and the in-situ LST and emissivity data. The SLSTR LST operational product algorithm and the different emissivity-dependent algorithms evaluated in this study are
described in
Section 3
Section 4
presents the validation results for each algorithm, and a discussion is provided in
Section 5
. Conclusions are drawn in
Section 6
2. Study Site and Ground Data
2.1. Site
The study site is a 100-km
rice paddy area located near Valencia, Spain (39.274°N, −0.317°E; WGS-84). This extensive area is bordered by the city of Valencia in the north, the Mediterranean Sea in the east, and tree crops and
small urban areas in the south and west. Due to rice phenology, over the year, three different homogeneous land covers alternate (
Figure 1
). Full vegetation covers July to mid-September; flooded surface (i.e., water) in December, January, and June; and bare soil from February to May, which is wet during February, and dry from March to
May. These seasonal changes allow us to validate over three different homogeneous land covers at a single site (i.e., as if we were observing three different sites). The SLSTR L2 fraction of
vegetation cover data in
Figure 2
show the typical seasonal changes. The composition of the bare soil found at the rice paddy site is: 14% sand, 50% silt, and 37% clay, with 4.5 % of organic matter (further soil details are provided
in [
]). Based on SLSTR Level 1 (L1) auxiliary data (See
Section 2.3
), over the year the atmospheric WVC at the study site varies between 0.5 and 4 cm.
This site has been extensively used for LST validation purposes [
]. Previous studies demonstrated a high thermal homogeneity for this site at different spatial resolutions [
] and concluded that it is suitable for validating satellite LST with in-situ measurements. For full vegetation cover, these studies found a standard deviation (SD) lower than 0.5 K for 33 × 33 ASTER
pixels (~9 km
) centered on the study area and for a Landsat TM5 scene (~16 km
). In [
], the authors analyzed the variability of 11 × 11 ASTER pixels (1 km
) centered on the study area, and obtained a SD < 0.3 K. In [
], the thermal variability of the area was studied for the three land covers present at the site with hand-held radiometer measurements along transects (~300 m long) through the station parcel on
different dates: the SD values obtained were 0.5 K, 0.4 K, and 0.9 K for full vegetation, flooded soil, and bare soil, respectively.
2.2. Ground Data
2.2.1. SI-121 Radiometer
The Apogee SI-121 radiometer of the LST validation station took measurements during five periods: 5 days in 2016; July 2017 (full month); from April to August 2018; August 2019 (full month), and from
November 2019 to April 2020. This instrument measures radiance in the TIR spectral region (8–14 µm) and has a field of view of 36° and an uncertainty of 0.2 K (manufacturer specification,
(accessed on 1 March 2021)). The SI-121 was installed at three meter height and observed the ground at nadir view, which resulted in a footprint of ~3 m
. A second SI-121 radiometer was set up at 53° from zenith to provide measurements representative of the downwelling hemispheric radiance [
]. Measurements were taken from both SI-121 radiometers every 4 s; the two radiometers were periodically cleaned and calibrated against a Landcal blackbody source P80P for temperatures ranging
between 273 K and 313 K. The uncertainty obtained for both SI-121 radiometers was less than ±0.1 K. The manufacturer specification uncertainty (±0.2 K) was used instead the calibration uncertainty.
During the Fiducial Reference Measurements for validation of surface temperature from satellites (FRM4STS) experiment in June 2016, the blackbody source was calibrated against the National Physics
Laboratory (NPL) reference radiometer (AMBER), characterized with an uncertainty of 0.053 K [
]. The blackbody showed good agreement in the temperature range from 273 to 323 K with a root mean square difference (RMSD) of 0.05 K [
Only measurements of the SI-121 radiometers acquired 3 min before and after a satellite overpass were retained to have enough measurements (i.e., 90) for statistical analyses, but avoiding
significant changes due to the daily trends in the LSTs within the temporal acquisition window [
]. The SD of the measurements within the 3 min was used in the estimation of the in-situ LST uncertainty. Then, the brightness temperatures (Ti) were corrected for emissivity and reflected sky
radiance (atmospheric transmittance and path radiance were negligible). The sky radiance was measured with the radiometer pointing to sky, which approximates downwelling atmospheric irradiance
divided by π. The emissivity values were known from previous characterizations of the site (see
Section 2.2.3
). The above corrections are described by Equation (1):
$B i ( T ) = L i − ( 1 − ε i ) L i , a ↓ ε i$
is land surface temperature,
$B i$
is Planck function integrated with the channel
filter function of the radiometer,
$L i$
is the radiance measured by the sensor and estimated from Ti as Li = Bi(Ti),
$ε i$
is surface emissivity in channel
, and
$L i , a ↓$
is the sky radiance in this channel. After retrieving
via inverting the Planck function, the in-situ LST used for validation was estimated as the average of the
values acquired concurrently to the SLSTR overpasses. The final dataset selected for validation was obtained by removing cloudy data with the cloud mask of the SLSTR LST product.
2.2.2. CIMEL Electronique CE-312 Radiometers
Two multiband CIMEL Electronique CE-312 radiometers [
] were used to acquire daytime LST concurrently to Sentinel-3A satellite overpasses and SI-121 radiometer measurements. The CE-312 radiometer has a field of view of 10° and six channels in the 8–13
µm TIR spectral range, i.e., one wide channel and five narrow bands (channel 1: 8–13.3 µm; channel 2: 10.9–11.7 µm; channel 3: 10.2–11.0 µm; channel 4: 9.0–9.3 µm; channel 5: 8.5–8.9 µm; channel 6:
8.3–8.6 µm). During the FRM4STS calibration campaign [
], both CE-312 radiometers were calibrated against the NPL ammonia heat-pipe reference blackbody. For a range of temperatures between 273 and 318 K, a RMSD between 0.06 and 0.1 K was obtained for
channels 1 to 3 and between 0.13 and 0.23 K for channels 4 to 6 [
The two handheld instruments were carried along ~300 m transects (150 m in opposite directions starting from the SI-121 station radiometer position) over the site on twelve cloudless days: four days
corresponded to flooded soil and eight to full vegetation. Sky radiance was measured just at the beginning and end of the transects. For flooded soil, the sky radiance was directly measured at zenith
due to the specular reflectance feature of the water. Vegetated and bare soil covers were considered as near-Lambertian surfaces and sky radiance for these land covers was measured using an Infragold
Reflectance Target (IRT-94-100) made by Labsphere [
], which is a highly diffuse gold panel with a reflectivity close to 0.92 in the 8–14 µm region [
Ground LSTs were estimated from the CE-312 radiometer measurements using Equation (1). Finally, average LSTs for the transect measurements, three minutes before and after the satellite overpasses,
were calculated.
2.2.3. In-Situ Land Surface Emissivity
Surface emissivity is a key parameter for accurate LST retrievals [
]. For the studied land covers, in-situ emissivity values were obtained with different techniques (i.e., temperature-emissivity separation (TES) method, box method, and relative emissivity
measurements) for the CE-312 radiometers.
The TES method [
] was used to obtain water and bare soil emissivity. The TES method requires at-surface radiances of the five CE-312 radiometer narrow bands (see [
] for details). These at-surface radiances were then used to obtain the relative spectral contrast. Minimum emissivity is retrieved via an empirical relationship between maximum-minimum difference
(MMD) of relative emissivity and absolute minimum emissivity. Minimum absolute emissivity was used to obtain absolute emissivity of the other four channels using the temperature-independent index.
LST can then be retrieved using any of the channel-specific emissivity values. The retrieved LST was used to obtain the emissivity of the CE-312′s broadband channel. Bare soil TES measurements from [
] were used to obtain wet and dry bare soil emissivity values. For obtaining the wet bare soil emissivity, the emissivities of a soil sample collected at the site, with different moisture contents,
with an average value of 0.41 m
, were used. For dry bare soil, emissivity values of the soil sample with soil moisture of 0.03 m
were used.
In the case of vegetation, emissivity was estimated with the box method [
]. The box consisted of four inner aluminum walls and three different lids: two aluminum cold lids (one of them with a little hole for the radiometer measurements to be used as a top lid and the
other one to be used as bottom lid), and a third non-reflecting hot lid with a temperature of around 60 °C (also used as a top lid). Moreover, the outside of the box walls and the lids were covered
by a thermally insulating material. Emissivity could then be obtained by combining four radiance measurements: (1) cold top lid—sample at bottom; (2) hot top lid—sample at bottom; (3) hot top
lid—cold bottom lid; (4) cold top lid—cold bottom lid [
The emissivities used for each land cover at the site are provided in
Table 1
along with the associated uncertainty for each spectral channel of the CE-312 radiometers. The emissivities used for the SI-121 radiometer were the same as those for the broadband channel 1 (8–13.3
µm) of the CE-312.
Since SLSTR’s view zenith angle can reach up to 60°, the angular variation of emissivity was taken into account: for flooded soil, i.e., water, the emissivity relationship in [
] was used, which directly estimates the emissivity in MODIS spectral channels 31 and 32 (11 µm and 12 µm). Due to the similarity between MODIS and SLSTR spectral channels, the same relationship
could be used here. For wet and dry bare soil, the emissivity values were measured with two CE-312 radiometers under view angles from 0° to 70° in steps of 10° in order to obtain relative to nadir
measurements [
]. These values were interpolated to SLSTR sensor view zenith angles. The emissivity values for flooded soil and bare soil at different view angles are shown in
Figure 3
2.3. SLSTR Level 1 Data
The SLSTR onboard Sentinel-3A (launched in February 2016) and Sentinel-3B (launched in April 2018) have nine spectral channels between 0.5 and 12 µm (three visible and near infrared channels, VNIR;
three short wave infrared channels, SWIR; three TIR channels). The SLSTR L1 product (baseline 003) was used in this study for the period from August 2016 to January 2020. LST was retrieved from
SLSTR’s TIR channels located at 11 and 12 µm (SLSTR channels 8 and 9, respectively). Brightness temperatures for these channels were provided by the SLSTR L1 product in K; auxiliary data were also
provided, e.g., cloud information or pixels filled with cosmetic values (i.e., copies of the closest adjacent valid pixels). All SLSTR level 1 pixels were analyzed with different cloud tests (i.e.,
VNIR and SWIR thresholds tests, and TIR histogram tests). Cloudy and cosmetic pixels were filtered out from the SLSTR dataset used in this study.
WVC is an input parameter of SWAs and is included in SLSTR L1 auxiliary data (obtained from the European Centre for Medium-Range Weather Forecasts (ECWMF) analysis data). Different authors compared
the WVC from ECWMF analyses with WVC obtained from Global Positioning System (GPS) data, radiosonde data, and flight measurements [
]. The studies showed a good performance of ECWMF WVC, although it was reported to overestimate WVC over dry areas [
] and underestimate it over humid areas [
]. In this paper, the ECMWF WVC provided in the SLSTR L1 data was used as input for the SWAs. In order to check its consistency over our study site, the WVC obtained from 12 SLSTR scenes concurrent
with the two radiometer transects (CE312) and permanent station acquisitions (Apogee SI-121) were compared with those obtained from National Center for Environmental Prediction (NCEP) atmospheric
profiles. Since NCEP atmospheric profiles are provided every 6 h on a grid of 1° × 1°, the four closest profiles before and after a Sentinel-3 overpass were interpolated temporally and spatially to
the time of the SLSTR data acquisition and site coordinates. The comparison showed that the bias between NCEP and ECWMF WVC was 0.26 cm and the SD was 0.22 cm. The corresponding RMSD was 0.34 cm,
which was lower than the uncertainty associated with the WVC (±0.5 cm; [
]). The mean WVC of the ECWMF WVC for the twelve days coincident with the transects measurements was 2.4 cm and had an SD of 0.7 cm.
3. LST Retrieval Algorithms
3.1. Operational SLSTR LST Product
The operational SLSTR LST L2 product is retrieved with the SWA described by Equation (2) [
$T = a f , i , w v c + b f , i ( T 11 − T 12 ) sec ( θ / m ) + ( b f , i + c f , i ) T 12$
is the LST,
$T 11$
$T 12$
are the brightness temperatures at 11 and 12 µm, respectively, θ is the satellite viewing angle, m is a parameter related to the view angle, and
$a f , i , w v c$
$b f , i$
$c f , i$
are algorithm coefficients, which depend on vegetation fraction (
), surface biome (
), WVC and day/night time. Algorithm coefficients are given for the 27 land cover classes of the Globcover classification scheme, which provides global classification maps with a resolution of 300 m
]. Each coefficient is subdivided into a vegetation and a soil coefficient, which are weighted by vegetation cover fraction. However, for some biomes, these vegetation and bare soil coefficients have
the same values, e.g., for irrigated cropland (biome 1), which is the biome assigned to the study area, but also for rainfed cropland (biome 2), needle leaved evergreen forest (biome 8), grassland
(biome 14), sparse vegetation (biome 15), vegetation on regularly flooded or waterlogged soil (biome 18), urban areas (biome 19), bare areas (biomes from 20 to 25), water bodies (biome 26), and
permanent snow and ice (biome 27). Moreover, day and night coefficients are equal for most of the biomes, except for those of water or flooded surfaces, as it is the case of forests regularly flooded
(biomes 16 and 17) and biomes 1, 18, and 26. The study area consists exclusively of biome 1, which corresponds to a post-flooding or irrigated croplands land classification. While this (constant)
classification of the station pixel is correct for the full vegetation period, it does not account for changes of surface type; therefore, the flooded and bare soil land covers encountered during
other parts of the year are misclassified.
The SLSTR LST L2 product (baseline collection 003) was used for the period from August 2016 to 13 January 2020. From the latter date onwards, the SLSTR LST product baseline collection changed to
version 004 (changes in product data format and re-gridding). Version 004 was to complete the Sentinel-3B database with bare soil covers.
The SLSTR operational product was validated in previous studies. The ESA validation report [
] showed that over most sites the accuracy threshold was achieved at daytime and nighttime. However, fewer sites met the precision threshold, especially in the case of the SURFRAD stations, likely
mainly due to the heterogeneity of the surroundings [
]. In contrast to the SURFRAD stations, KIT’s stations are located in specifically selected, homogenous areas, and use narrow band Heitronics KT15.85 IIP (9.6–11.5 µm) radiometers. For KIT’s Evora
site (Portugal, temperate evergreen vegetation) an SLSTR LST accuracy of −0.8 K and precision of 0.7 K was obtained for daytime, and an accuracy of −0.4 K and a precision of 0.3 K was found for
nighttime. For KIT’s Kalahari site (Namibia, Kalahari bush), an accuracy of 0.7 K (1.1 K) and a precision of 0.7 K (0.3 K) for daytime (nighttime) were obtained. For KIT’s Gobabeb site (Namibia,
gravel plains), an accuracy of 1.8 K (−0.9) and a precision of 0.8 K (1.1 K) for daytime (nighttime) were obtained. For the ARM station (cattle pasture), a high accuracy for both daytime (0.17 K) and
nighttime (−0.02 K) was obtained. However, precision was low, with values of 1.9 K and 2.1 K for daytime and nighttime, respectively. In [
], a pyrgeometer and a thermal infrared (TIR) wide band radiometer for validating the SLSTR LST product over a forest site in the Amazon basin were used. An accuracy of −0.1 K and a precision of 0.6
K were estimated from the comparison with the wide band radiometer, while an accuracy and precision of 1.0 K were estimated from the comparison with the pyrgeometer, thereby reaching the GCOS
thresholds. In [
], the SLSTR LST product over two desert sites (Dalad Banner and Wuhai, China) was validated using wide band radiometers. The accuracies obtained at these sites were 1.0 and 1.1 K, with precisions of
1.7 and 0.9 K for Dalad Banner and Wuhai, respectively. In [
], the SLSTR product was validated against in-situ LST from two KIT sites (Namib gravel plains near Gobabeb and Lake Constance): the product achieved an accuracy (RMSD) of 1.6 K (2.4 K) and 0.4 K
(0.7 K) over the Namib gravel plains and Lake Constance, respectively.
3.2. Proposal of Two Algorithms Adapted to SLSTR
We propose two alternative SLSTR algorithms that are based on the split-window and the dual angle technique, respectively. The three main differences between the SWA proposed here and the Sobrino16,
Zhang19, and Zheng19 SWAs are: (1) the use of the Cloudless Land Atmosphere Radiosounding (CLAR) database to calculate the coefficients of the proposed algorithms [
]; (2) the dependence of the LST retrieval algorithm on view angle; and (3) the independence of its coefficients from emissivity.
3.2.1. CLAR Database and Simulation Dataset
The CLAR database is composed of 382 clear-sky atmospheric profiles selected from radiosoundings compiled by the University of Wyoming [
]. These atmospheric profiles are relatively evenly distributed over the latitudes and, therefore, well suited to generate global algorithms: 40% of the radiosoundings belong to latitudes between
0°and 30°, 40% belong to latitudes between 30° and 60°, and 20% to latitudes higher than 60°. The WVC values of these profiles are distributed between nearly 0 and 7 cm. The temperatures of the
lowest layer of the atmosphere range from 253 to 313 K.
Gaussian angles from 0° to 65° (0°, 11.6°, 26.1°, 40.3°, 53.7°, and 65°) were chosen to generate the dataset for training the SWA. Input
values were set to:
− 6 K,
− 2 K,
+ 1 K,
+ 3 K,
+ 5 K,
+ 8 K, and
+ 12 K following the global analysis performed in [
]. The dataset contained a total of 16,044 different cases and was used to obtain the algorithm coefficients.
For a SLSTR dual-angle algorithm (DAA), we used the same range of input temperatures T and two pairs of viewing angles: 0–53.7° and 11.6–53.7°. In this case, the total number of simulations in the
dataset was 5348.
3.2.2. Split-Window Algorithm
The SWA presented in this work is based on the algorithm of Niclòs et al. in [
] for the Spinning Enhanced Visible and Infrared (SEVIRI) sensor on board METEOSAT Second Generation 2 (MSG-2), which is given by Equation (3):
$T = T 11 + a 0 + a 1 ( sec ( θ ) − 1 ) + ( a 2 + a 3 ( sec ( θ ) − 1 ) ) ( T 11 − T 12 ) + ( a 4 + a 5 ( sec ( θ ) − 1 ) ) ( T 11 − T 12 ) 2 + α ( 1 − ε ) − β Δ ε$
is LST and
$T 11$
$T 12$
are at-sensor brightness temperatures in K for the SLSTR channels at 11 µm and 12 µm, respectively;
$ε = 0.5 ( ε 11 + ε 12 )$
is the mean emissivity for the SLSTR channels at 11 µm and 12 µm and
$Δ ε$
$ε 11$
$ε 12$
is the difference between them; θ is the sensor viewing angle; and
$α = ( a 6 + a 7 W + a 8 W 2 )$
$β = ( a 9 + a 10 W )$
determine the emissivity correction term, with W defined as the WVC divided by the cosine of the viewing angle. The values of the algorithm coefficients
are given in
Table 2
. Emissivities obtained for each land cover (
Table 1
) and WVC from SLSTR L1 auxiliary data were used for the application of the algorithm.
The atmospheric coefficients (from
) in Equation (3) were obtained from regression analyses between LST–
Figure 4
, using the blackbody approach (
= 1 and Δ
= 0 [
]) and, therefore, the obtained coefficients are independent from emissivity. The emissivity correction term is controlled by α and β [
], which depend on atmospheric parameters (i.e., atmospheric transmissivity, at-surface brightness temperature, and effective atmospheric temperature).
The uncertainties of the LST retrieved with the algorithm were obtained as the square root of the quadratic sum of model fitting uncertainty,
$δ ( T ) M$
, and propagated input parameter uncertainties,
$δ ( T ) p$
$δ ( T ) = [ δ ( T ) M 2 + δ ( T ) p 2 ] 1 / 2$
where both error sources are considered independent and were defined as follows:
$δ ( T ) M = [ σ A C 2 + [ ( 1 − ε ) σ α ] 2 + [ Δ ε σ β ] 2 ] 1 / 2$
$δ ( T ) p = [ ∑ i [ ∂ T ∂ x i δ x i ] 2 ] 1 / 2$
is the fitting error associated with the atmospheric coefficients (from
) and
are the fitting errors associated with
, respectively. The fitting error was defined as the standard error obtained from the regression analyses for each set of coefficients. The regression standard error was estimated by minimizing the
sum of squared deviations from the predictions over the simulation dataset. The propagation uncertainty of the input parameters is expressed by Equation (6), where the partial derivative of
with respect to each input parameter
$x i$
(i.e., emissivity, WVC, brightness temperatures) is estimated and multiplied by uncertainty
$δ x i$
. The experimental emissivity uncertainties in
Table 1
were assigned and WVC uncertainty was assumed to be ±0.5 cm, which is considered to be a representative value [
]. Brightness temperature uncertainty is the noise equivalent error of the instrument, which is about ±0.05 K for the SLSTR thermal bands at 11 and 12 µm for a temperature of 270 K [
]. As the latter is a random uncertainty element, it must be divided by the square root of the number of pixels used to average the LST [
]. The mean and SD of the LST uncertainty contributions from each parameter are given in
Table 3
. Full vegetation and flooded soil were grouped together due to their similar emissivity values and were assigned the same emissivity uncertainties. For all cases, the main uncertainty sources were
modeling and emissivity.
3.2.3. Dual-Angle Algorithm
SLSTR’s dual view also allows retrieving LSTs with DAAs. However, over land surfaces DAAs perform worse than SWAs, which is mainly due to differences in footprints and observation geometries between
the two views [
]. Here, we analyzed the specific dual-view capability of the SLSTR TIR channels to retrieve LST. The DAA used here was adapted from [
] and is given by Equation (7):
$T = T n + c 1 ( T n − T b ) + c 2 ( T n − T b ) 2 + c 0 + α ( 1 − ε ) − β Δ ε$
is the LST,
are the atmospheric coefficients,
$T n$
$T b$
are the brightness temperatures corresponding to nadir view (
) and backward view (
is the mean emissivity for SLSTR nadir and backward views (
$ε = 0.5 ( ε n + ε b ) )$
and ∆
is the emissivity difference between nadir and backward views (
$Δ ε = ε n − ε b )$
$α = ( c 3 + c 4 W + c 5 W 2 )$
) and
$β = ( c 6 + c 7 W )$
are functions modifying the impact of emissivity on LST retrieval, where
is the water vapor content. The coefficients determined for the two DAAs (one for each channel) are given in
Table 2
LST uncertainty for the DAA was estimated in analogy to the SWA with Equations (4)–(6) for the simulation dataset. The same input parameters uncertainties were used to estimate the dual-angle LST
uncertainty. The mean uncertainty contribution of each input parameter, the algorithm fitting errors, and the mean LST uncertainty for the two DAAs are shown in
Table 3
. For the DAA at 12 µm (DAA12), the main uncertainty sources are the fitting error and the emissivity, as for the SWA. However, for the dual-angle algorithm at 11 µm (DAA11), the fitting error is
lower than for the SWA, as it was found for AATSR [
3.3. Alternative Split-Window Algorithms
Various SWAs with explicit emissivity dependence were proposed as alternatives to the operational AATSR/SLSTR LST product algorithm. These alternative SWAs used the same input parameters (i.e.,
emissivity, WVC, and brightness temperatures) as the adapted SWA (
Section 3.2.2
3.3.1. Sobrino16 Split-Window Algorithm
The Sobrino16 SWA [
] employed the algorithm given by Equation (8) for the retrieval of LST from AATSR:
$T = T 11 + d 1 ( T 11 − T 12 ) + d 2 ( T 11 − T 12 ) 2 + d 0 + ( d 3 + d 4 W ) ( 1 − ε ) + ( d 5 + d 6 W ) Δ ε$
is LST and
$T 11$
$T 12$
are at-sensor brightness temperatures in K for the SLSTR channels at 11 and 12 µm, respectively; W is the WVC divided by the cosine of the viewing angle,
$ε = 0.5 ( ε 11 + ε 12 )$
is the mean emissivity for the SLSTR channels at 11 and 12 µm, and
$Δ ε = ε 11 − ε 12$
is the corresponding difference between them;
$d i$
, for
from 0 to 6, are the coefficients of the Sobrino16 SWA.
In order to obtain coefficients
$d i$
, a broad range of
$T 11$
$T 12$
were simulated with the MODTRANv4 radiative transfer code [
] for 61 atmospheric profiles selected from the Thermodynamic Initial Guess Retrieval version 1 (TIGR-1) database and 108 emissivity spectra obtained from the ASTER Spectral Library [
]. For each atmospheric profile, five
values were simulated:
− 5 K,
+ 5 K,
+ 10 K,
+ 20 K, where
is the air temperature of the lowest level of the atmospheric profile. Additionally, five viewing angles (0°, 10°, 20°, 30°, and 40°) were simulated. Based on the uncertainties of the input
parameters (emissivity, WVC and brightness temperatures) and model regression uncertainty, a final algorithm uncertainty of ±1.6 K was estimated [
3.3.2. Zhang19 Split-Window Algorithm
The Zhang19 SWA [
] was developed to improve LST retrieval over barren surfaces. This algorithm is given by Equation (9):
$T = d 1 T 11 + d 2 ( T 11 − T 12 ) + d 3 ( T 11 − T 12 ) 2 + d 0 + ( d 4 − d 5 W ) ( 1 − ε ) + ( d 6 − d 7 W ) Δ ε$
where the variables represent the same quantities as in Equation (8). For the simulation dataset, 60 clear-sky atmospheric profiles were selected from the TIGR2000 database. These atmospheric
profiles were used as input to the MODTRANv5.2 code to obtain simulated values of
$T 11$
$T 12$
. For each atmospheric profile, the input
varied with
− 5 K,
+ 5 K,
+ 10 K,
+ 20 K when
> 280 K, and
− 5 K,
+ 5 K when
≤ 280 K. Average emissivity (
) was varied from 0.9 to 1.0 in steps of 0.02, and the difference in emissivity (∆
) varied from 0.02 to −0.02 in steps of 0.005. Simulations were performed for two viewing angles and yielded a total dataset of 30,456 simulated cases. According to [
], the SWA uncertainty ranges between ±0.5 K and ±1 K, depending on WVC. These values represent the uncertainty of the algorithm and do not consider input parameters uncertainties.
3.3.3. Zheng19 Split-Window Algorithm
Based on the refined form of the generalized split-window algorithm [
] proposed in [
], Zheng et al. in [
] adjusted the algorithm to match the spectral channels of SLSTR. The Zheng19 algorithm is described by Equation (10):
$T = d 0 + ( d 1 + d 2 1 − ε ε + d 3 Δ ε ε 2 ) T 11 + T 12 2 + ( d 4 + d 5 1 − ε ε + d 6 Δ ε ε 2 ) T 11 − T 12 2 + d 7 ( T 11 − T 12 ) 2$
where the variables are the same as in Equation (8). MODTRANv5.2 [
] was used to obtain a simulated dataset of
$T 11$
$T 12$
from 946 clear-sky atmospheric profiles selected from the TIGR-3 database. The input
values varied from
, ranging from
− 10 K to
+ 30 K in steps of 5 K. Sixty emissivity spectra from the ASTER spectral library and the University of California–Santa Barbara Emissivity Library were used to simulate a dataset under five viewing
angles (0°, 15°, 25°, 35°, and 45°), which resulted in a total of 2,550,200 different cases.
The simulation dataset was divided into 160 groups to obtain coefficients stratified by WVC, brightness temperature
and five viewing angles. However, only the coefficients for nadir view and four brightness temperature and WVC subranges were published in [
]. For these subranges, an algorithm uncertainty ranging from ±0.6 K to ±2.1 K was estimated by propagating model regression uncertainty and emissivity uncertainty.
4. Validation of Satellite LST Products
4.1. Analysis of In-Situ Measurements
The number of transect measurements (12) was too limited for a statistical analysis while we had a sufficient number of permanent station matchups for validating SLSTR LST retrieved with the proposed
adapted algorithms (201), the operational LST product (194), and LST retrieved with the other emissivity-dependent algorithms (201) from SLSTR L1 data. Based on the above, it was decided to use the
transect measurements to analyze the spatial representativeness of the station measurements.
Figure 5
compares the simultaneous measurements obtained with the (mobile) CE-312 and the (fixed) SI-121 radiometers.
The comparison between fixed station measurements and transect measurements shown in
Figure 5
yielded a RMSD less than 0.4 K. These results indicate that the permanent station LST values are representative of the site, since they are in good agreement with the LST along the transects. The
homogeneity of the area allows us to use the permanent station measurements for validating satellite LST, which is in agreement with previous studies (e.g., [
The uncertainties shown in
Figure 5
were estimated from the average of the propagated uncertainty for each measured variable and its standard deviation (i.e., within 3 min before and after the satellite overpasses). Mean uncertainty
values of ±0.7 K and ±0.3 K were obtained from the CE-312 measurements and the SI-121 measurements, respectively; the values for the CE-312 are larger due to the spatial variability along the
4.2. Operational SLSTR LST Product
The operational SLSTR LST product was evaluated against in-situ LSTs. Average LST weighted by the inverse of the squared distance to the site coordinates was obtained for the 2 × 2 closest satellite
pixels. The statistical parameters are the median of the differences (T
), robust standard deviation (RSD, given by Equation (11)), and robust root mean squared difference (R-RMSD), which is obtained as the square root of the quadratic sum of the median and the RSD.
$RSD = median | ( T SLSTR − T ground ) i − median ( T SLSTR − T ground ) i | · 1.483$
Figure 6
shows the comparison of the operational Sentinel-3A SLSTR LST product on cloudless days against the corresponding in-situ LST obtained from SI-121 measurements (daytime and nighttime data; land cover
types in different colors). Robust statistics were used in this analysis to avoid outlier effects [
The validation statistics for the operational SLSTR LST product averaged over the three surfaces yield a median of 1.3 K, an RSD of 1.3 K, and an R-RMSD of 1.8 K.
Table 4
details the statistics for all data together as well as separated by daytime, nighttime, and land cover.
A similar number of data were obtained at daytime and nighttime (98 points and 96 points, respectively). As the full vegetation data represent 65% (48%) of daytime (nighttime) data, the statistics
for all surfaces combined were similar to those obtained for fully vegetated surfaces.
As the Sentinel-3B satellite was launched two years after the Sentinel-3A satellite, fewer data (107) were available. The operational Sentinel-3B SLSTR LST product was evaluated with ground data
concurrently acquired with satellite overpasses during the following periods: July–August 2019 (full vegetation cover), November 2019–January 2020 (flooded soil), and February–April 2020 (bare soil).
All data were selected and cloud filtered as for Sentinel-3A. The statistical analysis yielded a median of 1.5 K and an RSD of 1.2 K for all surfaces combined. The validation statistics are
summarized in
Table 5
for all data as well as separately for daytime and nighttime.
4.3. LST Retrieved with Explicit Emissivity-Dependent Algorithms
The proposed SWA with angular dependence and explicit dependence on surface emissivity (adapted from [
]; see
Section 3.2
) was analyzed using the larger dataset of Sentinel-3A SLSTR measurements. Additionally, the alternative SWAs discussed in
Section 3.3
were evaluated.
Figure 7
shows the LSTs obtained with the SWAs against the in-situ LSTs obtained from the SI-121 measurements at the permanent validation station.
Figure 7
, LSTs range from 277 to 315 K, covering a wide range of values. Data for bare soil and flooded soil cover larger LST ranges, while full vegetation covers a smaller range (i.e., between 290 and 306
K). A median (RSD) of −0.4 K (1.1 K) was obtained for the proposed SWA. Similar statistical results were obtained for the other emissivity-dependent SWA: median (RSD) of −0.8 K (0.9 K) for Sobrino16,
−0.7 K (1.1 K) for Zhang19, and 0.4 K (1.1 K) for Zheng19. In total 198 points were used (32 flooded soil, 38 bare soil and 128 full vegetation). The statistics for all validation results are
summarized in
Table 6
Medians are lower than the RSDs, except for a few cases. The results obtained over flooded and bare soils are slightly better than those over full vegetation, in terms of both bias and RSD, for all
algorithms. Considering all surfaces, the SWA proposed here obtains the lowest R-RMSD.
The same statistical analysis was repeated for the full dataset for daytime and nighttime cases separately.
Table 7
shows the corresponding median, RSD, and R-RMSD. The total data used for each surface at daytime (nighttime) are 99 (99) points for all surfaces, 16 (16) points for flooded soil, 14 (24) for bare
soil, and 69 (59) points for full vegetation. Better results are obtained in general for nighttime cases than for daytime, especially over bare soil and full vegetation surfaces.
4.4. Proposed Dual-Angle Algorithms
Two DAAs with coefficients generated using the CLAR database were proposed for the SLSTR TIR channels at 11 µm and 12 µm.
Figure 8
shows the LST retrieved by DAA11 and DAA12 (for the station pixel) against the in-situ LST obtained from the SI-121 measurements at the station.
Both DAAs overestimated in-situ LST: DAA11 yielded better statistics for all surfaces combined with median (RSD) of 1.7 K (1.6 K) than DAA12, for which a median (RSD) of 2.2 K (1.7 K) was obtained.
The validation statistics for both DAAs are summarized in
Table 8
5. Discussion
An explicit emissivity and angle dependent SWA and two DAAs for Sentinel-3 SLSTR (using the channels centered at 11 µm and 12 µm) were proposed and validated. The SWA was adapted from the SWA
proposed in [
] for SEVIRI sensor, while the DAAs were adapted from an algorithm developed for AATSR in [
]. Although [
] determined a better performance for the AATSR SWA, the double view capability of SLSTR (i.e., its nadir and backward views) for LST retrieval should be analyzed in order to identify possible
differences to AATSR. Furthermore, the operational Sentinel-3A and Sentinel-3B SLSTR L2 LST products and three explicit emissivity-dependent SWAs (i.e., Sobrino16 [
], Zhang19 [
], and Zheng19 [
] SWAs) were validated.
The validation used in-situ LSTs from a rice paddy site close to Valencia, Spain, which represents three seasonal homogeneous land cover types with different spectral features. These in-situ data
were collected by two Apogee SI-121 wideband (8–14 µm) radiometers installed on a permanent station at the site. The narrower viewing geometry and spectral range makes TIR radiometers (e.g., Apogee
SI-121, Heitronics KT15.85) more suitable for LST validation purposes than broadband hemispherical pyrgeometers (3–50 µm), which are commonly used [
]. Additionally, the uncertainty of typically used radiometers (e.g., ±0.2 K for Apogee SI-121) is lower than for pyrgeometers, which is around 1 K [
]. When considering the uncertainties in upwelling and downwelling radiance measurements and emissivity, the uncertainty of in-situ LST obtained with pyrgeometers results in a typical uncertainty of
±1 to ±2 K [
]. In [
], compared simultaneous measurements with wideband radiometers and a pyrgeometer over asphalt and four grassland sites. From this comparison, they observed a standard deviation of up to 2 K at the
grassland sites and a general underestimation for the pyrgeometer data. This is in agreement with LST validations performed for various satellite sensors, e.g., MODIS [
], VIIRS [
], and Landsat-8 [
], which used pyrgeometer measurements as reference: especially at daytime, these studies obtained similar standard deviations of around 2 K at grassland sites.
The GlobCover classification map, which is based on a static global classification, is used for generating the SLSTR LST product. In order to consider surface changes due to vegetation, seasonal
changes or cropland harvest, each coefficient of the operational SLSTR LST algorithm is obtained as a combination of a vegetation coefficient and a bare soil coefficient, weighted by their cover
fractions. However, for flooded soil at the study site, the vegetation fraction is higher than 0.3 (
Figure 2
): while this may be plausible for the last few days considered as flooded soil, when the rice starts growing, it is implausible at the beginning of the flooding, when there is only water. According
to agricultural laborers, changes on the surface should be more marked, since the site is flooded in a few days and is then covered entirely by water. However, for 15 out of 27 land cover types, the
vegetation and bare soil coefficients provided in the SLSTR auxiliary data are the same, as is the case for the biome assigned to the study site (i.e., weighting by cover fraction has no effect).
Different coefficients for daytime and nighttime are provided only for water and flooded surface biomes (i.e., post-flooding or irrigated cropland). However, for most land cover types, e.g., bare
soils, non-flooded forests, scrubland or grassland areas, the coefficients are the same for daytime and nighttime.
In the SLSTR LST algorithm, coefficients for irrigated cropland areas were obtained as an average of the coefficients for water, winter wheat, and broadleaf-deciduous trees according to the land
cover classification given in [
]. Since the land cover of the study site changes over the year, only the period of full vegetation matches with the assigned biome. However, the best validation results were obtained for the bare
soil cover at daytime (R-RMSD = 1.6 K) and nighttime (R-RMSD = 1.0 K). Similar results were obtained for the SLSTR LST product over arid areas by other authors. In [
], a RMSD of 1.9 K at the Gobabeb (Namibia) station was obtained, with a bias of 1.8 K and a SD of 0.8 K. In [
], a bias of 1.1 K and a SD of 0.9 K were obtained, leading to a RMSD of 1.4 K. In these two cases, as well as in this study, SLSTR LST had a good precision, i.e., lower than or equal to 1.0 K, but
an accuracy larger than the GCOS threshold (>1 K). Yang et al. in [
] obtained a systematic uncertainty of 1.6 K and a RMSD of 2.4 K for Sentinel-3 SLSTR LST at the Gobabeb (Namibia) site. It should be noted that the biomes assigned to each validation site differ, so
discrepancies due to different coefficients are possible.
For the Valencia rice paddy site, the validation over full vegetation cover shows considerably better results at nighttime, with median and RSD around 1K. However, for the daytime data, the median
and RSD increase to 1.7 and 1.5 K, respectively. Due to the higher thermal heterogeneity at daytime, a slight increase in RSD is expected, but not the large increase observed for the median
difference, which causes the daytime accuracy to miss the GCOS threshold. It is suspected that the increased median difference is mainly caused by different day and night retrieval coefficients.
These results cannot be directly compared with results obtained over other vegetated areas, e.g., the Amazon site [
] and Evora [
]. The Amazon site [
] was classified as closed to open broadleaved evergreen and/or semi-deciduous forest (biome 5) and yielded a SLSTR LST bias of −0.1 and a SD of 0.6 K for daytime and nighttime data. For Evora [
], the assigned biome was rainfed croplands (biome 2) and the SLSTR LST bias was −0.8 (−0.4) K and SD 0.7 (0.3) K for daytime (nighttime). The biases obtained for these validation sites were
relatively small and showed a slight LST underestimation, while an overestimation was found at the Valencia site, which was misclassified as biome 1 (irrigated cropland) with very different
characteristics to a rice paddy.
For the evaluation of the operational Sentinel-3B SLSTR product, a total of 107 scenes (43 over flooded soil, 31 over bare soil, and 28 over full vegetation) were used. Compared to the validation
results for Sentinel-3A, the obtained accuracy for full vegetation and bare soil was slightly better, while the precision was similar for full vegetation and worse for bare soil. For flooded soil,
the validation results for Sentinel-3B were less accurate and more precise than for Sentinel-3A. As for Sentinel-3A, better results were observed for Sentinel-3B nighttime data, mainly because of the
higher thermal homogeneity. For both sensors, large systematic uncertainty was observed over flooded soil (around 2 K for both daytime and nighttime). In contrast, over the deep and large water body
of Lake Constance (classified as water body, biome 26), Yang et al. in [
] reported a considerably smaller systematic uncertainty of 0.4 K and a RMSD of 0.7 K for the operational Sentinel-3 SLSTR LST product.
The proposed SWA with explicit emissivity and angular dependence and the three published emissivity-dependent SWAs were validated under identical conditions. Generally, all investigated algorithms
performed well, with median and RSD lower than 1.5 K over all surfaces. For all surfaces combined, the proposed algorithm yielded median (RSD) values of −0.4 K (1.1 (K): together with the Zheng19
SWA, it showed the lowest median (best accuracy). However, all SWAs obtained similar RSD values between 0.9 and 1.1 K. The better accuracy of the Zheng19 algorithm is mainly linked to its
exceptionally low median over full vegetation cover, which also represented most data; the other SWA proposed here showed more consistently low median values for all three land covers.
The coefficients of the proposed SWA were based on a simulated dataset produced for LST ranging between −6 K and +12 K around the lowest level of air temperature (
). These values were determined in [
] from statistical analysis of MODIS products MOD08 and MOD11 for air temperature and LST values, respectively. This statistical analysis showed that the range of temperatures used for the simulation
dataset covers most of the cases found over natural surfaces [
]. A maximum increment of up to +20 K was used to produce Sobrino16 and Zhang19, although these increments were only for
< 280 K on the latter. The Zheng19 SWA was produced with even larger increments of up to +30 K: this can be interesting for some applications (e.g., urban heat island, analyses of extreme
temperatures), but can also cause an overfitting of retrieval coefficients, which in turn can increase retrieval uncertainty, particularly over the most common natural surfaces [
The similarity of the results could be linked to the moderate WVCs at the site (ranging from 0.5 to 4.4 cm, with a mean value of 2.4 ± 0.9 cm and only 3% of data >4 cm), which implies small
atmospheric effects and a small dependence on viewing angle. The effect of the differential absorption in the atmosphere in the regression of the proposed SWA per viewing angle was shown in
Figure 4
. For low to moderate brightness temperature differences, there is a minor angular dependence of the regression coefficients. However, for high brightness temperature differences, there is
considerable angular dependence of the coefficients, corresponding to high WVCs (up to 7 cm) in the CLAR atmospheric database used for the regressions (brightness temperature differences were up to
6.4 K). For comparison, the largest values in the database for the brightness temperature difference in SLSTR channels 8 and 9 were around 4.0 K (with a mean of 1.5 ± 0.8 K). Thus, further validation
experiments in tropical atmospheres and over regions with WVCs exceeding 4 cm should be performed to evaluate the algorithms in such extreme cases. Although there is a slight WVC seasonality (i.e.,
higher in the summer, lower in the winter), no significant differences observed in the results were unrelated to WVC, since no extreme WVC values were found at the site. Moreover, the uncertainty
introduced by WVC (~0.1 K) on the SWA is negligible compared to the uncertainty introduced by emissivity (~0.5 K) or the retrieval algorithm (~1.4 K), as shown in
Table 3
. The difference in the accuracy obtained with the proposed SWA for viewing angles lower and higher than 40° was 0.3 K, with an associated difference in precision of 0.6 K. Based on the simulations
shown in
Figure 4
, a decrease of precision with viewing angle was expected, since the atmospheric absorption increases considerably with viewing angle and, thus, also, the regression error. However, the relatively
small change in accuracy indicates a good performance of the algorithm also at larger viewing angles.
At nighttime, for the three land covers at the rice paddy site, the algorithms showed good performance with accuracies and precisions better than the GCOS threshold (<1 K). In contrast, at daytime,
the larger thermal heterogeneity caused an increase of RSDs for bare soil and full vegetation covers, with values of about 1.5 K. However, generally, similar accuracies were obtained at daytime and
nighttime over bare soil and full vegetation cover and most values met the GCOS accuracy threshold.
For all algorithms the results were in agreement with previous validations performed by other authors at different sites: in [
] obtained a bias of −1.4 K and a SD of 1.2 K for a cropland area in Oklahoma, which is comparable to our site with full vegetation cover. Although they had few data points, their results showed a
similar precision to that obtained for the rice paddy site. Similar results were also obtained for SLSTR LST at an Amazon site in [
], who obtained a bias of −1.3 K and a SD of 0.9 K. Zhang et al. in [
] obtained a bias of −0.4 K and a SD of 0.9 K for a desert area in Wuhai, which are similar results to those obtained here with the same algorithm for bare soil. Yang et al. in [
] trained nine SWAs to retrieved SLSTR LST. These SWAs were evaluated over the gravel plains at Gobabeb (Namibia) and Lake Constance (Germany, Switzerland and Austria); a bias from −0.2 K to −0.3 K
(from −0.2 K to 0.3 K) and an RMSD of 1.6 K (0.5 K) were obtained at Gobabeb (Lake Constance). Finally, Zheng et al. in [
] validated their proposed SWA using pyrgeometers and radiometers over cropland and grassland sites. Their overall results showed a bias of 0.6 K and SD of 2.2 K, which is higher than the
corresponding values obtained at our study site for all surfaces combined and for full vegetation cover. The underestimation reported for a station at Henan Hebi, China, with daytime data from a
radiometer over cropland, was similar to that for vegetation cover at our site, under similar conditions. The bias obtained in [
] was the same as the median obtained here, while SD deviation was 2.4 K at the Henan Hebi site. The RSD found here was 1.3 K and, therefore, the Zheng19 algorithm performed much better at our study
site. The proposed DAAs, which use SLSTR’s nadir and backward views, showed better results for the version applied to the 11 µm channel (DAA11), which over flooded soil yielded an accuracy and
precision better than the GCOS threshold. For bare soil, the accuracy and precision were also close to the GCOS threshold. However, the accuracy was worse for full vegetation cover. DAA12 yielded
R-RMSD values between 1.8 and 3 K for all land covers. These findings are in agreement with results for previous sensors (i.e., AATSR, [
]), where, regardless of land cover, DAAs also performed worse than SWAs, probably due to differences in sensor footprint between the views and directional effects on radiometric temperatures [
6. Conclusions
The operational SLSTR LST algorithm depends on biome, day/nighttime, vegetation fraction, and viewing zenith angle. From the validation results it is concluded that the operational Sentinel-3A SLSTR
LST product is accurate for nighttime data, with an accuracy (systematic uncertainty, i.e., median) of 1.0 K and a precision (random uncertainty, i.e., RSD) of 1.0 K for the three investigated
surfaces combined. In contrast, for daytime data an accuracy of 1.8 K and precision 1.2 K was determined. The increase in daytime RSD is attributed to the typically larger thermal heterogeneity of
the land surface. In contrast, the increase in bias is thought to be caused by wrongly assigned biomes, i.e., the same coefficients were used for the three investigated land cover types.
Additionally, the validation for the Sentinel-3B SLSTR LST product is of relevance since no robust validations were published for this platform. An accuracy of 1.5 K and a precision of 1.2 K were
obtained, yielding to similar results to those obtained for the Sentinel-3A SLSTR LST product for all data combined.
The angular and emissivity-dependent algorithm proposed by Niclòs et al. in [
] for MSG SEVIRI was adapted to Sentinel-3 SLSTR. The adapted SLSTR SWA was evaluated together with three emissivity-dependent algorithms proposed by Sobrino et al. in [
], Zhang et al. in [
] and Zheng et al. in [
] using Sentinel-3A SLSTR L1 data. For all data combined (i.e., the three land cover types), the differences between LST obtained with the proposed algorithm and in-situ LST had a median (RSD) of
−0.4 K (1.1 K); the respective values were −0.8 K (0.9 K) for Sobrino16, −0.7 K (1.1 K) for Zhang19, and 0.4 K (1.1 K) for Zheng19. While Zheng19 and the SWA proposed here achieved the overall best
accuracies, the latter showed a more consistent performance for the three investigated land covers. These cover a wide range of natural surface emissivities, i.e., from low values for dry bare soil,
to medium values for wet bare soil, and high emissivity values for vegetation and water surfaces. Additionally, the explicit angular dependence of the proposed SWA will have higher benefits over
areas with higher WVC, which is also illustrated by simulation data).
The overall accuracy improvements of the proposed SWA compared to the operational product is of 0.9 K, while it is 0.4 and 0.3 K compared to Sobrino16 and Zhang19 SWAs, respectively. The achieved
improvements are highly significant, e.g., for climatological studies: when performing LST trend analyses, a global LST increase of 0.27 K/decade was observed from satellite data [
], i.e., the observed trends per decade are still smaller than the accuracy improvement achieved by the proposed algorithm.
Furthermore, a DAA was proposed to investigate the usefulness of SLSTR’s dual-view capability for LST retrieval and separate sets of coefficients were determined for the 11 and 12 µm channels. While
DAA11 performed better than DAA12, the dual-view algorithms still performed worse than the SWAs. However, an acceptable accuracy and precision of DAA11 was found over flooded soil and bare soil at
the Valencia rice paddy site.
Over the rice paddy site, the explicitly emissivity-dependent SWAs were found to perform better than the operational Sentinel-3 SLSTR algorithm with biome-dependent coefficients. Among the
emissivity-dependent SWAs, the proposed algorithm with explicit angular dependence showed a slightly better performance at the three land covers. The results of this algorithm are expected to improve
for more humid atmospheres (i.e., WCV > 4 cm), where the impact of the angular effect is higher due to the increased atmospheric absorption.
Author Contributions
Conceptualization, R.N. and L.P.-P.; methodology, R.N. and L.P.-P.; validation, L.P.-P.; formal analysis, R.N. and L.P; resources and data curation, J.A.V. and J.P.; software, J.P., J.M.G. and
L.P.-P.; writing—original draft preparation, L.P.-P. and R.N.; writing—review and editing, C.C., F.-M.G. and E.V.; funding acquisition, R.N., E.V., C.C. All authors have read and agreed to the
published version of the manuscript.
This work was funded by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (FEDER) through project CGL2015-64268-R (MINECO/FEDER, UE), by the grant
supported by the regional program of training of technicians for R&D&I—Youth Guarantee GJIDI-2018-A-142, and by the Spanish Ministry of Science and Innovation funded contract Torres Quevedo
PTQ2018-010068. The validation campaign of 2016 was also funded by the Spanish Ministry of Economy and Competitiveness under the project CGL2013-46862-C2-1-P.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1. RGB true color compositions (R-G-B 4-3-2; top) and false color compositions (R-G-B 8-4-3; bottom) for three Sentinel-2 Multispectral Instrument (MSI) scenes. The three land covers at the
site are: bare soil (April, left), flooded soil, i.e., water (May, center), full vegetation (August, right). The location of the validation site is shown in the composition.
Figure 2. Fraction of vegetation cover given by the SLSTR L2 product as a function of day of year. A representative photo for each land cover is also shown.
Figure 3. Angular emissivity variation of the bare soil (left) and flooded soil (right) for the CE-312 channels centered on 11 and 12 µm.
Figure 4. LST–T[11] against T[11]–T[12] simulated from the CLAR database at the different view angles for the SLSTR SWA atmospheric coefficients retrieval. The regression functions corresponding to
each angular dataset are plotted as lines in the same color as their corresponding data.
Figure 5. LST obtained with the fixed SI-121 radiometer compared to LST obtained along the transects with mobile CE-312 radiometers.
Figure 6. Operational Sentinel-3A SLSTR LST product against ground LST obtained from the SI-121 radiometer over the three seasonal land cover types at the Valencia rice paddy site. The dark grey and
light grey shadows show 1-RSD and 3-RSD around the regression (dashed line).
Figure 7. LST retrieved from Sentinel-3A with emissivity-dependent algorithms against in-situ LST obtained from the SI-121 radiometer. Top left: Sobrino16. Top right: Zhang19. Bottom left: Zheng19.
Bottom right: the proposed algorithm.
Figure 8. SLSTR LST retrieved with the dual-angle algorithms for the 11 µm channel (left; DAA11) and 12 µm channel (right; DAA12) against in-situ LST for the three seasonal land covers at the
Valencia rice paddy site.
Land Cover 8–13.3 µm 10.9–11.7 µm 10.2–11.0 µm
Flooded soil 0.986 ± 0.005 0.991 ± 0.004 0.990 ± 0.004
Wet bare soil 0.973 ± 0.012 0.977 ± 0.008 0.972 ± 0.011
Dry bare soil 0.967 ± 0.016 0.972 ± 0.004 0.970 ± 0.005
Full vegetation soil 0.983 ± 0.004 0.980 ± 0.005 0.985 ± 0.004
Table 2. Coefficients of the proposed split-window algorithm (Equation (3)) and the dual-angle algorithms (Equation (7)).
Coefficient Split-Window Coefficient Dual-Angle 11 µm Dual-Angle 12 µm
a[0] (K) 0.052 ± 0.013 c[0] (K) −0.18 ± 0.02 −0.27 ± 0.04
a[1] (K) 0.15 ± 0.02 c[1] 2.03 ± 0.02 2.28 ± 0.04
a[2] 0.95 ± 0.02 c[2] (K^−1) 0.114 ± 0.005 0.198 ± 0.007
a[3] −0.30 ± 0.03 c[3] (K) 57.56 ± 0.15 66.02 ± 0.19
a[4] (K^−1) 0.305 ± 0.004 c[4] (K cm^−1) 1.85 ± 0.11 −4.35 ± 0.14
a[5] (K^−1) 0.202 ± 0.007 c[5] (K cm^−2) −1.278 ± 0.018 −0.81 ± 0.02
a[6] (K) 52.51 ± 0.18 c[6] (K) 132.2 ± 0.3 139.4 ± 0.4
a[7] (K cm^−1) −0.11 ± 0.12 c[7] (K cm^−1) −21.80 ± 0.07 −26.05 ± 0.11
a[8] (K cm^−2) −1.004 ± 0.018 − − −
a[9] (K) 75.7 ± 0.2 − − −
a[10] (K cm^−1) −11.21 ± 0.06 − − −
Table 3. Mean and SD of the uncertainty contributions obtained for the simulation dataset. The different uncertainty sources (modeling uncertainty and input parameters: emissivity, δ(T)[ε]; WVC, δ(T)
[W]; brightness temperature, δ(T)[BT]), and total SLSTR LST retrieval uncertainty are shown.
Surface $δ$ Split-Window Dual-Angle 11 µm Dual-Angle 12 µm
Mean (K) SD (K) Mean (K) SD (K) Mean (K) SD (K)
$δ ( T ) ε$ 0.50 0.14 0.74 0.10 0.76 0.12
$δ ( T ) W$ 0.09 0.04 0.03 0.02 0.04 0.02
Dry/Wet Bare Soil $δ ( T ) B T$ 0.08 0.02 0.101 0.010 0.114 0.013
$δ ( T ) p$ 0.52 0.13 0.75 0.10 0.77 0.11
$δ ( T ) M$ 1.4441 0.0009 0.9203 0.0002 1.4996 0.0002
$δ ( T )$ 1.54 0.05 1.19 0.06 1.69 0.05
$δ ( T ) ε$ 0.32 0.09 0.53 0.10 0.48 0.12
$δ ( T ) W$ 0.04 0.03 0.05 0.02 0.10 0.02
Water / Full vegetation $δ ( T ) B T$ 0.09 0.02 0.109 0.006 0.131 0.010
$δ ( T ) p$ 0.36 0.08 0.54 0.10 0.51 0.11
$δ ( T ) M$ 1.4362 0.0012 0.909 0.003 1.492 0.009
$δ ( T )$ 1.49 0.02 1.06 0.05 1.58 0.04
Table 4. Validation statistics for the operational Sentinel-3A SLSTR LST product against in-situ LST for the three land covers at the Valencia rice paddy site. All values are in Kelvin (K) and N is
the number of data points.
All Data Daytime Nighttime
MEDIAN RSD R-RMSD N MEDIAN RSD R-RMSD N MEDIAN RSD R-RMSD N
All Surfaces 1.3 1.3 1.8 194 1.8 1.2 2.2 98 1.0 1.0 1.4 96
Flooded soil 1.8 1.1 2.2 44 2.2 0.7 2.3 19 1.8 1.3 2.2 25
Bare soil 1.1 0.7 1.3 37 1.3 0.9 1.6 16 0.8 0.6 1.0 21
Vegetation 1.3 1.4 1.9 113 1.7 1.5 2.2 63 1.0 0.9 1.3 50
Table 5. Validation statistics for the operational Sentinel-3B SLSTR LST product against in-situ LST for the three land covers at the Valencia rice paddy site. All the statistics are in Kelvin (K)
and N is the number of data points.
All Data Daytime Nighttime
MEDIAN RSD R-RMSD N MEDIAN RSD R-RMSD N MEDIAN RSD R-RMSD N
All Surfaces 1.5 1.2 1.9 107 1.6 1.3 2.0 41 1.3 1.0 1.7 66
Flooded soil 2.1 0.6 2.2 48 2.5 1.1 2.7 15 1.9 0.7 2.0 33
Bare soil 0.8 1.1 1.3 31 0.8 1.7 1.8 13 0.8 1.0 1.3 18
Vegetation 1.0 1.3 1.6 28 1.4 0.9 1.7 13 0.5 1.2 1.3 15
Table 6. Validation statistics for the four emissivity-dependent split-window algorithms for the three land covers at the Valencia rice paddy site. All values are in Kelvin (K) and N is the number of
data points.
MEDIAN RSD R−RMSD N
Sobrino16 −0.8 0.9 1.2 198
All Surfaces Zhang19 −0.7 1.1 1.3 198
Zheng19 0.4 1.1 1.2 198
Proposed SWA −0.4 1.1 1.1 198
Sobrino16 −0.4 0.6 0.7 32
Flooded Soil Zhang19 −0.5 1.0 1.1 32
Zheng19 1.0 0.7 1.2 32
Proposed SWA 0.0 0.6 0.6 32
Sobrino16 −0.4 0.9 0.9 38
Bare Soil Zhang19 −0.5 0.6 0.8 38
Zheng19 0.9 0.7 1.2 38
Proposed SWA −0.2 0.9 0.9 38
Sobrino16 −1.0 1.0 1.4 128
Full Vegetation Zhang19 −0.9 1.2 1.5 128
Zheng19 −0.1 1.3 1.3 128
Proposed SWA −0.7 1.2 1.4 128
Table 7. Validation statistics for the different emissivity-dependent split-window algorithms. Results are shown for all data and separately for flooded soil, bare soil, and full vegetation cover.
All values are in Kelvin (K).
Daytime Nighttime
MEDIAN RSD R−RMSD MEDIAN RSD R−RMSD
Sobrino16 −0.8 1.2 1.5 −0.9 0.8 1.2
All Surfaces Zhang19 −0.5 1.3 1.4 −0.9 0.8 1.2
Zheng19 0.5 1.4 1.5 0.2 1.1 1.1
Proposed SWA −0.3 1.5 1.5 −0.5 0.8 0.9
Sobrino16 −0.3 0.7 0.7 −0.5 0.6 0.8
Flooded Soil Zhang19 −0.4 0.8 0.9 −0.9 0.9 1.2
Zheng19 1.1 0.6 1.3 0.9 0.9 1.3
Proposed SWA 0.2 0.7 0.8 −0.1 0.7 0.7
Sobrino16 −0.4 1.5 1.5 −0.4 0.8 0.9
Bare Soil Zhang19 0.6 1.5 1.6 −0.6 0.7 1.0
Zheng19 0.8 1.4 1.6 1.0 0.5 1.1
Proposed SWA 0.2 1.5 1.5 −0.2 0.6 0.6
Sobrino16 −1.3 1.5 1.9 −1.0 0.7 1.2
Full Vegetation Zhang19 −0.7 1.5 1.7 −0.9 0.8 1.2
Zheng19 0.0 1.4 1.4 −0.1 0.9 0.9
Proposed SWA −0.8 1.6 1.8 −0.7 0.8 1.0
Table 8. Validation statistics of the dual-angle algorithms for SLSTR 11 and 12 µm channels at the Valencia rice paddy site. All statistics are in Kelvin (K) and N is the number of data points.
Dual-Angle 11 µm Dual-Angle 12 µm
MEDIAN RSD R-RMSD N MEDIAN RSD R-RMSD N
All Surfaces 1.7 1.6 2.3 102 2.2 1.7 2.7 102
Flooded Soil 0.6 1.0 1.1 15 1.0 2.3 2.5 15
Bare soil 1.0 1.3 1.7 18 1.4 1.2 1.8 18
Vegetation 2.1 1.2 2.5 69 2.6 1.5 3.0 69
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Pérez-Planells, L.; Niclòs, R.; Puchades, J.; Coll, C.; Göttsche, F.-M.; Valiente, J.A.; Valor, E.; Galve, J.M. Validation of Sentinel-3 SLSTR Land Surface Temperature Retrieved by the Operational
Product and Comparison with Explicitly Emissivity-Dependent Algorithms. Remote Sens. 2021, 13, 2228. https://doi.org/10.3390/rs13112228
AMA Style
Pérez-Planells L, Niclòs R, Puchades J, Coll C, Göttsche F-M, Valiente JA, Valor E, Galve JM. Validation of Sentinel-3 SLSTR Land Surface Temperature Retrieved by the Operational Product and
Comparison with Explicitly Emissivity-Dependent Algorithms. Remote Sensing. 2021; 13(11):2228. https://doi.org/10.3390/rs13112228
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Pérez-Planells, Lluís, Raquel Niclòs, Jesús Puchades, César Coll, Frank-M. Göttsche, José A. Valiente, Enric Valor, and Joan M. Galve. 2021. "Validation of Sentinel-3 SLSTR Land Surface Temperature
Retrieved by the Operational Product and Comparison with Explicitly Emissivity-Dependent Algorithms" Remote Sensing 13, no. 11: 2228. https://doi.org/10.3390/rs13112228
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