idx int64 1 56k | question stringlengths 15 155 | answer stringlengths 2 29.2k ⌀ | question_cut stringlengths 15 100 | answer_cut stringlengths 2 200 ⌀ | conversation stringlengths 47 29.3k | conversation_cut stringlengths 47 301 |
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6,501 | Random walk on the edges of a cube | Let $x^*$ be the number of expected steps. Let $x_1$ be the number of expected steps from any corner adjacent to the origin of the spider and $x_0$ ditto for the ant.
Then $x^* = 1 + x_1$ and $x_0 = 1 + \frac{2}{3}x_1$. Since
$$x_1 = 1 + \frac{2}{3}x_0 + \frac{1}{3}x^*= 1 + \frac{2}{3}x_0 + \frac{1}{3} + \frac{1}{3}x_1... | Random walk on the edges of a cube | Let $x^*$ be the number of expected steps. Let $x_1$ be the number of expected steps from any corner adjacent to the origin of the spider and $x_0$ ditto for the ant.
Then $x^* = 1 + x_1$ and $x_0 = 1 | Random walk on the edges of a cube
Let $x^*$ be the number of expected steps. Let $x_1$ be the number of expected steps from any corner adjacent to the origin of the spider and $x_0$ ditto for the ant.
Then $x^* = 1 + x_1$ and $x_0 = 1 + \frac{2}{3}x_1$. Since
$$x_1 = 1 + \frac{2}{3}x_0 + \frac{1}{3}x^*= 1 + \frac{2}{3... | Random walk on the edges of a cube
Let $x^*$ be the number of expected steps. Let $x_1$ be the number of expected steps from any corner adjacent to the origin of the spider and $x_0$ ditto for the ant.
Then $x^* = 1 + x_1$ and $x_0 = 1 |
6,502 | Random walk on the edges of a cube | One nice abstraction to think of it is this:
Think of the Position of the Ant as $(0,0,0)$ and Spider $(1,1,1)$, now each move the spider can make will essentially switch exactly one of the three components from $1\to0$ or $0\to1$. So the question becomes:
If I randomly switch bits in (1,1,1) after how many steps in av... | Random walk on the edges of a cube | One nice abstraction to think of it is this:
Think of the Position of the Ant as $(0,0,0)$ and Spider $(1,1,1)$, now each move the spider can make will essentially switch exactly one of the three comp | Random walk on the edges of a cube
One nice abstraction to think of it is this:
Think of the Position of the Ant as $(0,0,0)$ and Spider $(1,1,1)$, now each move the spider can make will essentially switch exactly one of the three components from $1\to0$ or $0\to1$. So the question becomes:
If I randomly switch bits in... | Random walk on the edges of a cube
One nice abstraction to think of it is this:
Think of the Position of the Ant as $(0,0,0)$ and Spider $(1,1,1)$, now each move the spider can make will essentially switch exactly one of the three comp |
6,503 | Random walk on the edges of a cube | Just to compliment tiagotvv's answer:
I don't naturally think of these kinds of problems as matrices (even though they are). I have to draw it out, which I've done below. You can see that there are 3 places to move from S, all of which are As. From any A, you can either return to the S, or move to one of two Bs. From a... | Random walk on the edges of a cube | Just to compliment tiagotvv's answer:
I don't naturally think of these kinds of problems as matrices (even though they are). I have to draw it out, which I've done below. You can see that there are 3 | Random walk on the edges of a cube
Just to compliment tiagotvv's answer:
I don't naturally think of these kinds of problems as matrices (even though they are). I have to draw it out, which I've done below. You can see that there are 3 places to move from S, all of which are As. From any A, you can either return to the ... | Random walk on the edges of a cube
Just to compliment tiagotvv's answer:
I don't naturally think of these kinds of problems as matrices (even though they are). I have to draw it out, which I've done below. You can see that there are 3 |
6,504 | Random walk on the edges of a cube | Parity considerations give a very clean solution, using surprisingly simple machinery: no Markov chains, no iterated expectations, and only high school level summations. The basic idea is that if the spider has moved an even number of times in the $x$ direction, it has returned to its original $x$ coordinate so can't b... | Random walk on the edges of a cube | Parity considerations give a very clean solution, using surprisingly simple machinery: no Markov chains, no iterated expectations, and only high school level summations. The basic idea is that if the | Random walk on the edges of a cube
Parity considerations give a very clean solution, using surprisingly simple machinery: no Markov chains, no iterated expectations, and only high school level summations. The basic idea is that if the spider has moved an even number of times in the $x$ direction, it has returned to its... | Random walk on the edges of a cube
Parity considerations give a very clean solution, using surprisingly simple machinery: no Markov chains, no iterated expectations, and only high school level summations. The basic idea is that if the |
6,505 | Random walk on the edges of a cube | I have written a short Java program to answer your question numerically. The traversing of the spider is truly random, meaning that it can also traverse in cycles before getting to the ant.
However, you did not defined the term "opposite corner", so I have two different scenarios. Opposite as in across the same plane o... | Random walk on the edges of a cube | I have written a short Java program to answer your question numerically. The traversing of the spider is truly random, meaning that it can also traverse in cycles before getting to the ant.
However, y | Random walk on the edges of a cube
I have written a short Java program to answer your question numerically. The traversing of the spider is truly random, meaning that it can also traverse in cycles before getting to the ant.
However, you did not defined the term "opposite corner", so I have two different scenarios. Opp... | Random walk on the edges of a cube
I have written a short Java program to answer your question numerically. The traversing of the spider is truly random, meaning that it can also traverse in cycles before getting to the ant.
However, y |
6,506 | Random walk on the edges of a cube | I solved your conundrum via Monte Carlo simulations ($n = 10^4$) and obtained $\mathtt{mean(steps)} \approx 10$.
Here is the R code I used:
ant = c(0,0,0) # ant's coordinates
sim = 1e4 # number of MC simulations
steps = numeric() # initialize array of steps
for (i in 1:sim)
{
spider = c(1,1,1) # spider's coordin... | Random walk on the edges of a cube | I solved your conundrum via Monte Carlo simulations ($n = 10^4$) and obtained $\mathtt{mean(steps)} \approx 10$.
Here is the R code I used:
ant = c(0,0,0) # ant's coordinates
sim = 1e4 # number of | Random walk on the edges of a cube
I solved your conundrum via Monte Carlo simulations ($n = 10^4$) and obtained $\mathtt{mean(steps)} \approx 10$.
Here is the R code I used:
ant = c(0,0,0) # ant's coordinates
sim = 1e4 # number of MC simulations
steps = numeric() # initialize array of steps
for (i in 1:sim)
{
s... | Random walk on the edges of a cube
I solved your conundrum via Monte Carlo simulations ($n = 10^4$) and obtained $\mathtt{mean(steps)} \approx 10$.
Here is the R code I used:
ant = c(0,0,0) # ant's coordinates
sim = 1e4 # number of |
6,507 | Random walk on the edges of a cube | I believe alesc is on the right track when mentioning "However, you did not defined the term "opposite corner"
Unless I am missing something in the question, there is no correct answer, just answers based on assumptions.
The cube size is not defined I.E. 10 cubic ft, 1000 cubic ft etc.
Ant size is not defined I.E. sma... | Random walk on the edges of a cube | I believe alesc is on the right track when mentioning "However, you did not defined the term "opposite corner"
Unless I am missing something in the question, there is no correct answer, just answers b | Random walk on the edges of a cube
I believe alesc is on the right track when mentioning "However, you did not defined the term "opposite corner"
Unless I am missing something in the question, there is no correct answer, just answers based on assumptions.
The cube size is not defined I.E. 10 cubic ft, 1000 cubic ft et... | Random walk on the edges of a cube
I believe alesc is on the right track when mentioning "However, you did not defined the term "opposite corner"
Unless I am missing something in the question, there is no correct answer, just answers b |
6,508 | What is the difference between McNemar's test and the chi-squared test, and how do you know when to use each? | It is very unfortunate that McNemar's test is so difficult for people to understand. I even notice that at the top of its Wikipedia page it states that the explanation on the page is difficult for people to understand. The typical short explanation for McNemar's test is either that it is: 'a within-subjects chi-squar... | What is the difference between McNemar's test and the chi-squared test, and how do you know when to | It is very unfortunate that McNemar's test is so difficult for people to understand. I even notice that at the top of its Wikipedia page it states that the explanation on the page is difficult for pe | What is the difference between McNemar's test and the chi-squared test, and how do you know when to use each?
It is very unfortunate that McNemar's test is so difficult for people to understand. I even notice that at the top of its Wikipedia page it states that the explanation on the page is difficult for people to un... | What is the difference between McNemar's test and the chi-squared test, and how do you know when to
It is very unfortunate that McNemar's test is so difficult for people to understand. I even notice that at the top of its Wikipedia page it states that the explanation on the page is difficult for pe |
6,509 | What is the difference between McNemar's test and the chi-squared test, and how do you know when to use each? | Well, it seems I've made a hash of this. Let me try to explain this again, in a different way and we'll see if it might help clear things up.
The traditional way to explain McNemar's test vs. the chi-squared test is to ask if the data are "paired" and to recommend McNemar's test if the data are paired and the chi-sq... | What is the difference between McNemar's test and the chi-squared test, and how do you know when to | Well, it seems I've made a hash of this. Let me try to explain this again, in a different way and we'll see if it might help clear things up.
The traditional way to explain McNemar's test vs. the c | What is the difference between McNemar's test and the chi-squared test, and how do you know when to use each?
Well, it seems I've made a hash of this. Let me try to explain this again, in a different way and we'll see if it might help clear things up.
The traditional way to explain McNemar's test vs. the chi-squared... | What is the difference between McNemar's test and the chi-squared test, and how do you know when to
Well, it seems I've made a hash of this. Let me try to explain this again, in a different way and we'll see if it might help clear things up.
The traditional way to explain McNemar's test vs. the c |
6,510 | What is the difference between McNemar's test and the chi-squared test, and how do you know when to use each? | The question of which test to use, contingency table $\chi^{2}$ versus McNemar's $\chi^{2}$ of a null hypothesis of no association between two binary variables is simply a question of whether your data are paired/dependent, or unpaired/independent:
Binary Data in Two Independent Samples
In this case, you would use a co... | What is the difference between McNemar's test and the chi-squared test, and how do you know when to | The question of which test to use, contingency table $\chi^{2}$ versus McNemar's $\chi^{2}$ of a null hypothesis of no association between two binary variables is simply a question of whether your dat | What is the difference between McNemar's test and the chi-squared test, and how do you know when to use each?
The question of which test to use, contingency table $\chi^{2}$ versus McNemar's $\chi^{2}$ of a null hypothesis of no association between two binary variables is simply a question of whether your data are pair... | What is the difference between McNemar's test and the chi-squared test, and how do you know when to
The question of which test to use, contingency table $\chi^{2}$ versus McNemar's $\chi^{2}$ of a null hypothesis of no association between two binary variables is simply a question of whether your dat |
6,511 | What is the difference between McNemar's test and the chi-squared test, and how do you know when to use each? | My understanding of McNemar's test is as follows: It is used to see whether an intervention has made a significant difference to a binary outcome. In your example, a group of subjects are checked for infection and the response is recorded as yes or no. All subjects are then given some intervention, say an antibiotic d... | What is the difference between McNemar's test and the chi-squared test, and how do you know when to | My understanding of McNemar's test is as follows: It is used to see whether an intervention has made a significant difference to a binary outcome. In your example, a group of subjects are checked for | What is the difference between McNemar's test and the chi-squared test, and how do you know when to use each?
My understanding of McNemar's test is as follows: It is used to see whether an intervention has made a significant difference to a binary outcome. In your example, a group of subjects are checked for infection ... | What is the difference between McNemar's test and the chi-squared test, and how do you know when to
My understanding of McNemar's test is as follows: It is used to see whether an intervention has made a significant difference to a binary outcome. In your example, a group of subjects are checked for |
6,512 | How to create an arbitrary covariance matrix | Create an $n\times n$ matrix $A$ with arbitrary values
and then use $\Sigma = A^T A$ as your covariance matrix.
For example
n <- 4
A <- matrix(runif(n^2)*2-1, ncol=n)
Sigma <- t(A) %*% A | How to create an arbitrary covariance matrix | Create an $n\times n$ matrix $A$ with arbitrary values
and then use $\Sigma = A^T A$ as your covariance matrix.
For example
n <- 4
A <- matrix(runif(n^2)*2-1, ncol=n)
Sigma <- t(A) %*% A | How to create an arbitrary covariance matrix
Create an $n\times n$ matrix $A$ with arbitrary values
and then use $\Sigma = A^T A$ as your covariance matrix.
For example
n <- 4
A <- matrix(runif(n^2)*2-1, ncol=n)
Sigma <- t(A) %*% A | How to create an arbitrary covariance matrix
Create an $n\times n$ matrix $A$ with arbitrary values
and then use $\Sigma = A^T A$ as your covariance matrix.
For example
n <- 4
A <- matrix(runif(n^2)*2-1, ncol=n)
Sigma <- t(A) %*% A |
6,513 | How to create an arbitrary covariance matrix | I like to have control over the objects I create, even when they might be arbitrary.
Consider, then, that all possible $n\times n$ covariance matrices $\Sigma$ can be expressed in the form
$$\Sigma= P^\prime\ \text{Diagonal}(\sigma_1,\sigma_2,\ldots, \sigma_n)\ P$$
where $P$ is an orthogonal matrix and $\sigma_1 \ge \s... | How to create an arbitrary covariance matrix | I like to have control over the objects I create, even when they might be arbitrary.
Consider, then, that all possible $n\times n$ covariance matrices $\Sigma$ can be expressed in the form
$$\Sigma= P | How to create an arbitrary covariance matrix
I like to have control over the objects I create, even when they might be arbitrary.
Consider, then, that all possible $n\times n$ covariance matrices $\Sigma$ can be expressed in the form
$$\Sigma= P^\prime\ \text{Diagonal}(\sigma_1,\sigma_2,\ldots, \sigma_n)\ P$$
where $P$... | How to create an arbitrary covariance matrix
I like to have control over the objects I create, even when they might be arbitrary.
Consider, then, that all possible $n\times n$ covariance matrices $\Sigma$ can be expressed in the form
$$\Sigma= P |
6,514 | How to create an arbitrary covariance matrix | You can simulate random positive definite matrices from the Wishart distribution using the function "rWishart" from stats (included in base)
n <- 4
rWishart(1,n,diag(n))
From the documentation for rWishart:
Usage should be:
rWishart(n, df, Sigma)
where,
n: the number of samples.
df: the degrees of freedom, i.e. the ... | How to create an arbitrary covariance matrix | You can simulate random positive definite matrices from the Wishart distribution using the function "rWishart" from stats (included in base)
n <- 4
rWishart(1,n,diag(n))
From the documentation for rW | How to create an arbitrary covariance matrix
You can simulate random positive definite matrices from the Wishart distribution using the function "rWishart" from stats (included in base)
n <- 4
rWishart(1,n,diag(n))
From the documentation for rWishart:
Usage should be:
rWishart(n, df, Sigma)
where,
n: the number of s... | How to create an arbitrary covariance matrix
You can simulate random positive definite matrices from the Wishart distribution using the function "rWishart" from stats (included in base)
n <- 4
rWishart(1,n,diag(n))
From the documentation for rW |
6,515 | How to create an arbitrary covariance matrix | There is a package specifically for that, clusterGeneration (written among other by Harry Joe, a big name in that field).
There are two main functions:
genPositiveDefMat generate a covariance matrix, 4 different methods
rcorrmatrix : generate a correlation matrix
Quick example:
library(clusterGeneration)
#> Loading ... | How to create an arbitrary covariance matrix | There is a package specifically for that, clusterGeneration (written among other by Harry Joe, a big name in that field).
There are two main functions:
genPositiveDefMat generate a covariance matrix | How to create an arbitrary covariance matrix
There is a package specifically for that, clusterGeneration (written among other by Harry Joe, a big name in that field).
There are two main functions:
genPositiveDefMat generate a covariance matrix, 4 different methods
rcorrmatrix : generate a correlation matrix
Quick ex... | How to create an arbitrary covariance matrix
There is a package specifically for that, clusterGeneration (written among other by Harry Joe, a big name in that field).
There are two main functions:
genPositiveDefMat generate a covariance matrix |
6,516 | How to create an arbitrary covariance matrix | In my case I still want the eigenvalues to be randomly drawn so, to control the condition number of the covariance matrix does not exceed, say, $\kappa > 1$, one can generate $\sigma_k = 1 + (\kappa - 1) \cdot \mathcal{U}_k$ where $\mathcal{U}_k$ is uniformly sampled in $[0,1]$ for all $k=1,\dots,n$.
Following @whuber ... | How to create an arbitrary covariance matrix | In my case I still want the eigenvalues to be randomly drawn so, to control the condition number of the covariance matrix does not exceed, say, $\kappa > 1$, one can generate $\sigma_k = 1 + (\kappa - | How to create an arbitrary covariance matrix
In my case I still want the eigenvalues to be randomly drawn so, to control the condition number of the covariance matrix does not exceed, say, $\kappa > 1$, one can generate $\sigma_k = 1 + (\kappa - 1) \cdot \mathcal{U}_k$ where $\mathcal{U}_k$ is uniformly sampled in $[0,... | How to create an arbitrary covariance matrix
In my case I still want the eigenvalues to be randomly drawn so, to control the condition number of the covariance matrix does not exceed, say, $\kappa > 1$, one can generate $\sigma_k = 1 + (\kappa - |
6,517 | How to prove that the radial basis function is a kernel? | Zen used method 1. Here is method 2: Map $x$ to a spherically symmetric Gaussian distribution centered at $x$ in the Hilbert space $L^2$. The standard deviation and a constant factor have to be tweaked for this to work exactly. For example, in one dimension,
$$ \int_{-\infty}^\infty \frac{\exp[-(x-z)^2/(2\sigma^2)]}{\... | How to prove that the radial basis function is a kernel? | Zen used method 1. Here is method 2: Map $x$ to a spherically symmetric Gaussian distribution centered at $x$ in the Hilbert space $L^2$. The standard deviation and a constant factor have to be tweake | How to prove that the radial basis function is a kernel?
Zen used method 1. Here is method 2: Map $x$ to a spherically symmetric Gaussian distribution centered at $x$ in the Hilbert space $L^2$. The standard deviation and a constant factor have to be tweaked for this to work exactly. For example, in one dimension,
$$ ... | How to prove that the radial basis function is a kernel?
Zen used method 1. Here is method 2: Map $x$ to a spherically symmetric Gaussian distribution centered at $x$ in the Hilbert space $L^2$. The standard deviation and a constant factor have to be tweake |
6,518 | How to prove that the radial basis function is a kernel? | I'll add a third method, just for variety: building up the kernel from a sequence of general steps known to create pd kernels. Let $\mathcal X$ denote the domain of the kernels below and $\varphi$ the feature maps.
Scalings:
If $\kappa$ is a pd kernel, so is $\gamma \kappa$ for any constant $\gamma > 0$.
Proof: if $\v... | How to prove that the radial basis function is a kernel? | I'll add a third method, just for variety: building up the kernel from a sequence of general steps known to create pd kernels. Let $\mathcal X$ denote the domain of the kernels below and $\varphi$ the | How to prove that the radial basis function is a kernel?
I'll add a third method, just for variety: building up the kernel from a sequence of general steps known to create pd kernels. Let $\mathcal X$ denote the domain of the kernels below and $\varphi$ the feature maps.
Scalings:
If $\kappa$ is a pd kernel, so is $\g... | How to prove that the radial basis function is a kernel?
I'll add a third method, just for variety: building up the kernel from a sequence of general steps known to create pd kernels. Let $\mathcal X$ denote the domain of the kernels below and $\varphi$ the |
6,519 | How to prove that the radial basis function is a kernel? | I will use method 1. Check Douglas Zare's answer for a proof using method 2.
I will prove the case when $x,y$ are real numbers, so $k(x,y)=\exp(-(x-y)^2/2\sigma^2)$. The general case follows mutatis mutandis from the same argument, and is worth doing.
Without loss of generality, suppose that $\sigma^2=1$.
Write $k(x,y)... | How to prove that the radial basis function is a kernel? | I will use method 1. Check Douglas Zare's answer for a proof using method 2.
I will prove the case when $x,y$ are real numbers, so $k(x,y)=\exp(-(x-y)^2/2\sigma^2)$. The general case follows mutatis m | How to prove that the radial basis function is a kernel?
I will use method 1. Check Douglas Zare's answer for a proof using method 2.
I will prove the case when $x,y$ are real numbers, so $k(x,y)=\exp(-(x-y)^2/2\sigma^2)$. The general case follows mutatis mutandis from the same argument, and is worth doing.
Without los... | How to prove that the radial basis function is a kernel?
I will use method 1. Check Douglas Zare's answer for a proof using method 2.
I will prove the case when $x,y$ are real numbers, so $k(x,y)=\exp(-(x-y)^2/2\sigma^2)$. The general case follows mutatis m |
6,520 | Why use colormap viridis over jet? | See this video. You could also google it because there are a lot of (reasonable) jet-bashing everywhere.
Jet is very pleasing because it is flashy, colorful, and it does not require you to think about your color scale: even if you have just a few outliers, you still get "all the features" in your plot. You said it your... | Why use colormap viridis over jet? | See this video. You could also google it because there are a lot of (reasonable) jet-bashing everywhere.
Jet is very pleasing because it is flashy, colorful, and it does not require you to think about | Why use colormap viridis over jet?
See this video. You could also google it because there are a lot of (reasonable) jet-bashing everywhere.
Jet is very pleasing because it is flashy, colorful, and it does not require you to think about your color scale: even if you have just a few outliers, you still get "all the featu... | Why use colormap viridis over jet?
See this video. You could also google it because there are a lot of (reasonable) jet-bashing everywhere.
Jet is very pleasing because it is flashy, colorful, and it does not require you to think about |
6,521 | Why use colormap viridis over jet? | You need the plot because you need to show data and you need a colormap because you know that the color you show will not be seen equally by all persons: any color is an interpretation through our visual perception.
Indeed, colors are subjective in the sense that they are interpreted by the brain (in the sense that a s... | Why use colormap viridis over jet? | You need the plot because you need to show data and you need a colormap because you know that the color you show will not be seen equally by all persons: any color is an interpretation through our vis | Why use colormap viridis over jet?
You need the plot because you need to show data and you need a colormap because you know that the color you show will not be seen equally by all persons: any color is an interpretation through our visual perception.
Indeed, colors are subjective in the sense that they are interpreted ... | Why use colormap viridis over jet?
You need the plot because you need to show data and you need a colormap because you know that the color you show will not be seen equally by all persons: any color is an interpretation through our vis |
6,522 | Why use colormap viridis over jet? | There's several nice answers here already, but I think it's still pertinent to add another viewpoint, from the excellent paper
Good Colour Maps: How to Design Them. Peter Kovesi. arXiv:1509.03700 (2015). Software available here.
which lays out in a very clear fashion the principles of colour-map design, and provides ... | Why use colormap viridis over jet? | There's several nice answers here already, but I think it's still pertinent to add another viewpoint, from the excellent paper
Good Colour Maps: How to Design Them. Peter Kovesi. arXiv:1509.03700 (20 | Why use colormap viridis over jet?
There's several nice answers here already, but I think it's still pertinent to add another viewpoint, from the excellent paper
Good Colour Maps: How to Design Them. Peter Kovesi. arXiv:1509.03700 (2015). Software available here.
which lays out in a very clear fashion the principles ... | Why use colormap viridis over jet?
There's several nice answers here already, but I think it's still pertinent to add another viewpoint, from the excellent paper
Good Colour Maps: How to Design Them. Peter Kovesi. arXiv:1509.03700 (20 |
6,523 | Why use colormap viridis over jet? | The issue with using any kind of color scale to visually represent ordinal data is that of luminance monotonicity: that is to say, if you have data that satisfy some kind of ordering relationship, that relationship should be reflected not just by changes in hue, but by luminance. The problem with the "jet" color mapp... | Why use colormap viridis over jet? | The issue with using any kind of color scale to visually represent ordinal data is that of luminance monotonicity: that is to say, if you have data that satisfy some kind of ordering relationship, th | Why use colormap viridis over jet?
The issue with using any kind of color scale to visually represent ordinal data is that of luminance monotonicity: that is to say, if you have data that satisfy some kind of ordering relationship, that relationship should be reflected not just by changes in hue, but by luminance. Th... | Why use colormap viridis over jet?
The issue with using any kind of color scale to visually represent ordinal data is that of luminance monotonicity: that is to say, if you have data that satisfy some kind of ordering relationship, th |
6,524 | Does statistical independence mean lack of causation? | So if that's the case, does statistical independence automatically
mean lack of causation?
No, and here's a simple counter example with a multivariate normal,
set.seed(100)
n <- 1e6
a <- 0.2
b <- 0.1
c <- 0.5
z <- rnorm(n)
x <- a*z + sqrt(1-a^2)*rnorm(n)
y <- b*x - c*z + sqrt(1- b^2 - c^2 +2*a*b*c)*rnorm(n)
cor(x, y... | Does statistical independence mean lack of causation? | So if that's the case, does statistical independence automatically
mean lack of causation?
No, and here's a simple counter example with a multivariate normal,
set.seed(100)
n <- 1e6
a <- 0.2
b <- 0 | Does statistical independence mean lack of causation?
So if that's the case, does statistical independence automatically
mean lack of causation?
No, and here's a simple counter example with a multivariate normal,
set.seed(100)
n <- 1e6
a <- 0.2
b <- 0.1
c <- 0.5
z <- rnorm(n)
x <- a*z + sqrt(1-a^2)*rnorm(n)
y <- b*x... | Does statistical independence mean lack of causation?
So if that's the case, does statistical independence automatically
mean lack of causation?
No, and here's a simple counter example with a multivariate normal,
set.seed(100)
n <- 1e6
a <- 0.2
b <- 0 |
6,525 | Does statistical independence mean lack of causation? | Suppose we have a lightbulb controlled by two switches. Let $S_1$ and $S_2$ denote the state of the switches, which can be either 0 or 1. Let $L$ denote the state of the lighbulb, which can be either 0 (off) or 1 (on). We set up the circuit such that the lighbulb is on when the two switches are in different states, and... | Does statistical independence mean lack of causation? | Suppose we have a lightbulb controlled by two switches. Let $S_1$ and $S_2$ denote the state of the switches, which can be either 0 or 1. Let $L$ denote the state of the lighbulb, which can be either | Does statistical independence mean lack of causation?
Suppose we have a lightbulb controlled by two switches. Let $S_1$ and $S_2$ denote the state of the switches, which can be either 0 or 1. Let $L$ denote the state of the lighbulb, which can be either 0 (off) or 1 (on). We set up the circuit such that the lighbulb is... | Does statistical independence mean lack of causation?
Suppose we have a lightbulb controlled by two switches. Let $S_1$ and $S_2$ denote the state of the switches, which can be either 0 or 1. Let $L$ denote the state of the lighbulb, which can be either |
6,526 | Does statistical independence mean lack of causation? | Based on your question, you can think like this:
$ P(A B) = P(A) P(B)$ when $A$ and $B$ are independent. You can similarly imply
$P(AB)/P(A) = P(B|A) = P(B)$. Also,
$P(AB)/P(B) = P(A|B) = P(A)$.
In this regards, I believe that independence means a lack of causation. However, dependence doesn't necessarily imply causati... | Does statistical independence mean lack of causation? | Based on your question, you can think like this:
$ P(A B) = P(A) P(B)$ when $A$ and $B$ are independent. You can similarly imply
$P(AB)/P(A) = P(B|A) = P(B)$. Also,
$P(AB)/P(B) = P(A|B) = P(A)$.
In th | Does statistical independence mean lack of causation?
Based on your question, you can think like this:
$ P(A B) = P(A) P(B)$ when $A$ and $B$ are independent. You can similarly imply
$P(AB)/P(A) = P(B|A) = P(B)$. Also,
$P(AB)/P(B) = P(A|B) = P(A)$.
In this regards, I believe that independence means a lack of causation.... | Does statistical independence mean lack of causation?
Based on your question, you can think like this:
$ P(A B) = P(A) P(B)$ when $A$ and $B$ are independent. You can similarly imply
$P(AB)/P(A) = P(B|A) = P(B)$. Also,
$P(AB)/P(B) = P(A|B) = P(A)$.
In th |
6,527 | Difference between generalized linear models & generalized linear mixed models | The advent of generalized linear models has allowed us to build regression-type models of data when the distribution of the response variable is non-normal--for example, when your DV is binary. (If you would like to know a little more about GLiMs, I wrote a fairly extensive answer here, which may be useful although th... | Difference between generalized linear models & generalized linear mixed models | The advent of generalized linear models has allowed us to build regression-type models of data when the distribution of the response variable is non-normal--for example, when your DV is binary. (If y | Difference between generalized linear models & generalized linear mixed models
The advent of generalized linear models has allowed us to build regression-type models of data when the distribution of the response variable is non-normal--for example, when your DV is binary. (If you would like to know a little more about... | Difference between generalized linear models & generalized linear mixed models
The advent of generalized linear models has allowed us to build regression-type models of data when the distribution of the response variable is non-normal--for example, when your DV is binary. (If y |
6,528 | Difference between generalized linear models & generalized linear mixed models | The key is the introduction of random effects. Gung's link mentions it. But I think it should have been mentioned directly. That is the main difference. | Difference between generalized linear models & generalized linear mixed models | The key is the introduction of random effects. Gung's link mentions it. But I think it should have been mentioned directly. That is the main difference. | Difference between generalized linear models & generalized linear mixed models
The key is the introduction of random effects. Gung's link mentions it. But I think it should have been mentioned directly. That is the main difference. | Difference between generalized linear models & generalized linear mixed models
The key is the introduction of random effects. Gung's link mentions it. But I think it should have been mentioned directly. That is the main difference. |
6,529 | Difference between generalized linear models & generalized linear mixed models | I suggest you also examine answers of a question I asked some time ago:
General Linear Model vs. Generalized Linear Model (with an identity link function?) | Difference between generalized linear models & generalized linear mixed models | I suggest you also examine answers of a question I asked some time ago:
General Linear Model vs. Generalized Linear Model (with an identity link function?) | Difference between generalized linear models & generalized linear mixed models
I suggest you also examine answers of a question I asked some time ago:
General Linear Model vs. Generalized Linear Model (with an identity link function?) | Difference between generalized linear models & generalized linear mixed models
I suggest you also examine answers of a question I asked some time ago:
General Linear Model vs. Generalized Linear Model (with an identity link function?) |
6,530 | Recall and precision in classification | Whether a classifier is “good” really depends on
What else is available for your particular problem. Obviously, you want a classifier to be better than random or naive guesses (e.g. classifying everything as belonging to the most common category) but some things are easier to classify than others.
The cost of differe... | Recall and precision in classification | Whether a classifier is “good” really depends on
What else is available for your particular problem. Obviously, you want a classifier to be better than random or naive guesses (e.g. classifying ever | Recall and precision in classification
Whether a classifier is “good” really depends on
What else is available for your particular problem. Obviously, you want a classifier to be better than random or naive guesses (e.g. classifying everything as belonging to the most common category) but some things are easier to cl... | Recall and precision in classification
Whether a classifier is “good” really depends on
What else is available for your particular problem. Obviously, you want a classifier to be better than random or naive guesses (e.g. classifying ever |
6,531 | Recall and precision in classification | In the context of binary classification, examples are either positive or negative.
The recall addresses the question: "Given a positive example, will the classifier detect it ?"
The precision addresses the question: "Given a positive prediction from the classifier, how likely is it to be correct ?"
So it depends if t... | Recall and precision in classification | In the context of binary classification, examples are either positive or negative.
The recall addresses the question: "Given a positive example, will the classifier detect it ?"
The precision address | Recall and precision in classification
In the context of binary classification, examples are either positive or negative.
The recall addresses the question: "Given a positive example, will the classifier detect it ?"
The precision addresses the question: "Given a positive prediction from the classifier, how likely is ... | Recall and precision in classification
In the context of binary classification, examples are either positive or negative.
The recall addresses the question: "Given a positive example, will the classifier detect it ?"
The precision address |
6,532 | Recall and precision in classification | Moving from continuous predictions, as used in computing ROC area (concordance probability; c-index) to a discontinuous improper scoring rule (forced-choice classification accuracy) results in all kinds of anomalies and will mislead the analyst to choose the wrong predictors and/or the wrong model. It is much better t... | Recall and precision in classification | Moving from continuous predictions, as used in computing ROC area (concordance probability; c-index) to a discontinuous improper scoring rule (forced-choice classification accuracy) results in all kin | Recall and precision in classification
Moving from continuous predictions, as used in computing ROC area (concordance probability; c-index) to a discontinuous improper scoring rule (forced-choice classification accuracy) results in all kinds of anomalies and will mislead the analyst to choose the wrong predictors and/o... | Recall and precision in classification
Moving from continuous predictions, as used in computing ROC area (concordance probability; c-index) to a discontinuous improper scoring rule (forced-choice classification accuracy) results in all kin |
6,533 | Recall and precision in classification | Neither precision nor recall tell the full story, and it is hard to compare a predictor with, say, 90% recall and 60% precision to a predictor with, say, 85% precision and 65% recall - unless, of course, you have cost/benefit associated with each of the 4 cells (tp/fp/tn/fn) in the confusion matrix.
An interesting way ... | Recall and precision in classification | Neither precision nor recall tell the full story, and it is hard to compare a predictor with, say, 90% recall and 60% precision to a predictor with, say, 85% precision and 65% recall - unless, of cour | Recall and precision in classification
Neither precision nor recall tell the full story, and it is hard to compare a predictor with, say, 90% recall and 60% precision to a predictor with, say, 85% precision and 65% recall - unless, of course, you have cost/benefit associated with each of the 4 cells (tp/fp/tn/fn) in th... | Recall and precision in classification
Neither precision nor recall tell the full story, and it is hard to compare a predictor with, say, 90% recall and 60% precision to a predictor with, say, 85% precision and 65% recall - unless, of cour |
6,534 | Effect size as the hypothesis for significance testing | As far as significance testing goes (or anything else that does essentially the same thing as significance testing), I have long thought that the best approach in most situations is likely to be estimating a standardized effect size, with a 95% confidence interval about that effect size. There's nothing really new the... | Effect size as the hypothesis for significance testing | As far as significance testing goes (or anything else that does essentially the same thing as significance testing), I have long thought that the best approach in most situations is likely to be estim | Effect size as the hypothesis for significance testing
As far as significance testing goes (or anything else that does essentially the same thing as significance testing), I have long thought that the best approach in most situations is likely to be estimating a standardized effect size, with a 95% confidence interval ... | Effect size as the hypothesis for significance testing
As far as significance testing goes (or anything else that does essentially the same thing as significance testing), I have long thought that the best approach in most situations is likely to be estim |
6,535 | Effect size as the hypothesis for significance testing | Why do we insist on any form of hypothesis test in statistics?
In the wonderful book Statistics as Principled Argument Robert Abelson argues that statistical analysis is part of a principled argument about the subject in question. He says that, rather than be evaluated as hypotheses to be rejected or not rejected (or e... | Effect size as the hypothesis for significance testing | Why do we insist on any form of hypothesis test in statistics?
In the wonderful book Statistics as Principled Argument Robert Abelson argues that statistical analysis is part of a principled argument | Effect size as the hypothesis for significance testing
Why do we insist on any form of hypothesis test in statistics?
In the wonderful book Statistics as Principled Argument Robert Abelson argues that statistical analysis is part of a principled argument about the subject in question. He says that, rather than be evalu... | Effect size as the hypothesis for significance testing
Why do we insist on any form of hypothesis test in statistics?
In the wonderful book Statistics as Principled Argument Robert Abelson argues that statistical analysis is part of a principled argument |
6,536 | Effect size as the hypothesis for significance testing | Your last question not only makes sense: nowadays sensible industrial statisticians do not test for significant difference but for significant equivalence, that is, testing a null hypothesis of the form $H_0\colon \{|\mu_1-\mu_2|>\epsilon\}$ where $\epsilon$ is set by the user and is indeed related to the notion of "ef... | Effect size as the hypothesis for significance testing | Your last question not only makes sense: nowadays sensible industrial statisticians do not test for significant difference but for significant equivalence, that is, testing a null hypothesis of the fo | Effect size as the hypothesis for significance testing
Your last question not only makes sense: nowadays sensible industrial statisticians do not test for significant difference but for significant equivalence, that is, testing a null hypothesis of the form $H_0\colon \{|\mu_1-\mu_2|>\epsilon\}$ where $\epsilon$ is set... | Effect size as the hypothesis for significance testing
Your last question not only makes sense: nowadays sensible industrial statisticians do not test for significant difference but for significant equivalence, that is, testing a null hypothesis of the fo |
6,537 | Effect size as the hypothesis for significance testing | Traditional hypothesis tests tell you whether there is statistically significant evidence for the existence of an effect, whereas what we often want to know about is the existence of evidence of a practically significant effect.
It is certainly possible to form Bayesian "hypothesis tests" with a minimum effect size (I... | Effect size as the hypothesis for significance testing | Traditional hypothesis tests tell you whether there is statistically significant evidence for the existence of an effect, whereas what we often want to know about is the existence of evidence of a pr | Effect size as the hypothesis for significance testing
Traditional hypothesis tests tell you whether there is statistically significant evidence for the existence of an effect, whereas what we often want to know about is the existence of evidence of a practically significant effect.
It is certainly possible to form Ba... | Effect size as the hypothesis for significance testing
Traditional hypothesis tests tell you whether there is statistically significant evidence for the existence of an effect, whereas what we often want to know about is the existence of evidence of a pr |
6,538 | Effect size as the hypothesis for significance testing | A lot of this comes down to what question you are actually asking, how you design your study, and even what you mean by equal.
I ran accros an interesting little insert in the British Medical Journal once that talked about what people interpreted certain phases to mean. It turns out that "always" can mean that somethi... | Effect size as the hypothesis for significance testing | A lot of this comes down to what question you are actually asking, how you design your study, and even what you mean by equal.
I ran accros an interesting little insert in the British Medical Journal | Effect size as the hypothesis for significance testing
A lot of this comes down to what question you are actually asking, how you design your study, and even what you mean by equal.
I ran accros an interesting little insert in the British Medical Journal once that talked about what people interpreted certain phases to ... | Effect size as the hypothesis for significance testing
A lot of this comes down to what question you are actually asking, how you design your study, and even what you mean by equal.
I ran accros an interesting little insert in the British Medical Journal |
6,539 | Effect size as the hypothesis for significance testing | From an organisational perspective, be it government with policy options or a company looking to roll out a new process/product, the use of a simple cost-benefit analysis can help too. I have argued in the past that (ignoring political reasons) given the known cost of a new initiative, what is the break even point for ... | Effect size as the hypothesis for significance testing | From an organisational perspective, be it government with policy options or a company looking to roll out a new process/product, the use of a simple cost-benefit analysis can help too. I have argued i | Effect size as the hypothesis for significance testing
From an organisational perspective, be it government with policy options or a company looking to roll out a new process/product, the use of a simple cost-benefit analysis can help too. I have argued in the past that (ignoring political reasons) given the known cost... | Effect size as the hypothesis for significance testing
From an organisational perspective, be it government with policy options or a company looking to roll out a new process/product, the use of a simple cost-benefit analysis can help too. I have argued i |
6,540 | What is the intuition behind the formula for conditional probability? | A good intuition is that we are now in the universe in which B occurred, the full circle. Of that circle, how much is also A?
This intuitive explanation was offered in a class by Marc Herman
https://people.math.rochester.edu/faculty/herman/ | What is the intuition behind the formula for conditional probability? | A good intuition is that we are now in the universe in which B occurred, the full circle. Of that circle, how much is also A?
This intuitive explanation was offered in a class by Marc Herman
https:/ | What is the intuition behind the formula for conditional probability?
A good intuition is that we are now in the universe in which B occurred, the full circle. Of that circle, how much is also A?
This intuitive explanation was offered in a class by Marc Herman
https://people.math.rochester.edu/faculty/herman/ | What is the intuition behind the formula for conditional probability?
A good intuition is that we are now in the universe in which B occurred, the full circle. Of that circle, how much is also A?
This intuitive explanation was offered in a class by Marc Herman
https:/ |
6,541 | What is the intuition behind the formula for conditional probability? | I would think of it like this:
I take for granted that you understand the intuition until:
Given that B has occurred, the only way for A to occur is for the even to fall in the intersection of A & B.
and I am going to comment the second image you posted:
Imagine that the entire white rectangle is your sample space $... | What is the intuition behind the formula for conditional probability? | I would think of it like this:
I take for granted that you understand the intuition until:
Given that B has occurred, the only way for A to occur is for the even to fall in the intersection of A & B. | What is the intuition behind the formula for conditional probability?
I would think of it like this:
I take for granted that you understand the intuition until:
Given that B has occurred, the only way for A to occur is for the even to fall in the intersection of A & B.
and I am going to comment the second image you p... | What is the intuition behind the formula for conditional probability?
I would think of it like this:
I take for granted that you understand the intuition until:
Given that B has occurred, the only way for A to occur is for the even to fall in the intersection of A & B. |
6,542 | What is the intuition behind the formula for conditional probability? | You'll see intuition easily thinking about the following problem.
Suppose, you have 10 balls: 6 Black and 4 red. Of Black balls 3 are Awesome and of red balls only 1 is Awesome. How likely it is that a Black ball is also Awesome?
The answer is very easy: it's 50%, because we have 3 Awesome Black balls out of total 6 Bl... | What is the intuition behind the formula for conditional probability? | You'll see intuition easily thinking about the following problem.
Suppose, you have 10 balls: 6 Black and 4 red. Of Black balls 3 are Awesome and of red balls only 1 is Awesome. How likely it is that | What is the intuition behind the formula for conditional probability?
You'll see intuition easily thinking about the following problem.
Suppose, you have 10 balls: 6 Black and 4 red. Of Black balls 3 are Awesome and of red balls only 1 is Awesome. How likely it is that a Black ball is also Awesome?
The answer is very e... | What is the intuition behind the formula for conditional probability?
You'll see intuition easily thinking about the following problem.
Suppose, you have 10 balls: 6 Black and 4 red. Of Black balls 3 are Awesome and of red balls only 1 is Awesome. How likely it is that |
6,543 | What is the intuition behind the formula for conditional probability? | For a basic intuition of the conditional probability formula, I always like using a two way table. Let's say there are 150 students in a yeargroup, of whom 80 are female and 70 male, each of whom must study exactly one language course. The two-way table of students taking different courses is:
| French German... | What is the intuition behind the formula for conditional probability? | For a basic intuition of the conditional probability formula, I always like using a two way table. Let's say there are 150 students in a yeargroup, of whom 80 are female and 70 male, each of whom must | What is the intuition behind the formula for conditional probability?
For a basic intuition of the conditional probability formula, I always like using a two way table. Let's say there are 150 students in a yeargroup, of whom 80 are female and 70 male, each of whom must study exactly one language course. The two-way ta... | What is the intuition behind the formula for conditional probability?
For a basic intuition of the conditional probability formula, I always like using a two way table. Let's say there are 150 students in a yeargroup, of whom 80 are female and 70 male, each of whom must |
6,544 | What is the intuition behind the formula for conditional probability? | I would reverse the logic. The probability that both $A$ and $B$ is either:
The probability $B$ happened, and that given that $A$ happened.
Same but reverse roles for $A$ and $B$
This will give you
$p(A\cap B)=p(B)p(A\mid B)$
If you're looking for a negative to your suggestion, it's while it's true the probabilit... | What is the intuition behind the formula for conditional probability? | I would reverse the logic. The probability that both $A$ and $B$ is either:
The probability $B$ happened, and that given that $A$ happened.
Same but reverse roles for $A$ and $B$
This will give y | What is the intuition behind the formula for conditional probability?
I would reverse the logic. The probability that both $A$ and $B$ is either:
The probability $B$ happened, and that given that $A$ happened.
Same but reverse roles for $A$ and $B$
This will give you
$p(A\cap B)=p(B)p(A\mid B)$
If you're looking ... | What is the intuition behind the formula for conditional probability?
I would reverse the logic. The probability that both $A$ and $B$ is either:
The probability $B$ happened, and that given that $A$ happened.
Same but reverse roles for $A$ and $B$
This will give y |
6,545 | What is the intuition behind the formula for conditional probability? | The Venn diagram doesn't represent probability, it represents the measure of subsets of the event space. A probability is the ratio between two measures; the probability of X is the size of "everything that constitutes X" divided the size of "all the events being considered". Any time you're calculating a probability, ... | What is the intuition behind the formula for conditional probability? | The Venn diagram doesn't represent probability, it represents the measure of subsets of the event space. A probability is the ratio between two measures; the probability of X is the size of "everythin | What is the intuition behind the formula for conditional probability?
The Venn diagram doesn't represent probability, it represents the measure of subsets of the event space. A probability is the ratio between two measures; the probability of X is the size of "everything that constitutes X" divided the size of "all the... | What is the intuition behind the formula for conditional probability?
The Venn diagram doesn't represent probability, it represents the measure of subsets of the event space. A probability is the ratio between two measures; the probability of X is the size of "everythin |
6,546 | What is the intuition behind the formula for conditional probability? | I think the best way to think about this is drawing step-by-step paths.
Let's describe Event B as rolling a $4$ on a fair die - this can be easily shown to have probability $\frac{1}{6}$.
Now let's describe Event A as drawing an Ace from a standard 52-card deck of cards - this can be easily shown to have probability $\... | What is the intuition behind the formula for conditional probability? | I think the best way to think about this is drawing step-by-step paths.
Let's describe Event B as rolling a $4$ on a fair die - this can be easily shown to have probability $\frac{1}{6}$.
Now let's de | What is the intuition behind the formula for conditional probability?
I think the best way to think about this is drawing step-by-step paths.
Let's describe Event B as rolling a $4$ on a fair die - this can be easily shown to have probability $\frac{1}{6}$.
Now let's describe Event A as drawing an Ace from a standard 5... | What is the intuition behind the formula for conditional probability?
I think the best way to think about this is drawing step-by-step paths.
Let's describe Event B as rolling a $4$ on a fair die - this can be easily shown to have probability $\frac{1}{6}$.
Now let's de |
6,547 | What is the intuition behind the formula for conditional probability? | Think of it on terms of counts. Marginal probability is how many times A occurred divided by sample size. Joint probability of A and B is how many times A occurred together with B divided by sample size. Conditional probability of A given B is how many times A occurred together with B divided by how many times B occurr... | What is the intuition behind the formula for conditional probability? | Think of it on terms of counts. Marginal probability is how many times A occurred divided by sample size. Joint probability of A and B is how many times A occurred together with B divided by sample si | What is the intuition behind the formula for conditional probability?
Think of it on terms of counts. Marginal probability is how many times A occurred divided by sample size. Joint probability of A and B is how many times A occurred together with B divided by sample size. Conditional probability of A given B is how ma... | What is the intuition behind the formula for conditional probability?
Think of it on terms of counts. Marginal probability is how many times A occurred divided by sample size. Joint probability of A and B is how many times A occurred together with B divided by sample si |
6,548 | What is the intuition behind the formula for conditional probability? | At the time of writing there is about 10 answers which seem to all miss the most important point: you are essentially right.
In that case, wouldn't the probability of P(A | B) simply be equal to the probability of A intersection B, since that's the only way the event could happen?
This is definitely true. This explai... | What is the intuition behind the formula for conditional probability? | At the time of writing there is about 10 answers which seem to all miss the most important point: you are essentially right.
In that case, wouldn't the probability of P(A | B) simply be equal to the | What is the intuition behind the formula for conditional probability?
At the time of writing there is about 10 answers which seem to all miss the most important point: you are essentially right.
In that case, wouldn't the probability of P(A | B) simply be equal to the probability of A intersection B, since that's the ... | What is the intuition behind the formula for conditional probability?
At the time of writing there is about 10 answers which seem to all miss the most important point: you are essentially right.
In that case, wouldn't the probability of P(A | B) simply be equal to the |
6,549 | What is the intuition behind the formula for conditional probability? | I feel it is more intuitive when we have a concrete data to estimate the probabilities.
Let's use mtcars data as an example, the data looks like this (we only use number of cylinders and transmission type.)
> mtcars[,c("am","cyl")]
am cyl
Mazda RX4 1 6
Mazda RX4 Wag 1 6
Datsun ... | What is the intuition behind the formula for conditional probability? | I feel it is more intuitive when we have a concrete data to estimate the probabilities.
Let's use mtcars data as an example, the data looks like this (we only use number of cylinders and transmission | What is the intuition behind the formula for conditional probability?
I feel it is more intuitive when we have a concrete data to estimate the probabilities.
Let's use mtcars data as an example, the data looks like this (we only use number of cylinders and transmission type.)
> mtcars[,c("am","cyl")]
... | What is the intuition behind the formula for conditional probability?
I feel it is more intuitive when we have a concrete data to estimate the probabilities.
Let's use mtcars data as an example, the data looks like this (we only use number of cylinders and transmission |
6,550 | What is the intuition behind the formula for conditional probability? | If A were a superset of B the probability that A happens is always 1 given that B happened, i.e. P(A|B) = 1. However, B itself may have a probability much smaller than 1.
Consider the following example:
given x is a natural number in 1..100,
A is 'x is an even number'
B is 'x is divisible by 10'
we then have:
P... | What is the intuition behind the formula for conditional probability? | If A were a superset of B the probability that A happens is always 1 given that B happened, i.e. P(A|B) = 1. However, B itself may have a probability much smaller than 1.
Consider the following exam | What is the intuition behind the formula for conditional probability?
If A were a superset of B the probability that A happens is always 1 given that B happened, i.e. P(A|B) = 1. However, B itself may have a probability much smaller than 1.
Consider the following example:
given x is a natural number in 1..100,
A i... | What is the intuition behind the formula for conditional probability?
If A were a superset of B the probability that A happens is always 1 given that B happened, i.e. P(A|B) = 1. However, B itself may have a probability much smaller than 1.
Consider the following exam |
6,551 | What is the intuition behind the formula for conditional probability? | I think your initial statement may be a misunderstanding.
You wrote:
The formula for conditional probability of A happening, once B has happened is:
From your phrasing, it may sound as if there are 2 events "First B happened, and then we want to calculate the probability that A will happen".
This is not the case. (Th... | What is the intuition behind the formula for conditional probability? | I think your initial statement may be a misunderstanding.
You wrote:
The formula for conditional probability of A happening, once B has happened is:
From your phrasing, it may sound as if there are | What is the intuition behind the formula for conditional probability?
I think your initial statement may be a misunderstanding.
You wrote:
The formula for conditional probability of A happening, once B has happened is:
From your phrasing, it may sound as if there are 2 events "First B happened, and then we want to ca... | What is the intuition behind the formula for conditional probability?
I think your initial statement may be a misunderstanding.
You wrote:
The formula for conditional probability of A happening, once B has happened is:
From your phrasing, it may sound as if there are |
6,552 | What is the intuition behind the formula for conditional probability? | The conditioning probability is NOT equal to the probability of intersection. Here is an intuitive answer:
1) $P(B \mid A)$: "We know that $A$ happened. What is the probability that $B$ will happen?"
2: $P(A \cap B)$ : "We don't know if $A$ or $B$ did happen. What is the probability that both will happen?
The differen... | What is the intuition behind the formula for conditional probability? | The conditioning probability is NOT equal to the probability of intersection. Here is an intuitive answer:
1) $P(B \mid A)$: "We know that $A$ happened. What is the probability that $B$ will happen?"
| What is the intuition behind the formula for conditional probability?
The conditioning probability is NOT equal to the probability of intersection. Here is an intuitive answer:
1) $P(B \mid A)$: "We know that $A$ happened. What is the probability that $B$ will happen?"
2: $P(A \cap B)$ : "We don't know if $A$ or $B$ di... | What is the intuition behind the formula for conditional probability?
The conditioning probability is NOT equal to the probability of intersection. Here is an intuitive answer:
1) $P(B \mid A)$: "We know that $A$ happened. What is the probability that $B$ will happen?"
|
6,553 | What is the intuition behind the formula for conditional probability? | You could see P(A∩B)/P(B) as P(A∩B)/(1/x) where x is the average amount for event B, then simplify to x • P(A∩B), which means on average it will skip over events that aren’t B | What is the intuition behind the formula for conditional probability? | You could see P(A∩B)/P(B) as P(A∩B)/(1/x) where x is the average amount for event B, then simplify to x • P(A∩B), which means on average it will skip over events that aren’t B | What is the intuition behind the formula for conditional probability?
You could see P(A∩B)/P(B) as P(A∩B)/(1/x) where x is the average amount for event B, then simplify to x • P(A∩B), which means on average it will skip over events that aren’t B | What is the intuition behind the formula for conditional probability?
You could see P(A∩B)/P(B) as P(A∩B)/(1/x) where x is the average amount for event B, then simplify to x • P(A∩B), which means on average it will skip over events that aren’t B |
6,554 | What is the intuition behind the formula for conditional probability? | The way I think about it is this:
$$P(A|B)=\frac{P(A \cap B)}{P(B)}$$
The key is to understand what each term means and what division by $P(B)$ means in this case (it's a little more confusing because we are dividing by a number which is $\le 1$Also, Do not think about this using a Venn diagram. As far as I know, this ... | What is the intuition behind the formula for conditional probability? | The way I think about it is this:
$$P(A|B)=\frac{P(A \cap B)}{P(B)}$$
The key is to understand what each term means and what division by $P(B)$ means in this case (it's a little more confusing because | What is the intuition behind the formula for conditional probability?
The way I think about it is this:
$$P(A|B)=\frac{P(A \cap B)}{P(B)}$$
The key is to understand what each term means and what division by $P(B)$ means in this case (it's a little more confusing because we are dividing by a number which is $\le 1$Also,... | What is the intuition behind the formula for conditional probability?
The way I think about it is this:
$$P(A|B)=\frac{P(A \cap B)}{P(B)}$$
The key is to understand what each term means and what division by $P(B)$ means in this case (it's a little more confusing because |
6,555 | What is the intuition behind the formula for conditional probability? | Let add a bit of philosophy and heuristic.
The question ends at : wouldn't the probability of P(A|B) simply be equal to the probability of A intersection B.
In line with proof by contradiction;
Let say Answer to your question is YES.
Then, how should we calculate p(A|B).
Is this equal to n(A|B) / n(B) ---- (1)
Wait,... | What is the intuition behind the formula for conditional probability? | Let add a bit of philosophy and heuristic.
The question ends at : wouldn't the probability of P(A|B) simply be equal to the probability of A intersection B.
In line with proof by contradiction;
Let | What is the intuition behind the formula for conditional probability?
Let add a bit of philosophy and heuristic.
The question ends at : wouldn't the probability of P(A|B) simply be equal to the probability of A intersection B.
In line with proof by contradiction;
Let say Answer to your question is YES.
Then, how shou... | What is the intuition behind the formula for conditional probability?
Let add a bit of philosophy and heuristic.
The question ends at : wouldn't the probability of P(A|B) simply be equal to the probability of A intersection B.
In line with proof by contradiction;
Let |
6,556 | How is the cost function from Logistic Regression differentiated | Adapted from the notes in the course, which I don't see available (including this derivation) outside the notes contributed by students within the page of Andrew Ng's Coursera Machine Learning course.
In what follows, the superscript $(i)$ denotes individual measurements or training "examples."
$\small
\frac{\partial ... | How is the cost function from Logistic Regression differentiated | Adapted from the notes in the course, which I don't see available (including this derivation) outside the notes contributed by students within the page of Andrew Ng's Coursera Machine Learning course. | How is the cost function from Logistic Regression differentiated
Adapted from the notes in the course, which I don't see available (including this derivation) outside the notes contributed by students within the page of Andrew Ng's Coursera Machine Learning course.
In what follows, the superscript $(i)$ denotes indivi... | How is the cost function from Logistic Regression differentiated
Adapted from the notes in the course, which I don't see available (including this derivation) outside the notes contributed by students within the page of Andrew Ng's Coursera Machine Learning course. |
6,557 | How is the cost function from Logistic Regression differentiated | To avoid impression of excessive complexity of the matter, let us just see the structure of solution.
With simplification and some abuse of notation, let $G(\theta)$ be a term in sum of $J(\theta)$, and $h = 1/(1+e^{-z})$ is a function of $z(\theta)= x \theta $:
$$ G = y \cdot \log(h)+(1-y)\cdot \log(1-h) $$
We may use... | How is the cost function from Logistic Regression differentiated | To avoid impression of excessive complexity of the matter, let us just see the structure of solution.
With simplification and some abuse of notation, let $G(\theta)$ be a term in sum of $J(\theta)$, a | How is the cost function from Logistic Regression differentiated
To avoid impression of excessive complexity of the matter, let us just see the structure of solution.
With simplification and some abuse of notation, let $G(\theta)$ be a term in sum of $J(\theta)$, and $h = 1/(1+e^{-z})$ is a function of $z(\theta)= x \t... | How is the cost function from Logistic Regression differentiated
To avoid impression of excessive complexity of the matter, let us just see the structure of solution.
With simplification and some abuse of notation, let $G(\theta)$ be a term in sum of $J(\theta)$, a |
6,558 | How is the cost function from Logistic Regression differentiated | For those of us who are not so strong at calculus, but would like to play around with adjusting the cost function and need to find a way to calculate derivatives... a short cut to re-learning calculus is this online tool to automatically provide the derivation, with step by step explanations of the rule.
https://www.de... | How is the cost function from Logistic Regression differentiated | For those of us who are not so strong at calculus, but would like to play around with adjusting the cost function and need to find a way to calculate derivatives... a short cut to re-learning calculus | How is the cost function from Logistic Regression differentiated
For those of us who are not so strong at calculus, but would like to play around with adjusting the cost function and need to find a way to calculate derivatives... a short cut to re-learning calculus is this online tool to automatically provide the deriv... | How is the cost function from Logistic Regression differentiated
For those of us who are not so strong at calculus, but would like to play around with adjusting the cost function and need to find a way to calculate derivatives... a short cut to re-learning calculus |
6,559 | How is the cost function from Logistic Regression differentiated | The credit for this answer goes to Antoni Parellada from the comments, which I think deserves a more prominent place on this page (as it helped me out when many other answers did not). Also, this is not a full derivation but more of a clear statement of $\frac{\partial J(\theta)}{\partial \theta}$. (For full derivation... | How is the cost function from Logistic Regression differentiated | The credit for this answer goes to Antoni Parellada from the comments, which I think deserves a more prominent place on this page (as it helped me out when many other answers did not). Also, this is n | How is the cost function from Logistic Regression differentiated
The credit for this answer goes to Antoni Parellada from the comments, which I think deserves a more prominent place on this page (as it helped me out when many other answers did not). Also, this is not a full derivation but more of a clear statement of $... | How is the cost function from Logistic Regression differentiated
The credit for this answer goes to Antoni Parellada from the comments, which I think deserves a more prominent place on this page (as it helped me out when many other answers did not). Also, this is n |
6,560 | How is the cost function from Logistic Regression differentiated | Another presentation, with matrix notation.
Preparation: $\sigma(t)=\frac{1}{1+e^{-t}}$ has $\frac{d \ln \sigma(t)}{dt}=\sigma(-t)=1-\sigma(t)$ hence $\frac{d \sigma}{dt}=\sigma(1-\sigma)$
and hence $\frac{d \ln (1- \sigma)}{dt}=\sigma$.
We use the convention in which all vectors are column vectors. Let $X$ be the data... | How is the cost function from Logistic Regression differentiated | Another presentation, with matrix notation.
Preparation: $\sigma(t)=\frac{1}{1+e^{-t}}$ has $\frac{d \ln \sigma(t)}{dt}=\sigma(-t)=1-\sigma(t)$ hence $\frac{d \sigma}{dt}=\sigma(1-\sigma)$
and hence $ | How is the cost function from Logistic Regression differentiated
Another presentation, with matrix notation.
Preparation: $\sigma(t)=\frac{1}{1+e^{-t}}$ has $\frac{d \ln \sigma(t)}{dt}=\sigma(-t)=1-\sigma(t)$ hence $\frac{d \sigma}{dt}=\sigma(1-\sigma)$
and hence $\frac{d \ln (1- \sigma)}{dt}=\sigma$.
We use the conven... | How is the cost function from Logistic Regression differentiated
Another presentation, with matrix notation.
Preparation: $\sigma(t)=\frac{1}{1+e^{-t}}$ has $\frac{d \ln \sigma(t)}{dt}=\sigma(-t)=1-\sigma(t)$ hence $\frac{d \sigma}{dt}=\sigma(1-\sigma)$
and hence $ |
6,561 | Under what conditions should one use multilevel/hierarchical analysis? | When the structure of your data is naturally hierarchical or nested, multilevel modeling is a good candidate. More generally, it's one method to model interactions.
A natural example is when your data is from an organized structure such as country, state, districts, where you want to examine effects at those levels. ... | Under what conditions should one use multilevel/hierarchical analysis? | When the structure of your data is naturally hierarchical or nested, multilevel modeling is a good candidate. More generally, it's one method to model interactions.
A natural example is when your dat | Under what conditions should one use multilevel/hierarchical analysis?
When the structure of your data is naturally hierarchical or nested, multilevel modeling is a good candidate. More generally, it's one method to model interactions.
A natural example is when your data is from an organized structure such as country,... | Under what conditions should one use multilevel/hierarchical analysis?
When the structure of your data is naturally hierarchical or nested, multilevel modeling is a good candidate. More generally, it's one method to model interactions.
A natural example is when your dat |
6,562 | Under what conditions should one use multilevel/hierarchical analysis? | The Centre for Multilevel Modelling has some good free online tutorials for multi-level modeling, and they have software tutorials for fitting models in both their MLwiN software and STATA.
Take this as heresy, because I have not read more than a chapter in the book, but Hierarchical linear models: applications and dat... | Under what conditions should one use multilevel/hierarchical analysis? | The Centre for Multilevel Modelling has some good free online tutorials for multi-level modeling, and they have software tutorials for fitting models in both their MLwiN software and STATA.
Take this | Under what conditions should one use multilevel/hierarchical analysis?
The Centre for Multilevel Modelling has some good free online tutorials for multi-level modeling, and they have software tutorials for fitting models in both their MLwiN software and STATA.
Take this as heresy, because I have not read more than a ch... | Under what conditions should one use multilevel/hierarchical analysis?
The Centre for Multilevel Modelling has some good free online tutorials for multi-level modeling, and they have software tutorials for fitting models in both their MLwiN software and STATA.
Take this |
6,563 | Under what conditions should one use multilevel/hierarchical analysis? | Here's another perspective on using multilevel vs. regression models: In an interesting paper by Afshartous and de Leeuw, they show that if the purpose of the modeling is predictive (that is, to predict new observations), the choice of model is different from when the goal is inference (where you try to match the model... | Under what conditions should one use multilevel/hierarchical analysis? | Here's another perspective on using multilevel vs. regression models: In an interesting paper by Afshartous and de Leeuw, they show that if the purpose of the modeling is predictive (that is, to predi | Under what conditions should one use multilevel/hierarchical analysis?
Here's another perspective on using multilevel vs. regression models: In an interesting paper by Afshartous and de Leeuw, they show that if the purpose of the modeling is predictive (that is, to predict new observations), the choice of model is diff... | Under what conditions should one use multilevel/hierarchical analysis?
Here's another perspective on using multilevel vs. regression models: In an interesting paper by Afshartous and de Leeuw, they show that if the purpose of the modeling is predictive (that is, to predi |
6,564 | Under what conditions should one use multilevel/hierarchical analysis? | Here's an example where a multilevel model might be "essential." Suppose you want to rate the "quality" of the education provided by a set of schools using students' test scores. One way to define school quality is in terms of average test performance after taking student characteristics into account. You could conce... | Under what conditions should one use multilevel/hierarchical analysis? | Here's an example where a multilevel model might be "essential." Suppose you want to rate the "quality" of the education provided by a set of schools using students' test scores. One way to define sc | Under what conditions should one use multilevel/hierarchical analysis?
Here's an example where a multilevel model might be "essential." Suppose you want to rate the "quality" of the education provided by a set of schools using students' test scores. One way to define school quality is in terms of average test performa... | Under what conditions should one use multilevel/hierarchical analysis?
Here's an example where a multilevel model might be "essential." Suppose you want to rate the "quality" of the education provided by a set of schools using students' test scores. One way to define sc |
6,565 | Under what conditions should one use multilevel/hierarchical analysis? | Multi-level modelling is appropriate, as the name suggests, when your data have influences occurring at different levels (individual, over time, over domains, etc). Single level modeling assumes everything is occurring at the lowest level. Another thing that a multi-level model does is to introduce correlations among... | Under what conditions should one use multilevel/hierarchical analysis? | Multi-level modelling is appropriate, as the name suggests, when your data have influences occurring at different levels (individual, over time, over domains, etc). Single level modeling assumes ever | Under what conditions should one use multilevel/hierarchical analysis?
Multi-level modelling is appropriate, as the name suggests, when your data have influences occurring at different levels (individual, over time, over domains, etc). Single level modeling assumes everything is occurring at the lowest level. Another... | Under what conditions should one use multilevel/hierarchical analysis?
Multi-level modelling is appropriate, as the name suggests, when your data have influences occurring at different levels (individual, over time, over domains, etc). Single level modeling assumes ever |
6,566 | Under what conditions should one use multilevel/hierarchical analysis? | Generally, speaking a hierarchical bayesian (HB) analysis will lead to efficient and stable individual level estimates unless your data is such that individual level effects are completely homogeneous (an unrealistic scenario). The efficiency and stable parameter estimates of HB models becomes really important when you... | Under what conditions should one use multilevel/hierarchical analysis? | Generally, speaking a hierarchical bayesian (HB) analysis will lead to efficient and stable individual level estimates unless your data is such that individual level effects are completely homogeneous | Under what conditions should one use multilevel/hierarchical analysis?
Generally, speaking a hierarchical bayesian (HB) analysis will lead to efficient and stable individual level estimates unless your data is such that individual level effects are completely homogeneous (an unrealistic scenario). The efficiency and st... | Under what conditions should one use multilevel/hierarchical analysis?
Generally, speaking a hierarchical bayesian (HB) analysis will lead to efficient and stable individual level estimates unless your data is such that individual level effects are completely homogeneous |
6,567 | Under what conditions should one use multilevel/hierarchical analysis? | I learned from Snijders and Bosker, Multilevel Analysis: An introduction to basic and advanced multilevel modeling. It is very well pitched at the beginner I think, it must be because I am a thicko where these things are concerned and it made sense to me.
I second the Gelman and Hill as well, a truly brilliant book. | Under what conditions should one use multilevel/hierarchical analysis? | I learned from Snijders and Bosker, Multilevel Analysis: An introduction to basic and advanced multilevel modeling. It is very well pitched at the beginner I think, it must be because I am a thicko wh | Under what conditions should one use multilevel/hierarchical analysis?
I learned from Snijders and Bosker, Multilevel Analysis: An introduction to basic and advanced multilevel modeling. It is very well pitched at the beginner I think, it must be because I am a thicko where these things are concerned and it made sense ... | Under what conditions should one use multilevel/hierarchical analysis?
I learned from Snijders and Bosker, Multilevel Analysis: An introduction to basic and advanced multilevel modeling. It is very well pitched at the beginner I think, it must be because I am a thicko wh |
6,568 | Under what conditions should one use multilevel/hierarchical analysis? | Multi-level models should be employed when the data are nested in a hierarchical structure, particularly when there are significant differences between higher level units in the dependent variable (e.g., student achievement orientation varies between students, and also between the classes with which the students are ne... | Under what conditions should one use multilevel/hierarchical analysis? | Multi-level models should be employed when the data are nested in a hierarchical structure, particularly when there are significant differences between higher level units in the dependent variable (e. | Under what conditions should one use multilevel/hierarchical analysis?
Multi-level models should be employed when the data are nested in a hierarchical structure, particularly when there are significant differences between higher level units in the dependent variable (e.g., student achievement orientation varies betwee... | Under what conditions should one use multilevel/hierarchical analysis?
Multi-level models should be employed when the data are nested in a hierarchical structure, particularly when there are significant differences between higher level units in the dependent variable (e. |
6,569 | What is the difference between censoring and truncation? | Definitions vary, and the two terms are sometimes used interchangeably. I'll try to explain the most common uses using the following data set:
$$ 1\qquad 1.25\qquad 2\qquad 4 \qquad 5$$
Censoring: some observations will be censored, meaning that we only know that they are below (or above) some bound. This can for insta... | What is the difference between censoring and truncation? | Definitions vary, and the two terms are sometimes used interchangeably. I'll try to explain the most common uses using the following data set:
$$ 1\qquad 1.25\qquad 2\qquad 4 \qquad 5$$
Censoring: som | What is the difference between censoring and truncation?
Definitions vary, and the two terms are sometimes used interchangeably. I'll try to explain the most common uses using the following data set:
$$ 1\qquad 1.25\qquad 2\qquad 4 \qquad 5$$
Censoring: some observations will be censored, meaning that we only know that... | What is the difference between censoring and truncation?
Definitions vary, and the two terms are sometimes used interchangeably. I'll try to explain the most common uses using the following data set:
$$ 1\qquad 1.25\qquad 2\qquad 4 \qquad 5$$
Censoring: som |
6,570 | What is the difference between censoring and truncation? | Just as a perspective from another field (programming), censoring and truncating are two distinct operations.
When working with a sensitive dataset, for example social security numbers and telephone numbers, I might censor it or have it censored prior to access being granted:
123-12-1234 => 999-99-9999
567-56-5678 => 9... | What is the difference between censoring and truncation? | Just as a perspective from another field (programming), censoring and truncating are two distinct operations.
When working with a sensitive dataset, for example social security numbers and telephone n | What is the difference between censoring and truncation?
Just as a perspective from another field (programming), censoring and truncating are two distinct operations.
When working with a sensitive dataset, for example social security numbers and telephone numbers, I might censor it or have it censored prior to access b... | What is the difference between censoring and truncation?
Just as a perspective from another field (programming), censoring and truncating are two distinct operations.
When working with a sensitive dataset, for example social security numbers and telephone n |
6,571 | Examples of PCA where PCs with low variance are "useful" | Here's a cool excerpt from Jolliffe (1982) that I didn't include in my previous answer to the very similar question, "Low variance components in PCA, are they really just noise? Is there any way to test for it?" I find it pretty intuitive.
$\quad$Suppose that it is required to predict the height of the cloud-base, $H$... | Examples of PCA where PCs with low variance are "useful" | Here's a cool excerpt from Jolliffe (1982) that I didn't include in my previous answer to the very similar question, "Low variance components in PCA, are they really just noise? Is there any way to te | Examples of PCA where PCs with low variance are "useful"
Here's a cool excerpt from Jolliffe (1982) that I didn't include in my previous answer to the very similar question, "Low variance components in PCA, are they really just noise? Is there any way to test for it?" I find it pretty intuitive.
$\quad$Suppose that it... | Examples of PCA where PCs with low variance are "useful"
Here's a cool excerpt from Jolliffe (1982) that I didn't include in my previous answer to the very similar question, "Low variance components in PCA, are they really just noise? Is there any way to te |
6,572 | Examples of PCA where PCs with low variance are "useful" | If you have R, there is a good example in the crabs data in the MASS package.
> library(MASS)
> data(crabs)
> head(crabs)
sp sex index FL RW CL CW BD
1 B M 1 8.1 6.7 16.1 19.0 7.0
2 B M 2 8.8 7.7 18.1 20.8 7.4
3 B M 3 9.2 7.8 19.0 22.4 7.7
4 B M 4 9.6 7.9 20.1 23.1 8.2
5 B ... | Examples of PCA where PCs with low variance are "useful" | If you have R, there is a good example in the crabs data in the MASS package.
> library(MASS)
> data(crabs)
> head(crabs)
sp sex index FL RW CL CW BD
1 B M 1 8.1 6.7 16.1 19.0 7.0 | Examples of PCA where PCs with low variance are "useful"
If you have R, there is a good example in the crabs data in the MASS package.
> library(MASS)
> data(crabs)
> head(crabs)
sp sex index FL RW CL CW BD
1 B M 1 8.1 6.7 16.1 19.0 7.0
2 B M 2 8.8 7.7 18.1 20.8 7.4
3 B M 3 9.2 7.8 ... | Examples of PCA where PCs with low variance are "useful"
If you have R, there is a good example in the crabs data in the MASS package.
> library(MASS)
> data(crabs)
> head(crabs)
sp sex index FL RW CL CW BD
1 B M 1 8.1 6.7 16.1 19.0 7.0 |
6,573 | Examples of PCA where PCs with low variance are "useful" | Here are two examples from my experience (chemometrics, optical/vibrational/Raman spectroscopy):
I recently had optical spectroscopy data, where > 99% of the total variance of the raw data was due to changes in the background light (spotlight more or less intense on the measured point, fluorescent lamps switched on/of... | Examples of PCA where PCs with low variance are "useful" | Here are two examples from my experience (chemometrics, optical/vibrational/Raman spectroscopy):
I recently had optical spectroscopy data, where > 99% of the total variance of the raw data was due to | Examples of PCA where PCs with low variance are "useful"
Here are two examples from my experience (chemometrics, optical/vibrational/Raman spectroscopy):
I recently had optical spectroscopy data, where > 99% of the total variance of the raw data was due to changes in the background light (spotlight more or less intens... | Examples of PCA where PCs with low variance are "useful"
Here are two examples from my experience (chemometrics, optical/vibrational/Raman spectroscopy):
I recently had optical spectroscopy data, where > 99% of the total variance of the raw data was due to |
6,574 | Examples of PCA where PCs with low variance are "useful" | I have noticed that PCs with low variance are most helpful when performing a PCA on a covariance matrix where the underlying data are clustered or grouped in some way. If one of the groups has a substantially lower average variance than the other groups, then the smallest PCs would be dominated by that group. However, ... | Examples of PCA where PCs with low variance are "useful" | I have noticed that PCs with low variance are most helpful when performing a PCA on a covariance matrix where the underlying data are clustered or grouped in some way. If one of the groups has a subst | Examples of PCA where PCs with low variance are "useful"
I have noticed that PCs with low variance are most helpful when performing a PCA on a covariance matrix where the underlying data are clustered or grouped in some way. If one of the groups has a substantially lower average variance than the other groups, then the... | Examples of PCA where PCs with low variance are "useful"
I have noticed that PCs with low variance are most helpful when performing a PCA on a covariance matrix where the underlying data are clustered or grouped in some way. If one of the groups has a subst |
6,575 | Examples of PCA where PCs with low variance are "useful" | In this talk (slides) the presenters discuss their use of PCA to discriminate between high variability and low variability features.
They actually prefer the low variability features for anomaly detection, since a significant shift in a low variability dimension is a strong indicator of anomalous behavior. The motivat... | Examples of PCA where PCs with low variance are "useful" | In this talk (slides) the presenters discuss their use of PCA to discriminate between high variability and low variability features.
They actually prefer the low variability features for anomaly detec | Examples of PCA where PCs with low variance are "useful"
In this talk (slides) the presenters discuss their use of PCA to discriminate between high variability and low variability features.
They actually prefer the low variability features for anomaly detection, since a significant shift in a low variability dimension ... | Examples of PCA where PCs with low variance are "useful"
In this talk (slides) the presenters discuss their use of PCA to discriminate between high variability and low variability features.
They actually prefer the low variability features for anomaly detec |
6,576 | Open Source statistical textbooks? | Try IPSUR, Introduction to Probability and Statistics Using R by G. Jay Kerns. It's "free, in the GNU sense of the word".
http://ipsur.r-forge.r-project.org/book/
It's definitely open source - on the download page you can download the LaTeX source or the lyx source used to generate this. | Open Source statistical textbooks? | Try IPSUR, Introduction to Probability and Statistics Using R by G. Jay Kerns. It's "free, in the GNU sense of the word".
http://ipsur.r-forge.r-project.org/book/
It's definitely open source - on the | Open Source statistical textbooks?
Try IPSUR, Introduction to Probability and Statistics Using R by G. Jay Kerns. It's "free, in the GNU sense of the word".
http://ipsur.r-forge.r-project.org/book/
It's definitely open source - on the download page you can download the LaTeX source or the lyx source used to generate t... | Open Source statistical textbooks?
Try IPSUR, Introduction to Probability and Statistics Using R by G. Jay Kerns. It's "free, in the GNU sense of the word".
http://ipsur.r-forge.r-project.org/book/
It's definitely open source - on the |
6,577 | Open Source statistical textbooks? | Michael Lavine: Introduction to Statistical Thought, licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. | Open Source statistical textbooks? | Michael Lavine: Introduction to Statistical Thought, licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. | Open Source statistical textbooks?
Michael Lavine: Introduction to Statistical Thought, licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. | Open Source statistical textbooks?
Michael Lavine: Introduction to Statistical Thought, licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. |
6,578 | Open Source statistical textbooks? | Multivariate statistics with R | Open Source statistical textbooks? | Multivariate statistics with R | Open Source statistical textbooks?
Multivariate statistics with R | Open Source statistical textbooks?
Multivariate statistics with R |
6,579 | Open Source statistical textbooks? | The "Statistics" book on wikibooks | Open Source statistical textbooks? | The "Statistics" book on wikibooks | Open Source statistical textbooks?
The "Statistics" book on wikibooks | Open Source statistical textbooks?
The "Statistics" book on wikibooks |
6,580 | Open Source statistical textbooks? | OpenIntro Statistics is available via CC BY-SA. The LaTeX source code plus the R code to generate every figure in the textbook is also readily available in a single download.
OpenIntro's website also highlights several other freely available statistics textbooks at the beginner, intermediate, and advanced levels. | Open Source statistical textbooks? | OpenIntro Statistics is available via CC BY-SA. The LaTeX source code plus the R code to generate every figure in the textbook is also readily available in a single download.
OpenIntro's website also | Open Source statistical textbooks?
OpenIntro Statistics is available via CC BY-SA. The LaTeX source code plus the R code to generate every figure in the textbook is also readily available in a single download.
OpenIntro's website also highlights several other freely available statistics textbooks at the beginner, inter... | Open Source statistical textbooks?
OpenIntro Statistics is available via CC BY-SA. The LaTeX source code plus the R code to generate every figure in the textbook is also readily available in a single download.
OpenIntro's website also |
6,581 | Open Source statistical textbooks? | Statistical Analysis with the General Linear Model
It covers basic linear models (ANOVA, ANCOVA, multiple regression). I can tell by personal experience that it is really really good book to get into the general framework of linear models, which are very useful in many advanced approaches (e.g., hierarchical modeling). | Open Source statistical textbooks? | Statistical Analysis with the General Linear Model
It covers basic linear models (ANOVA, ANCOVA, multiple regression). I can tell by personal experience that it is really really good book to get into | Open Source statistical textbooks?
Statistical Analysis with the General Linear Model
It covers basic linear models (ANOVA, ANCOVA, multiple regression). I can tell by personal experience that it is really really good book to get into the general framework of linear models, which are very useful in many advanced approa... | Open Source statistical textbooks?
Statistical Analysis with the General Linear Model
It covers basic linear models (ANOVA, ANCOVA, multiple regression). I can tell by personal experience that it is really really good book to get into |
6,582 | Open Source statistical textbooks? | Street-Fighting Mathematics. The Art of Educated Guessing and Opportunistic Problem Solving by Sanjoy Mahajan from MIT. Available under a Creative Commons Noncommercial Share Alike license.
Available as a free download on the MIT Press website (but not from the author's website). | Open Source statistical textbooks? | Street-Fighting Mathematics. The Art of Educated Guessing and Opportunistic Problem Solving by Sanjoy Mahajan from MIT. Available under a Creative Commons Noncommercial Share Alike license.
Available | Open Source statistical textbooks?
Street-Fighting Mathematics. The Art of Educated Guessing and Opportunistic Problem Solving by Sanjoy Mahajan from MIT. Available under a Creative Commons Noncommercial Share Alike license.
Available as a free download on the MIT Press website (but not from the author's website). | Open Source statistical textbooks?
Street-Fighting Mathematics. The Art of Educated Guessing and Opportunistic Problem Solving by Sanjoy Mahajan from MIT. Available under a Creative Commons Noncommercial Share Alike license.
Available |
6,583 | Open Source statistical textbooks? | R programming wiki book | Open Source statistical textbooks? | R programming wiki book | Open Source statistical textbooks?
R programming wiki book | Open Source statistical textbooks?
R programming wiki book |
6,584 | Open Source statistical textbooks? | Collaborative Statistics is CC BY: http://cnx.org/content/col10522/latest/ | Open Source statistical textbooks? | Collaborative Statistics is CC BY: http://cnx.org/content/col10522/latest/ | Open Source statistical textbooks?
Collaborative Statistics is CC BY: http://cnx.org/content/col10522/latest/ | Open Source statistical textbooks?
Collaborative Statistics is CC BY: http://cnx.org/content/col10522/latest/ |
6,585 | Open Source statistical textbooks? | Some googling found Statistics & Probability on CollegeOpenTextbooks.org. Still, be aware that most of CC-ed material is share-aliked (so you must also publish your work on CC) or at least attributed (so you must add info that certain part was copied and from whom). The same works with GFDL (both SA & A), it is even wo... | Open Source statistical textbooks? | Some googling found Statistics & Probability on CollegeOpenTextbooks.org. Still, be aware that most of CC-ed material is share-aliked (so you must also publish your work on CC) or at least attributed | Open Source statistical textbooks?
Some googling found Statistics & Probability on CollegeOpenTextbooks.org. Still, be aware that most of CC-ed material is share-aliked (so you must also publish your work on CC) or at least attributed (so you must add info that certain part was copied and from whom). The same works wit... | Open Source statistical textbooks?
Some googling found Statistics & Probability on CollegeOpenTextbooks.org. Still, be aware that most of CC-ed material is share-aliked (so you must also publish your work on CC) or at least attributed |
6,586 | Open Source statistical textbooks? | Look at Statistics Topics ebook on Amazon by Mehta, and his free web log Statistics Ideas that has lecture slides. Nearly free and better in some pedagogical topics, than the ones you cite on your list of resources. | Open Source statistical textbooks? | Look at Statistics Topics ebook on Amazon by Mehta, and his free web log Statistics Ideas that has lecture slides. Nearly free and better in some pedagogical topics, than the ones you cite on your li | Open Source statistical textbooks?
Look at Statistics Topics ebook on Amazon by Mehta, and his free web log Statistics Ideas that has lecture slides. Nearly free and better in some pedagogical topics, than the ones you cite on your list of resources. | Open Source statistical textbooks?
Look at Statistics Topics ebook on Amazon by Mehta, and his free web log Statistics Ideas that has lecture slides. Nearly free and better in some pedagogical topics, than the ones you cite on your li |
6,587 | How to translate the results from lm() to an equation? | Consider this example:
set.seed(5) # this line will allow you to run these commands on your
# own computer & get *exactly* the same output
x = rnorm(50)
y = rnorm(50)
fit = lm(y~x)
summary(fit)
# Call:
# lm(formula = y ~ x)
#
# Residuals:
# Min 1Q Median 3Q ... | How to translate the results from lm() to an equation? | Consider this example:
set.seed(5) # this line will allow you to run these commands on your
# own computer & get *exactly* the same output
x = rnorm(50)
y = rnorm(5 | How to translate the results from lm() to an equation?
Consider this example:
set.seed(5) # this line will allow you to run these commands on your
# own computer & get *exactly* the same output
x = rnorm(50)
y = rnorm(50)
fit = lm(y~x)
summary(fit)
# Call:
# lm(formula = y ~ x)
#
#... | How to translate the results from lm() to an equation?
Consider this example:
set.seed(5) # this line will allow you to run these commands on your
# own computer & get *exactly* the same output
x = rnorm(50)
y = rnorm(5 |
6,588 | How to translate the results from lm() to an equation? | If what you want is to predict scores using your resulting regression equation, you can construct the equation by hand by typing summary(fit) (if your regression analysis is stored in a variable called fit, for example), and looking at the estimates for each coefficient included in your model.
For example, if you have... | How to translate the results from lm() to an equation? | If what you want is to predict scores using your resulting regression equation, you can construct the equation by hand by typing summary(fit) (if your regression analysis is stored in a variable calle | How to translate the results from lm() to an equation?
If what you want is to predict scores using your resulting regression equation, you can construct the equation by hand by typing summary(fit) (if your regression analysis is stored in a variable called fit, for example), and looking at the estimates for each coeffi... | How to translate the results from lm() to an equation?
If what you want is to predict scores using your resulting regression equation, you can construct the equation by hand by typing summary(fit) (if your regression analysis is stored in a variable calle |
6,589 | How to translate the results from lm() to an equation? | Building on @keithpjolley's answer, this replaces the '+' signs used in the separator with the actual sign of the co-efficient and replaces the 'y' with whatever the model's dependent variable actually is.
The function accepts arguments to 'format', such as 'digits' and 'trim'.
library(dplyr)
model_equation <- functio... | How to translate the results from lm() to an equation? | Building on @keithpjolley's answer, this replaces the '+' signs used in the separator with the actual sign of the co-efficient and replaces the 'y' with whatever the model's dependent variable actuall | How to translate the results from lm() to an equation?
Building on @keithpjolley's answer, this replaces the '+' signs used in the separator with the actual sign of the co-efficient and replaces the 'y' with whatever the model's dependent variable actually is.
The function accepts arguments to 'format', such as 'digits... | How to translate the results from lm() to an equation?
Building on @keithpjolley's answer, this replaces the '+' signs used in the separator with the actual sign of the co-efficient and replaces the 'y' with whatever the model's dependent variable actuall |
6,590 | How to translate the results from lm() to an equation? | If you want to show the equation, like to cut/paste into a doc, but don't want to fuss with putting the entire equation together:
R> library(MASS)
R> crime.lm <- lm(y~., UScrime)
R> cc <- crime.lm$coefficients
R> (eqn <- paste("Y =", paste(round(cc[1],2), paste(round(cc[-1],2), names(cc[-1]), sep=" * ", collapse=" + ")... | How to translate the results from lm() to an equation? | If you want to show the equation, like to cut/paste into a doc, but don't want to fuss with putting the entire equation together:
R> library(MASS)
R> crime.lm <- lm(y~., UScrime)
R> cc <- crime.lm$coe | How to translate the results from lm() to an equation?
If you want to show the equation, like to cut/paste into a doc, but don't want to fuss with putting the entire equation together:
R> library(MASS)
R> crime.lm <- lm(y~., UScrime)
R> cc <- crime.lm$coefficients
R> (eqn <- paste("Y =", paste(round(cc[1],2), paste(rou... | How to translate the results from lm() to an equation?
If you want to show the equation, like to cut/paste into a doc, but don't want to fuss with putting the entire equation together:
R> library(MASS)
R> crime.lm <- lm(y~., UScrime)
R> cc <- crime.lm$coe |
6,591 | How to translate the results from lm() to an equation? | You can use the equatiomatic package to solve many challenges with extracting and reporting equations. https://github.com/datalorax/equatiomatic
Basic Example
Inserting Model Coefficients
You can also include the coefficients. | How to translate the results from lm() to an equation? | You can use the equatiomatic package to solve many challenges with extracting and reporting equations. https://github.com/datalorax/equatiomatic
Basic Example
Inserting Model Coefficients
You can als | How to translate the results from lm() to an equation?
You can use the equatiomatic package to solve many challenges with extracting and reporting equations. https://github.com/datalorax/equatiomatic
Basic Example
Inserting Model Coefficients
You can also include the coefficients. | How to translate the results from lm() to an equation?
You can use the equatiomatic package to solve many challenges with extracting and reporting equations. https://github.com/datalorax/equatiomatic
Basic Example
Inserting Model Coefficients
You can als |
6,592 | What is the difference between a stationary test and a unit root test? | I don't know how those tests work in detail, but one difference is that ADF test uses null hypothesis that a series contains a unit root, while KPSS test uses null hypothesis that the series is stationary.
Here is wikipedia passage that might be useful:
In econometrics, Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests a... | What is the difference between a stationary test and a unit root test? | I don't know how those tests work in detail, but one difference is that ADF test uses null hypothesis that a series contains a unit root, while KPSS test uses null hypothesis that the series is statio | What is the difference between a stationary test and a unit root test?
I don't know how those tests work in detail, but one difference is that ADF test uses null hypothesis that a series contains a unit root, while KPSS test uses null hypothesis that the series is stationary.
Here is wikipedia passage that might be use... | What is the difference between a stationary test and a unit root test?
I don't know how those tests work in detail, but one difference is that ADF test uses null hypothesis that a series contains a unit root, while KPSS test uses null hypothesis that the series is statio |
6,593 | What is the difference between a stationary test and a unit root test? | The Concepts and examples of Unit-root tests and stationarity tests
Concept of Unit-root tests:
Null hypothesis: Unit-root
Alternative hypothesis: Process has root outside the unit circle, which is usually equivalent to stationarity or trend stationarity
Concept of Stationarity tests
Null hypothesis: (Trend) Stationar... | What is the difference between a stationary test and a unit root test? | The Concepts and examples of Unit-root tests and stationarity tests
Concept of Unit-root tests:
Null hypothesis: Unit-root
Alternative hypothesis: Process has root outside the unit circle, which is u | What is the difference between a stationary test and a unit root test?
The Concepts and examples of Unit-root tests and stationarity tests
Concept of Unit-root tests:
Null hypothesis: Unit-root
Alternative hypothesis: Process has root outside the unit circle, which is usually equivalent to stationarity or trend statio... | What is the difference between a stationary test and a unit root test?
The Concepts and examples of Unit-root tests and stationarity tests
Concept of Unit-root tests:
Null hypothesis: Unit-root
Alternative hypothesis: Process has root outside the unit circle, which is u |
6,594 | What is the difference between a stationary test and a unit root test? | I don't totally agree with the accepted answer: the null hypothesis of the KPSS test is not stationarity, but trend stationarity, which is quite a different concept.
To summarize:
KPSS test:
Null Hypothesis: the process is trend-stationary
Alternative Hypothesis: the process has a unit root (this is how the authors of... | What is the difference between a stationary test and a unit root test? | I don't totally agree with the accepted answer: the null hypothesis of the KPSS test is not stationarity, but trend stationarity, which is quite a different concept.
To summarize:
KPSS test:
Null Hyp | What is the difference between a stationary test and a unit root test?
I don't totally agree with the accepted answer: the null hypothesis of the KPSS test is not stationarity, but trend stationarity, which is quite a different concept.
To summarize:
KPSS test:
Null Hypothesis: the process is trend-stationary
Alternat... | What is the difference between a stationary test and a unit root test?
I don't totally agree with the accepted answer: the null hypothesis of the KPSS test is not stationarity, but trend stationarity, which is quite a different concept.
To summarize:
KPSS test:
Null Hyp |
6,595 | What is the difference between a stationary test and a unit root test? | I don't know the specifics of the two tests you mentioned but I can address the general question posed in the title of your question and maybe that applies to these specific tests. Stationarity is a property of stochastic processes (or time series in particular) where the joint distribution of any k consecutive observ... | What is the difference between a stationary test and a unit root test? | I don't know the specifics of the two tests you mentioned but I can address the general question posed in the title of your question and maybe that applies to these specific tests. Stationarity is a | What is the difference between a stationary test and a unit root test?
I don't know the specifics of the two tests you mentioned but I can address the general question posed in the title of your question and maybe that applies to these specific tests. Stationarity is a property of stochastic processes (or time series ... | What is the difference between a stationary test and a unit root test?
I don't know the specifics of the two tests you mentioned but I can address the general question posed in the title of your question and maybe that applies to these specific tests. Stationarity is a |
6,596 | What exactly is a seed in a random number generator? | Most pseudo-random number generators (PRNGs) are build on algorithms involving some kind of recursive method starting from a base value that is determined by an input called the "seed". The default PRNG in most statistical software (R, Python, Stata, etc.) is the Mersenne Twister algorithm MT19937, which is set out in... | What exactly is a seed in a random number generator? | Most pseudo-random number generators (PRNGs) are build on algorithms involving some kind of recursive method starting from a base value that is determined by an input called the "seed". The default P | What exactly is a seed in a random number generator?
Most pseudo-random number generators (PRNGs) are build on algorithms involving some kind of recursive method starting from a base value that is determined by an input called the "seed". The default PRNG in most statistical software (R, Python, Stata, etc.) is the Me... | What exactly is a seed in a random number generator?
Most pseudo-random number generators (PRNGs) are build on algorithms involving some kind of recursive method starting from a base value that is determined by an input called the "seed". The default P |
6,597 | What exactly is a seed in a random number generator? | First, there is no true randomness in today's computer-generated "random numbers." All pseudorandom generators use
deterministic methods. (Possibly, quantum computers will change that.)
The difficult task is to contrive algorithms
that produce output that cannot meaningfully be distinguished from
data coming from a tru... | What exactly is a seed in a random number generator? | First, there is no true randomness in today's computer-generated "random numbers." All pseudorandom generators use
deterministic methods. (Possibly, quantum computers will change that.)
The difficult | What exactly is a seed in a random number generator?
First, there is no true randomness in today's computer-generated "random numbers." All pseudorandom generators use
deterministic methods. (Possibly, quantum computers will change that.)
The difficult task is to contrive algorithms
that produce output that cannot mean... | What exactly is a seed in a random number generator?
First, there is no true randomness in today's computer-generated "random numbers." All pseudorandom generators use
deterministic methods. (Possibly, quantum computers will change that.)
The difficult |
6,598 | What exactly is a seed in a random number generator? | TL;DR;
A seed usually enables you to reproduce the sequence of random numbers. In that sense they are not true random numbers but "pseudo random numbers", hence a PNR Generator (PNRG). These are a real help in real life!
A bit more detail:
Virtually all "random" number generators implemented in computer languages are p... | What exactly is a seed in a random number generator? | TL;DR;
A seed usually enables you to reproduce the sequence of random numbers. In that sense they are not true random numbers but "pseudo random numbers", hence a PNR Generator (PNRG). These are a rea | What exactly is a seed in a random number generator?
TL;DR;
A seed usually enables you to reproduce the sequence of random numbers. In that sense they are not true random numbers but "pseudo random numbers", hence a PNR Generator (PNRG). These are a real help in real life!
A bit more detail:
Virtually all "random" numb... | What exactly is a seed in a random number generator?
TL;DR;
A seed usually enables you to reproduce the sequence of random numbers. In that sense they are not true random numbers but "pseudo random numbers", hence a PNR Generator (PNRG). These are a rea |
6,599 | Why is the mean function in Gaussian Process uninteresting? | I think I know what the speaker was getting at. Personally I don't completely agree with her/him, and there's a lot of people who don't. But to be fair, there are also many who do :) First of all, note that specifying the covariance function (kernel) implies specifying a prior distribution over functions. Just by chang... | Why is the mean function in Gaussian Process uninteresting? | I think I know what the speaker was getting at. Personally I don't completely agree with her/him, and there's a lot of people who don't. But to be fair, there are also many who do :) First of all, not | Why is the mean function in Gaussian Process uninteresting?
I think I know what the speaker was getting at. Personally I don't completely agree with her/him, and there's a lot of people who don't. But to be fair, there are also many who do :) First of all, note that specifying the covariance function (kernel) implies s... | Why is the mean function in Gaussian Process uninteresting?
I think I know what the speaker was getting at. Personally I don't completely agree with her/him, and there's a lot of people who don't. But to be fair, there are also many who do :) First of all, not |
6,600 | Why is the mean function in Gaussian Process uninteresting? | Well one very good reason is that the mean function may not live in the the space of functions you wish to model. each input point, $x_i$, may have a corresponding posterior mean, $\mu(x_i)$. However, these posterior mean points are the expectation before you see any other data. So there are many cases where no situati... | Why is the mean function in Gaussian Process uninteresting? | Well one very good reason is that the mean function may not live in the the space of functions you wish to model. each input point, $x_i$, may have a corresponding posterior mean, $\mu(x_i)$. However, | Why is the mean function in Gaussian Process uninteresting?
Well one very good reason is that the mean function may not live in the the space of functions you wish to model. each input point, $x_i$, may have a corresponding posterior mean, $\mu(x_i)$. However, these posterior mean points are the expectation before you ... | Why is the mean function in Gaussian Process uninteresting?
Well one very good reason is that the mean function may not live in the the space of functions you wish to model. each input point, $x_i$, may have a corresponding posterior mean, $\mu(x_i)$. However, |
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