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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15,601 | What is/are the "mechanical" difference between multiple linear regression with lags and time series? | Why create a whole new method, i.e., time series (ARIMA), instead of using multiple linear regression and adding lagged variables to it (with the order of lags determined using ACF and PACF)?
One immediate point is that a linear regression only works with observed variables while ARIMA incorporates unobserved variable... | What is/are the "mechanical" difference between multiple linear regression with lags and time series | Why create a whole new method, i.e., time series (ARIMA), instead of using multiple linear regression and adding lagged variables to it (with the order of lags determined using ACF and PACF)?
One imm | What is/are the "mechanical" difference between multiple linear regression with lags and time series?
Why create a whole new method, i.e., time series (ARIMA), instead of using multiple linear regression and adding lagged variables to it (with the order of lags determined using ACF and PACF)?
One immediate point is th... | What is/are the "mechanical" difference between multiple linear regression with lags and time series
Why create a whole new method, i.e., time series (ARIMA), instead of using multiple linear regression and adding lagged variables to it (with the order of lags determined using ACF and PACF)?
One imm |
15,602 | What is/are the "mechanical" difference between multiple linear regression with lags and time series? | That's a great question. The real difference between ARIMA models and multiple linear regression lies in your error structure. You can manipulate the independent variables in a multiple linear regression model so that they fit your time series data, which is what @IrishStat is saying. However, after that, you need to i... | What is/are the "mechanical" difference between multiple linear regression with lags and time series | That's a great question. The real difference between ARIMA models and multiple linear regression lies in your error structure. You can manipulate the independent variables in a multiple linear regress | What is/are the "mechanical" difference between multiple linear regression with lags and time series?
That's a great question. The real difference between ARIMA models and multiple linear regression lies in your error structure. You can manipulate the independent variables in a multiple linear regression model so that ... | What is/are the "mechanical" difference between multiple linear regression with lags and time series
That's a great question. The real difference between ARIMA models and multiple linear regression lies in your error structure. You can manipulate the independent variables in a multiple linear regress |
15,603 | What is/are the "mechanical" difference between multiple linear regression with lags and time series? | Good question, I actually have built both in my day job as a Data Scientist. Time series models are easy to build (forecast package in R lets you build one in less in 5 seconds), the same or more accurate than regression models, etc. Generally, one should always build time series, then regression. There are Philosophic... | What is/are the "mechanical" difference between multiple linear regression with lags and time series | Good question, I actually have built both in my day job as a Data Scientist. Time series models are easy to build (forecast package in R lets you build one in less in 5 seconds), the same or more accu | What is/are the "mechanical" difference between multiple linear regression with lags and time series?
Good question, I actually have built both in my day job as a Data Scientist. Time series models are easy to build (forecast package in R lets you build one in less in 5 seconds), the same or more accurate than regressi... | What is/are the "mechanical" difference between multiple linear regression with lags and time series
Good question, I actually have built both in my day job as a Data Scientist. Time series models are easy to build (forecast package in R lets you build one in less in 5 seconds), the same or more accu |
15,604 | What is/are the "mechanical" difference between multiple linear regression with lags and time series? | In think the deepest difference between transfer functions and multipe linear regression (in its usual use) lies in their objectives, multiple regressions is oriented to find the main causal observable determinants of the dependent variable while transfer functions just want to forecasts the effect on a dependent varia... | What is/are the "mechanical" difference between multiple linear regression with lags and time series | In think the deepest difference between transfer functions and multipe linear regression (in its usual use) lies in their objectives, multiple regressions is oriented to find the main causal observabl | What is/are the "mechanical" difference between multiple linear regression with lags and time series?
In think the deepest difference between transfer functions and multipe linear regression (in its usual use) lies in their objectives, multiple regressions is oriented to find the main causal observable determinants of ... | What is/are the "mechanical" difference between multiple linear regression with lags and time series
In think the deepest difference between transfer functions and multipe linear regression (in its usual use) lies in their objectives, multiple regressions is oriented to find the main causal observabl |
15,605 | Can we use categorical independent variable in discriminant analysis? | Discriminant analysis assumes a multivariate normal distribution because what we usually consider to be predictors are really a multivariate dependent variable, and the grouping variable is considered to be a predictor. This means that categorical variables that are to be treated as predictors in the sense you wish ar... | Can we use categorical independent variable in discriminant analysis? | Discriminant analysis assumes a multivariate normal distribution because what we usually consider to be predictors are really a multivariate dependent variable, and the grouping variable is considered | Can we use categorical independent variable in discriminant analysis?
Discriminant analysis assumes a multivariate normal distribution because what we usually consider to be predictors are really a multivariate dependent variable, and the grouping variable is considered to be a predictor. This means that categorical v... | Can we use categorical independent variable in discriminant analysis?
Discriminant analysis assumes a multivariate normal distribution because what we usually consider to be predictors are really a multivariate dependent variable, and the grouping variable is considered |
15,606 | Can we use categorical independent variable in discriminant analysis? | The short answer is rather no than yes.
One preliminary note. It is difficult to say whether the variables which produce discriminant functions out of themselves should be called "independent" or "dependent". LDA is basically a specific case of Canonical correlation analysis, and therefore it is ambidirectional. It can... | Can we use categorical independent variable in discriminant analysis? | The short answer is rather no than yes.
One preliminary note. It is difficult to say whether the variables which produce discriminant functions out of themselves should be called "independent" or "dep | Can we use categorical independent variable in discriminant analysis?
The short answer is rather no than yes.
One preliminary note. It is difficult to say whether the variables which produce discriminant functions out of themselves should be called "independent" or "dependent". LDA is basically a specific case of Canon... | Can we use categorical independent variable in discriminant analysis?
The short answer is rather no than yes.
One preliminary note. It is difficult to say whether the variables which produce discriminant functions out of themselves should be called "independent" or "dep |
15,607 | When doing a t-test for the significance of a regression coefficient, why is the number of degrees of freedom $n-p-1$? | You lose one degree of freedom for each estimated mean parameter. For an ordinary t-test that's 1 (the mean). For regression, each predictor costs you a degree of freedom. The extra one is for the intercept.
More specifically, the degrees of freedom come from the denominator in the t-test, which is based on the residua... | When doing a t-test for the significance of a regression coefficient, why is the number of degrees o | You lose one degree of freedom for each estimated mean parameter. For an ordinary t-test that's 1 (the mean). For regression, each predictor costs you a degree of freedom. The extra one is for the int | When doing a t-test for the significance of a regression coefficient, why is the number of degrees of freedom $n-p-1$?
You lose one degree of freedom for each estimated mean parameter. For an ordinary t-test that's 1 (the mean). For regression, each predictor costs you a degree of freedom. The extra one is for the inte... | When doing a t-test for the significance of a regression coefficient, why is the number of degrees o
You lose one degree of freedom for each estimated mean parameter. For an ordinary t-test that's 1 (the mean). For regression, each predictor costs you a degree of freedom. The extra one is for the int |
15,608 | When doing a t-test for the significance of a regression coefficient, why is the number of degrees of freedom $n-p-1$? | This is not true in general. The number of degrees of freedom of t-test depends on a specific model. They're talking about linear regression. So, t-test for an estimator has $n-p-1$ degrees of freedom where $p$ is number of explanatory parameters in the model. | When doing a t-test for the significance of a regression coefficient, why is the number of degrees o | This is not true in general. The number of degrees of freedom of t-test depends on a specific model. They're talking about linear regression. So, t-test for an estimator has $n-p-1$ degrees of freedom | When doing a t-test for the significance of a regression coefficient, why is the number of degrees of freedom $n-p-1$?
This is not true in general. The number of degrees of freedom of t-test depends on a specific model. They're talking about linear regression. So, t-test for an estimator has $n-p-1$ degrees of freedom ... | When doing a t-test for the significance of a regression coefficient, why is the number of degrees o
This is not true in general. The number of degrees of freedom of t-test depends on a specific model. They're talking about linear regression. So, t-test for an estimator has $n-p-1$ degrees of freedom |
15,609 | When doing a t-test for the significance of a regression coefficient, why is the number of degrees of freedom $n-p-1$? | Degrees of freedom is the number of independent values or quantities which can be assigned to a statistical distribution.
So in this case its n−p−1 because:
n is number of training samples.
p is number of predictors.
1 is for intercept. | When doing a t-test for the significance of a regression coefficient, why is the number of degrees o | Degrees of freedom is the number of independent values or quantities which can be assigned to a statistical distribution.
So in this case its n−p−1 because:
n is number of training samples.
p is numbe | When doing a t-test for the significance of a regression coefficient, why is the number of degrees of freedom $n-p-1$?
Degrees of freedom is the number of independent values or quantities which can be assigned to a statistical distribution.
So in this case its n−p−1 because:
n is number of training samples.
p is number... | When doing a t-test for the significance of a regression coefficient, why is the number of degrees o
Degrees of freedom is the number of independent values or quantities which can be assigned to a statistical distribution.
So in this case its n−p−1 because:
n is number of training samples.
p is numbe |
15,610 | Overdispersion in logistic regression | A binomial random variable with $N$ trials and probability of success $p$ can take more than two values. The binomial random variable represents the number of successes in those $N$ trials, and can in fact take $N+1$ different values ($0,1,2,3,...,N$). So if the variance of that distribution is greater than too be ex... | Overdispersion in logistic regression | A binomial random variable with $N$ trials and probability of success $p$ can take more than two values. The binomial random variable represents the number of successes in those $N$ trials, and can i | Overdispersion in logistic regression
A binomial random variable with $N$ trials and probability of success $p$ can take more than two values. The binomial random variable represents the number of successes in those $N$ trials, and can in fact take $N+1$ different values ($0,1,2,3,...,N$). So if the variance of that ... | Overdispersion in logistic regression
A binomial random variable with $N$ trials and probability of success $p$ can take more than two values. The binomial random variable represents the number of successes in those $N$ trials, and can i |
15,611 | Overdispersion in logistic regression | As already noted by others, overdispersion doesn't apply in the case of a Bernoulli (0/1) variable, since in that case, the mean necessarily determines the variance. In the context of logistic regression, this means that if your outcome is binary, you can't estimate a dispersion parameter. (N.B. This does not mean th... | Overdispersion in logistic regression | As already noted by others, overdispersion doesn't apply in the case of a Bernoulli (0/1) variable, since in that case, the mean necessarily determines the variance. In the context of logistic regres | Overdispersion in logistic regression
As already noted by others, overdispersion doesn't apply in the case of a Bernoulli (0/1) variable, since in that case, the mean necessarily determines the variance. In the context of logistic regression, this means that if your outcome is binary, you can't estimate a dispersion p... | Overdispersion in logistic regression
As already noted by others, overdispersion doesn't apply in the case of a Bernoulli (0/1) variable, since in that case, the mean necessarily determines the variance. In the context of logistic regres |
15,612 | Performing PCA with only a distance matrix | Update: I entirely removed my original answer, because it was based on a confusion between Euclidean distances and scalar products. This is a new version of my answer. Apologies.
If by pairwise distances you mean Euclidean distances, then yes, there is a way to perform PCA and to find principal components. I describe t... | Performing PCA with only a distance matrix | Update: I entirely removed my original answer, because it was based on a confusion between Euclidean distances and scalar products. This is a new version of my answer. Apologies.
If by pairwise distan | Performing PCA with only a distance matrix
Update: I entirely removed my original answer, because it was based on a confusion between Euclidean distances and scalar products. This is a new version of my answer. Apologies.
If by pairwise distances you mean Euclidean distances, then yes, there is a way to perform PCA and... | Performing PCA with only a distance matrix
Update: I entirely removed my original answer, because it was based on a confusion between Euclidean distances and scalar products. This is a new version of my answer. Apologies.
If by pairwise distan |
15,613 | Performing PCA with only a distance matrix | PCA with a distance matrix exists, and it is called Multi-dimensional scaling (MDS). You can learn more on wikipedia or in this book.
You can do it in R with mds function cmdscale. For a sample x, you can check that prcomp(x) and cmdscale(dist(x)) give the same result (where prcomp does PCA and dist just computes eucli... | Performing PCA with only a distance matrix | PCA with a distance matrix exists, and it is called Multi-dimensional scaling (MDS). You can learn more on wikipedia or in this book.
You can do it in R with mds function cmdscale. For a sample x, you | Performing PCA with only a distance matrix
PCA with a distance matrix exists, and it is called Multi-dimensional scaling (MDS). You can learn more on wikipedia or in this book.
You can do it in R with mds function cmdscale. For a sample x, you can check that prcomp(x) and cmdscale(dist(x)) give the same result (where p... | Performing PCA with only a distance matrix
PCA with a distance matrix exists, and it is called Multi-dimensional scaling (MDS). You can learn more on wikipedia or in this book.
You can do it in R with mds function cmdscale. For a sample x, you |
15,614 | Performing PCA with only a distance matrix | This looks like a problem that spectral clustering could be applied to. Since you have the pairwise distance matrix, you can define a fully connected graph where each node has N connections, corresponding to its distance from every other node in the graph. From this, you can compute the graph Laplacian (if this sounds ... | Performing PCA with only a distance matrix | This looks like a problem that spectral clustering could be applied to. Since you have the pairwise distance matrix, you can define a fully connected graph where each node has N connections, correspon | Performing PCA with only a distance matrix
This looks like a problem that spectral clustering could be applied to. Since you have the pairwise distance matrix, you can define a fully connected graph where each node has N connections, corresponding to its distance from every other node in the graph. From this, you can c... | Performing PCA with only a distance matrix
This looks like a problem that spectral clustering could be applied to. Since you have the pairwise distance matrix, you can define a fully connected graph where each node has N connections, correspon |
15,615 | Performing PCA with only a distance matrix | The pairwise distances also form a square matrix just like the co-variance matrix. PCA is just SVD (http://en.wikipedia.org/wiki/Singular_value_decomposition) applied to the co-variance matrix. You should still be able to do dimension reduction using SVD on your data. I'm not exactly sure how to interpret your output... | Performing PCA with only a distance matrix | The pairwise distances also form a square matrix just like the co-variance matrix. PCA is just SVD (http://en.wikipedia.org/wiki/Singular_value_decomposition) applied to the co-variance matrix. You s | Performing PCA with only a distance matrix
The pairwise distances also form a square matrix just like the co-variance matrix. PCA is just SVD (http://en.wikipedia.org/wiki/Singular_value_decomposition) applied to the co-variance matrix. You should still be able to do dimension reduction using SVD on your data. I'm no... | Performing PCA with only a distance matrix
The pairwise distances also form a square matrix just like the co-variance matrix. PCA is just SVD (http://en.wikipedia.org/wiki/Singular_value_decomposition) applied to the co-variance matrix. You s |
15,616 | Understanding Metropolis-Hastings with asymmetric proposal distribution | The bibliography states that if q is a symmetric distribution the ratio q(x|y)/q(y|x) becomes 1 and the algorithm is called Metropolis. Is that correct?
Yes, this is correct. The Metropolis algorithm is a special case of the MH algorithm.
What about "Random Walk" Metropolis(-Hastings)? How does it differ from the ot... | Understanding Metropolis-Hastings with asymmetric proposal distribution | The bibliography states that if q is a symmetric distribution the ratio q(x|y)/q(y|x) becomes 1 and the algorithm is called Metropolis. Is that correct?
Yes, this is correct. The Metropolis algorith | Understanding Metropolis-Hastings with asymmetric proposal distribution
The bibliography states that if q is a symmetric distribution the ratio q(x|y)/q(y|x) becomes 1 and the algorithm is called Metropolis. Is that correct?
Yes, this is correct. The Metropolis algorithm is a special case of the MH algorithm.
What a... | Understanding Metropolis-Hastings with asymmetric proposal distribution
The bibliography states that if q is a symmetric distribution the ratio q(x|y)/q(y|x) becomes 1 and the algorithm is called Metropolis. Is that correct?
Yes, this is correct. The Metropolis algorith |
15,617 | Generate normally distributed random numbers with non positive-definite covariance matrix | Solution Method A:
If C is not symmetric, then symmetrize it. D <-- $0.5(C + C^T)$
Add a multiple of the Identity matrix to the symmetrized C sufficient to make it positive definite with whatever margin, m, is desired, i.e., such that smallest eigenvalue of new matrix has minimum eigenvalue = m. Specifically, D <-- $... | Generate normally distributed random numbers with non positive-definite covariance matrix | Solution Method A:
If C is not symmetric, then symmetrize it. D <-- $0.5(C + C^T)$
Add a multiple of the Identity matrix to the symmetrized C sufficient to make it positive definite with whatever mar | Generate normally distributed random numbers with non positive-definite covariance matrix
Solution Method A:
If C is not symmetric, then symmetrize it. D <-- $0.5(C + C^T)$
Add a multiple of the Identity matrix to the symmetrized C sufficient to make it positive definite with whatever margin, m, is desired, i.e., such... | Generate normally distributed random numbers with non positive-definite covariance matrix
Solution Method A:
If C is not symmetric, then symmetrize it. D <-- $0.5(C + C^T)$
Add a multiple of the Identity matrix to the symmetrized C sufficient to make it positive definite with whatever mar |
15,618 | Generate normally distributed random numbers with non positive-definite covariance matrix | The question concerns how to generate random variates from a multivariate Normal distribution with a (possibly) singular covariance matrix $\mathbb{C}$. This answer explains one way that will work for any covariance matrix. It provides an R implementation that tests its accuracy.
Algebraic analysis of the covariance... | Generate normally distributed random numbers with non positive-definite covariance matrix | The question concerns how to generate random variates from a multivariate Normal distribution with a (possibly) singular covariance matrix $\mathbb{C}$. This answer explains one way that will work fo | Generate normally distributed random numbers with non positive-definite covariance matrix
The question concerns how to generate random variates from a multivariate Normal distribution with a (possibly) singular covariance matrix $\mathbb{C}$. This answer explains one way that will work for any covariance matrix. It p... | Generate normally distributed random numbers with non positive-definite covariance matrix
The question concerns how to generate random variates from a multivariate Normal distribution with a (possibly) singular covariance matrix $\mathbb{C}$. This answer explains one way that will work fo |
15,619 | Generate normally distributed random numbers with non positive-definite covariance matrix | One way would be to compute the matrix from an eigenvalue decomposition. Now I'll admit I don't know too much of the Math behind these processes but from my research it seems fruitful to look at this help file:
http://stat.ethz.ch/R-manual/R-patched/library/Matrix/html/chol.html
and some other related commands in R.
Al... | Generate normally distributed random numbers with non positive-definite covariance matrix | One way would be to compute the matrix from an eigenvalue decomposition. Now I'll admit I don't know too much of the Math behind these processes but from my research it seems fruitful to look at this | Generate normally distributed random numbers with non positive-definite covariance matrix
One way would be to compute the matrix from an eigenvalue decomposition. Now I'll admit I don't know too much of the Math behind these processes but from my research it seems fruitful to look at this help file:
http://stat.ethz.ch... | Generate normally distributed random numbers with non positive-definite covariance matrix
One way would be to compute the matrix from an eigenvalue decomposition. Now I'll admit I don't know too much of the Math behind these processes but from my research it seems fruitful to look at this |
15,620 | Generate normally distributed random numbers with non positive-definite covariance matrix | I would begin by thinking about the model you are estimating.
If a covariance matrix is not positive semi-definite, it may indicate that you have a colinearity problem in your variables which would indicate a problem with the model and should not necessarily be solved by numerical methods.
If the matrix is not positi... | Generate normally distributed random numbers with non positive-definite covariance matrix | I would begin by thinking about the model you are estimating.
If a covariance matrix is not positive semi-definite, it may indicate that you have a colinearity problem in your variables which would i | Generate normally distributed random numbers with non positive-definite covariance matrix
I would begin by thinking about the model you are estimating.
If a covariance matrix is not positive semi-definite, it may indicate that you have a colinearity problem in your variables which would indicate a problem with the mod... | Generate normally distributed random numbers with non positive-definite covariance matrix
I would begin by thinking about the model you are estimating.
If a covariance matrix is not positive semi-definite, it may indicate that you have a colinearity problem in your variables which would i |
15,621 | Generate normally distributed random numbers with non positive-definite covariance matrix | You can get the results from the nearPD function in the Matrix package in R. This will give you a real valued matrix back.
library(Matrix)
A <- matrix(1, 3,3); A[1,3] <- A[3,1] <- 0
n.A <- nearPD(A, corr=T, do2eigen=FALSE)
n.A$mat
# 3 x 3 Matrix of class "dpoMatrix"
# [,1] [,2] [,3]
# [1,] 1.000000... | Generate normally distributed random numbers with non positive-definite covariance matrix | You can get the results from the nearPD function in the Matrix package in R. This will give you a real valued matrix back.
library(Matrix)
A <- matrix(1, 3,3); A[1,3] <- A[3,1] <- 0
n.A <- nearPD(A, c | Generate normally distributed random numbers with non positive-definite covariance matrix
You can get the results from the nearPD function in the Matrix package in R. This will give you a real valued matrix back.
library(Matrix)
A <- matrix(1, 3,3); A[1,3] <- A[3,1] <- 0
n.A <- nearPD(A, corr=T, do2eigen=FALSE)
n.A$mat... | Generate normally distributed random numbers with non positive-definite covariance matrix
You can get the results from the nearPD function in the Matrix package in R. This will give you a real valued matrix back.
library(Matrix)
A <- matrix(1, 3,3); A[1,3] <- A[3,1] <- 0
n.A <- nearPD(A, c |
15,622 | What is the variance of the maximum of a sample? | For any $n$ random variables $X_i$ , the best general bound is
$\newcommand{\Var}{\mathrm{Var}}\Var(\max X_i) \le \sum_i \Var(X_i)$ as stated in the original question.
Here is a proof sketch: If X,Y are IID then $E[(X-Y)^2] =2\Var(X)$. Given a vector of possibly dependent variables $(X_1,\ldots ,X_n)$, let $(Y_1,\ld... | What is the variance of the maximum of a sample? | For any $n$ random variables $X_i$ , the best general bound is
$\newcommand{\Var}{\mathrm{Var}}\Var(\max X_i) \le \sum_i \Var(X_i)$ as stated in the original question.
Here is a proof sketch: If X,Y | What is the variance of the maximum of a sample?
For any $n$ random variables $X_i$ , the best general bound is
$\newcommand{\Var}{\mathrm{Var}}\Var(\max X_i) \le \sum_i \Var(X_i)$ as stated in the original question.
Here is a proof sketch: If X,Y are IID then $E[(X-Y)^2] =2\Var(X)$. Given a vector of possibly depen... | What is the variance of the maximum of a sample?
For any $n$ random variables $X_i$ , the best general bound is
$\newcommand{\Var}{\mathrm{Var}}\Var(\max X_i) \le \sum_i \Var(X_i)$ as stated in the original question.
Here is a proof sketch: If X,Y |
15,623 | What is the variance of the maximum of a sample? | A question on MathOverflow is related to this question.
For IID random variables, the $k$th highest is called an order statistic.
Even for IID Bernoulli random variables, the variance of any order statistic other than the median can be greater than the variance of the population. For example, if $X_i$ is $1$ with prob... | What is the variance of the maximum of a sample? | A question on MathOverflow is related to this question.
For IID random variables, the $k$th highest is called an order statistic.
Even for IID Bernoulli random variables, the variance of any order st | What is the variance of the maximum of a sample?
A question on MathOverflow is related to this question.
For IID random variables, the $k$th highest is called an order statistic.
Even for IID Bernoulli random variables, the variance of any order statistic other than the median can be greater than the variance of the p... | What is the variance of the maximum of a sample?
A question on MathOverflow is related to this question.
For IID random variables, the $k$th highest is called an order statistic.
Even for IID Bernoulli random variables, the variance of any order st |
15,624 | Is there a standard method to deal with label switching problem in MCMC estimation of mixture models? | There is a nice and reasonably recent discussion of this problem here:
Christian P. Robert Multimodality and label switching: a
discussion. Workshop on mixtures, ICMS March 3, 2010.
Essentially, there are several standard strategies, and each has pros and cons. The most obvious thing to do is to formulate the prior... | Is there a standard method to deal with label switching problem in MCMC estimation of mixture models | There is a nice and reasonably recent discussion of this problem here:
Christian P. Robert Multimodality and label switching: a
discussion. Workshop on mixtures, ICMS March 3, 2010.
Essentially, t | Is there a standard method to deal with label switching problem in MCMC estimation of mixture models?
There is a nice and reasonably recent discussion of this problem here:
Christian P. Robert Multimodality and label switching: a
discussion. Workshop on mixtures, ICMS March 3, 2010.
Essentially, there are several s... | Is there a standard method to deal with label switching problem in MCMC estimation of mixture models
There is a nice and reasonably recent discussion of this problem here:
Christian P. Robert Multimodality and label switching: a
discussion. Workshop on mixtures, ICMS March 3, 2010.
Essentially, t |
15,625 | Is there a standard method to deal with label switching problem in MCMC estimation of mixture models? | Gilles Celeux also worked on the problem of label switching, e.g.
G. Celeux, Bayesian inference for
Mixture: the label switching problem.
Proceedings Compstat 98, pp. 227-232, Physica-Verlag (1998).
As a complement to @darrenjw's fine answer, here are two online papers that reviewed alternative strategies:
Jasra et ... | Is there a standard method to deal with label switching problem in MCMC estimation of mixture models | Gilles Celeux also worked on the problem of label switching, e.g.
G. Celeux, Bayesian inference for
Mixture: the label switching problem.
Proceedings Compstat 98, pp. 227-232, Physica-Verlag (1998).
| Is there a standard method to deal with label switching problem in MCMC estimation of mixture models?
Gilles Celeux also worked on the problem of label switching, e.g.
G. Celeux, Bayesian inference for
Mixture: the label switching problem.
Proceedings Compstat 98, pp. 227-232, Physica-Verlag (1998).
As a complement t... | Is there a standard method to deal with label switching problem in MCMC estimation of mixture models
Gilles Celeux also worked on the problem of label switching, e.g.
G. Celeux, Bayesian inference for
Mixture: the label switching problem.
Proceedings Compstat 98, pp. 227-232, Physica-Verlag (1998).
|
15,626 | Is there a standard method to deal with label switching problem in MCMC estimation of mixture models? | With the R-Package "label.switching" (https://cran.r-project.org/web/packages/label.switching/index.html), the following algorithms can be compared:
ECR algorithm (default version), ECR algorithm (two iterative versions), PRA
algorithm, Stephens’ algorithm, Artificial Identifiability Constraint (AIC), Data-Based relabe... | Is there a standard method to deal with label switching problem in MCMC estimation of mixture models | With the R-Package "label.switching" (https://cran.r-project.org/web/packages/label.switching/index.html), the following algorithms can be compared:
ECR algorithm (default version), ECR algorithm (two | Is there a standard method to deal with label switching problem in MCMC estimation of mixture models?
With the R-Package "label.switching" (https://cran.r-project.org/web/packages/label.switching/index.html), the following algorithms can be compared:
ECR algorithm (default version), ECR algorithm (two iterative version... | Is there a standard method to deal with label switching problem in MCMC estimation of mixture models
With the R-Package "label.switching" (https://cran.r-project.org/web/packages/label.switching/index.html), the following algorithms can be compared:
ECR algorithm (default version), ECR algorithm (two |
15,627 | Purpose of Dirichlet noise in the AlphaZero paper | Question 1 is straightforward, here $\alpha$ is a vector of repetitions of the given value. (As answered by Max S.)
Question 2 is more interesting: The Dirichlet distribution has the following interpretation relevant in this context: When $\alpha$ is the observed vector of outcome-counts drawn from some (unknown) categ... | Purpose of Dirichlet noise in the AlphaZero paper | Question 1 is straightforward, here $\alpha$ is a vector of repetitions of the given value. (As answered by Max S.)
Question 2 is more interesting: The Dirichlet distribution has the following interpr | Purpose of Dirichlet noise in the AlphaZero paper
Question 1 is straightforward, here $\alpha$ is a vector of repetitions of the given value. (As answered by Max S.)
Question 2 is more interesting: The Dirichlet distribution has the following interpretation relevant in this context: When $\alpha$ is the observed vector... | Purpose of Dirichlet noise in the AlphaZero paper
Question 1 is straightforward, here $\alpha$ is a vector of repetitions of the given value. (As answered by Max S.)
Question 2 is more interesting: The Dirichlet distribution has the following interpr |
15,628 | Purpose of Dirichlet noise in the AlphaZero paper | For question number 1 the answer is yes, $\alpha$ is a vector, but in this case all values are the same. According to wikipedia this is called a symmetric Dirichlet distribution, and is used when "there is no prior knowledge favoring one component over another". In this case this means that you don't want to add more n... | Purpose of Dirichlet noise in the AlphaZero paper | For question number 1 the answer is yes, $\alpha$ is a vector, but in this case all values are the same. According to wikipedia this is called a symmetric Dirichlet distribution, and is used when "the | Purpose of Dirichlet noise in the AlphaZero paper
For question number 1 the answer is yes, $\alpha$ is a vector, but in this case all values are the same. According to wikipedia this is called a symmetric Dirichlet distribution, and is used when "there is no prior knowledge favoring one component over another". In this... | Purpose of Dirichlet noise in the AlphaZero paper
For question number 1 the answer is yes, $\alpha$ is a vector, but in this case all values are the same. According to wikipedia this is called a symmetric Dirichlet distribution, and is used when "the |
15,629 | Layman's explanation of censoring in survival analysis | Censoring is often described in comparison with truncation. Nice description of the two processes is provided by Gelman et al (2005, p. 235):
Truncated data differs from censored data that no count of
observations beyond truncation point is available. With censoring the
values of observations beyond the truncation... | Layman's explanation of censoring in survival analysis | Censoring is often described in comparison with truncation. Nice description of the two processes is provided by Gelman et al (2005, p. 235):
Truncated data differs from censored data that no count o | Layman's explanation of censoring in survival analysis
Censoring is often described in comparison with truncation. Nice description of the two processes is provided by Gelman et al (2005, p. 235):
Truncated data differs from censored data that no count of
observations beyond truncation point is available. With censo... | Layman's explanation of censoring in survival analysis
Censoring is often described in comparison with truncation. Nice description of the two processes is provided by Gelman et al (2005, p. 235):
Truncated data differs from censored data that no count o |
15,630 | Layman's explanation of censoring in survival analysis | Censoring is central to survival analysis.
The basic idea is that information is censored, it is invisible to you. Simply explained, a censored distribution of life times is obtained if you record the life times before everyone in the sample has died. If you think of time moving "rightwards" on the X-axis, this can be ... | Layman's explanation of censoring in survival analysis | Censoring is central to survival analysis.
The basic idea is that information is censored, it is invisible to you. Simply explained, a censored distribution of life times is obtained if you record the | Layman's explanation of censoring in survival analysis
Censoring is central to survival analysis.
The basic idea is that information is censored, it is invisible to you. Simply explained, a censored distribution of life times is obtained if you record the life times before everyone in the sample has died. If you think ... | Layman's explanation of censoring in survival analysis
Censoring is central to survival analysis.
The basic idea is that information is censored, it is invisible to you. Simply explained, a censored distribution of life times is obtained if you record the |
15,631 | Confidence intervals vs. standard deviation | There are two things here :
The "2 sigma rule" where sigma refers to standard deviation is a way to construct tolerance intervals for normally distributed data, not confidence intervals (see this link to learn about the difference). Said shortly, tolerance intervals refer to the distribution inside the population, whe... | Confidence intervals vs. standard deviation | There are two things here :
The "2 sigma rule" where sigma refers to standard deviation is a way to construct tolerance intervals for normally distributed data, not confidence intervals (see this lin | Confidence intervals vs. standard deviation
There are two things here :
The "2 sigma rule" where sigma refers to standard deviation is a way to construct tolerance intervals for normally distributed data, not confidence intervals (see this link to learn about the difference). Said shortly, tolerance intervals refer to... | Confidence intervals vs. standard deviation
There are two things here :
The "2 sigma rule" where sigma refers to standard deviation is a way to construct tolerance intervals for normally distributed data, not confidence intervals (see this lin |
15,632 | Confidence intervals vs. standard deviation | May be, it will be easier to explain, to avoid confusion.
Standard deviation:
With probability about 95% we will find every new sample in interval
(x_mean - 2 * sigma; x_mean + 2 * sigma)
what says us where to expect the location of new samples.
Confidence interval:
With probability of f.e. 95% the real x_mean value wi... | Confidence intervals vs. standard deviation | May be, it will be easier to explain, to avoid confusion.
Standard deviation:
With probability about 95% we will find every new sample in interval
(x_mean - 2 * sigma; x_mean + 2 * sigma)
what says us | Confidence intervals vs. standard deviation
May be, it will be easier to explain, to avoid confusion.
Standard deviation:
With probability about 95% we will find every new sample in interval
(x_mean - 2 * sigma; x_mean + 2 * sigma)
what says us where to expect the location of new samples.
Confidence interval:
With prob... | Confidence intervals vs. standard deviation
May be, it will be easier to explain, to avoid confusion.
Standard deviation:
With probability about 95% we will find every new sample in interval
(x_mean - 2 * sigma; x_mean + 2 * sigma)
what says us |
15,633 | Confidence intervals vs. standard deviation | This equation relies on the assumption that the errors are Gaussian. Also, the factor of 2 in front of the SE(β1) term will vary slightly depending on the number of observations n in the linear regression. To be precise, rather
than the number 2, the equation should contain the 97.5 % quantile of a t-distribution with ... | Confidence intervals vs. standard deviation | This equation relies on the assumption that the errors are Gaussian. Also, the factor of 2 in front of the SE(β1) term will vary slightly depending on the number of observations n in the linear regres | Confidence intervals vs. standard deviation
This equation relies on the assumption that the errors are Gaussian. Also, the factor of 2 in front of the SE(β1) term will vary slightly depending on the number of observations n in the linear regression. To be precise, rather
than the number 2, the equation should contain t... | Confidence intervals vs. standard deviation
This equation relies on the assumption that the errors are Gaussian. Also, the factor of 2 in front of the SE(β1) term will vary slightly depending on the number of observations n in the linear regres |
15,634 | Why is optimizing a mixture of Gaussian directly computationally hard? | First, GMM is a particular algorithm for clustering, where you try to find the optimal labelling of your $n$ observations. Having $k$ possible classes, it means that there are $k^n$ possible labellings of your training data. This becomes already huge for moderate values of $k$ and $n$.
Second, the functional you are tr... | Why is optimizing a mixture of Gaussian directly computationally hard? | First, GMM is a particular algorithm for clustering, where you try to find the optimal labelling of your $n$ observations. Having $k$ possible classes, it means that there are $k^n$ possible labelling | Why is optimizing a mixture of Gaussian directly computationally hard?
First, GMM is a particular algorithm for clustering, where you try to find the optimal labelling of your $n$ observations. Having $k$ possible classes, it means that there are $k^n$ possible labellings of your training data. This becomes already hug... | Why is optimizing a mixture of Gaussian directly computationally hard?
First, GMM is a particular algorithm for clustering, where you try to find the optimal labelling of your $n$ observations. Having $k$ possible classes, it means that there are $k^n$ possible labelling |
15,635 | Why is optimizing a mixture of Gaussian directly computationally hard? | In addition to juampa's points, let me signal those difficulties:
The function $l(\theta|S_n)$ is unbounded, so the true maximum is $+\infty$ and corresponds to $\hat\mu^{(i)}=x_1$ (for instance) and $\hat\sigma_i=0$. A true maximiser should therefore end up with this solution, which is not useful for estimation purpo... | Why is optimizing a mixture of Gaussian directly computationally hard? | In addition to juampa's points, let me signal those difficulties:
The function $l(\theta|S_n)$ is unbounded, so the true maximum is $+\infty$ and corresponds to $\hat\mu^{(i)}=x_1$ (for instance) and | Why is optimizing a mixture of Gaussian directly computationally hard?
In addition to juampa's points, let me signal those difficulties:
The function $l(\theta|S_n)$ is unbounded, so the true maximum is $+\infty$ and corresponds to $\hat\mu^{(i)}=x_1$ (for instance) and $\hat\sigma_i=0$. A true maximiser should theref... | Why is optimizing a mixture of Gaussian directly computationally hard?
In addition to juampa's points, let me signal those difficulties:
The function $l(\theta|S_n)$ is unbounded, so the true maximum is $+\infty$ and corresponds to $\hat\mu^{(i)}=x_1$ (for instance) and |
15,636 | Check memoryless property of a Markov chain | I wonder if the following would give a valid Pearson $\chi^2$ test for proportions as follows.
Estimate the one-step transition probabilities -- you've done that.
Obtain the two-step model probabilities:
$$
\hat p_{U,V} = {\rm Prob}[X_{i+2}=U|X_i=V] = \sum_{W\in\{A,B,C,D\}} {\rm Prob}[X_{i+2}=U|X_{i+1}=W]{\rm Prob}[X_... | Check memoryless property of a Markov chain | I wonder if the following would give a valid Pearson $\chi^2$ test for proportions as follows.
Estimate the one-step transition probabilities -- you've done that.
Obtain the two-step model probabilit | Check memoryless property of a Markov chain
I wonder if the following would give a valid Pearson $\chi^2$ test for proportions as follows.
Estimate the one-step transition probabilities -- you've done that.
Obtain the two-step model probabilities:
$$
\hat p_{U,V} = {\rm Prob}[X_{i+2}=U|X_i=V] = \sum_{W\in\{A,B,C,D\}} ... | Check memoryless property of a Markov chain
I wonder if the following would give a valid Pearson $\chi^2$ test for proportions as follows.
Estimate the one-step transition probabilities -- you've done that.
Obtain the two-step model probabilit |
15,637 | Check memoryless property of a Markov chain | Markov property might be hard to test directly. But it might be enough to fit a model which assumes Markov property and then test whether the model holds. It may turn out that the fitted model is a good approximation which is useful for you in practice, and you need not to be concerned whether Markov property really ho... | Check memoryless property of a Markov chain | Markov property might be hard to test directly. But it might be enough to fit a model which assumes Markov property and then test whether the model holds. It may turn out that the fitted model is a go | Check memoryless property of a Markov chain
Markov property might be hard to test directly. But it might be enough to fit a model which assumes Markov property and then test whether the model holds. It may turn out that the fitted model is a good approximation which is useful for you in practice, and you need not to be... | Check memoryless property of a Markov chain
Markov property might be hard to test directly. But it might be enough to fit a model which assumes Markov property and then test whether the model holds. It may turn out that the fitted model is a go |
15,638 | Check memoryless property of a Markov chain | To concretize the suggestion of the previous reply, you first want to estimate the Markov probabilities - assuming it's Markov. See the reply here Estimating Markov Chain Probabilities
You should get a 4 x 4 matrix based on the proportion of transitions from state A to A, A to B, etc. Call this matrix $M$. $M^2$ shoul... | Check memoryless property of a Markov chain | To concretize the suggestion of the previous reply, you first want to estimate the Markov probabilities - assuming it's Markov. See the reply here Estimating Markov Chain Probabilities
You should get | Check memoryless property of a Markov chain
To concretize the suggestion of the previous reply, you first want to estimate the Markov probabilities - assuming it's Markov. See the reply here Estimating Markov Chain Probabilities
You should get a 4 x 4 matrix based on the proportion of transitions from state A to A, A ... | Check memoryless property of a Markov chain
To concretize the suggestion of the previous reply, you first want to estimate the Markov probabilities - assuming it's Markov. See the reply here Estimating Markov Chain Probabilities
You should get |
15,639 | Check memoryless property of a Markov chain | Beyond Markov Property (MP), a further property is Time
Homogeneity (TH): $X_t$ can be Markov but with its transition matrix
$\mathbf{P}(t)$ depending on time $t$. E.g., it may depend on
the weekday at $t$ if observations are daily, and then a dependence
$X_t$ on $X_{t-7}$ conditional on $X_{t-1}$ may be diagnosed if T... | Check memoryless property of a Markov chain | Beyond Markov Property (MP), a further property is Time
Homogeneity (TH): $X_t$ can be Markov but with its transition matrix
$\mathbf{P}(t)$ depending on time $t$. E.g., it may depend on
the weekday a | Check memoryless property of a Markov chain
Beyond Markov Property (MP), a further property is Time
Homogeneity (TH): $X_t$ can be Markov but with its transition matrix
$\mathbf{P}(t)$ depending on time $t$. E.g., it may depend on
the weekday at $t$ if observations are daily, and then a dependence
$X_t$ on $X_{t-7}$ co... | Check memoryless property of a Markov chain
Beyond Markov Property (MP), a further property is Time
Homogeneity (TH): $X_t$ can be Markov but with its transition matrix
$\mathbf{P}(t)$ depending on time $t$. E.g., it may depend on
the weekday a |
15,640 | Check memoryless property of a Markov chain | I think placida and mpiktas have both given very thoughtful and excellent approaches.
I am answering because I just want to add that one could construct a test to see if $P(X_i=x|X_{i-1}=y)$ is different from $P(X_i=x|X_{i-1}=y \text{ and } X_{i-2}=z)$.
I would pick values for $x$, $y$ and $z$ for which there are a lar... | Check memoryless property of a Markov chain | I think placida and mpiktas have both given very thoughtful and excellent approaches.
I am answering because I just want to add that one could construct a test to see if $P(X_i=x|X_{i-1}=y)$ is differ | Check memoryless property of a Markov chain
I think placida and mpiktas have both given very thoughtful and excellent approaches.
I am answering because I just want to add that one could construct a test to see if $P(X_i=x|X_{i-1}=y)$ is different from $P(X_i=x|X_{i-1}=y \text{ and } X_{i-2}=z)$.
I would pick values fo... | Check memoryless property of a Markov chain
I think placida and mpiktas have both given very thoughtful and excellent approaches.
I am answering because I just want to add that one could construct a test to see if $P(X_i=x|X_{i-1}=y)$ is differ |
15,641 | Check memoryless property of a Markov chain | You could bin the data into evenly spaced intervals, then compute the unbiased sample variances of subsets $\{X_{n+1}:X_n=x_1,X_{n-k}=x_2\}$. By the law of total variance, $$\mathrm{Var}[E(X_{n+1}|X_n,X_{n-k})|X_n] = \mathrm{Var}[X_{n+1}|X_n]-E(\mathrm{Var}[X_{n+1}|X_n])$$
The LHS, if it is almost zero, provides eviden... | Check memoryless property of a Markov chain | You could bin the data into evenly spaced intervals, then compute the unbiased sample variances of subsets $\{X_{n+1}:X_n=x_1,X_{n-k}=x_2\}$. By the law of total variance, $$\mathrm{Var}[E(X_{n+1}|X_n | Check memoryless property of a Markov chain
You could bin the data into evenly spaced intervals, then compute the unbiased sample variances of subsets $\{X_{n+1}:X_n=x_1,X_{n-k}=x_2\}$. By the law of total variance, $$\mathrm{Var}[E(X_{n+1}|X_n,X_{n-k})|X_n] = \mathrm{Var}[X_{n+1}|X_n]-E(\mathrm{Var}[X_{n+1}|X_n])$$
Th... | Check memoryless property of a Markov chain
You could bin the data into evenly spaced intervals, then compute the unbiased sample variances of subsets $\{X_{n+1}:X_n=x_1,X_{n-k}=x_2\}$. By the law of total variance, $$\mathrm{Var}[E(X_{n+1}|X_n |
15,642 | How to minimize residual sum of squares of an exponential fit? | A (negative) exponential law takes the form $y=-\exp(-x)$. When you allow for changes of units in the $x$ and $y$ values, though, say to $y = \alpha y' + \beta$ and $x = \gamma x' + \delta$, then the law will be expressed as
$$\alpha y' + \beta = y = -\exp(-x) = -\exp(-\gamma x' - \delta),$$
which algebraically is equi... | How to minimize residual sum of squares of an exponential fit? | A (negative) exponential law takes the form $y=-\exp(-x)$. When you allow for changes of units in the $x$ and $y$ values, though, say to $y = \alpha y' + \beta$ and $x = \gamma x' + \delta$, then the | How to minimize residual sum of squares of an exponential fit?
A (negative) exponential law takes the form $y=-\exp(-x)$. When you allow for changes of units in the $x$ and $y$ values, though, say to $y = \alpha y' + \beta$ and $x = \gamma x' + \delta$, then the law will be expressed as
$$\alpha y' + \beta = y = -\exp(... | How to minimize residual sum of squares of an exponential fit?
A (negative) exponential law takes the form $y=-\exp(-x)$. When you allow for changes of units in the $x$ and $y$ values, though, say to $y = \alpha y' + \beta$ and $x = \gamma x' + \delta$, then the |
15,643 | Best way to perform multiclass SVM | There are a lot of methods for multi-class classification. Two classic options, which are not SVM-specific are:
One-vs-all (OVA) classification:
Suppose you have classes A, B, C, and D. Instead of doing a four way classification, train up four binary classifiers: A vs. not-A, B vs. not-B, C vs. not-C, and D vs. not-D.... | Best way to perform multiclass SVM | There are a lot of methods for multi-class classification. Two classic options, which are not SVM-specific are:
One-vs-all (OVA) classification:
Suppose you have classes A, B, C, and D. Instead of do | Best way to perform multiclass SVM
There are a lot of methods for multi-class classification. Two classic options, which are not SVM-specific are:
One-vs-all (OVA) classification:
Suppose you have classes A, B, C, and D. Instead of doing a four way classification, train up four binary classifiers: A vs. not-A, B vs. n... | Best way to perform multiclass SVM
There are a lot of methods for multi-class classification. Two classic options, which are not SVM-specific are:
One-vs-all (OVA) classification:
Suppose you have classes A, B, C, and D. Instead of do |
15,644 | Best way to perform multiclass SVM | Let me add that there is work on extending SVMs to multiple classes (as opposed to the methods Matt Krause describes that are decomposition into several binary classification tasks). One important work is: On the Algorithmic Implementation of Multiclass Kernel-based Vector Machine | Best way to perform multiclass SVM | Let me add that there is work on extending SVMs to multiple classes (as opposed to the methods Matt Krause describes that are decomposition into several binary classification tasks). One important wor | Best way to perform multiclass SVM
Let me add that there is work on extending SVMs to multiple classes (as opposed to the methods Matt Krause describes that are decomposition into several binary classification tasks). One important work is: On the Algorithmic Implementation of Multiclass Kernel-based Vector Machine | Best way to perform multiclass SVM
Let me add that there is work on extending SVMs to multiple classes (as opposed to the methods Matt Krause describes that are decomposition into several binary classification tasks). One important wor |
15,645 | Name of mean absolute error analogue to Brier score? | Answer seems to be: no, because MAE doesn't lead to a proper scoring rule.
See Loss Functions for Binary Class Probability Estimation and Classification: Structure and Applications where the MAE is discussed under "Counterexamples of proper scoring rules". | Name of mean absolute error analogue to Brier score? | Answer seems to be: no, because MAE doesn't lead to a proper scoring rule.
See Loss Functions for Binary Class Probability Estimation and Classification: Structure and Applications where the MAE is di | Name of mean absolute error analogue to Brier score?
Answer seems to be: no, because MAE doesn't lead to a proper scoring rule.
See Loss Functions for Binary Class Probability Estimation and Classification: Structure and Applications where the MAE is discussed under "Counterexamples of proper scoring rules". | Name of mean absolute error analogue to Brier score?
Answer seems to be: no, because MAE doesn't lead to a proper scoring rule.
See Loss Functions for Binary Class Probability Estimation and Classification: Structure and Applications where the MAE is di |
15,646 | Plotting sparklines in R | I initially managed to produce something approaching your original picture with some quick and dirty R code (see this gist), until I discovered that the sparkTable package should do this very much better, provided you are willing to use $\LaTeX$. (In the meantime, it has also been pointed out by @Bernd!)
Here is an exa... | Plotting sparklines in R | I initially managed to produce something approaching your original picture with some quick and dirty R code (see this gist), until I discovered that the sparkTable package should do this very much bet | Plotting sparklines in R
I initially managed to produce something approaching your original picture with some quick and dirty R code (see this gist), until I discovered that the sparkTable package should do this very much better, provided you are willing to use $\LaTeX$. (In the meantime, it has also been pointed out b... | Plotting sparklines in R
I initially managed to produce something approaching your original picture with some quick and dirty R code (see this gist), until I discovered that the sparkTable package should do this very much bet |
15,647 | Fitting an Orthogonal Grid to Noisy Points | Because the streets are on an irregular orthogonal grid, the Fourier Transform solutions--although clever--will likely fail. The following approach exploits two ideas:
Using the Hough transform to identify sets of points that tend to line up.
Exploiting the assumed orthogonality of the grid to augment the data.
Th... | Fitting an Orthogonal Grid to Noisy Points | Because the streets are on an irregular orthogonal grid, the Fourier Transform solutions--although clever--will likely fail. The following approach exploits two ideas:
Using the Hough transform to i | Fitting an Orthogonal Grid to Noisy Points
Because the streets are on an irregular orthogonal grid, the Fourier Transform solutions--although clever--will likely fail. The following approach exploits two ideas:
Using the Hough transform to identify sets of points that tend to line up.
Exploiting the assumed orthogon... | Fitting an Orthogonal Grid to Noisy Points
Because the streets are on an irregular orthogonal grid, the Fourier Transform solutions--although clever--will likely fail. The following approach exploits two ideas:
Using the Hough transform to i |
15,648 | Fitting an Orthogonal Grid to Noisy Points | How about this approach:
Your grid can be described by seven parameters: Consider the lower left rectangle in your grid. You need five parameters to describe the four corners of this rectangle, and two parameters for the number of rectangles in both directions.
Next, you design a cost function that takes as input those... | Fitting an Orthogonal Grid to Noisy Points | How about this approach:
Your grid can be described by seven parameters: Consider the lower left rectangle in your grid. You need five parameters to describe the four corners of this rectangle, and tw | Fitting an Orthogonal Grid to Noisy Points
How about this approach:
Your grid can be described by seven parameters: Consider the lower left rectangle in your grid. You need five parameters to describe the four corners of this rectangle, and two parameters for the number of rectangles in both directions.
Next, you desig... | Fitting an Orthogonal Grid to Noisy Points
How about this approach:
Your grid can be described by seven parameters: Consider the lower left rectangle in your grid. You need five parameters to describe the four corners of this rectangle, and tw |
15,649 | Fitting an Orthogonal Grid to Noisy Points | This answer is not a complete answer to the exact question but I leave it as it does explain some principles and works well in another related situation.
This answer is based on the assumption that the grid is regular (equal distance between grid lines). In the solution this assumption is necessary when we apply the f... | Fitting an Orthogonal Grid to Noisy Points | This answer is not a complete answer to the exact question but I leave it as it does explain some principles and works well in another related situation.
This answer is based on the assumption that t | Fitting an Orthogonal Grid to Noisy Points
This answer is not a complete answer to the exact question but I leave it as it does explain some principles and works well in another related situation.
This answer is based on the assumption that the grid is regular (equal distance between grid lines). In the solution this ... | Fitting an Orthogonal Grid to Noisy Points
This answer is not a complete answer to the exact question but I leave it as it does explain some principles and works well in another related situation.
This answer is based on the assumption that t |
15,650 | Fitting an Orthogonal Grid to Noisy Points | We can turn the problem of finding the points in a noisy grid into the problem of finding the peaks of a smooth function, which can be much easier.
If your points are $p_i$, define the function $f$ of 2D vector $k$ by $f(k)=|\sum_{i=1}^n\exp(i k\cdot p_i)|$. For your example set of points you'll get the function whose ... | Fitting an Orthogonal Grid to Noisy Points | We can turn the problem of finding the points in a noisy grid into the problem of finding the peaks of a smooth function, which can be much easier.
If your points are $p_i$, define the function $f$ of | Fitting an Orthogonal Grid to Noisy Points
We can turn the problem of finding the points in a noisy grid into the problem of finding the peaks of a smooth function, which can be much easier.
If your points are $p_i$, define the function $f$ of 2D vector $k$ by $f(k)=|\sum_{i=1}^n\exp(i k\cdot p_i)|$. For your example s... | Fitting an Orthogonal Grid to Noisy Points
We can turn the problem of finding the points in a noisy grid into the problem of finding the peaks of a smooth function, which can be much easier.
If your points are $p_i$, define the function $f$ of |
15,651 | Fitting an Orthogonal Grid to Noisy Points | As an adjustable approach, you could try
Rotate all the points by a fixed angle (around the origin or centre of mass, whatever suits).
Project the rotated points to the x-axis and y-axis. This is just the x-values and y-values of the rotated points.
Use k-means or a method of clustering points on a 1d line. K-means le... | Fitting an Orthogonal Grid to Noisy Points | As an adjustable approach, you could try
Rotate all the points by a fixed angle (around the origin or centre of mass, whatever suits).
Project the rotated points to the x-axis and y-axis. This is jus | Fitting an Orthogonal Grid to Noisy Points
As an adjustable approach, you could try
Rotate all the points by a fixed angle (around the origin or centre of mass, whatever suits).
Project the rotated points to the x-axis and y-axis. This is just the x-values and y-values of the rotated points.
Use k-means or a method of... | Fitting an Orthogonal Grid to Noisy Points
As an adjustable approach, you could try
Rotate all the points by a fixed angle (around the origin or centre of mass, whatever suits).
Project the rotated points to the x-axis and y-axis. This is jus |
15,652 | Suppose $Y_1, \dots, Y_n \overset{\text{iid}}{\sim} \text{Exp}(1)$. Show $\sum_{i=1}^{n}(Y_i - Y_{(1)}) \sim \text{Gamma}(n-1, 1)$ | The proof is given in the Mother of All Random Generation Books, Devroye's Non-uniform Random Variate Generation, on p.211 (and it is a very elegant one!):
Theorem 2.3 (Sukhatme, 1937) If we define $E_{(0)}=0$ then the normalised exponential spacings $$(n-i+1)(E_{(i)}-E_{(i-1)})$$
derived from the order statistics $... | Suppose $Y_1, \dots, Y_n \overset{\text{iid}}{\sim} \text{Exp}(1)$. Show $\sum_{i=1}^{n}(Y_i - Y_{(1 | The proof is given in the Mother of All Random Generation Books, Devroye's Non-uniform Random Variate Generation, on p.211 (and it is a very elegant one!):
Theorem 2.3 (Sukhatme, 1937) If we define $ | Suppose $Y_1, \dots, Y_n \overset{\text{iid}}{\sim} \text{Exp}(1)$. Show $\sum_{i=1}^{n}(Y_i - Y_{(1)}) \sim \text{Gamma}(n-1, 1)$
The proof is given in the Mother of All Random Generation Books, Devroye's Non-uniform Random Variate Generation, on p.211 (and it is a very elegant one!):
Theorem 2.3 (Sukhatme, 1937) If ... | Suppose $Y_1, \dots, Y_n \overset{\text{iid}}{\sim} \text{Exp}(1)$. Show $\sum_{i=1}^{n}(Y_i - Y_{(1
The proof is given in the Mother of All Random Generation Books, Devroye's Non-uniform Random Variate Generation, on p.211 (and it is a very elegant one!):
Theorem 2.3 (Sukhatme, 1937) If we define $ |
15,653 | Suppose $Y_1, \dots, Y_n \overset{\text{iid}}{\sim} \text{Exp}(1)$. Show $\sum_{i=1}^{n}(Y_i - Y_{(1)}) \sim \text{Gamma}(n-1, 1)$ | I lay out here what has been suggested in comments by @jbowman.
Let a constant $a\geq 0$. Let $Y_i$ follow an $\text{Exp(1)}$ and consider $Z_i = Y_i-a$. Then
$$\Pr(Z_i\leq z_i \mid Y_i \geq a) = \Pr(Y_i-a\leq z_i \mid Y_i \geq a)$$
$$\implies \Pr(Y_i\leq z_i+a \mid Y_i \geq a) = \frac {\Pr(Y_i\leq z_i+a,Y_i \geq a... | Suppose $Y_1, \dots, Y_n \overset{\text{iid}}{\sim} \text{Exp}(1)$. Show $\sum_{i=1}^{n}(Y_i - Y_{(1 | I lay out here what has been suggested in comments by @jbowman.
Let a constant $a\geq 0$. Let $Y_i$ follow an $\text{Exp(1)}$ and consider $Z_i = Y_i-a$. Then
$$\Pr(Z_i\leq z_i \mid Y_i \geq a) = | Suppose $Y_1, \dots, Y_n \overset{\text{iid}}{\sim} \text{Exp}(1)$. Show $\sum_{i=1}^{n}(Y_i - Y_{(1)}) \sim \text{Gamma}(n-1, 1)$
I lay out here what has been suggested in comments by @jbowman.
Let a constant $a\geq 0$. Let $Y_i$ follow an $\text{Exp(1)}$ and consider $Z_i = Y_i-a$. Then
$$\Pr(Z_i\leq z_i \mid Y_i... | Suppose $Y_1, \dots, Y_n \overset{\text{iid}}{\sim} \text{Exp}(1)$. Show $\sum_{i=1}^{n}(Y_i - Y_{(1
I lay out here what has been suggested in comments by @jbowman.
Let a constant $a\geq 0$. Let $Y_i$ follow an $\text{Exp(1)}$ and consider $Z_i = Y_i-a$. Then
$$\Pr(Z_i\leq z_i \mid Y_i \geq a) = |
15,654 | simulating random samples with a given MLE | One option would be to use a constrained HMC variant as described in A Family of MCMC Methods on Implicitly Defined Manifolds by Brubaker et al (1). This requires that we can express the condition that the maximum-likelihood estimate of the location parameter is equal to some fixed $\mu_0$ as some implicitly defined (a... | simulating random samples with a given MLE | One option would be to use a constrained HMC variant as described in A Family of MCMC Methods on Implicitly Defined Manifolds by Brubaker et al (1). This requires that we can express the condition tha | simulating random samples with a given MLE
One option would be to use a constrained HMC variant as described in A Family of MCMC Methods on Implicitly Defined Manifolds by Brubaker et al (1). This requires that we can express the condition that the maximum-likelihood estimate of the location parameter is equal to some ... | simulating random samples with a given MLE
One option would be to use a constrained HMC variant as described in A Family of MCMC Methods on Implicitly Defined Manifolds by Brubaker et al (1). This requires that we can express the condition tha |
15,655 | Why can't ridge regression provide better interpretability than LASSO? | If you order 1 million ridge-shrunk, scaled, but non-zero features, you will have to make some kind of decision: you will look at the n best predictors, but what is n? The LASSO solves this problem in a principled, objective way, because for every step on the path (and often, you'd settle on one point via e.g. cross va... | Why can't ridge regression provide better interpretability than LASSO? | If you order 1 million ridge-shrunk, scaled, but non-zero features, you will have to make some kind of decision: you will look at the n best predictors, but what is n? The LASSO solves this problem in | Why can't ridge regression provide better interpretability than LASSO?
If you order 1 million ridge-shrunk, scaled, but non-zero features, you will have to make some kind of decision: you will look at the n best predictors, but what is n? The LASSO solves this problem in a principled, objective way, because for every s... | Why can't ridge regression provide better interpretability than LASSO?
If you order 1 million ridge-shrunk, scaled, but non-zero features, you will have to make some kind of decision: you will look at the n best predictors, but what is n? The LASSO solves this problem in |
15,656 | Why can't ridge regression provide better interpretability than LASSO? | Interpretability decreases if the target is dependent on lot of features. It increases if we can reduce the number of features as well maintain the accuracy. Ridge regularization does not have the ability to reduce number of features. But Lasso has the ability. How this happens is explained visually in the following li... | Why can't ridge regression provide better interpretability than LASSO? | Interpretability decreases if the target is dependent on lot of features. It increases if we can reduce the number of features as well maintain the accuracy. Ridge regularization does not have the abi | Why can't ridge regression provide better interpretability than LASSO?
Interpretability decreases if the target is dependent on lot of features. It increases if we can reduce the number of features as well maintain the accuracy. Ridge regularization does not have the ability to reduce number of features. But Lasso has ... | Why can't ridge regression provide better interpretability than LASSO?
Interpretability decreases if the target is dependent on lot of features. It increases if we can reduce the number of features as well maintain the accuracy. Ridge regularization does not have the abi |
15,657 | Why should one use EM vs. say, Gradient Descent with MLE? | I think there's some crossed wires here. The MLE, as referred to in the statistical literature, is the Maximum Likelihood Estimate. This is an estimator. The EM algorithm is, as the name implies, an algorithm which is often used to compute the MLE. These are apples and oranges.
When the MLE is not in closed form, a co... | Why should one use EM vs. say, Gradient Descent with MLE? | I think there's some crossed wires here. The MLE, as referred to in the statistical literature, is the Maximum Likelihood Estimate. This is an estimator. The EM algorithm is, as the name implies, an a | Why should one use EM vs. say, Gradient Descent with MLE?
I think there's some crossed wires here. The MLE, as referred to in the statistical literature, is the Maximum Likelihood Estimate. This is an estimator. The EM algorithm is, as the name implies, an algorithm which is often used to compute the MLE. These are app... | Why should one use EM vs. say, Gradient Descent with MLE?
I think there's some crossed wires here. The MLE, as referred to in the statistical literature, is the Maximum Likelihood Estimate. This is an estimator. The EM algorithm is, as the name implies, an a |
15,658 | What is the problem with post-hoc testing? | "You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won't believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance that I ... | What is the problem with post-hoc testing? | "You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won't believe what happened. I saw a car with the | What is the problem with post-hoc testing?
"You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won't believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plate... | What is the problem with post-hoc testing?
"You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won't believe what happened. I saw a car with the |
15,659 | What is the problem with post-hoc testing? | If you first collect data and then construct a theory based on the data, you are in danger of fitting a story to your observations. The problem is that we humans are extremely good at writing stories. Put another way: any data can be "explained" by a story, if the story is just convoluted enough.
This process provides ... | What is the problem with post-hoc testing? | If you first collect data and then construct a theory based on the data, you are in danger of fitting a story to your observations. The problem is that we humans are extremely good at writing stories. | What is the problem with post-hoc testing?
If you first collect data and then construct a theory based on the data, you are in danger of fitting a story to your observations. The problem is that we humans are extremely good at writing stories. Put another way: any data can be "explained" by a story, if the story is jus... | What is the problem with post-hoc testing?
If you first collect data and then construct a theory based on the data, you are in danger of fitting a story to your observations. The problem is that we humans are extremely good at writing stories. |
15,660 | What is the problem with post-hoc testing? | Science operates by forming hypotheses (which are of course are motivated by experience), making predictions based on those hypotheses and then testing them. Would it make sense to observe something in the past, generalize this observation into a theory, but then treat the past itself as a kind of retroactive experime... | What is the problem with post-hoc testing? | Science operates by forming hypotheses (which are of course are motivated by experience), making predictions based on those hypotheses and then testing them. Would it make sense to observe something | What is the problem with post-hoc testing?
Science operates by forming hypotheses (which are of course are motivated by experience), making predictions based on those hypotheses and then testing them. Would it make sense to observe something in the past, generalize this observation into a theory, but then treat the pa... | What is the problem with post-hoc testing?
Science operates by forming hypotheses (which are of course are motivated by experience), making predictions based on those hypotheses and then testing them. Would it make sense to observe something |
15,661 | What is the problem with post-hoc testing? | Your professor and the other answers are right that post-hoc analysis have problems. However, you are also right that a lot of good science comes from post-hoc analysis. The key point is that properly designed experiments should be preferred and that post-hoc analysis should be dealt with caution and with special tools... | What is the problem with post-hoc testing? | Your professor and the other answers are right that post-hoc analysis have problems. However, you are also right that a lot of good science comes from post-hoc analysis. The key point is that properly | What is the problem with post-hoc testing?
Your professor and the other answers are right that post-hoc analysis have problems. However, you are also right that a lot of good science comes from post-hoc analysis. The key point is that properly designed experiments should be preferred and that post-hoc analysis should b... | What is the problem with post-hoc testing?
Your professor and the other answers are right that post-hoc analysis have problems. However, you are also right that a lot of good science comes from post-hoc analysis. The key point is that properly |
15,662 | What is the problem with post-hoc testing? | If you don't have a theory backing your propositions, then even if your proposition is validated, it could be through coincidence and does not prove anything. For example, I find that I do potty when the sun rises and have been doing that for the past 10 years - based on this data, a post-hoc analysis tells me that the... | What is the problem with post-hoc testing? | If you don't have a theory backing your propositions, then even if your proposition is validated, it could be through coincidence and does not prove anything. For example, I find that I do potty when | What is the problem with post-hoc testing?
If you don't have a theory backing your propositions, then even if your proposition is validated, it could be through coincidence and does not prove anything. For example, I find that I do potty when the sun rises and have been doing that for the past 10 years - based on this ... | What is the problem with post-hoc testing?
If you don't have a theory backing your propositions, then even if your proposition is validated, it could be through coincidence and does not prove anything. For example, I find that I do potty when |
15,663 | What is the problem with post-hoc testing? | Here is an intuition that you may find useful. If you are bored and count cars, you still have to remember that what you see is the result of some random process. In particular, the cars could have been different colors.
Therefore if you form the hypothesis that the most frequent color is white, if may be because it ac... | What is the problem with post-hoc testing? | Here is an intuition that you may find useful. If you are bored and count cars, you still have to remember that what you see is the result of some random process. In particular, the cars could have be | What is the problem with post-hoc testing?
Here is an intuition that you may find useful. If you are bored and count cars, you still have to remember that what you see is the result of some random process. In particular, the cars could have been different colors.
Therefore if you form the hypothesis that the most frequ... | What is the problem with post-hoc testing?
Here is an intuition that you may find useful. If you are bored and count cars, you still have to remember that what you see is the result of some random process. In particular, the cars could have be |
15,664 | Real examples of Correlation confused with Causation | For many years large observational epidemiological studies interpreted by researchers using Bradford Hill-style heuristic criteria for inferring causation asserted evidence that hormone replacement therapy (HRT) in females decreased risk of coronary heart disease, and it was only after two large scale randomized trials... | Real examples of Correlation confused with Causation | For many years large observational epidemiological studies interpreted by researchers using Bradford Hill-style heuristic criteria for inferring causation asserted evidence that hormone replacement th | Real examples of Correlation confused with Causation
For many years large observational epidemiological studies interpreted by researchers using Bradford Hill-style heuristic criteria for inferring causation asserted evidence that hormone replacement therapy (HRT) in females decreased risk of coronary heart disease, an... | Real examples of Correlation confused with Causation
For many years large observational epidemiological studies interpreted by researchers using Bradford Hill-style heuristic criteria for inferring causation asserted evidence that hormone replacement th |
15,665 | Real examples of Correlation confused with Causation | Not the most glamorous topic, but Nora T. Gedgaudas (Ch. 18) summarizes very nicely the turnaround in findings about fiber's role in preventing colon cancer. Fiber, widely thought for 25 years to be an important preventative factor (based on correlation), was shown through the 16-year, 88,000-subject Nurses' Study to ... | Real examples of Correlation confused with Causation | Not the most glamorous topic, but Nora T. Gedgaudas (Ch. 18) summarizes very nicely the turnaround in findings about fiber's role in preventing colon cancer. Fiber, widely thought for 25 years to be | Real examples of Correlation confused with Causation
Not the most glamorous topic, but Nora T. Gedgaudas (Ch. 18) summarizes very nicely the turnaround in findings about fiber's role in preventing colon cancer. Fiber, widely thought for 25 years to be an important preventative factor (based on correlation), was shown ... | Real examples of Correlation confused with Causation
Not the most glamorous topic, but Nora T. Gedgaudas (Ch. 18) summarizes very nicely the turnaround in findings about fiber's role in preventing colon cancer. Fiber, widely thought for 25 years to be |
15,666 | Real examples of Correlation confused with Causation | Pellagra
According to this book chapter, pellagra, a disease characterized by dizziness, lethargy, running sores, vomiting, and severe diarrhea that had reached epidemic proportions in the US South by the early 1900s, was widely attributed to an unknown pathogen on the basis of a correlation with unsanitary living cond... | Real examples of Correlation confused with Causation | Pellagra
According to this book chapter, pellagra, a disease characterized by dizziness, lethargy, running sores, vomiting, and severe diarrhea that had reached epidemic proportions in the US South by | Real examples of Correlation confused with Causation
Pellagra
According to this book chapter, pellagra, a disease characterized by dizziness, lethargy, running sores, vomiting, and severe diarrhea that had reached epidemic proportions in the US South by the early 1900s, was widely attributed to an unknown pathogen on t... | Real examples of Correlation confused with Causation
Pellagra
According to this book chapter, pellagra, a disease characterized by dizziness, lethargy, running sores, vomiting, and severe diarrhea that had reached epidemic proportions in the US South by |
15,667 | How to perform a bootstrap test to compare the means of two samples? | I would just do a regular bootstrap test:
compute the t-statistic in your data and store it
change the data such that the null-hypothesis is true. In this case, subtract the mean in group 1 for group 1 and add the overall mean, and do the same for group 2, that way the means in both group will be the overall mean.
Tak... | How to perform a bootstrap test to compare the means of two samples? | I would just do a regular bootstrap test:
compute the t-statistic in your data and store it
change the data such that the null-hypothesis is true. In this case, subtract the mean in group 1 for group | How to perform a bootstrap test to compare the means of two samples?
I would just do a regular bootstrap test:
compute the t-statistic in your data and store it
change the data such that the null-hypothesis is true. In this case, subtract the mean in group 1 for group 1 and add the overall mean, and do the same for gr... | How to perform a bootstrap test to compare the means of two samples?
I would just do a regular bootstrap test:
compute the t-statistic in your data and store it
change the data such that the null-hypothesis is true. In this case, subtract the mean in group 1 for group |
15,668 | How to scale new observations for making predictions when the model was fitted with scaled data? | The short answer to your question is, yes - that expression for scaled.new is correct (except you wanted sd instead of std).
It may be worth noting that scale has optional arguments which you could use:
scaled.new <- scale(new, center = mean(data), scale = sd(data))
Also, the object returned by scale (scaled.data) has... | How to scale new observations for making predictions when the model was fitted with scaled data? | The short answer to your question is, yes - that expression for scaled.new is correct (except you wanted sd instead of std).
It may be worth noting that scale has optional arguments which you could us | How to scale new observations for making predictions when the model was fitted with scaled data?
The short answer to your question is, yes - that expression for scaled.new is correct (except you wanted sd instead of std).
It may be worth noting that scale has optional arguments which you could use:
scaled.new <- scale(... | How to scale new observations for making predictions when the model was fitted with scaled data?
The short answer to your question is, yes - that expression for scaled.new is correct (except you wanted sd instead of std).
It may be worth noting that scale has optional arguments which you could us |
15,669 | How to scale new observations for making predictions when the model was fitted with scaled data? | There are now simpler ways to do this.
For example, the preprocess function of the caret package
library(caret)
preproc <- preProcess(data, method = c("center", "scale")
scaled.new <- predict(preproc, newdata = new)
or scale_by in the standardize package
or using the receipes package
library(recipes); library(dplyr)
r... | How to scale new observations for making predictions when the model was fitted with scaled data? | There are now simpler ways to do this.
For example, the preprocess function of the caret package
library(caret)
preproc <- preProcess(data, method = c("center", "scale")
scaled.new <- predict(preproc, | How to scale new observations for making predictions when the model was fitted with scaled data?
There are now simpler ways to do this.
For example, the preprocess function of the caret package
library(caret)
preproc <- preProcess(data, method = c("center", "scale")
scaled.new <- predict(preproc, newdata = new)
or sca... | How to scale new observations for making predictions when the model was fitted with scaled data?
There are now simpler ways to do this.
For example, the preprocess function of the caret package
library(caret)
preproc <- preProcess(data, method = c("center", "scale")
scaled.new <- predict(preproc, |
15,670 | Impact of data-based bin boundaries on a chi-square goodness of fit test? | The basic results of chi-square goodness-of-fit testing can be understood hierarchically.
Level 0. The classical Pearson's chi-square test statistic for testing a multinomial sample against a fixed probability vector $p$ is
$$
X^2(p) = \sum_{i=1}^k \frac{(X^{(n)}_i - n p_i)^2}{n p_i} \stackrel{d}{\to} \chi_{k-1}^2 \>,
... | Impact of data-based bin boundaries on a chi-square goodness of fit test? | The basic results of chi-square goodness-of-fit testing can be understood hierarchically.
Level 0. The classical Pearson's chi-square test statistic for testing a multinomial sample against a fixed pr | Impact of data-based bin boundaries on a chi-square goodness of fit test?
The basic results of chi-square goodness-of-fit testing can be understood hierarchically.
Level 0. The classical Pearson's chi-square test statistic for testing a multinomial sample against a fixed probability vector $p$ is
$$
X^2(p) = \sum_{i=1}... | Impact of data-based bin boundaries on a chi-square goodness of fit test?
The basic results of chi-square goodness-of-fit testing can be understood hierarchically.
Level 0. The classical Pearson's chi-square test statistic for testing a multinomial sample against a fixed pr |
15,671 | Impact of data-based bin boundaries on a chi-square goodness of fit test? | I've found at least partial answers to my question, below. (I'd still like to give someone that bonus, so any further information appreciated.)
Moore (1971) said that Roy (1956) and Watson (1957,58,59) showed that when the cell boundaries
for a chi-square statistic are functions of best asymptotic normal estimated par... | Impact of data-based bin boundaries on a chi-square goodness of fit test? | I've found at least partial answers to my question, below. (I'd still like to give someone that bonus, so any further information appreciated.)
Moore (1971) said that Roy (1956) and Watson (1957,58,59 | Impact of data-based bin boundaries on a chi-square goodness of fit test?
I've found at least partial answers to my question, below. (I'd still like to give someone that bonus, so any further information appreciated.)
Moore (1971) said that Roy (1956) and Watson (1957,58,59) showed that when the cell boundaries
for a ... | Impact of data-based bin boundaries on a chi-square goodness of fit test?
I've found at least partial answers to my question, below. (I'd still like to give someone that bonus, so any further information appreciated.)
Moore (1971) said that Roy (1956) and Watson (1957,58,59 |
15,672 | Periods in history of statistics | These recent papers by Stigler, where he argues (convincingly I believe) for the types of periods you seem to have in mind.
Stigler, Stephen M. 2010. Darwin, Galton and the statistical
enlightenment. Journal of the Royal Statistical Society: Series A
173(3):469-482.
Stigler, Stephen M. 2012. Studies in the history ... | Periods in history of statistics | These recent papers by Stigler, where he argues (convincingly I believe) for the types of periods you seem to have in mind.
Stigler, Stephen M. 2010. Darwin, Galton and the statistical
enlightenme | Periods in history of statistics
These recent papers by Stigler, where he argues (convincingly I believe) for the types of periods you seem to have in mind.
Stigler, Stephen M. 2010. Darwin, Galton and the statistical
enlightenment. Journal of the Royal Statistical Society: Series A
173(3):469-482.
Stigler, Stephen... | Periods in history of statistics
These recent papers by Stigler, where he argues (convincingly I believe) for the types of periods you seem to have in mind.
Stigler, Stephen M. 2010. Darwin, Galton and the statistical
enlightenme |
15,673 | Periods in history of statistics | I think that "periods" in history are closely related to people and their developments. Of course one can expect "waves" in Toffler's sense, but even those waves are related to persons.
Anyway, wikipedia has an article in this regard. | Periods in history of statistics | I think that "periods" in history are closely related to people and their developments. Of course one can expect "waves" in Toffler's sense, but even those waves are related to persons.
Anyway, wikip | Periods in history of statistics
I think that "periods" in history are closely related to people and their developments. Of course one can expect "waves" in Toffler's sense, but even those waves are related to persons.
Anyway, wikipedia has an article in this regard. | Periods in history of statistics
I think that "periods" in history are closely related to people and their developments. Of course one can expect "waves" in Toffler's sense, but even those waves are related to persons.
Anyway, wikip |
15,674 | Periods in history of statistics | According to the webpage titled "Materials for the History of Statistics" by the Department of Mathematics at the University of York, a major text on this subject is:
Oscar Sheynin, Theory of Probability: A Historical Essay (published by NG Verlag 2005, ISBN 3-938417-15-3)
The book is packed full of names, dates, ideas... | Periods in history of statistics | According to the webpage titled "Materials for the History of Statistics" by the Department of Mathematics at the University of York, a major text on this subject is:
Oscar Sheynin, Theory of Probabil | Periods in history of statistics
According to the webpage titled "Materials for the History of Statistics" by the Department of Mathematics at the University of York, a major text on this subject is:
Oscar Sheynin, Theory of Probability: A Historical Essay (published by NG Verlag 2005, ISBN 3-938417-15-3)
The book is p... | Periods in history of statistics
According to the webpage titled "Materials for the History of Statistics" by the Department of Mathematics at the University of York, a major text on this subject is:
Oscar Sheynin, Theory of Probabil |
15,675 | Correlation of log-normal random variables | I assume that $X_1\sim N(0,\sigma_1^2)$ and $X_2\sim N(0,\sigma_2^2)$. Denote $Z_i=\exp(\sqrt{T}X_i)$. Then
\begin{align}
\log(Z_i)\sim N(0,T\sigma_i^2)
\end{align}
so $Z_i$ are log-normal. Thus
\begin{align}
EZ_i&=\exp\left(\frac{T\sigma_i^2}{2}\right)\\
var(Z_i)&=(\exp(T\sigma_i^2)-1)\exp(T\sigma_i^2)
\end{align}
an... | Correlation of log-normal random variables | I assume that $X_1\sim N(0,\sigma_1^2)$ and $X_2\sim N(0,\sigma_2^2)$. Denote $Z_i=\exp(\sqrt{T}X_i)$. Then
\begin{align}
\log(Z_i)\sim N(0,T\sigma_i^2)
\end{align}
so $Z_i$ are log-normal. Thus
\beg | Correlation of log-normal random variables
I assume that $X_1\sim N(0,\sigma_1^2)$ and $X_2\sim N(0,\sigma_2^2)$. Denote $Z_i=\exp(\sqrt{T}X_i)$. Then
\begin{align}
\log(Z_i)\sim N(0,T\sigma_i^2)
\end{align}
so $Z_i$ are log-normal. Thus
\begin{align}
EZ_i&=\exp\left(\frac{T\sigma_i^2}{2}\right)\\
var(Z_i)&=(\exp(T\si... | Correlation of log-normal random variables
I assume that $X_1\sim N(0,\sigma_1^2)$ and $X_2\sim N(0,\sigma_2^2)$. Denote $Z_i=\exp(\sqrt{T}X_i)$. Then
\begin{align}
\log(Z_i)\sim N(0,T\sigma_i^2)
\end{align}
so $Z_i$ are log-normal. Thus
\beg |
15,676 | Data partitioning for spatial data | After watching the video, I have become more confident that this
application is more like "data reproduction", where a random
partitioning is OK, rather than "data prediction".
To me, you justify your choice of using random CV for spatial ML models too much with "if I use it for data reproduction, it is ok".
The ... | Data partitioning for spatial data | After watching the video, I have become more confident that this
application is more like "data reproduction", where a random
partitioning is OK, rather than "data prediction".
To me, you justify | Data partitioning for spatial data
After watching the video, I have become more confident that this
application is more like "data reproduction", where a random
partitioning is OK, rather than "data prediction".
To me, you justify your choice of using random CV for spatial ML models too much with "if I use it for ... | Data partitioning for spatial data
After watching the video, I have become more confident that this
application is more like "data reproduction", where a random
partitioning is OK, rather than "data prediction".
To me, you justify |
15,677 | Data partitioning for spatial data | Nice question, and I fully agree with Roozbeh.
Spatial cross validation is relevant when you have spatial autocorrelation in your training data that usually occur when your data are clustered in space.
If you want to know how well your model is able to generalize (i.e. make predictions beyond training locations), you w... | Data partitioning for spatial data | Nice question, and I fully agree with Roozbeh.
Spatial cross validation is relevant when you have spatial autocorrelation in your training data that usually occur when your data are clustered in space | Data partitioning for spatial data
Nice question, and I fully agree with Roozbeh.
Spatial cross validation is relevant when you have spatial autocorrelation in your training data that usually occur when your data are clustered in space.
If you want to know how well your model is able to generalize (i.e. make prediction... | Data partitioning for spatial data
Nice question, and I fully agree with Roozbeh.
Spatial cross validation is relevant when you have spatial autocorrelation in your training data that usually occur when your data are clustered in space |
15,678 | Data partitioning for spatial data | Very interesting question! The importance of spatial/block cross-validation comes to play when you think your performance might be affected by spatial autocorrelation. This totally depends on the purpose of your study. If you are interested to assess the performance of your models for only those specific locations or n... | Data partitioning for spatial data | Very interesting question! The importance of spatial/block cross-validation comes to play when you think your performance might be affected by spatial autocorrelation. This totally depends on the purp | Data partitioning for spatial data
Very interesting question! The importance of spatial/block cross-validation comes to play when you think your performance might be affected by spatial autocorrelation. This totally depends on the purpose of your study. If you are interested to assess the performance of your models for... | Data partitioning for spatial data
Very interesting question! The importance of spatial/block cross-validation comes to play when you think your performance might be affected by spatial autocorrelation. This totally depends on the purp |
15,679 | How to measure the statistical "distance" between two frequency distributions? | You may be interested in the Earth mover's distance, also known as the Wasserstein metric. It is implemented in R (look at the emdist package) and in Python. We also have a number of threads on it.
The EMD works for both continuous and discrete distributions. The emdist package for R works on discrete distributions.
Th... | How to measure the statistical "distance" between two frequency distributions? | You may be interested in the Earth mover's distance, also known as the Wasserstein metric. It is implemented in R (look at the emdist package) and in Python. We also have a number of threads on it.
Th | How to measure the statistical "distance" between two frequency distributions?
You may be interested in the Earth mover's distance, also known as the Wasserstein metric. It is implemented in R (look at the emdist package) and in Python. We also have a number of threads on it.
The EMD works for both continuous and discr... | How to measure the statistical "distance" between two frequency distributions?
You may be interested in the Earth mover's distance, also known as the Wasserstein metric. It is implemented in R (look at the emdist package) and in Python. We also have a number of threads on it.
Th |
15,680 | How to measure the statistical "distance" between two frequency distributions? | If you randomly sample an individual from each of the two distributions, you can calculate a difference between them. If you repeat this (with replacement) a number of times, you can generate a distribution of differences that contains all the information you are after. You can plot this distribution and characterize i... | How to measure the statistical "distance" between two frequency distributions? | If you randomly sample an individual from each of the two distributions, you can calculate a difference between them. If you repeat this (with replacement) a number of times, you can generate a distri | How to measure the statistical "distance" between two frequency distributions?
If you randomly sample an individual from each of the two distributions, you can calculate a difference between them. If you repeat this (with replacement) a number of times, you can generate a distribution of differences that contains all t... | How to measure the statistical "distance" between two frequency distributions?
If you randomly sample an individual from each of the two distributions, you can calculate a difference between them. If you repeat this (with replacement) a number of times, you can generate a distri |
15,681 | How to measure the statistical "distance" between two frequency distributions? | One of the metric is Hellinger distance between two distributions which are characterised by means and standard deviations. The application can be found in the following article.
https://www.sciencedirect.com/science/article/pii/S1568494615005104 | How to measure the statistical "distance" between two frequency distributions? | One of the metric is Hellinger distance between two distributions which are characterised by means and standard deviations. The application can be found in the following article.
https://www.sciencedi | How to measure the statistical "distance" between two frequency distributions?
One of the metric is Hellinger distance between two distributions which are characterised by means and standard deviations. The application can be found in the following article.
https://www.sciencedirect.com/science/article/pii/S15684946150... | How to measure the statistical "distance" between two frequency distributions?
One of the metric is Hellinger distance between two distributions which are characterised by means and standard deviations. The application can be found in the following article.
https://www.sciencedi |
15,682 | How does a Bayesian update his belief when something with probability 0 happened? | Any posterior probability is valid in this case
This is an interesting question, which gets into the territory of the foundations of probability. There are a few possible approaches here, but for reasons that I will elaborate on soon, the approach I favour is to give a broader definition of conditional probability tha... | How does a Bayesian update his belief when something with probability 0 happened? | Any posterior probability is valid in this case
This is an interesting question, which gets into the territory of the foundations of probability. There are a few possible approaches here, but for rea | How does a Bayesian update his belief when something with probability 0 happened?
Any posterior probability is valid in this case
This is an interesting question, which gets into the territory of the foundations of probability. There are a few possible approaches here, but for reasons that I will elaborate on soon, th... | How does a Bayesian update his belief when something with probability 0 happened?
Any posterior probability is valid in this case
This is an interesting question, which gets into the territory of the foundations of probability. There are a few possible approaches here, but for rea |
15,683 | How does a Bayesian update his belief when something with probability 0 happened? | This is related to field of logic. in particular, a false statement implies all other statements, true or false. In your scenario $X$ is a false statement.This means we can write $X\implies S$ for any other proposition $S$. For example, we have $X\implies E$ (it implies tails) and also $X\implies E^c$ (it implies not t... | How does a Bayesian update his belief when something with probability 0 happened? | This is related to field of logic. in particular, a false statement implies all other statements, true or false. In your scenario $X$ is a false statement.This means we can write $X\implies S$ for any | How does a Bayesian update his belief when something with probability 0 happened?
This is related to field of logic. in particular, a false statement implies all other statements, true or false. In your scenario $X$ is a false statement.This means we can write $X\implies S$ for any other proposition $S$. For example, w... | How does a Bayesian update his belief when something with probability 0 happened?
This is related to field of logic. in particular, a false statement implies all other statements, true or false. In your scenario $X$ is a false statement.This means we can write $X\implies S$ for any |
15,684 | How does a Bayesian update his belief when something with probability 0 happened? | There's an implicit assumption in all reasoning, Bayesian or otherwise, that we know everything that could happen and accounted for it. If something happens which is impossible under the model, it just means that that assumption is false. The principled thing to do is to go back and expand the model, and start over. At... | How does a Bayesian update his belief when something with probability 0 happened? | There's an implicit assumption in all reasoning, Bayesian or otherwise, that we know everything that could happen and accounted for it. If something happens which is impossible under the model, it jus | How does a Bayesian update his belief when something with probability 0 happened?
There's an implicit assumption in all reasoning, Bayesian or otherwise, that we know everything that could happen and accounted for it. If something happens which is impossible under the model, it just means that that assumption is false.... | How does a Bayesian update his belief when something with probability 0 happened?
There's an implicit assumption in all reasoning, Bayesian or otherwise, that we know everything that could happen and accounted for it. If something happens which is impossible under the model, it jus |
15,685 | Accuracy vs. area under the ROC curve | It's indeed possible. The key is to remember that the accuracy is highly affected by class imbalance. E.g., in your case, you have more negative samples than positive samples, since when the FPR ($=\frac{FP}{FP+TN}$) is close to 0, and TPR (= $\frac{TP}{TP+FN}$) is 0.5, your accuracy ($= \frac{TP+TN}{TP+FN+FP+TN}$) is ... | Accuracy vs. area under the ROC curve | It's indeed possible. The key is to remember that the accuracy is highly affected by class imbalance. E.g., in your case, you have more negative samples than positive samples, since when the FPR ($=\f | Accuracy vs. area under the ROC curve
It's indeed possible. The key is to remember that the accuracy is highly affected by class imbalance. E.g., in your case, you have more negative samples than positive samples, since when the FPR ($=\frac{FP}{FP+TN}$) is close to 0, and TPR (= $\frac{TP}{TP+FN}$) is 0.5, your accura... | Accuracy vs. area under the ROC curve
It's indeed possible. The key is to remember that the accuracy is highly affected by class imbalance. E.g., in your case, you have more negative samples than positive samples, since when the FPR ($=\f |
15,686 | Accuracy vs. area under the ROC curve | Okay, remember the relation between the $FPR$ (False Positive Rate), $TPR$ (True Positive Rate) and $ACC$ (Accuracy):
$$TPR = \frac{\sum \text{True positive}}{\sum \text{Positive cases}}$$
$$FPR = \frac{\sum \text{False positive}}{\sum \text{Negative cases}}$$
$$ACC = \frac{TPR \cdot \sum \text{Positive cases} + (1-FPR... | Accuracy vs. area under the ROC curve | Okay, remember the relation between the $FPR$ (False Positive Rate), $TPR$ (True Positive Rate) and $ACC$ (Accuracy):
$$TPR = \frac{\sum \text{True positive}}{\sum \text{Positive cases}}$$
$$FPR = \fr | Accuracy vs. area under the ROC curve
Okay, remember the relation between the $FPR$ (False Positive Rate), $TPR$ (True Positive Rate) and $ACC$ (Accuracy):
$$TPR = \frac{\sum \text{True positive}}{\sum \text{Positive cases}}$$
$$FPR = \frac{\sum \text{False positive}}{\sum \text{Negative cases}}$$
$$ACC = \frac{TPR \cd... | Accuracy vs. area under the ROC curve
Okay, remember the relation between the $FPR$ (False Positive Rate), $TPR$ (True Positive Rate) and $ACC$ (Accuracy):
$$TPR = \frac{\sum \text{True positive}}{\sum \text{Positive cases}}$$
$$FPR = \fr |
15,687 | Is variance a more fundamental concept than standard deviation? | Robert's and Bey's answers do give part of the story (i.e. moments tend to be regarded as basic properties of distributions, and conventionally standard deviation is defined in terms of the second central moment rather than the other way around), but the extent to which those things are really fundamental depends partl... | Is variance a more fundamental concept than standard deviation? | Robert's and Bey's answers do give part of the story (i.e. moments tend to be regarded as basic properties of distributions, and conventionally standard deviation is defined in terms of the second cen | Is variance a more fundamental concept than standard deviation?
Robert's and Bey's answers do give part of the story (i.e. moments tend to be regarded as basic properties of distributions, and conventionally standard deviation is defined in terms of the second central moment rather than the other way around), but the e... | Is variance a more fundamental concept than standard deviation?
Robert's and Bey's answers do give part of the story (i.e. moments tend to be regarded as basic properties of distributions, and conventionally standard deviation is defined in terms of the second cen |
15,688 | Is variance a more fundamental concept than standard deviation? | Variance is defined by the first and second moments of a distribution. In contrast, the standard deviation is more like a "norm" than a moment. Moments are fundamental properties of a distribution, whereas norms are just ways to make a distinction. | Is variance a more fundamental concept than standard deviation? | Variance is defined by the first and second moments of a distribution. In contrast, the standard deviation is more like a "norm" than a moment. Moments are fundamental properties of a distribution, wh | Is variance a more fundamental concept than standard deviation?
Variance is defined by the first and second moments of a distribution. In contrast, the standard deviation is more like a "norm" than a moment. Moments are fundamental properties of a distribution, whereas norms are just ways to make a distinction. | Is variance a more fundamental concept than standard deviation?
Variance is defined by the first and second moments of a distribution. In contrast, the standard deviation is more like a "norm" than a moment. Moments are fundamental properties of a distribution, wh |
15,689 | Is variance a more fundamental concept than standard deviation? | The variance is more fundamental than the standard deviation because the standard deviation is defined as 'the square root of the variance', e.g. its definition depends completely on the variance.
Variance, on the other hand is defined - completely independently - as the 'the expectation of the squared difference betwe... | Is variance a more fundamental concept than standard deviation? | The variance is more fundamental than the standard deviation because the standard deviation is defined as 'the square root of the variance', e.g. its definition depends completely on the variance.
Var | Is variance a more fundamental concept than standard deviation?
The variance is more fundamental than the standard deviation because the standard deviation is defined as 'the square root of the variance', e.g. its definition depends completely on the variance.
Variance, on the other hand is defined - completely indepen... | Is variance a more fundamental concept than standard deviation?
The variance is more fundamental than the standard deviation because the standard deviation is defined as 'the square root of the variance', e.g. its definition depends completely on the variance.
Var |
15,690 | Is variance a more fundamental concept than standard deviation? | In addition to the answers given here, one could point out that variance is more 'fundamental' than standard deviation in some sense, if we consider estimation from a (e.g. normal) population. For a sample of size $n$ drawn from a population $X$ with $\mathrm{Var}[X] = \sigma^2$, it is known that the sample variance $S... | Is variance a more fundamental concept than standard deviation? | In addition to the answers given here, one could point out that variance is more 'fundamental' than standard deviation in some sense, if we consider estimation from a (e.g. normal) population. For a s | Is variance a more fundamental concept than standard deviation?
In addition to the answers given here, one could point out that variance is more 'fundamental' than standard deviation in some sense, if we consider estimation from a (e.g. normal) population. For a sample of size $n$ drawn from a population $X$ with $\mat... | Is variance a more fundamental concept than standard deviation?
In addition to the answers given here, one could point out that variance is more 'fundamental' than standard deviation in some sense, if we consider estimation from a (e.g. normal) population. For a s |
15,691 | Including Interaction Terms in Random Forest | Although feature engineering is very important in real life, trees (and random forests) are very good at finding interaction terms of the form x*y. Here is a toy example of a regression with a two-way interaction. A naive linear model is compared with a tree and a bag of trees (which is a simpler alternative to a rando... | Including Interaction Terms in Random Forest | Although feature engineering is very important in real life, trees (and random forests) are very good at finding interaction terms of the form x*y. Here is a toy example of a regression with a two-way | Including Interaction Terms in Random Forest
Although feature engineering is very important in real life, trees (and random forests) are very good at finding interaction terms of the form x*y. Here is a toy example of a regression with a two-way interaction. A naive linear model is compared with a tree and a bag of tre... | Including Interaction Terms in Random Forest
Although feature engineering is very important in real life, trees (and random forests) are very good at finding interaction terms of the form x*y. Here is a toy example of a regression with a two-way |
15,692 | Definition of normalized Euclidean distance | The normalized squared euclidean distance gives the squared distance between two vectors where there lengths have been scaled to have unit norm. This is helpful when the direction of the vector is meaningful but the magnitude is not. It's not related to Mahalanobis distance. | Definition of normalized Euclidean distance | The normalized squared euclidean distance gives the squared distance between two vectors where there lengths have been scaled to have unit norm. This is helpful when the direction of the vector is mea | Definition of normalized Euclidean distance
The normalized squared euclidean distance gives the squared distance between two vectors where there lengths have been scaled to have unit norm. This is helpful when the direction of the vector is meaningful but the magnitude is not. It's not related to Mahalanobis distance. | Definition of normalized Euclidean distance
The normalized squared euclidean distance gives the squared distance between two vectors where there lengths have been scaled to have unit norm. This is helpful when the direction of the vector is mea |
15,693 | Definition of normalized Euclidean distance | The weighted Minkowski distance of order $q$ between two real vectors $u, v \in \mathbb{R}^n$ is given by
$$d^{(q)} (u, v) = \left(\sum_{i=1}^n w_i (u_i - v_i)^q \right)^\frac{1}{q}$$
[See equation $3.1.7$, Clustering Methodology for Symbolic Data By Lynne Billard, Edwin Diday (2019)]
If we choose $w_i = \frac{1}{n}$ ... | Definition of normalized Euclidean distance | The weighted Minkowski distance of order $q$ between two real vectors $u, v \in \mathbb{R}^n$ is given by
$$d^{(q)} (u, v) = \left(\sum_{i=1}^n w_i (u_i - v_i)^q \right)^\frac{1}{q}$$
[See equation $ | Definition of normalized Euclidean distance
The weighted Minkowski distance of order $q$ between two real vectors $u, v \in \mathbb{R}^n$ is given by
$$d^{(q)} (u, v) = \left(\sum_{i=1}^n w_i (u_i - v_i)^q \right)^\frac{1}{q}$$
[See equation $3.1.7$, Clustering Methodology for Symbolic Data By Lynne Billard, Edwin Did... | Definition of normalized Euclidean distance
The weighted Minkowski distance of order $q$ between two real vectors $u, v \in \mathbb{R}^n$ is given by
$$d^{(q)} (u, v) = \left(\sum_{i=1}^n w_i (u_i - v_i)^q \right)^\frac{1}{q}$$
[See equation $ |
15,694 | Definition of normalized Euclidean distance | Here is one way of thinking about the Normalised Squared Euclidean Distance $NED^2$, defined as $$NED^2(u,v) = 0.5 \frac{ \text{Var}(u-v) }{ \text{Var}(u) + \text{Var}(v) }$$ for two vectors $u,v\in\mathbb{R}^k$.
This definition does not appear very much in the scientific literature. I can see at least two problems wit... | Definition of normalized Euclidean distance | Here is one way of thinking about the Normalised Squared Euclidean Distance $NED^2$, defined as $$NED^2(u,v) = 0.5 \frac{ \text{Var}(u-v) }{ \text{Var}(u) + \text{Var}(v) }$$ for two vectors $u,v\in\m | Definition of normalized Euclidean distance
Here is one way of thinking about the Normalised Squared Euclidean Distance $NED^2$, defined as $$NED^2(u,v) = 0.5 \frac{ \text{Var}(u-v) }{ \text{Var}(u) + \text{Var}(v) }$$ for two vectors $u,v\in\mathbb{R}^k$.
This definition does not appear very much in the scientific lit... | Definition of normalized Euclidean distance
Here is one way of thinking about the Normalised Squared Euclidean Distance $NED^2$, defined as $$NED^2(u,v) = 0.5 \frac{ \text{Var}(u-v) }{ \text{Var}(u) + \text{Var}(v) }$$ for two vectors $u,v\in\m |
15,695 | Definition of normalized Euclidean distance | I believe this is the correct implementation in pytorch (should be easy to translate to numpy etc):
import torch.nn as nn
def ned(x1, x2, dim=1, eps=1e-8):
ned_2 = 0.5 * ((x1 - x2).var(dim=dim) / (x1.var(dim=dim) + x2.var(dim=dim) + eps))
return ned_2 ** 0.5
def nes(x1, x2, dim=1, eps=1e-8):
return 1 - n... | Definition of normalized Euclidean distance | I believe this is the correct implementation in pytorch (should be easy to translate to numpy etc):
import torch.nn as nn
def ned(x1, x2, dim=1, eps=1e-8):
ned_2 = 0.5 * ((x1 - x2).var(dim=dim) | Definition of normalized Euclidean distance
I believe this is the correct implementation in pytorch (should be easy to translate to numpy etc):
import torch.nn as nn
def ned(x1, x2, dim=1, eps=1e-8):
ned_2 = 0.5 * ((x1 - x2).var(dim=dim) / (x1.var(dim=dim) + x2.var(dim=dim) + eps))
return ned_2 ** 0.5
def ne... | Definition of normalized Euclidean distance
I believe this is the correct implementation in pytorch (should be easy to translate to numpy etc):
import torch.nn as nn
def ned(x1, x2, dim=1, eps=1e-8):
ned_2 = 0.5 * ((x1 - x2).var(dim=dim) |
15,696 | Gelman and Rubin convergence diagnostic, how to generalise to work with vectors? | A recommendation: just compute the PSRF separately for each scalar component
The original article by Gelman & Rubin [1], as well as the Bayesian Data Analysis textbook of Gelman et al. [2], recommends calculating the potential scale reduction factor (PSRF) separately for each scalar parameter of interest. To deduce con... | Gelman and Rubin convergence diagnostic, how to generalise to work with vectors? | A recommendation: just compute the PSRF separately for each scalar component
The original article by Gelman & Rubin [1], as well as the Bayesian Data Analysis textbook of Gelman et al. [2], recommends | Gelman and Rubin convergence diagnostic, how to generalise to work with vectors?
A recommendation: just compute the PSRF separately for each scalar component
The original article by Gelman & Rubin [1], as well as the Bayesian Data Analysis textbook of Gelman et al. [2], recommends calculating the potential scale reduct... | Gelman and Rubin convergence diagnostic, how to generalise to work with vectors?
A recommendation: just compute the PSRF separately for each scalar component
The original article by Gelman & Rubin [1], as well as the Bayesian Data Analysis textbook of Gelman et al. [2], recommends |
15,697 | Are standardized betas in multiple linear regression partial correlations? [duplicate] | Longer answer.
If I have this right --
Partial correlation:
$$
r_{y1.2} = \frac{r_{y1}-r_{y2}r_{12}}{\sqrt{(1-r^2_{y2})(1-r^2_{12})}}
$$
equivalent standardized beta:
$$
\beta_1 = \frac{r_{y1}-r_{y2}r_{12}}{(1-r^2_{12})}
$$
As you see, the denominator is different.
Their relative size depends on whether $\sqrt{(1-r^2_... | Are standardized betas in multiple linear regression partial correlations? [duplicate] | Longer answer.
If I have this right --
Partial correlation:
$$
r_{y1.2} = \frac{r_{y1}-r_{y2}r_{12}}{\sqrt{(1-r^2_{y2})(1-r^2_{12})}}
$$
equivalent standardized beta:
$$
\beta_1 = \frac{r_{y1}-r_{y2}r | Are standardized betas in multiple linear regression partial correlations? [duplicate]
Longer answer.
If I have this right --
Partial correlation:
$$
r_{y1.2} = \frac{r_{y1}-r_{y2}r_{12}}{\sqrt{(1-r^2_{y2})(1-r^2_{12})}}
$$
equivalent standardized beta:
$$
\beta_1 = \frac{r_{y1}-r_{y2}r_{12}}{(1-r^2_{12})}
$$
As you se... | Are standardized betas in multiple linear regression partial correlations? [duplicate]
Longer answer.
If I have this right --
Partial correlation:
$$
r_{y1.2} = \frac{r_{y1}-r_{y2}r_{12}}{\sqrt{(1-r^2_{y2})(1-r^2_{12})}}
$$
equivalent standardized beta:
$$
\beta_1 = \frac{r_{y1}-r_{y2}r |
15,698 | Are standardized betas in multiple linear regression partial correlations? [duplicate] | I've in another question the following correlation matrix C for the three variables X,Y,Z given:
$$ \text{ C =} \small \left[ \begin{array} {rrr}
1&-0.286122&-0.448535\\
-0.286122&1&0.928251\\
-0.448535&0.928251&1
\end{array} \right] $$
From its cholesky-decomposition L
$$ \text{ L =} \small \left[ \begin{arr... | Are standardized betas in multiple linear regression partial correlations? [duplicate] | I've in another question the following correlation matrix C for the three variables X,Y,Z given:
$$ \text{ C =} \small \left[ \begin{array} {rrr}
1&-0.286122&-0.448535\\
-0.286122&1&0.928251\\
-0.44 | Are standardized betas in multiple linear regression partial correlations? [duplicate]
I've in another question the following correlation matrix C for the three variables X,Y,Z given:
$$ \text{ C =} \small \left[ \begin{array} {rrr}
1&-0.286122&-0.448535\\
-0.286122&1&0.928251\\
-0.448535&0.928251&1
\end{array} ... | Are standardized betas in multiple linear regression partial correlations? [duplicate]
I've in another question the following correlation matrix C for the three variables X,Y,Z given:
$$ \text{ C =} \small \left[ \begin{array} {rrr}
1&-0.286122&-0.448535\\
-0.286122&1&0.928251\\
-0.44 |
15,699 | Are truncated numbers from a random number generator still 'random'? | Yes, the truncated values are random. The distribution has changed from a continuous distribution to a discrete distribution. Random values with discrete distributions are often used.
There are senses in which this change to the distribution is very small. The maximum difference between the cumulative distribution func... | Are truncated numbers from a random number generator still 'random'? | Yes, the truncated values are random. The distribution has changed from a continuous distribution to a discrete distribution. Random values with discrete distributions are often used.
There are senses | Are truncated numbers from a random number generator still 'random'?
Yes, the truncated values are random. The distribution has changed from a continuous distribution to a discrete distribution. Random values with discrete distributions are often used.
There are senses in which this change to the distribution is very s... | Are truncated numbers from a random number generator still 'random'?
Yes, the truncated values are random. The distribution has changed from a continuous distribution to a discrete distribution. Random values with discrete distributions are often used.
There are senses |
15,700 | Why does the supremum of the Brownian bridge have the Kolmogorov–Smirnov distribution? | $\sqrt{n}\sup_x|F_n-F|=\sup_x|\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_i(x)| $
where $Z_i(x)=1_{X_i\leq x}-E[1_{X_i\leq x}]$
by CLT you have
$G_n=\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_i(x)\rightarrow \mathcal{N}(0,F(x)(1-F(x)))$
this is the intuition...
brownian bridge $B(t)$ has variance $t(1-t)$ http://en.wikipedia.org/wiki/Brow... | Why does the supremum of the Brownian bridge have the Kolmogorov–Smirnov distribution? | $\sqrt{n}\sup_x|F_n-F|=\sup_x|\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_i(x)| $
where $Z_i(x)=1_{X_i\leq x}-E[1_{X_i\leq x}]$
by CLT you have
$G_n=\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_i(x)\rightarrow \mathcal{N}(0 | Why does the supremum of the Brownian bridge have the Kolmogorov–Smirnov distribution?
$\sqrt{n}\sup_x|F_n-F|=\sup_x|\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_i(x)| $
where $Z_i(x)=1_{X_i\leq x}-E[1_{X_i\leq x}]$
by CLT you have
$G_n=\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_i(x)\rightarrow \mathcal{N}(0,F(x)(1-F(x)))$
this is the intui... | Why does the supremum of the Brownian bridge have the Kolmogorov–Smirnov distribution?
$\sqrt{n}\sup_x|F_n-F|=\sup_x|\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_i(x)| $
where $Z_i(x)=1_{X_i\leq x}-E[1_{X_i\leq x}]$
by CLT you have
$G_n=\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_i(x)\rightarrow \mathcal{N}(0 |
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