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General hints regarding the use of the binomial distribution in conditional probabilty problems
In this response I will limit myself to the general question of when to use the binomial. The binomial arises naturally as the count of successes in a sequence of independent Bernoulli trials with constant success probability $p$. It's useful to keep the conditions in mind (both for Bernoulli trials and the independent...
General hints regarding the use of the binomial distribution in conditional probabilty problems
In this response I will limit myself to the general question of when to use the binomial. The binomial arises naturally as the count of successes in a sequence of independent Bernoulli trials with con
General hints regarding the use of the binomial distribution in conditional probabilty problems In this response I will limit myself to the general question of when to use the binomial. The binomial arises naturally as the count of successes in a sequence of independent Bernoulli trials with constant success probabilit...
General hints regarding the use of the binomial distribution in conditional probabilty problems In this response I will limit myself to the general question of when to use the binomial. The binomial arises naturally as the count of successes in a sequence of independent Bernoulli trials with con
47,102
Bayesian and Frequentist linear regressions: different results
I think the frequentist linear regression results are going to be anticonservative in this case. We know those results are wrong (in the sense of an inadequate control of type 1 error) because the model assumptions are not met. The correct frequentist inference would be obtained by using sandwich standard errors. See t...
Bayesian and Frequentist linear regressions: different results
I think the frequentist linear regression results are going to be anticonservative in this case. We know those results are wrong (in the sense of an inadequate control of type 1 error) because the mod
Bayesian and Frequentist linear regressions: different results I think the frequentist linear regression results are going to be anticonservative in this case. We know those results are wrong (in the sense of an inadequate control of type 1 error) because the model assumptions are not met. The correct frequentist infer...
Bayesian and Frequentist linear regressions: different results I think the frequentist linear regression results are going to be anticonservative in this case. We know those results are wrong (in the sense of an inadequate control of type 1 error) because the mod
47,103
Bayesian and Frequentist linear regressions: different results
So there are a couple of issues here. As correctly pointed out above, you have calculation issues with the errors in the Frequentist model. But you also have another issue, it appears you may be interpreting the Bayesian interval incorrectly. That the interval contains zero is very important in Frequentist models bu...
Bayesian and Frequentist linear regressions: different results
So there are a couple of issues here. As correctly pointed out above, you have calculation issues with the errors in the Frequentist model. But you also have another issue, it appears you may be int
Bayesian and Frequentist linear regressions: different results So there are a couple of issues here. As correctly pointed out above, you have calculation issues with the errors in the Frequentist model. But you also have another issue, it appears you may be interpreting the Bayesian interval incorrectly. That the in...
Bayesian and Frequentist linear regressions: different results So there are a couple of issues here. As correctly pointed out above, you have calculation issues with the errors in the Frequentist model. But you also have another issue, it appears you may be int
47,104
Is there an opposite to Clustering or "Anti-"Clustering?
Based on the poster's response in one of the answers, I will rephrase the problem statement because it can be solved using an R package that I wrote: The task is to partition a set of elements into K groups such that the distances between clusters is minimized and the distance within clusters is maximized. This is math...
Is there an opposite to Clustering or "Anti-"Clustering?
Based on the poster's response in one of the answers, I will rephrase the problem statement because it can be solved using an R package that I wrote: The task is to partition a set of elements into K
Is there an opposite to Clustering or "Anti-"Clustering? Based on the poster's response in one of the answers, I will rephrase the problem statement because it can be solved using an R package that I wrote: The task is to partition a set of elements into K groups such that the distances between clusters is minimized an...
Is there an opposite to Clustering or "Anti-"Clustering? Based on the poster's response in one of the answers, I will rephrase the problem statement because it can be solved using an R package that I wrote: The task is to partition a set of elements into K
47,105
Is there an opposite to Clustering or "Anti-"Clustering?
There are several related domains: Outlier detection: find unusual points, rather than typical representatives as in clustering. Stratified sampling. Choose a random sample such that the samples correspond to different classes. For larger sets, so that the class distribution matches. Archetypal analysis. Summarize dat...
Is there an opposite to Clustering or "Anti-"Clustering?
There are several related domains: Outlier detection: find unusual points, rather than typical representatives as in clustering. Stratified sampling. Choose a random sample such that the samples corr
Is there an opposite to Clustering or "Anti-"Clustering? There are several related domains: Outlier detection: find unusual points, rather than typical representatives as in clustering. Stratified sampling. Choose a random sample such that the samples correspond to different classes. For larger sets, so that the class...
Is there an opposite to Clustering or "Anti-"Clustering? There are several related domains: Outlier detection: find unusual points, rather than typical representatives as in clustering. Stratified sampling. Choose a random sample such that the samples corr
47,106
Is there an opposite to Clustering or "Anti-"Clustering?
It's still clustering that you want, because you're trying to split the data into subsets, which is clustering. If you clustering algorithm is based on a distance measure, then simple negative of that measure or inverse should accomplish subsetting you request. Some algorithms rely on the distance measure being positiv...
Is there an opposite to Clustering or "Anti-"Clustering?
It's still clustering that you want, because you're trying to split the data into subsets, which is clustering. If you clustering algorithm is based on a distance measure, then simple negative of that
Is there an opposite to Clustering or "Anti-"Clustering? It's still clustering that you want, because you're trying to split the data into subsets, which is clustering. If you clustering algorithm is based on a distance measure, then simple negative of that measure or inverse should accomplish subsetting you request. S...
Is there an opposite to Clustering or "Anti-"Clustering? It's still clustering that you want, because you're trying to split the data into subsets, which is clustering. If you clustering algorithm is based on a distance measure, then simple negative of that
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Is there an opposite to Clustering or "Anti-"Clustering?
Building on the useful comments & answers by Nick Cox and others here: DBSCAN is a clustering algorithm that also identifies points that do not belong well to any particular cluster, and treats them as 'noise'. This might be one way to achieve what you are after, since it identifies points that are distant enough from...
Is there an opposite to Clustering or "Anti-"Clustering?
Building on the useful comments & answers by Nick Cox and others here: DBSCAN is a clustering algorithm that also identifies points that do not belong well to any particular cluster, and treats them
Is there an opposite to Clustering or "Anti-"Clustering? Building on the useful comments & answers by Nick Cox and others here: DBSCAN is a clustering algorithm that also identifies points that do not belong well to any particular cluster, and treats them as 'noise'. This might be one way to achieve what you are after...
Is there an opposite to Clustering or "Anti-"Clustering? Building on the useful comments & answers by Nick Cox and others here: DBSCAN is a clustering algorithm that also identifies points that do not belong well to any particular cluster, and treats them
47,108
Is there an opposite to Clustering or "Anti-"Clustering?
We have been working on unclusters (and anti-clusters and noclusters) and we have a few definitions that we have been exploring. the holes between clusters that illustrate where points can not exist. the density of points on the periphery of a cluster. the density of clusters (as compared to the density of points in ...
Is there an opposite to Clustering or "Anti-"Clustering?
We have been working on unclusters (and anti-clusters and noclusters) and we have a few definitions that we have been exploring. the holes between clusters that illustrate where points can not exist
Is there an opposite to Clustering or "Anti-"Clustering? We have been working on unclusters (and anti-clusters and noclusters) and we have a few definitions that we have been exploring. the holes between clusters that illustrate where points can not exist. the density of points on the periphery of a cluster. the dens...
Is there an opposite to Clustering or "Anti-"Clustering? We have been working on unclusters (and anti-clusters and noclusters) and we have a few definitions that we have been exploring. the holes between clusters that illustrate where points can not exist
47,109
Understanding simulation in the secretary problem
Why Simulate? Quick research into the problem will reveal that the optimal algorithm to solve this problem is to rank the first $\frac{1}{e}$ applicants, not hire them, and then choose the first candidate who is better than your previous ones. For example, if there are ten candidates each with unique rankings, you migh...
Understanding simulation in the secretary problem
Why Simulate? Quick research into the problem will reveal that the optimal algorithm to solve this problem is to rank the first $\frac{1}{e}$ applicants, not hire them, and then choose the first candi
Understanding simulation in the secretary problem Why Simulate? Quick research into the problem will reveal that the optimal algorithm to solve this problem is to rank the first $\frac{1}{e}$ applicants, not hire them, and then choose the first candidate who is better than your previous ones. For example, if there are ...
Understanding simulation in the secretary problem Why Simulate? Quick research into the problem will reveal that the optimal algorithm to solve this problem is to rank the first $\frac{1}{e}$ applicants, not hire them, and then choose the first candi
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Understanding simulation in the secretary problem
You can use your derivations to deduce the asymptotics of $k_n^*$ for large $n$. However, for smaller $n$, it's not trivial to derive the maximal $r$. You'll end up with a $n$'th degree polynomial in $r$, which is nasty to optimize over integers $r$, and may have surprising behavior for smaller $n$. Explicitely, for la...
Understanding simulation in the secretary problem
You can use your derivations to deduce the asymptotics of $k_n^*$ for large $n$. However, for smaller $n$, it's not trivial to derive the maximal $r$. You'll end up with a $n$'th degree polynomial in
Understanding simulation in the secretary problem You can use your derivations to deduce the asymptotics of $k_n^*$ for large $n$. However, for smaller $n$, it's not trivial to derive the maximal $r$. You'll end up with a $n$'th degree polynomial in $r$, which is nasty to optimize over integers $r$, and may have surpri...
Understanding simulation in the secretary problem You can use your derivations to deduce the asymptotics of $k_n^*$ for large $n$. However, for smaller $n$, it's not trivial to derive the maximal $r$. You'll end up with a $n$'th degree polynomial in
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Inference and predictive models
I see two somewhat unrelated questions in this question. Is it possible to draw reliable inference about individual coefficients in predictive models, especially if we have a large number of predictors and use some form of variable selection and/or regularization? Can the coefficients in a predictive model be interpre...
Inference and predictive models
I see two somewhat unrelated questions in this question. Is it possible to draw reliable inference about individual coefficients in predictive models, especially if we have a large number of predicto
Inference and predictive models I see two somewhat unrelated questions in this question. Is it possible to draw reliable inference about individual coefficients in predictive models, especially if we have a large number of predictors and use some form of variable selection and/or regularization? Can the coefficients i...
Inference and predictive models I see two somewhat unrelated questions in this question. Is it possible to draw reliable inference about individual coefficients in predictive models, especially if we have a large number of predicto
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Inference and predictive models
Frank Harrell, in his 'Regression Modeling Strategies' (2015) offers a range of possible modeling strategies (section 4.12, if you are able to obtain a copy), some of which may be considered facetious ('develop a black box model that performs poorly and is difficult to interpret'), but he then goes on to develop a stra...
Inference and predictive models
Frank Harrell, in his 'Regression Modeling Strategies' (2015) offers a range of possible modeling strategies (section 4.12, if you are able to obtain a copy), some of which may be considered facetious
Inference and predictive models Frank Harrell, in his 'Regression Modeling Strategies' (2015) offers a range of possible modeling strategies (section 4.12, if you are able to obtain a copy), some of which may be considered facetious ('develop a black box model that performs poorly and is difficult to interpret'), but h...
Inference and predictive models Frank Harrell, in his 'Regression Modeling Strategies' (2015) offers a range of possible modeling strategies (section 4.12, if you are able to obtain a copy), some of which may be considered facetious
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When I normalize the standardized values of data, why do I get the same values as just normalizing?
The easy answer is that standardization and normalization are linear transformations of the data and any line is determined by two distinct points on it. Since columns 3 and 4 are constructed to include the points $(\min(\text{data}), 0)$ and $(\max(\text{data}), 1)$ (namely, $(2,0)$ and $(95, 1)$), they must result f...
When I normalize the standardized values of data, why do I get the same values as just normalizing?
The easy answer is that standardization and normalization are linear transformations of the data and any line is determined by two distinct points on it. Since columns 3 and 4 are constructed to incl
When I normalize the standardized values of data, why do I get the same values as just normalizing? The easy answer is that standardization and normalization are linear transformations of the data and any line is determined by two distinct points on it. Since columns 3 and 4 are constructed to include the points $(\mi...
When I normalize the standardized values of data, why do I get the same values as just normalizing? The easy answer is that standardization and normalization are linear transformations of the data and any line is determined by two distinct points on it. Since columns 3 and 4 are constructed to incl
47,114
Machine learning algorithms as matrix factorization
Generalized Low Rank Models paper deals with exactly this. From the abstract: This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix completion, sparse and robust PCA, k-means, k-SVD, and maximum margin matrix factorization.
Machine learning algorithms as matrix factorization
Generalized Low Rank Models paper deals with exactly this. From the abstract: This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, ma
Machine learning algorithms as matrix factorization Generalized Low Rank Models paper deals with exactly this. From the abstract: This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix completion, sparse and robust PCA, k-means, k-SVD, and maximu...
Machine learning algorithms as matrix factorization Generalized Low Rank Models paper deals with exactly this. From the abstract: This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, ma
47,115
How to estimate the Poisson distribution with one event occurrence?
I would treat this as one observation of, assuming you started at midnight on Dec. 31, 2000, 4 years, and another observation censored at 12.956 years. In the case of the exponential distribution, this is a pretty easy problem to solve: $$p(x_1, x_2 | \lambda) = \left(\frac{1}{\lambda}e^{-x_1/\lambda}\right)\left(e^{...
How to estimate the Poisson distribution with one event occurrence?
I would treat this as one observation of, assuming you started at midnight on Dec. 31, 2000, 4 years, and another observation censored at 12.956 years. In the case of the exponential distribution, t
How to estimate the Poisson distribution with one event occurrence? I would treat this as one observation of, assuming you started at midnight on Dec. 31, 2000, 4 years, and another observation censored at 12.956 years. In the case of the exponential distribution, this is a pretty easy problem to solve: $$p(x_1, x_2 ...
How to estimate the Poisson distribution with one event occurrence? I would treat this as one observation of, assuming you started at midnight on Dec. 31, 2000, 4 years, and another observation censored at 12.956 years. In the case of the exponential distribution, t
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How to estimate the Poisson distribution with one event occurrence?
Looking more closely at your simulation, I can see a bit more clearly what you're doing. You've set up a data generating process with a random intensity taking a gamma distribution for each gamma intensity, there's a Poisson realization, you then turn to estimate the Poisson rate among a subsample with non-zero counts....
How to estimate the Poisson distribution with one event occurrence?
Looking more closely at your simulation, I can see a bit more clearly what you're doing. You've set up a data generating process with a random intensity taking a gamma distribution for each gamma inte
How to estimate the Poisson distribution with one event occurrence? Looking more closely at your simulation, I can see a bit more clearly what you're doing. You've set up a data generating process with a random intensity taking a gamma distribution for each gamma intensity, there's a Poisson realization, you then turn ...
How to estimate the Poisson distribution with one event occurrence? Looking more closely at your simulation, I can see a bit more clearly what you're doing. You've set up a data generating process with a random intensity taking a gamma distribution for each gamma inte
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Alternative to Chi-squared test to check if categorical distribution in two sets are the same
Tests for equivalence test the null hypothesis that quantities are different by a threshold of relevance—the smallest value that researchers, or regulators in the case of, for example, the FDA, consider to be meaningful—and rejection of this null hypothesis is to conclude that the quantities are equivalent within the b...
Alternative to Chi-squared test to check if categorical distribution in two sets are the same
Tests for equivalence test the null hypothesis that quantities are different by a threshold of relevance—the smallest value that researchers, or regulators in the case of, for example, the FDA, consid
Alternative to Chi-squared test to check if categorical distribution in two sets are the same Tests for equivalence test the null hypothesis that quantities are different by a threshold of relevance—the smallest value that researchers, or regulators in the case of, for example, the FDA, consider to be meaningful—and re...
Alternative to Chi-squared test to check if categorical distribution in two sets are the same Tests for equivalence test the null hypothesis that quantities are different by a threshold of relevance—the smallest value that researchers, or regulators in the case of, for example, the FDA, consid
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Alternative to Chi-squared test to check if categorical distribution in two sets are the same
I think identifying an appropriate threshold that indicates a meaningful rather than simply a statistically significant difference between your two samples would be a valuable step as described, in part, by @Alexis's answer. I would like to propose an alternative approach of sorts, though, one based on simulation. The...
Alternative to Chi-squared test to check if categorical distribution in two sets are the same
I think identifying an appropriate threshold that indicates a meaningful rather than simply a statistically significant difference between your two samples would be a valuable step as described, in pa
Alternative to Chi-squared test to check if categorical distribution in two sets are the same I think identifying an appropriate threshold that indicates a meaningful rather than simply a statistically significant difference between your two samples would be a valuable step as described, in part, by @Alexis's answer. ...
Alternative to Chi-squared test to check if categorical distribution in two sets are the same I think identifying an appropriate threshold that indicates a meaningful rather than simply a statistically significant difference between your two samples would be a valuable step as described, in pa
47,119
When is likelihood also a probability distribution?
The likelihood is a two variables function $L(\theta,x)$. For fixed $\theta$, this function can be seen as a function of $x$, and this is a distribution: the distribution of $x$ for this fixed $\theta$. For fixed $x$, this function can be seen as a function of $\theta$, and this should not be thought as a distribution...
When is likelihood also a probability distribution?
The likelihood is a two variables function $L(\theta,x)$. For fixed $\theta$, this function can be seen as a function of $x$, and this is a distribution: the distribution of $x$ for this fixed $\theta
When is likelihood also a probability distribution? The likelihood is a two variables function $L(\theta,x)$. For fixed $\theta$, this function can be seen as a function of $x$, and this is a distribution: the distribution of $x$ for this fixed $\theta$. For fixed $x$, this function can be seen as a function of $\theta...
When is likelihood also a probability distribution? The likelihood is a two variables function $L(\theta,x)$. For fixed $\theta$, this function can be seen as a function of $x$, and this is a distribution: the distribution of $x$ for this fixed $\theta
47,120
When is likelihood also a probability distribution?
On the generic and general distinction between likelihood and probability density, check this question on CV as it has fairly detailed and useful answers. Plus this other question on mathoverflow. For the likelihood $\ell(\theta|x)$ to be a probability distribution, or more precisely the density of a probability distri...
When is likelihood also a probability distribution?
On the generic and general distinction between likelihood and probability density, check this question on CV as it has fairly detailed and useful answers. Plus this other question on mathoverflow. For
When is likelihood also a probability distribution? On the generic and general distinction between likelihood and probability density, check this question on CV as it has fairly detailed and useful answers. Plus this other question on mathoverflow. For the likelihood $\ell(\theta|x)$ to be a probability distribution, o...
When is likelihood also a probability distribution? On the generic and general distinction between likelihood and probability density, check this question on CV as it has fairly detailed and useful answers. Plus this other question on mathoverflow. For
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Hyper-parameter estimation for Beta-Binomial Empirical Bayes
The hierarchical model You don't actually even need the marginal probability mass function $m()$, you actually only need the marginal moments of $Y$. In this tutorial, Casella (1992) is assuming the following hierarchical model for a response count $Y$: $$Y|p\sim\mbox{bin}(n,p)$$ and $$p \sim \mbox{Beta}(\lambda,\lambd...
Hyper-parameter estimation for Beta-Binomial Empirical Bayes
The hierarchical model You don't actually even need the marginal probability mass function $m()$, you actually only need the marginal moments of $Y$. In this tutorial, Casella (1992) is assuming the f
Hyper-parameter estimation for Beta-Binomial Empirical Bayes The hierarchical model You don't actually even need the marginal probability mass function $m()$, you actually only need the marginal moments of $Y$. In this tutorial, Casella (1992) is assuming the following hierarchical model for a response count $Y$: $$Y|p...
Hyper-parameter estimation for Beta-Binomial Empirical Bayes The hierarchical model You don't actually even need the marginal probability mass function $m()$, you actually only need the marginal moments of $Y$. In this tutorial, Casella (1992) is assuming the f
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Interpretation of an I(2) process?
One interpetation is that the rate of change is random walk. It's like a free fall where the gravitational force is stochastically changing. If you drop the body on earth, it's moving according to the law: $$\frac{d^2}{dt^2}x=g$$ where $g$ is a gravitational constant. This means that the body's acceleration is always ...
Interpretation of an I(2) process?
One interpetation is that the rate of change is random walk. It's like a free fall where the gravitational force is stochastically changing. If you drop the body on earth, it's moving according to th
Interpretation of an I(2) process? One interpetation is that the rate of change is random walk. It's like a free fall where the gravitational force is stochastically changing. If you drop the body on earth, it's moving according to the law: $$\frac{d^2}{dt^2}x=g$$ where $g$ is a gravitational constant. This means that...
Interpretation of an I(2) process? One interpetation is that the rate of change is random walk. It's like a free fall where the gravitational force is stochastically changing. If you drop the body on earth, it's moving according to th
47,123
Calculate entropy of sample
Your question, as I understand it, basically amounts to, "what are the best practices for calculating entropy for an empirically measured distribution of a continuous random variable?" That's the question I'll attempt to answer. The good news is that the entropy of a continuous random variable is well-defined: one sim...
Calculate entropy of sample
Your question, as I understand it, basically amounts to, "what are the best practices for calculating entropy for an empirically measured distribution of a continuous random variable?" That's the que
Calculate entropy of sample Your question, as I understand it, basically amounts to, "what are the best practices for calculating entropy for an empirically measured distribution of a continuous random variable?" That's the question I'll attempt to answer. The good news is that the entropy of a continuous random varia...
Calculate entropy of sample Your question, as I understand it, basically amounts to, "what are the best practices for calculating entropy for an empirically measured distribution of a continuous random variable?" That's the que
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Determining whether a Time series is white noise
You want to look at an autocorrelation function (ACF) plot. If no lags are significantly correlated, then you basically have white noise or a MA(q) process aka moving average. You can use this guide here to compare what your ACF plot looks like to determine if your time-series is "white noise" or not. Guide to ACF/PACF...
Determining whether a Time series is white noise
You want to look at an autocorrelation function (ACF) plot. If no lags are significantly correlated, then you basically have white noise or a MA(q) process aka moving average. You can use this guide h
Determining whether a Time series is white noise You want to look at an autocorrelation function (ACF) plot. If no lags are significantly correlated, then you basically have white noise or a MA(q) process aka moving average. You can use this guide here to compare what your ACF plot looks like to determine if your time-...
Determining whether a Time series is white noise You want to look at an autocorrelation function (ACF) plot. If no lags are significantly correlated, then you basically have white noise or a MA(q) process aka moving average. You can use this guide h
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In multinomial logistic regression, why do the decision boundaries tend to be parallel to each other?
In multinomial logistic regression, $$ p(k) = \frac{e^{x\beta_k}}{\sum_i e^{x\beta_i}} $$ where $i, k$ are possible class labels, $x$ - input data, $\beta_i$ - coefficient vector for the class $i$. Given class $k$ and base class $j$, log-odds are calculated as $$ \log\frac{p(k)}{p(j)}=\log\frac{e^{x\beta_k}}{e^{x\beta...
In multinomial logistic regression, why do the decision boundaries tend to be parallel to each other
In multinomial logistic regression, $$ p(k) = \frac{e^{x\beta_k}}{\sum_i e^{x\beta_i}} $$ where $i, k$ are possible class labels, $x$ - input data, $\beta_i$ - coefficient vector for the class $i$. G
In multinomial logistic regression, why do the decision boundaries tend to be parallel to each other? In multinomial logistic regression, $$ p(k) = \frac{e^{x\beta_k}}{\sum_i e^{x\beta_i}} $$ where $i, k$ are possible class labels, $x$ - input data, $\beta_i$ - coefficient vector for the class $i$. Given class $k$ and...
In multinomial logistic regression, why do the decision boundaries tend to be parallel to each other In multinomial logistic regression, $$ p(k) = \frac{e^{x\beta_k}}{\sum_i e^{x\beta_i}} $$ where $i, k$ are possible class labels, $x$ - input data, $\beta_i$ - coefficient vector for the class $i$. G
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Can the product of a Beta and some other distribution give an Exponential?
Based on multiple answers to Distribution of $XY$ if $X \sim$ Beta$(1,K-1)$ and $Y \sim$ chi-squared with $2K$ degrees , let $X \sim \mathrm{Beta} \left(1,K-1 \right) $ and $Y$ be chi-squared with $2K$ degrees of freedom. Then the product is exponential with parameter $\frac{1}{2}$ .
Can the product of a Beta and some other distribution give an Exponential?
Based on multiple answers to Distribution of $XY$ if $X \sim$ Beta$(1,K-1)$ and $Y \sim$ chi-squared with $2K$ degrees , let $X \sim \mathrm{Beta} \left(1,K-1 \right) $ and $Y$ be chi-squared with $2K
Can the product of a Beta and some other distribution give an Exponential? Based on multiple answers to Distribution of $XY$ if $X \sim$ Beta$(1,K-1)$ and $Y \sim$ chi-squared with $2K$ degrees , let $X \sim \mathrm{Beta} \left(1,K-1 \right) $ and $Y$ be chi-squared with $2K$ degrees of freedom. Then the product is ex...
Can the product of a Beta and some other distribution give an Exponential? Based on multiple answers to Distribution of $XY$ if $X \sim$ Beta$(1,K-1)$ and $Y \sim$ chi-squared with $2K$ degrees , let $X \sim \mathrm{Beta} \left(1,K-1 \right) $ and $Y$ be chi-squared with $2K
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Can the product of a Beta and some other distribution give an Exponential?
A thought: Since we are prescribing the distributions of $Z, X$, then $Z = XY \implies Y = Z/X$, so why not try to calculate $Y$ as a ratio distribution and see what we get? We have $$f_y(y) = \int_0^1 x f_z(yx)f_x(x)dx$$ We assume that $X$ is a $Beta(a,b)$ and $Z$ is an $Exp(\lambda)$ (rate parameter). So $$f_y(y) =...
Can the product of a Beta and some other distribution give an Exponential?
A thought: Since we are prescribing the distributions of $Z, X$, then $Z = XY \implies Y = Z/X$, so why not try to calculate $Y$ as a ratio distribution and see what we get? We have $$f_y(y) = \int_0
Can the product of a Beta and some other distribution give an Exponential? A thought: Since we are prescribing the distributions of $Z, X$, then $Z = XY \implies Y = Z/X$, so why not try to calculate $Y$ as a ratio distribution and see what we get? We have $$f_y(y) = \int_0^1 x f_z(yx)f_x(x)dx$$ We assume that $X$ is ...
Can the product of a Beta and some other distribution give an Exponential? A thought: Since we are prescribing the distributions of $Z, X$, then $Z = XY \implies Y = Z/X$, so why not try to calculate $Y$ as a ratio distribution and see what we get? We have $$f_y(y) = \int_0
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Interpretation when converting correlation of continuous data to Cohen's d
You've hit on a personal pet peeve of mine. I don't think that the interpretation given in the book (of an r-to-d transformed value of a correlation coefficient that is based on two continuous variables) makes any sense. There is no explicit or implicit dichotomization happening here (and of which variable, the first o...
Interpretation when converting correlation of continuous data to Cohen's d
You've hit on a personal pet peeve of mine. I don't think that the interpretation given in the book (of an r-to-d transformed value of a correlation coefficient that is based on two continuous variabl
Interpretation when converting correlation of continuous data to Cohen's d You've hit on a personal pet peeve of mine. I don't think that the interpretation given in the book (of an r-to-d transformed value of a correlation coefficient that is based on two continuous variables) makes any sense. There is no explicit or ...
Interpretation when converting correlation of continuous data to Cohen's d You've hit on a personal pet peeve of mine. I don't think that the interpretation given in the book (of an r-to-d transformed value of a correlation coefficient that is based on two continuous variabl
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How to interpret glm and ols with offset
You show four models, one of them is strange (the one marked OLS) so I will first discuss the three others. What is common is that a response variable (amount of pay for the Gamma, a count of days for the others) is modeled as a function of covariates, but what is modeled is value pr year, that is: $$ \DeclareMathOpera...
How to interpret glm and ols with offset
You show four models, one of them is strange (the one marked OLS) so I will first discuss the three others. What is common is that a response variable (amount of pay for the Gamma, a count of days for
How to interpret glm and ols with offset You show four models, one of them is strange (the one marked OLS) so I will first discuss the three others. What is common is that a response variable (amount of pay for the Gamma, a count of days for the others) is modeled as a function of covariates, but what is modeled is val...
How to interpret glm and ols with offset You show four models, one of them is strange (the one marked OLS) so I will first discuss the three others. What is common is that a response variable (amount of pay for the Gamma, a count of days for
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How to interpret glm and ols with offset
Your OLS model: lm(formula = payment_amt ~ offset(years) + as.factor(gender) + age, data = pm) Is the same as: lm(formula = payment_amt - years ~ as.factor(gender) + age, data = pm) With a log-link, you can use offsets to model rates because of how math works with logarithms, but for an identity link the...
How to interpret glm and ols with offset
Your OLS model: lm(formula = payment_amt ~ offset(years) + as.factor(gender) + age, data = pm) Is the same as: lm(formula = payment_amt - years ~ as.factor(gender) + age, data = pm) Wit
How to interpret glm and ols with offset Your OLS model: lm(formula = payment_amt ~ offset(years) + as.factor(gender) + age, data = pm) Is the same as: lm(formula = payment_amt - years ~ as.factor(gender) + age, data = pm) With a log-link, you can use offsets to model rates because of how math works with...
How to interpret glm and ols with offset Your OLS model: lm(formula = payment_amt ~ offset(years) + as.factor(gender) + age, data = pm) Is the same as: lm(formula = payment_amt - years ~ as.factor(gender) + age, data = pm) Wit
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Cross entropy versus Mean of Cross Entropy [duplicate]
For online training methods like stochastic gradient descent, the loss on each iteration reflects the contribution of a single data point. So, no summation is necessary in this case. For batch or minibatch training, it's necessary to combine the loss from each point in the batch/minibatch by taking the sum or mean. Whe...
Cross entropy versus Mean of Cross Entropy [duplicate]
For online training methods like stochastic gradient descent, the loss on each iteration reflects the contribution of a single data point. So, no summation is necessary in this case. For batch or mini
Cross entropy versus Mean of Cross Entropy [duplicate] For online training methods like stochastic gradient descent, the loss on each iteration reflects the contribution of a single data point. So, no summation is necessary in this case. For batch or minibatch training, it's necessary to combine the loss from each poin...
Cross entropy versus Mean of Cross Entropy [duplicate] For online training methods like stochastic gradient descent, the loss on each iteration reflects the contribution of a single data point. So, no summation is necessary in this case. For batch or mini
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How to set step-size in Hamiltonian Monte Carlo?
There's a section Radford Neal's Handbook on HMC that discusses how to set the discretisation length $\epsilon$ and the number of leapfrog steps $L$ appropriately: http://www.mcmchandbook.net/HandbookChapter5.pdf. Here I can summarise the key points: The overall distance moved is $\epsilon L$ so both have to be conside...
How to set step-size in Hamiltonian Monte Carlo?
There's a section Radford Neal's Handbook on HMC that discusses how to set the discretisation length $\epsilon$ and the number of leapfrog steps $L$ appropriately: http://www.mcmchandbook.net/Handbook
How to set step-size in Hamiltonian Monte Carlo? There's a section Radford Neal's Handbook on HMC that discusses how to set the discretisation length $\epsilon$ and the number of leapfrog steps $L$ appropriately: http://www.mcmchandbook.net/HandbookChapter5.pdf. Here I can summarise the key points: The overall distance...
How to set step-size in Hamiltonian Monte Carlo? There's a section Radford Neal's Handbook on HMC that discusses how to set the discretisation length $\epsilon$ and the number of leapfrog steps $L$ appropriately: http://www.mcmchandbook.net/Handbook
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If KL-divergence's asymmetrical should I minimize KL(P||Q) or KL(Q||P)?
You usually want $KL(Q||P)$. That's from $P$ to $Q$. I remember that it goes from right to left, just like the notation for conditional probabilities. You want the expectation being taken with respect to the true distribution $P$. That way, sample averages can be assumed to converge to the true expectations, by the la...
If KL-divergence's asymmetrical should I minimize KL(P||Q) or KL(Q||P)?
You usually want $KL(Q||P)$. That's from $P$ to $Q$. I remember that it goes from right to left, just like the notation for conditional probabilities. You want the expectation being taken with respec
If KL-divergence's asymmetrical should I minimize KL(P||Q) or KL(Q||P)? You usually want $KL(Q||P)$. That's from $P$ to $Q$. I remember that it goes from right to left, just like the notation for conditional probabilities. You want the expectation being taken with respect to the true distribution $P$. That way, sample...
If KL-divergence's asymmetrical should I minimize KL(P||Q) or KL(Q||P)? You usually want $KL(Q||P)$. That's from $P$ to $Q$. I remember that it goes from right to left, just like the notation for conditional probabilities. You want the expectation being taken with respec
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If KL-divergence's asymmetrical should I minimize KL(P||Q) or KL(Q||P)?
The KL divergence is not a distance this is why the alternative word divergence is used instead. If you want symmetry you can take the sum of $KL(Q||P)$ and $KL(P||Q)$ as mentioned in one of the answers in the linked post. The intuition is that you do not know the true distribution $Q$ so you make an estimate or a gue...
If KL-divergence's asymmetrical should I minimize KL(P||Q) or KL(Q||P)?
The KL divergence is not a distance this is why the alternative word divergence is used instead. If you want symmetry you can take the sum of $KL(Q||P)$ and $KL(P||Q)$ as mentioned in one of the answe
If KL-divergence's asymmetrical should I minimize KL(P||Q) or KL(Q||P)? The KL divergence is not a distance this is why the alternative word divergence is used instead. If you want symmetry you can take the sum of $KL(Q||P)$ and $KL(P||Q)$ as mentioned in one of the answers in the linked post. The intuition is that yo...
If KL-divergence's asymmetrical should I minimize KL(P||Q) or KL(Q||P)? The KL divergence is not a distance this is why the alternative word divergence is used instead. If you want symmetry you can take the sum of $KL(Q||P)$ and $KL(P||Q)$ as mentioned in one of the answe
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Simulating Responses from fitted Generalized Additive Model
I'll illustrate with the classic 4 term data set oft used to illustrate GAMs, but will only simulate data from the strongly nonlinear term $f(x_2)$ as it is easy to visualise the process with a single covariate. library('mgcv') set.seed(20) f2 <- function(x) 0.2 * x^11 * (10 * (1 - x))^6 + 10 * (10 * x)^3 * (1 - x)^10...
Simulating Responses from fitted Generalized Additive Model
I'll illustrate with the classic 4 term data set oft used to illustrate GAMs, but will only simulate data from the strongly nonlinear term $f(x_2)$ as it is easy to visualise the process with a single
Simulating Responses from fitted Generalized Additive Model I'll illustrate with the classic 4 term data set oft used to illustrate GAMs, but will only simulate data from the strongly nonlinear term $f(x_2)$ as it is easy to visualise the process with a single covariate. library('mgcv') set.seed(20) f2 <- function(x) ...
Simulating Responses from fitted Generalized Additive Model I'll illustrate with the classic 4 term data set oft used to illustrate GAMs, but will only simulate data from the strongly nonlinear term $f(x_2)$ as it is easy to visualise the process with a single
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Can an LS estimator be unbiased if there are infinite solutions?
If $\mathbf A$ doesn't have full rank, then the model is not identifiable, which implies that an unbiased estimator does not exist. Here is what the proof looks like in this specific scenario. If $\mathbf A$ doesn't have full rank, then there exist two distinct vectors $\mathbf x^{(1)}$ and $\mathbf x^{(2)}$ such that...
Can an LS estimator be unbiased if there are infinite solutions?
If $\mathbf A$ doesn't have full rank, then the model is not identifiable, which implies that an unbiased estimator does not exist. Here is what the proof looks like in this specific scenario. If $\m
Can an LS estimator be unbiased if there are infinite solutions? If $\mathbf A$ doesn't have full rank, then the model is not identifiable, which implies that an unbiased estimator does not exist. Here is what the proof looks like in this specific scenario. If $\mathbf A$ doesn't have full rank, then there exist two d...
Can an LS estimator be unbiased if there are infinite solutions? If $\mathbf A$ doesn't have full rank, then the model is not identifiable, which implies that an unbiased estimator does not exist. Here is what the proof looks like in this specific scenario. If $\m
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How can preprocessing with PCA but keeping the same dimensionality improve random forest results? [duplicate]
Random forest struggles when the decision boundary is "diagonal" in the feature space because RF has to approximate that diagonal with lots of "rectangular" splits. To the extent that PCA re-orients the data so that splits perpendicular to the rotated & rescaled axes align well with the decision boundary, PCA will help...
How can preprocessing with PCA but keeping the same dimensionality improve random forest results? [d
Random forest struggles when the decision boundary is "diagonal" in the feature space because RF has to approximate that diagonal with lots of "rectangular" splits. To the extent that PCA re-orients t
How can preprocessing with PCA but keeping the same dimensionality improve random forest results? [duplicate] Random forest struggles when the decision boundary is "diagonal" in the feature space because RF has to approximate that diagonal with lots of "rectangular" splits. To the extent that PCA re-orients the data so...
How can preprocessing with PCA but keeping the same dimensionality improve random forest results? [d Random forest struggles when the decision boundary is "diagonal" in the feature space because RF has to approximate that diagonal with lots of "rectangular" splits. To the extent that PCA re-orients t
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Variance condition for (weak) stationarity
Yes, weak stationarity requires both constant variance and constant mean (over time). To quote from wikipedia: A wide-sense stationary random processes only require that 1st moment (i.e. the mean) and autocovariance do not vary with respect to time.
Variance condition for (weak) stationarity
Yes, weak stationarity requires both constant variance and constant mean (over time). To quote from wikipedia: A wide-sense stationary random processes only require that 1st moment (i.e. the mean) and
Variance condition for (weak) stationarity Yes, weak stationarity requires both constant variance and constant mean (over time). To quote from wikipedia: A wide-sense stationary random processes only require that 1st moment (i.e. the mean) and autocovariance do not vary with respect to time.
Variance condition for (weak) stationarity Yes, weak stationarity requires both constant variance and constant mean (over time). To quote from wikipedia: A wide-sense stationary random processes only require that 1st moment (i.e. the mean) and
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Variance condition for (weak) stationarity
To give another view than that of Digio, I have actually only encountered the requirement for a finite second moment¹, and not for a constant one; at least in books and academic papers, as opposed to online resources (presentations, blogposts, etc.). I thus believe the formal definition for a weak (or wide-sense) stati...
Variance condition for (weak) stationarity
To give another view than that of Digio, I have actually only encountered the requirement for a finite second moment¹, and not for a constant one; at least in books and academic papers, as opposed to
Variance condition for (weak) stationarity To give another view than that of Digio, I have actually only encountered the requirement for a finite second moment¹, and not for a constant one; at least in books and academic papers, as opposed to online resources (presentations, blogposts, etc.). I thus believe the formal ...
Variance condition for (weak) stationarity To give another view than that of Digio, I have actually only encountered the requirement for a finite second moment¹, and not for a constant one; at least in books and academic papers, as opposed to
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Are the marginals of the multivariate t distribution univariate Student t distributions?
Yes. The proof is simple: The definition of a multivariate-t that you give is that it is a random variable that is the ratio of a multivariate Gaussian and the square-root of a Gamma variable The marginal distribution of that is the marginal distribution of the ratio of the Gaussian and the square-root of a gamma The...
Are the marginals of the multivariate t distribution univariate Student t distributions?
Yes. The proof is simple: The definition of a multivariate-t that you give is that it is a random variable that is the ratio of a multivariate Gaussian and the square-root of a Gamma variable The mar
Are the marginals of the multivariate t distribution univariate Student t distributions? Yes. The proof is simple: The definition of a multivariate-t that you give is that it is a random variable that is the ratio of a multivariate Gaussian and the square-root of a Gamma variable The marginal distribution of that is t...
Are the marginals of the multivariate t distribution univariate Student t distributions? Yes. The proof is simple: The definition of a multivariate-t that you give is that it is a random variable that is the ratio of a multivariate Gaussian and the square-root of a Gamma variable The mar
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Partial correlation in panda dataframe python [closed]
This will give you what you are asking for: from scipy import stats, linalg def partial_corr(C): """ Returns the sample linear partial correlation coefficients between pairs of variables in C, controlling for the remaining variables in C. Parameters ---------- C : array-like, shape (n, p) ...
Partial correlation in panda dataframe python [closed]
This will give you what you are asking for: from scipy import stats, linalg def partial_corr(C): """ Returns the sample linear partial correlation coefficients between pairs of variables in C
Partial correlation in panda dataframe python [closed] This will give you what you are asking for: from scipy import stats, linalg def partial_corr(C): """ Returns the sample linear partial correlation coefficients between pairs of variables in C, controlling for the remaining variables in C. Paramete...
Partial correlation in panda dataframe python [closed] This will give you what you are asking for: from scipy import stats, linalg def partial_corr(C): """ Returns the sample linear partial correlation coefficients between pairs of variables in C
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Partial correlation in panda dataframe python [closed]
AFAIU from your comment, you're talking about recursive feature elimination, specifically, using linear regression. As described here: Recursive feature elimination is based on the idea to repeatedly construct a model (for example an SVM or a regression model) and choose either the best or worst performing feature (fo...
Partial correlation in panda dataframe python [closed]
AFAIU from your comment, you're talking about recursive feature elimination, specifically, using linear regression. As described here: Recursive feature elimination is based on the idea to repeatedly
Partial correlation in panda dataframe python [closed] AFAIU from your comment, you're talking about recursive feature elimination, specifically, using linear regression. As described here: Recursive feature elimination is based on the idea to repeatedly construct a model (for example an SVM or a regression model) and...
Partial correlation in panda dataframe python [closed] AFAIU from your comment, you're talking about recursive feature elimination, specifically, using linear regression. As described here: Recursive feature elimination is based on the idea to repeatedly
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"report the statistical results of the analysis without the covariate"
At its face this requirement is quite absurd. The most important issue is having a pre-specified, sound, statistical analysis plan. Adjustment variables should usually be pre-specified as part of this plan, and the reasons for adjustment made clear. That makes the covariate-adjusted analysis the "gold standard" and ...
"report the statistical results of the analysis without the covariate"
At its face this requirement is quite absurd. The most important issue is having a pre-specified, sound, statistical analysis plan. Adjustment variables should usually be pre-specified as part of th
"report the statistical results of the analysis without the covariate" At its face this requirement is quite absurd. The most important issue is having a pre-specified, sound, statistical analysis plan. Adjustment variables should usually be pre-specified as part of this plan, and the reasons for adjustment made clea...
"report the statistical results of the analysis without the covariate" At its face this requirement is quite absurd. The most important issue is having a pre-specified, sound, statistical analysis plan. Adjustment variables should usually be pre-specified as part of th
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Standard Errors with Weighted Least Squares Regression
Essentially you already computed everything you need. The missing piece is just that the sig_i should be the residual standard error divided by the corresponding square root of the weight. In OLS this isn't necessary because all weights are 1. sig_i <- resid_var2 / sqrt(wts) var_betas2 <- solve(t(X) %*% W %*% X) %*% (t...
Standard Errors with Weighted Least Squares Regression
Essentially you already computed everything you need. The missing piece is just that the sig_i should be the residual standard error divided by the corresponding square root of the weight. In OLS this
Standard Errors with Weighted Least Squares Regression Essentially you already computed everything you need. The missing piece is just that the sig_i should be the residual standard error divided by the corresponding square root of the weight. In OLS this isn't necessary because all weights are 1. sig_i <- resid_var2 /...
Standard Errors with Weighted Least Squares Regression Essentially you already computed everything you need. The missing piece is just that the sig_i should be the residual standard error divided by the corresponding square root of the weight. In OLS this
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OLS with Time Series Data - yay or nay?
There are time series models (such as VAR, ARIMA, etc.) and there are estimation techniques (such as OLS, maximum likelihood (ML), etc.). Different models can be estimated by different techniques (sometimes more than one). E.g. a VAR can be estimated by OLS or ML while ARIMA (with a nonempty MA part) cannot be estimate...
OLS with Time Series Data - yay or nay?
There are time series models (such as VAR, ARIMA, etc.) and there are estimation techniques (such as OLS, maximum likelihood (ML), etc.). Different models can be estimated by different techniques (som
OLS with Time Series Data - yay or nay? There are time series models (such as VAR, ARIMA, etc.) and there are estimation techniques (such as OLS, maximum likelihood (ML), etc.). Different models can be estimated by different techniques (sometimes more than one). E.g. a VAR can be estimated by OLS or ML while ARIMA (wit...
OLS with Time Series Data - yay or nay? There are time series models (such as VAR, ARIMA, etc.) and there are estimation techniques (such as OLS, maximum likelihood (ML), etc.). Different models can be estimated by different techniques (som
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How does outlier impact logistic regression?
Outliers may have the same essential impact on a logistic regression as they have in linear regression: The deletion-diagnostic model, fit by deleting the outlying observation, may have DF-betas greater than the full-model coefficient; this means that the sigmoid-slope of association may be of opposite direction. Separ...
How does outlier impact logistic regression?
Outliers may have the same essential impact on a logistic regression as they have in linear regression: The deletion-diagnostic model, fit by deleting the outlying observation, may have DF-betas great
How does outlier impact logistic regression? Outliers may have the same essential impact on a logistic regression as they have in linear regression: The deletion-diagnostic model, fit by deleting the outlying observation, may have DF-betas greater than the full-model coefficient; this means that the sigmoid-slope of as...
How does outlier impact logistic regression? Outliers may have the same essential impact on a logistic regression as they have in linear regression: The deletion-diagnostic model, fit by deleting the outlying observation, may have DF-betas great
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How does outlier impact logistic regression?
Logistic regression is robust to concordant outliers, (with an extreme X value but an outcome that accords with the X value), in a way that the linear probability model is not. See http://teaching.sociology.ul.ie:3838/logitinfl for a simple illustration. It might be better to say that such observations are outliers for...
How does outlier impact logistic regression?
Logistic regression is robust to concordant outliers, (with an extreme X value but an outcome that accords with the X value), in a way that the linear probability model is not. See http://teaching.soc
How does outlier impact logistic regression? Logistic regression is robust to concordant outliers, (with an extreme X value but an outcome that accords with the X value), in a way that the linear probability model is not. See http://teaching.sociology.ul.ie:3838/logitinfl for a simple illustration. It might be better t...
How does outlier impact logistic regression? Logistic regression is robust to concordant outliers, (with an extreme X value but an outcome that accords with the X value), in a way that the linear probability model is not. See http://teaching.soc
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Error metric for regression with right-skewed data and outliers
I don't think the problem here is your metric; RMSE & R^2 are generally quite acceptable. And in general, deviations from normality are not a major problem (see discussion here). However, if you have a number of outliers, you will likely improve your model if you change the error distribution you are using to one that ...
Error metric for regression with right-skewed data and outliers
I don't think the problem here is your metric; RMSE & R^2 are generally quite acceptable. And in general, deviations from normality are not a major problem (see discussion here). However, if you have
Error metric for regression with right-skewed data and outliers I don't think the problem here is your metric; RMSE & R^2 are generally quite acceptable. And in general, deviations from normality are not a major problem (see discussion here). However, if you have a number of outliers, you will likely improve your model...
Error metric for regression with right-skewed data and outliers I don't think the problem here is your metric; RMSE & R^2 are generally quite acceptable. And in general, deviations from normality are not a major problem (see discussion here). However, if you have
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Error metric for regression with right-skewed data and outliers
Just spitballing here, but this seems like as much of an economics problem as a statistics one. Given your application, it seems like you want to pick the model that maximizes your "profit": what the user would spend without the voucher plus the additional spend due to the voucher less the cost of the voucher. I'm ass...
Error metric for regression with right-skewed data and outliers
Just spitballing here, but this seems like as much of an economics problem as a statistics one. Given your application, it seems like you want to pick the model that maximizes your "profit": what the
Error metric for regression with right-skewed data and outliers Just spitballing here, but this seems like as much of an economics problem as a statistics one. Given your application, it seems like you want to pick the model that maximizes your "profit": what the user would spend without the voucher plus the additional...
Error metric for regression with right-skewed data and outliers Just spitballing here, but this seems like as much of an economics problem as a statistics one. Given your application, it seems like you want to pick the model that maximizes your "profit": what the
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Error metric for regression with right-skewed data and outliers
Since your response variable is spending, it might be plausible to assume it's log normal and use a log transformation, which could remove the skewness. I believe the distribution of income is often fit well by a log normal, so maybe spending is as well. What do you mean by time sensitive? Do you have measurements thro...
Error metric for regression with right-skewed data and outliers
Since your response variable is spending, it might be plausible to assume it's log normal and use a log transformation, which could remove the skewness. I believe the distribution of income is often f
Error metric for regression with right-skewed data and outliers Since your response variable is spending, it might be plausible to assume it's log normal and use a log transformation, which could remove the skewness. I believe the distribution of income is often fit well by a log normal, so maybe spending is as well. W...
Error metric for regression with right-skewed data and outliers Since your response variable is spending, it might be plausible to assume it's log normal and use a log transformation, which could remove the skewness. I believe the distribution of income is often f
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Marginal independence does not imply joint independence
Your first question assumes that $X, Y$, and $Z$ are pairwise independent random variables and asks whether $$ X,Y, Z~~\text{pairwise independent} \implies (X,Y)~~\text{and}~~Z~~\text{independent}??\tag{1} $$ In general, the answer is NO, the implication $(1)$ does not hold in all cases. The proof of this is by a redu...
Marginal independence does not imply joint independence
Your first question assumes that $X, Y$, and $Z$ are pairwise independent random variables and asks whether $$ X,Y, Z~~\text{pairwise independent} \implies (X,Y)~~\text{and}~~Z~~\text{independent}??\
Marginal independence does not imply joint independence Your first question assumes that $X, Y$, and $Z$ are pairwise independent random variables and asks whether $$ X,Y, Z~~\text{pairwise independent} \implies (X,Y)~~\text{and}~~Z~~\text{independent}??\tag{1} $$ In general, the answer is NO, the implication $(1)$ do...
Marginal independence does not imply joint independence Your first question assumes that $X, Y$, and $Z$ are pairwise independent random variables and asks whether $$ X,Y, Z~~\text{pairwise independent} \implies (X,Y)~~\text{and}~~Z~~\text{independent}??\
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AIC and BIC criterion for Model selection, how is it used in this paper?
In my answer here I show that in a case like the present one, in which we test nested models against each other, the minimum AIC rule selects the larger model (i.e., rejects the null) if the likelihood ratio statistic $$ \mathcal{LR}=n[\log(\widehat{\sigma}^2_1)-\log(\widehat{\sigma}^2_2)], $$ with $\widehat{\sigma}^2_...
AIC and BIC criterion for Model selection, how is it used in this paper?
In my answer here I show that in a case like the present one, in which we test nested models against each other, the minimum AIC rule selects the larger model (i.e., rejects the null) if the likelihoo
AIC and BIC criterion for Model selection, how is it used in this paper? In my answer here I show that in a case like the present one, in which we test nested models against each other, the minimum AIC rule selects the larger model (i.e., rejects the null) if the likelihood ratio statistic $$ \mathcal{LR}=n[\log(\wideh...
AIC and BIC criterion for Model selection, how is it used in this paper? In my answer here I show that in a case like the present one, in which we test nested models against each other, the minimum AIC rule selects the larger model (i.e., rejects the null) if the likelihoo
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Covariance between a normally distributed variable and its exponent
First solving the following integral by completing the square (4th equality) and recognising the resulting integral as the expected value of a normal variate with an expected value of $\mu+\sigma^2$ (last equality), \begin{align} E(Xe^X) &=\int_{-\infty}^\infty xe^x\frac1{\sqrt{2\pi}\sigma}e^{-\frac1{2\sigma^2 }(x-\mu)...
Covariance between a normally distributed variable and its exponent
First solving the following integral by completing the square (4th equality) and recognising the resulting integral as the expected value of a normal variate with an expected value of $\mu+\sigma^2$ (
Covariance between a normally distributed variable and its exponent First solving the following integral by completing the square (4th equality) and recognising the resulting integral as the expected value of a normal variate with an expected value of $\mu+\sigma^2$ (last equality), \begin{align} E(Xe^X) &=\int_{-\inft...
Covariance between a normally distributed variable and its exponent First solving the following integral by completing the square (4th equality) and recognising the resulting integral as the expected value of a normal variate with an expected value of $\mu+\sigma^2$ (
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Is the following inequality correct? How to prove it?
Expand using product rule: $$P(X \ge a_0 \cap Y \le b_0) = P(X \ge a_0 | Y \le b_0) P(Y \le b_0)$$ Assuming $P(Y \le b_0)$ is nonzero, you can use the first inequality to obtain: $$P(X \ge a_0 | Y \le b_0) P(Y \le b_0) \le f(y\le b_0) P(Y \le b_0)$$ Because $f(y)$ is increasing in $y$, $f(y \le b_0) \le f(b_0)$ so the ...
Is the following inequality correct? How to prove it?
Expand using product rule: $$P(X \ge a_0 \cap Y \le b_0) = P(X \ge a_0 | Y \le b_0) P(Y \le b_0)$$ Assuming $P(Y \le b_0)$ is nonzero, you can use the first inequality to obtain: $$P(X \ge a_0 | Y \le
Is the following inequality correct? How to prove it? Expand using product rule: $$P(X \ge a_0 \cap Y \le b_0) = P(X \ge a_0 | Y \le b_0) P(Y \le b_0)$$ Assuming $P(Y \le b_0)$ is nonzero, you can use the first inequality to obtain: $$P(X \ge a_0 | Y \le b_0) P(Y \le b_0) \le f(y\le b_0) P(Y \le b_0)$$ Because $f(y)$ i...
Is the following inequality correct? How to prove it? Expand using product rule: $$P(X \ge a_0 \cap Y \le b_0) = P(X \ge a_0 | Y \le b_0) P(Y \le b_0)$$ Assuming $P(Y \le b_0)$ is nonzero, you can use the first inequality to obtain: $$P(X \ge a_0 | Y \le
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Is the following inequality correct? How to prove it?
First, we have $$\eqalign{ \textbf{Pr}(X \ge a_0|Y = y) &=\int_{a_0}^{+\infty}p_{X}(x|Y= y)dx \\ &=\int_{a_0}^{+\infty}\frac{p_{X,Y}(x,y)}{p_Y(y)}dx\\ &=\frac{1}{p_Y(y)}\int_{a_0}^{+\infty}{p_{X,Y}(x,y)}dx, }$$ where $p_{X,Y}(x, y)$ is the joint PDF of $(X,Y)$, $p_{X}(x| Y=y)$ is the PDF of $X$ conditional on $Y=y$, a...
Is the following inequality correct? How to prove it?
First, we have $$\eqalign{ \textbf{Pr}(X \ge a_0|Y = y) &=\int_{a_0}^{+\infty}p_{X}(x|Y= y)dx \\ &=\int_{a_0}^{+\infty}\frac{p_{X,Y}(x,y)}{p_Y(y)}dx\\ &=\frac{1}{p_Y(y)}\int_{a_0}^{+\infty}{p_{X,Y}(x,
Is the following inequality correct? How to prove it? First, we have $$\eqalign{ \textbf{Pr}(X \ge a_0|Y = y) &=\int_{a_0}^{+\infty}p_{X}(x|Y= y)dx \\ &=\int_{a_0}^{+\infty}\frac{p_{X,Y}(x,y)}{p_Y(y)}dx\\ &=\frac{1}{p_Y(y)}\int_{a_0}^{+\infty}{p_{X,Y}(x,y)}dx, }$$ where $p_{X,Y}(x, y)$ is the joint PDF of $(X,Y)$, $p_{...
Is the following inequality correct? How to prove it? First, we have $$\eqalign{ \textbf{Pr}(X \ge a_0|Y = y) &=\int_{a_0}^{+\infty}p_{X}(x|Y= y)dx \\ &=\int_{a_0}^{+\infty}\frac{p_{X,Y}(x,y)}{p_Y(y)}dx\\ &=\frac{1}{p_Y(y)}\int_{a_0}^{+\infty}{p_{X,Y}(x,
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Why is power to detect interactions less than that for main effects?
It's signal and noise. When looking for an interaction, you're looking for it against more noise. Consider looking for a mean in a single group. Only the noise of that group affects your estimate of the mean. Now consider looking for a difference of means between two groups. The estimated difference is affected by the ...
Why is power to detect interactions less than that for main effects?
It's signal and noise. When looking for an interaction, you're looking for it against more noise. Consider looking for a mean in a single group. Only the noise of that group affects your estimate of t
Why is power to detect interactions less than that for main effects? It's signal and noise. When looking for an interaction, you're looking for it against more noise. Consider looking for a mean in a single group. Only the noise of that group affects your estimate of the mean. Now consider looking for a difference of m...
Why is power to detect interactions less than that for main effects? It's signal and noise. When looking for an interaction, you're looking for it against more noise. Consider looking for a mean in a single group. Only the noise of that group affects your estimate of t
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Why is power to detect interactions less than that for main effects?
Here's the pure intuition. Consider these two functions: $$f(x_1,x_2)=\beta_0+\beta_1x_1+\beta_2x_2$$ vs. $$g(x_1,x_2)=\beta_0+\beta_1x_1+\beta_2x_2+\beta_{12}x_1x_2$$ You need just three observations to solve for the parameters of $f(x_1,x_2|\beta_0,\beta_1,\beta_2)$, but you need four observations of function $g(x_1,...
Why is power to detect interactions less than that for main effects?
Here's the pure intuition. Consider these two functions: $$f(x_1,x_2)=\beta_0+\beta_1x_1+\beta_2x_2$$ vs. $$g(x_1,x_2)=\beta_0+\beta_1x_1+\beta_2x_2+\beta_{12}x_1x_2$$ You need just three observations
Why is power to detect interactions less than that for main effects? Here's the pure intuition. Consider these two functions: $$f(x_1,x_2)=\beta_0+\beta_1x_1+\beta_2x_2$$ vs. $$g(x_1,x_2)=\beta_0+\beta_1x_1+\beta_2x_2+\beta_{12}x_1x_2$$ You need just three observations to solve for the parameters of $f(x_1,x_2|\beta_0,...
Why is power to detect interactions less than that for main effects? Here's the pure intuition. Consider these two functions: $$f(x_1,x_2)=\beta_0+\beta_1x_1+\beta_2x_2$$ vs. $$g(x_1,x_2)=\beta_0+\beta_1x_1+\beta_2x_2+\beta_{12}x_1x_2$$ You need just three observations
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Why is power to detect interactions less than that for main effects?
I don't think this is a statistical question as much as a question of the state of the world. In most realistic situations, differences between effects are smaller than the effects themselves. However, this is not statistically or logically necessary.
Why is power to detect interactions less than that for main effects?
I don't think this is a statistical question as much as a question of the state of the world. In most realistic situations, differences between effects are smaller than the effects themselves. However
Why is power to detect interactions less than that for main effects? I don't think this is a statistical question as much as a question of the state of the world. In most realistic situations, differences between effects are smaller than the effects themselves. However, this is not statistically or logically necessary.
Why is power to detect interactions less than that for main effects? I don't think this is a statistical question as much as a question of the state of the world. In most realistic situations, differences between effects are smaller than the effects themselves. However
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Why is power to detect interactions less than that for main effects?
Power is less for interactions, because interactions are generally quite small. Theoretically, this can be explained by the Piranha Problem: Statistically observable interactions are small because either: 1) Only small interactions are possible, or 2) Large interactions are possible, but then they wash each other out, ...
Why is power to detect interactions less than that for main effects?
Power is less for interactions, because interactions are generally quite small. Theoretically, this can be explained by the Piranha Problem: Statistically observable interactions are small because eit
Why is power to detect interactions less than that for main effects? Power is less for interactions, because interactions are generally quite small. Theoretically, this can be explained by the Piranha Problem: Statistically observable interactions are small because either: 1) Only small interactions are possible, or 2)...
Why is power to detect interactions less than that for main effects? Power is less for interactions, because interactions are generally quite small. Theoretically, this can be explained by the Piranha Problem: Statistically observable interactions are small because eit
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Scikit-Learn - Adding Weights to Features
Generally, it's not a great idea to try to meddle with feature weights - RF (and machine learning algorithms in general) works out the importance of features by itself. See also: https://stackoverflow.com/questions/38034702/how-to-put-more-weight-on-certain-features-in-machine-learning
Scikit-Learn - Adding Weights to Features
Generally, it's not a great idea to try to meddle with feature weights - RF (and machine learning algorithms in general) works out the importance of features by itself. See also: https://stackoverflow
Scikit-Learn - Adding Weights to Features Generally, it's not a great idea to try to meddle with feature weights - RF (and machine learning algorithms in general) works out the importance of features by itself. See also: https://stackoverflow.com/questions/38034702/how-to-put-more-weight-on-certain-features-in-machine-...
Scikit-Learn - Adding Weights to Features Generally, it's not a great idea to try to meddle with feature weights - RF (and machine learning algorithms in general) works out the importance of features by itself. See also: https://stackoverflow
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Model Stacking - Gives poor performance
It sounds like you may not be generating the "probabilities" (aka "level-one" data) correctly. These predicted values should be cross-validated predicted values from the base learners (or sometimes people use a separate hold-out set to generate these predicted values). My guess is that you are using predictions gener...
Model Stacking - Gives poor performance
It sounds like you may not be generating the "probabilities" (aka "level-one" data) correctly. These predicted values should be cross-validated predicted values from the base learners (or sometimes p
Model Stacking - Gives poor performance It sounds like you may not be generating the "probabilities" (aka "level-one" data) correctly. These predicted values should be cross-validated predicted values from the base learners (or sometimes people use a separate hold-out set to generate these predicted values). My guess...
Model Stacking - Gives poor performance It sounds like you may not be generating the "probabilities" (aka "level-one" data) correctly. These predicted values should be cross-validated predicted values from the base learners (or sometimes p
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Model Stacking - Gives poor performance
Stacking can give poor performance relative to the base models if a lot of overlap exists in the correct predictions of the ensembled models. Also, stacking tends to do better with a larger number of input models than with a smaller number of ensembled models.
Model Stacking - Gives poor performance
Stacking can give poor performance relative to the base models if a lot of overlap exists in the correct predictions of the ensembled models. Also, stacking tends to do better with a larger number of
Model Stacking - Gives poor performance Stacking can give poor performance relative to the base models if a lot of overlap exists in the correct predictions of the ensembled models. Also, stacking tends to do better with a larger number of input models than with a smaller number of ensembled models.
Model Stacking - Gives poor performance Stacking can give poor performance relative to the base models if a lot of overlap exists in the correct predictions of the ensembled models. Also, stacking tends to do better with a larger number of
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Model Stacking - Gives poor performance
It is quite easy to mess up the first stage model or to fail to see the leakage of informations when working with large blends of models. As stated by @Erin LeDell you should make sure that the second stage is learned from cross validated predictions of the first stage. I wrote the following tutorials regarding blendin...
Model Stacking - Gives poor performance
It is quite easy to mess up the first stage model or to fail to see the leakage of informations when working with large blends of models. As stated by @Erin LeDell you should make sure that the second
Model Stacking - Gives poor performance It is quite easy to mess up the first stage model or to fail to see the leakage of informations when working with large blends of models. As stated by @Erin LeDell you should make sure that the second stage is learned from cross validated predictions of the first stage. I wrote t...
Model Stacking - Gives poor performance It is quite easy to mess up the first stage model or to fail to see the leakage of informations when working with large blends of models. As stated by @Erin LeDell you should make sure that the second
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What is information-theoric about the Kullback-Leibler divergence?
This question attempts to explain (1) the information-theoretic interpretation of KL divergence, (2) how such an application lends itself to Bayesian analysis. What follows is directly quoted from pp. 148-150, Section 6.6. of Stone's Information Theory, a very good book which I recommend. Kullback-Leibler divergence (...
What is information-theoric about the Kullback-Leibler divergence?
This question attempts to explain (1) the information-theoretic interpretation of KL divergence, (2) how such an application lends itself to Bayesian analysis. What follows is directly quoted from pp.
What is information-theoric about the Kullback-Leibler divergence? This question attempts to explain (1) the information-theoretic interpretation of KL divergence, (2) how such an application lends itself to Bayesian analysis. What follows is directly quoted from pp. 148-150, Section 6.6. of Stone's Information Theory,...
What is information-theoric about the Kullback-Leibler divergence? This question attempts to explain (1) the information-theoretic interpretation of KL divergence, (2) how such an application lends itself to Bayesian analysis. What follows is directly quoted from pp.
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What is information-theoric about the Kullback-Leibler divergence?
There is a lot of material I found in Wikipedia, but to sum up, one way to look at the Kullback-Liebler divergence is as following: Let's say you designed a code that is optimal for a source with distribution q. Now you need to use the same code for another source with distribution p. The Kullback-Liebler divergence re...
What is information-theoric about the Kullback-Leibler divergence?
There is a lot of material I found in Wikipedia, but to sum up, one way to look at the Kullback-Liebler divergence is as following: Let's say you designed a code that is optimal for a source with dist
What is information-theoric about the Kullback-Leibler divergence? There is a lot of material I found in Wikipedia, but to sum up, one way to look at the Kullback-Liebler divergence is as following: Let's say you designed a code that is optimal for a source with distribution q. Now you need to use the same code for ano...
What is information-theoric about the Kullback-Leibler divergence? There is a lot of material I found in Wikipedia, but to sum up, one way to look at the Kullback-Liebler divergence is as following: Let's say you designed a code that is optimal for a source with dist
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Bayesian MCMC when a likelihood function cannot be written
The situation you describe where $p(x|\theta)$ cannot be computed but simulations from $p(\cdot|\theta)$ can be produced is call a generative model. It leads to likelihood-free resolutions like ABC (Approximate Bayesian computation), which is indeed properly introduced in the Wikipedia page: Approximate Bayesian compu...
Bayesian MCMC when a likelihood function cannot be written
The situation you describe where $p(x|\theta)$ cannot be computed but simulations from $p(\cdot|\theta)$ can be produced is call a generative model. It leads to likelihood-free resolutions like ABC (
Bayesian MCMC when a likelihood function cannot be written The situation you describe where $p(x|\theta)$ cannot be computed but simulations from $p(\cdot|\theta)$ can be produced is call a generative model. It leads to likelihood-free resolutions like ABC (Approximate Bayesian computation), which is indeed properly i...
Bayesian MCMC when a likelihood function cannot be written The situation you describe where $p(x|\theta)$ cannot be computed but simulations from $p(\cdot|\theta)$ can be produced is call a generative model. It leads to likelihood-free resolutions like ABC (
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Why odds ratio in logistic regression
You are right. If the book said that, it is wrong. I do wonder if it is a typo or a poorly phrased passage that lends itself to misunderstanding, though. As you show, $\exp(\beta_0 + \beta_1X)$ is the odds of 'success' predicted by the model. The odds ratio associated with a $1$-unit change in $X$ is $\exp(\beta)$....
Why odds ratio in logistic regression
You are right. If the book said that, it is wrong. I do wonder if it is a typo or a poorly phrased passage that lends itself to misunderstanding, though. As you show, $\exp(\beta_0 + \beta_1X)$ is
Why odds ratio in logistic regression You are right. If the book said that, it is wrong. I do wonder if it is a typo or a poorly phrased passage that lends itself to misunderstanding, though. As you show, $\exp(\beta_0 + \beta_1X)$ is the odds of 'success' predicted by the model. The odds ratio associated with a $1...
Why odds ratio in logistic regression You are right. If the book said that, it is wrong. I do wonder if it is a typo or a poorly phrased passage that lends itself to misunderstanding, though. As you show, $\exp(\beta_0 + \beta_1X)$ is
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visualizing snow depth
The snow-depth maps presented by the US National Weather Service seem to work well, with a color scale starting with a silver-gray (not pure white) running into blues, purples and browns. This overlays nicely onto their mapping of elevation. Here's an example: I don't think that this corresponds to any of the standard...
visualizing snow depth
The snow-depth maps presented by the US National Weather Service seem to work well, with a color scale starting with a silver-gray (not pure white) running into blues, purples and browns. This overlay
visualizing snow depth The snow-depth maps presented by the US National Weather Service seem to work well, with a color scale starting with a silver-gray (not pure white) running into blues, purples and browns. This overlays nicely onto their mapping of elevation. Here's an example: I don't think that this corresponds...
visualizing snow depth The snow-depth maps presented by the US National Weather Service seem to work well, with a color scale starting with a silver-gray (not pure white) running into blues, purples and browns. This overlay
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Data Distribution and Feature Scaling Techniques
I cannot speak in terms of machine learning, but I can speak in terms of scaling. From our tag wiki: tl;dr version first: normalization refers to scaling all numeric variables in the range [0,1], such as using the formula: $$x_{new}=\frac{x-x_{min}}{x_{max}-x_{min}}$$ standardization refers to a transform to the d...
Data Distribution and Feature Scaling Techniques
I cannot speak in terms of machine learning, but I can speak in terms of scaling. From our tag wiki: tl;dr version first: normalization refers to scaling all numeric variables in the range [0,1], s
Data Distribution and Feature Scaling Techniques I cannot speak in terms of machine learning, but I can speak in terms of scaling. From our tag wiki: tl;dr version first: normalization refers to scaling all numeric variables in the range [0,1], such as using the formula: $$x_{new}=\frac{x-x_{min}}{x_{max}-x_{min}}...
Data Distribution and Feature Scaling Techniques I cannot speak in terms of machine learning, but I can speak in terms of scaling. From our tag wiki: tl;dr version first: normalization refers to scaling all numeric variables in the range [0,1], s
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Data Distribution and Feature Scaling Techniques
It's not depend on Gaussian distribution,its depend on the MODEL that used this features. The result of standardization (or Z-score normalization) is that the features will be rescaled so that they’ll have the properties of a standard normal distribution with μ=0 and σ=1 that help in different cases such as when you wa...
Data Distribution and Feature Scaling Techniques
It's not depend on Gaussian distribution,its depend on the MODEL that used this features. The result of standardization (or Z-score normalization) is that the features will be rescaled so that they’ll
Data Distribution and Feature Scaling Techniques It's not depend on Gaussian distribution,its depend on the MODEL that used this features. The result of standardization (or Z-score normalization) is that the features will be rescaled so that they’ll have the properties of a standard normal distribution with μ=0 and σ=1...
Data Distribution and Feature Scaling Techniques It's not depend on Gaussian distribution,its depend on the MODEL that used this features. The result of standardization (or Z-score normalization) is that the features will be rescaled so that they’ll
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Statistical language in "The Avengers"
To "recognise" something it must first exist, so you will be using a supervised algorithm while clustering is an unsupervised class of machine learning methods. Clustering algorithms group in terms of similarity, rather then recognize known patterns. So I'd say it sounds like another example where there is less science...
Statistical language in "The Avengers"
To "recognise" something it must first exist, so you will be using a supervised algorithm while clustering is an unsupervised class of machine learning methods. Clustering algorithms group in terms of
Statistical language in "The Avengers" To "recognise" something it must first exist, so you will be using a supervised algorithm while clustering is an unsupervised class of machine learning methods. Clustering algorithms group in terms of similarity, rather then recognize known patterns. So I'd say it sounds like anot...
Statistical language in "The Avengers" To "recognise" something it must first exist, so you will be using a supervised algorithm while clustering is an unsupervised class of machine learning methods. Clustering algorithms group in terms of
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Statistical language in "The Avengers"
Clusters of detected gamma photons are being studied in order to find potential sources of gamma rays The cube or tesseract is emitting gamma radiation. In order to find sources of gamma radiation (and thus a potential location of the tesseract) one can use algorithms to detect clusters in the detected locations of gam...
Statistical language in "The Avengers"
Clusters of detected gamma photons are being studied in order to find potential sources of gamma rays The cube or tesseract is emitting gamma radiation. In order to find sources of gamma radiation (an
Statistical language in "The Avengers" Clusters of detected gamma photons are being studied in order to find potential sources of gamma rays The cube or tesseract is emitting gamma radiation. In order to find sources of gamma radiation (and thus a potential location of the tesseract) one can use algorithms to detect cl...
Statistical language in "The Avengers" Clusters of detected gamma photons are being studied in order to find potential sources of gamma rays The cube or tesseract is emitting gamma radiation. In order to find sources of gamma radiation (an
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Statistical language in "The Avengers"
this is so funny. I was re-watching the Avengers and heard Banner say that. I have been learning ML for the past year, so I was wondering if anyone else caught that. Clustering analysis is the process of minimizing distance between data points and maximizing centroids (clusters). As Tim suggested, this is unsupervised ...
Statistical language in "The Avengers"
this is so funny. I was re-watching the Avengers and heard Banner say that. I have been learning ML for the past year, so I was wondering if anyone else caught that. Clustering analysis is the process
Statistical language in "The Avengers" this is so funny. I was re-watching the Avengers and heard Banner say that. I have been learning ML for the past year, so I was wondering if anyone else caught that. Clustering analysis is the process of minimizing distance between data points and maximizing centroids (clusters). ...
Statistical language in "The Avengers" this is so funny. I was re-watching the Avengers and heard Banner say that. I have been learning ML for the past year, so I was wondering if anyone else caught that. Clustering analysis is the process
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Whats the difference between applying Correlation and DTW in a Time Series
Dynamic Time Warping (DTW) and correlation capture very different aspects of similarity between two time-series. Which one to choose depends on what you are interested in - an information that you did not provide in your question. Yet I will give an example which might help to clarify the difference for you. Assume y...
Whats the difference between applying Correlation and DTW in a Time Series
Dynamic Time Warping (DTW) and correlation capture very different aspects of similarity between two time-series. Which one to choose depends on what you are interested in - an information that you did
Whats the difference between applying Correlation and DTW in a Time Series Dynamic Time Warping (DTW) and correlation capture very different aspects of similarity between two time-series. Which one to choose depends on what you are interested in - an information that you did not provide in your question. Yet I will gi...
Whats the difference between applying Correlation and DTW in a Time Series Dynamic Time Warping (DTW) and correlation capture very different aspects of similarity between two time-series. Which one to choose depends on what you are interested in - an information that you did
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What distribution to fit if the log of log is still convex?
It is plausible these data follow a Zipf distribution. Here, for comparison, are random data generated according to a Zipf (power-law) distribution with power near $-1.4$ and plotted as in the question and the linked discussion. I have tuned the power and the total frequencies to match the figures in the question--the...
What distribution to fit if the log of log is still convex?
It is plausible these data follow a Zipf distribution. Here, for comparison, are random data generated according to a Zipf (power-law) distribution with power near $-1.4$ and plotted as in the questio
What distribution to fit if the log of log is still convex? It is plausible these data follow a Zipf distribution. Here, for comparison, are random data generated according to a Zipf (power-law) distribution with power near $-1.4$ and plotted as in the question and the linked discussion. I have tuned the power and the...
What distribution to fit if the log of log is still convex? It is plausible these data follow a Zipf distribution. Here, for comparison, are random data generated according to a Zipf (power-law) distribution with power near $-1.4$ and plotted as in the questio
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Metric as straightforward as R^2 for Bayesian models
See also Andrew Gelman's paper literally titled "R-squared for Bayesian regression models": http://www.stat.columbia.edu/~gelman/research/unpublished/bayes_R2.pdf In that paper, he proposes $R^2 = Var(\hat{y}) / (Var(\hat{y}) + Var(e)) $ which has a similar interpretation to traditional $R^2$. However, since this is ba...
Metric as straightforward as R^2 for Bayesian models
See also Andrew Gelman's paper literally titled "R-squared for Bayesian regression models": http://www.stat.columbia.edu/~gelman/research/unpublished/bayes_R2.pdf In that paper, he proposes $R^2 = Var
Metric as straightforward as R^2 for Bayesian models See also Andrew Gelman's paper literally titled "R-squared for Bayesian regression models": http://www.stat.columbia.edu/~gelman/research/unpublished/bayes_R2.pdf In that paper, he proposes $R^2 = Var(\hat{y}) / (Var(\hat{y}) + Var(e)) $ which has a similar interpret...
Metric as straightforward as R^2 for Bayesian models See also Andrew Gelman's paper literally titled "R-squared for Bayesian regression models": http://www.stat.columbia.edu/~gelman/research/unpublished/bayes_R2.pdf In that paper, he proposes $R^2 = Var
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Metric as straightforward as R^2 for Bayesian models
To check model adequacy, hierarchical Bayesian models are usually evaluated exactly like in DHARMa - you simulated from the fitted model and calculate the quantile residuals. The only difference to DHARMa is that you vary parameters as well while doing the simulations. The approach is explained in many textbooks. Keyw...
Metric as straightforward as R^2 for Bayesian models
To check model adequacy, hierarchical Bayesian models are usually evaluated exactly like in DHARMa - you simulated from the fitted model and calculate the quantile residuals. The only difference to DH
Metric as straightforward as R^2 for Bayesian models To check model adequacy, hierarchical Bayesian models are usually evaluated exactly like in DHARMa - you simulated from the fitted model and calculate the quantile residuals. The only difference to DHARMa is that you vary parameters as well while doing the simulation...
Metric as straightforward as R^2 for Bayesian models To check model adequacy, hierarchical Bayesian models are usually evaluated exactly like in DHARMa - you simulated from the fitted model and calculate the quantile residuals. The only difference to DH
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How to pool c-statistic/AUROC (or any bounded variable) after using multiple imputation techniques?
The c-index is a useful measure of predictive discrimination because it is easy to interpret and at least moderately sensitive. It is not a full-information proper accuracy scoring rule. It is not sensitive enough for comparing two models. So I suggest you obtain the best model using all the partial information avai...
How to pool c-statistic/AUROC (or any bounded variable) after using multiple imputation techniques?
The c-index is a useful measure of predictive discrimination because it is easy to interpret and at least moderately sensitive. It is not a full-information proper accuracy scoring rule. It is not s
How to pool c-statistic/AUROC (or any bounded variable) after using multiple imputation techniques? The c-index is a useful measure of predictive discrimination because it is easy to interpret and at least moderately sensitive. It is not a full-information proper accuracy scoring rule. It is not sensitive enough for ...
How to pool c-statistic/AUROC (or any bounded variable) after using multiple imputation techniques? The c-index is a useful measure of predictive discrimination because it is easy to interpret and at least moderately sensitive. It is not a full-information proper accuracy scoring rule. It is not s
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How to pool c-statistic/AUROC (or any bounded variable) after using multiple imputation techniques?
After asking and looking around, I've been pointed at the following reference concerning the meta-analyses of prediction models (in biomedical research) by Debray TPA et al in the British Medical Journal 2016. In appendix 9 the authors provide an explanation of how to pool multiple c-indices across different studies an...
How to pool c-statistic/AUROC (or any bounded variable) after using multiple imputation techniques?
After asking and looking around, I've been pointed at the following reference concerning the meta-analyses of prediction models (in biomedical research) by Debray TPA et al in the British Medical Jour
How to pool c-statistic/AUROC (or any bounded variable) after using multiple imputation techniques? After asking and looking around, I've been pointed at the following reference concerning the meta-analyses of prediction models (in biomedical research) by Debray TPA et al in the British Medical Journal 2016. In appendi...
How to pool c-statistic/AUROC (or any bounded variable) after using multiple imputation techniques? After asking and looking around, I've been pointed at the following reference concerning the meta-analyses of prediction models (in biomedical research) by Debray TPA et al in the British Medical Jour
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Exotic distribution
Writing $$x\exp(-ax^2+bx) = \exp\left(\frac{b^2}{4a}\right)x^{2-1}\exp\left(-\left(\frac{x-b/(2a)}{1/\sqrt{a}}\right)^2\right)$$ exhibits this distribution as a Generalized Gamma with scale parameter $1/\sqrt{a}$ and shape parameters $d=2, p=2$ that has been shifted by $\mu=b/(2a)$ and truncated at the left at $b/(2a...
Exotic distribution
Writing $$x\exp(-ax^2+bx) = \exp\left(\frac{b^2}{4a}\right)x^{2-1}\exp\left(-\left(\frac{x-b/(2a)}{1/\sqrt{a}}\right)^2\right)$$ exhibits this distribution as a Generalized Gamma with scale paramete
Exotic distribution Writing $$x\exp(-ax^2+bx) = \exp\left(\frac{b^2}{4a}\right)x^{2-1}\exp\left(-\left(\frac{x-b/(2a)}{1/\sqrt{a}}\right)^2\right)$$ exhibits this distribution as a Generalized Gamma with scale parameter $1/\sqrt{a}$ and shape parameters $d=2, p=2$ that has been shifted by $\mu=b/(2a)$ and truncated a...
Exotic distribution Writing $$x\exp(-ax^2+bx) = \exp\left(\frac{b^2}{4a}\right)x^{2-1}\exp\left(-\left(\frac{x-b/(2a)}{1/\sqrt{a}}\right)^2\right)$$ exhibits this distribution as a Generalized Gamma with scale paramete
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Difference Between IV and Control Function in a Non-Linear Model
I would still like others to maybe contribute if they have something substantive to say, but I think this personally cleared the issue up for me: http://www.nber.org/WNE/lect_6_controlfuncs.pdf So, it appears in the non-linear setting the CF approach imposes a linear conditional expectation on the endogenous variable,...
Difference Between IV and Control Function in a Non-Linear Model
I would still like others to maybe contribute if they have something substantive to say, but I think this personally cleared the issue up for me: http://www.nber.org/WNE/lect_6_controlfuncs.pdf So, i
Difference Between IV and Control Function in a Non-Linear Model I would still like others to maybe contribute if they have something substantive to say, but I think this personally cleared the issue up for me: http://www.nber.org/WNE/lect_6_controlfuncs.pdf So, it appears in the non-linear setting the CF approach imp...
Difference Between IV and Control Function in a Non-Linear Model I would still like others to maybe contribute if they have something substantive to say, but I think this personally cleared the issue up for me: http://www.nber.org/WNE/lect_6_controlfuncs.pdf So, i
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Difference Between IV and Control Function in a Non-Linear Model
Petrin and Train (2009) is one example of endogeneity being handled with a control function. E.g. a WP here: http://eml.berkeley.edu/~train/petrintrain.pdf I have always found the distinction a bit tricky. I guess in CF you typically invoke more specific assumptions about how two error terms are related. In IV, you rel...
Difference Between IV and Control Function in a Non-Linear Model
Petrin and Train (2009) is one example of endogeneity being handled with a control function. E.g. a WP here: http://eml.berkeley.edu/~train/petrintrain.pdf I have always found the distinction a bit tr
Difference Between IV and Control Function in a Non-Linear Model Petrin and Train (2009) is one example of endogeneity being handled with a control function. E.g. a WP here: http://eml.berkeley.edu/~train/petrintrain.pdf I have always found the distinction a bit tricky. I guess in CF you typically invoke more specific ...
Difference Between IV and Control Function in a Non-Linear Model Petrin and Train (2009) is one example of endogeneity being handled with a control function. E.g. a WP here: http://eml.berkeley.edu/~train/petrintrain.pdf I have always found the distinction a bit tr
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Is this a valid way to think about p-values?
Would it be accurate to say that a p-value is a random variable whose null distribution is Unif$(0,1)$ which stochastically dominates its distribution under the alternative hypothesis? No, that dominance would be a desirable property, not a definition. Many good (and widely used) tests are biased under some alternati...
Is this a valid way to think about p-values?
Would it be accurate to say that a p-value is a random variable whose null distribution is Unif$(0,1)$ which stochastically dominates its distribution under the alternative hypothesis? No, that domi
Is this a valid way to think about p-values? Would it be accurate to say that a p-value is a random variable whose null distribution is Unif$(0,1)$ which stochastically dominates its distribution under the alternative hypothesis? No, that dominance would be a desirable property, not a definition. Many good (and widel...
Is this a valid way to think about p-values? Would it be accurate to say that a p-value is a random variable whose null distribution is Unif$(0,1)$ which stochastically dominates its distribution under the alternative hypothesis? No, that domi
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Is this a valid way to think about p-values?
$p$ can indeed be regarded as a random variable (in fact, it's a statistic), and is required to be uniformity distributed on $[0, 1]$ under the null hypothesis. However, no guarantees are made about the distribution of $p$ under the alternative hypothesis. This makes sense when you consider that the alternative hypothe...
Is this a valid way to think about p-values?
$p$ can indeed be regarded as a random variable (in fact, it's a statistic), and is required to be uniformity distributed on $[0, 1]$ under the null hypothesis. However, no guarantees are made about t
Is this a valid way to think about p-values? $p$ can indeed be regarded as a random variable (in fact, it's a statistic), and is required to be uniformity distributed on $[0, 1]$ under the null hypothesis. However, no guarantees are made about the distribution of $p$ under the alternative hypothesis. This makes sense w...
Is this a valid way to think about p-values? $p$ can indeed be regarded as a random variable (in fact, it's a statistic), and is required to be uniformity distributed on $[0, 1]$ under the null hypothesis. However, no guarantees are made about t
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Is this a valid way to think about p-values?
There have been many different definitions of P-values and there will, no doubt, be many more. I do not find yours to be satisfactory because the decision theory-based concept of stochastic dominance seems to fit more with the dichotomous hypothesis test framework than with the significance testing framework that yield...
Is this a valid way to think about p-values?
There have been many different definitions of P-values and there will, no doubt, be many more. I do not find yours to be satisfactory because the decision theory-based concept of stochastic dominance
Is this a valid way to think about p-values? There have been many different definitions of P-values and there will, no doubt, be many more. I do not find yours to be satisfactory because the decision theory-based concept of stochastic dominance seems to fit more with the dichotomous hypothesis test framework than with ...
Is this a valid way to think about p-values? There have been many different definitions of P-values and there will, no doubt, be many more. I do not find yours to be satisfactory because the decision theory-based concept of stochastic dominance
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Varied Activation Functions in Neural Networks
Clearly you can use different activations in a neural network. An MLP with any activation and a softmax readout layer is one example (for example, multi-class classification). An RNN with LSTM units has at least two activation functions (logistic, tanh and any activations used elsewhere). ReLU activations in the hidden...
Varied Activation Functions in Neural Networks
Clearly you can use different activations in a neural network. An MLP with any activation and a softmax readout layer is one example (for example, multi-class classification). An RNN with LSTM units h
Varied Activation Functions in Neural Networks Clearly you can use different activations in a neural network. An MLP with any activation and a softmax readout layer is one example (for example, multi-class classification). An RNN with LSTM units has at least two activation functions (logistic, tanh and any activations ...
Varied Activation Functions in Neural Networks Clearly you can use different activations in a neural network. An MLP with any activation and a softmax readout layer is one example (for example, multi-class classification). An RNN with LSTM units h
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Varied Activation Functions in Neural Networks
I believe what is meant by the question is: can we mix different activation functions in a single layer. So imagine we have only one hidden layer with 3 nodes, can I set the first node to have sigmoid, second node to have ReLU, and third node to have tanh? I just thought about this as well, and I believe this should be...
Varied Activation Functions in Neural Networks
I believe what is meant by the question is: can we mix different activation functions in a single layer. So imagine we have only one hidden layer with 3 nodes, can I set the first node to have sigmoid
Varied Activation Functions in Neural Networks I believe what is meant by the question is: can we mix different activation functions in a single layer. So imagine we have only one hidden layer with 3 nodes, can I set the first node to have sigmoid, second node to have ReLU, and third node to have tanh? I just thought a...
Varied Activation Functions in Neural Networks I believe what is meant by the question is: can we mix different activation functions in a single layer. So imagine we have only one hidden layer with 3 nodes, can I set the first node to have sigmoid
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Quantile regression for non linear regression analysis?
Unlike the answer from @Arne Jonas Warnke , I see no need to restrict attention to non-parametric estimators for nonlinear quantile regression. Simply use whatever form of nonlinear function of the parameter vector $\beta$ , namely $f(\beta)$, you have in place of $X\beta$ in https://en.wikipedia.org/wiki/Quantile_regr...
Quantile regression for non linear regression analysis?
Unlike the answer from @Arne Jonas Warnke , I see no need to restrict attention to non-parametric estimators for nonlinear quantile regression. Simply use whatever form of nonlinear function of the pa
Quantile regression for non linear regression analysis? Unlike the answer from @Arne Jonas Warnke , I see no need to restrict attention to non-parametric estimators for nonlinear quantile regression. Simply use whatever form of nonlinear function of the parameter vector $\beta$ , namely $f(\beta)$, you have in place of...
Quantile regression for non linear regression analysis? Unlike the answer from @Arne Jonas Warnke , I see no need to restrict attention to non-parametric estimators for nonlinear quantile regression. Simply use whatever form of nonlinear function of the pa
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Quantile regression for non linear regression analysis?
Yes, of course, there are non-parametric estimator for quantile regression, see for example Horrowitz and Lee (2004). But I think there may be some confusion about the meaning of the term linear. See this nice answer here at CrossValidated. The models in the blog post are indeed additive and linear.
Quantile regression for non linear regression analysis?
Yes, of course, there are non-parametric estimator for quantile regression, see for example Horrowitz and Lee (2004). But I think there may be some confusion about the meaning of the term linear. See
Quantile regression for non linear regression analysis? Yes, of course, there are non-parametric estimator for quantile regression, see for example Horrowitz and Lee (2004). But I think there may be some confusion about the meaning of the term linear. See this nice answer here at CrossValidated. The models in the blog ...
Quantile regression for non linear regression analysis? Yes, of course, there are non-parametric estimator for quantile regression, see for example Horrowitz and Lee (2004). But I think there may be some confusion about the meaning of the term linear. See
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Partial exchangeability - Theory and results
This is described by Diaconis (1988; see also Diaconis and Freedman, 1980): In 1938 de Finetti broaden the concept of exchangeability. Consider first the special case with two observations $X_1,X_2,\dots;Y_1,Y_2,\dots$. The $X_i$ might represent binary outcomes for a group of men and $Y_i$ might represent binary...
Partial exchangeability - Theory and results
This is described by Diaconis (1988; see also Diaconis and Freedman, 1980): In 1938 de Finetti broaden the concept of exchangeability. Consider first the special case with two observations $X_1,X
Partial exchangeability - Theory and results This is described by Diaconis (1988; see also Diaconis and Freedman, 1980): In 1938 de Finetti broaden the concept of exchangeability. Consider first the special case with two observations $X_1,X_2,\dots;Y_1,Y_2,\dots$. The $X_i$ might represent binary outcomes for a ...
Partial exchangeability - Theory and results This is described by Diaconis (1988; see also Diaconis and Freedman, 1980): In 1938 de Finetti broaden the concept of exchangeability. Consider first the special case with two observations $X_1,X
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Partial exchangeability - Theory and results
To complete Tim's answer, from the useful references given there: Suppose the quantities $\{X_i\} := \{X_1, X_2, \dotsc\}$ and $\{Y_j\}$ can have values $\{x\}$ and $\{y\}$ from two discrete sets. The assumption of infinite partial exchangeability for their joint probabilities means $$ \mathrm{P}(X_1=x_1, \dotsc, X_n=x...
Partial exchangeability - Theory and results
To complete Tim's answer, from the useful references given there: Suppose the quantities $\{X_i\} := \{X_1, X_2, \dotsc\}$ and $\{Y_j\}$ can have values $\{x\}$ and $\{y\}$ from two discrete sets. The
Partial exchangeability - Theory and results To complete Tim's answer, from the useful references given there: Suppose the quantities $\{X_i\} := \{X_1, X_2, \dotsc\}$ and $\{Y_j\}$ can have values $\{x\}$ and $\{y\}$ from two discrete sets. The assumption of infinite partial exchangeability for their joint probabiliti...
Partial exchangeability - Theory and results To complete Tim's answer, from the useful references given there: Suppose the quantities $\{X_i\} := \{X_1, X_2, \dotsc\}$ and $\{Y_j\}$ can have values $\{x\}$ and $\{y\}$ from two discrete sets. The
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Can chi-square test be used on non-integer observed frequencies?
observed count have decimal points. If you have fractions, you don't have observed counts but something else. Counts actually count things, 0, 1, 2... . The real data is the prediction of males and females according to the job roles and City from Bureau of Labor Statistics Predictions don't have the same propertie...
Can chi-square test be used on non-integer observed frequencies?
observed count have decimal points. If you have fractions, you don't have observed counts but something else. Counts actually count things, 0, 1, 2... . The real data is the prediction of males an
Can chi-square test be used on non-integer observed frequencies? observed count have decimal points. If you have fractions, you don't have observed counts but something else. Counts actually count things, 0, 1, 2... . The real data is the prediction of males and females according to the job roles and City from Bure...
Can chi-square test be used on non-integer observed frequencies? observed count have decimal points. If you have fractions, you don't have observed counts but something else. Counts actually count things, 0, 1, 2... . The real data is the prediction of males an
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Can chi-square test be used on non-integer observed frequencies?
Even before rounding, parts of your question describes things which seem to me problematic. IMHO, it pays to consider them, as they relate to the cause of rounding. Scaling Say my model just divides everything by 3 The rationale behind this test involves the multinomial distribution, and contains combinatorical term...
Can chi-square test be used on non-integer observed frequencies?
Even before rounding, parts of your question describes things which seem to me problematic. IMHO, it pays to consider them, as they relate to the cause of rounding. Scaling Say my model just divides
Can chi-square test be used on non-integer observed frequencies? Even before rounding, parts of your question describes things which seem to me problematic. IMHO, it pays to consider them, as they relate to the cause of rounding. Scaling Say my model just divides everything by 3 The rationale behind this test involv...
Can chi-square test be used on non-integer observed frequencies? Even before rounding, parts of your question describes things which seem to me problematic. IMHO, it pays to consider them, as they relate to the cause of rounding. Scaling Say my model just divides
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Can chi-square test be used on non-integer observed frequencies?
What you are doing here is in a manner of speaking called weighting of cases. Let's assume that you are doing a study where you want to find out the prevalence of child abuse in high school population. In the population you have 50% boys and 50% girls, however your sample is 60% boys and 40% girls due to the sampling e...
Can chi-square test be used on non-integer observed frequencies?
What you are doing here is in a manner of speaking called weighting of cases. Let's assume that you are doing a study where you want to find out the prevalence of child abuse in high school population
Can chi-square test be used on non-integer observed frequencies? What you are doing here is in a manner of speaking called weighting of cases. Let's assume that you are doing a study where you want to find out the prevalence of child abuse in high school population. In the population you have 50% boys and 50% girls, ho...
Can chi-square test be used on non-integer observed frequencies? What you are doing here is in a manner of speaking called weighting of cases. Let's assume that you are doing a study where you want to find out the prevalence of child abuse in high school population
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Lagrange Multipler KKT condition
A key concept with optimization in general and the KKT conditions in particular is distinguishing: necessary conditions for an optimum sufficient conditions for an optimum In its most basic form, the KKT conditions are necessary conditions for an optimum. If an optimum exists and certain regularity conditions are sat...
Lagrange Multipler KKT condition
A key concept with optimization in general and the KKT conditions in particular is distinguishing: necessary conditions for an optimum sufficient conditions for an optimum In its most basic form, th
Lagrange Multipler KKT condition A key concept with optimization in general and the KKT conditions in particular is distinguishing: necessary conditions for an optimum sufficient conditions for an optimum In its most basic form, the KKT conditions are necessary conditions for an optimum. If an optimum exists and cert...
Lagrange Multipler KKT condition A key concept with optimization in general and the KKT conditions in particular is distinguishing: necessary conditions for an optimum sufficient conditions for an optimum In its most basic form, th
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LDA: Why am I getting the same topic for all points in test set?
My first bet would be that the function words in a corpus of source code differ vastly from those of standard stop lists, and that your model's first topic is indeed capturing standard programming fare: if, int, new, while, etc. Besides building a custom stop list—seeing which words have high probability under the most...
LDA: Why am I getting the same topic for all points in test set?
My first bet would be that the function words in a corpus of source code differ vastly from those of standard stop lists, and that your model's first topic is indeed capturing standard programming far
LDA: Why am I getting the same topic for all points in test set? My first bet would be that the function words in a corpus of source code differ vastly from those of standard stop lists, and that your model's first topic is indeed capturing standard programming fare: if, int, new, while, etc. Besides building a custom ...
LDA: Why am I getting the same topic for all points in test set? My first bet would be that the function words in a corpus of source code differ vastly from those of standard stop lists, and that your model's first topic is indeed capturing standard programming far
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LDA: Why am I getting the same topic for all points in test set?
I found the problem. As indicated by several comments and the answer, it seemed LDA was doing fine and it was. The only problem was the data was too much for LDA and too noisy. I was training on 750K+ docs and everything was noisy. Upon reducing the data to 30k relevant docs, I was able to achieve much better results.
LDA: Why am I getting the same topic for all points in test set?
I found the problem. As indicated by several comments and the answer, it seemed LDA was doing fine and it was. The only problem was the data was too much for LDA and too noisy. I was training on 750K+
LDA: Why am I getting the same topic for all points in test set? I found the problem. As indicated by several comments and the answer, it seemed LDA was doing fine and it was. The only problem was the data was too much for LDA and too noisy. I was training on 750K+ docs and everything was noisy. Upon reducing the data ...
LDA: Why am I getting the same topic for all points in test set? I found the problem. As indicated by several comments and the answer, it seemed LDA was doing fine and it was. The only problem was the data was too much for LDA and too noisy. I was training on 750K+
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Predicting future values with hurdle Poisson model
The basic idea is 'success probability of binomial distribution' * 'lambda (= an expected value) of poisson distribution'. But you have to consider that the poisson model in count part never returns 0. I supposed your example data and predicted when width is c(23, 26, 29). library(pscl) data <- data.frame(y = c(8, 0, 3...
Predicting future values with hurdle Poisson model
The basic idea is 'success probability of binomial distribution' * 'lambda (= an expected value) of poisson distribution'. But you have to consider that the poisson model in count part never returns 0
Predicting future values with hurdle Poisson model The basic idea is 'success probability of binomial distribution' * 'lambda (= an expected value) of poisson distribution'. But you have to consider that the poisson model in count part never returns 0. I supposed your example data and predicted when width is c(23, 26, ...
Predicting future values with hurdle Poisson model The basic idea is 'success probability of binomial distribution' * 'lambda (= an expected value) of poisson distribution'. But you have to consider that the poisson model in count part never returns 0
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Predicting future values with hurdle Poisson model
Hurdle models are also called two-part models. They are useful to implement count data with many zeros. In the first step the probability is computed, that the dependent variable is zero, or a positive number. This table should be interpreted just like a logit regression. Zero hurdle model coefficients (binomial with ...
Predicting future values with hurdle Poisson model
Hurdle models are also called two-part models. They are useful to implement count data with many zeros. In the first step the probability is computed, that the dependent variable is zero, or a positiv
Predicting future values with hurdle Poisson model Hurdle models are also called two-part models. They are useful to implement count data with many zeros. In the first step the probability is computed, that the dependent variable is zero, or a positive number. This table should be interpreted just like a logit regressi...
Predicting future values with hurdle Poisson model Hurdle models are also called two-part models. They are useful to implement count data with many zeros. In the first step the probability is computed, that the dependent variable is zero, or a positiv
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Different ways of performing the Wilcoxon rank sum test and the interpretation of the resulting W-statistic
The Mann-Whitney U statistic counts 1 every time an observation in one sample is less than an observation in the other sample, across all cross-sample pairs of observations. However, it's arbitrary which sample is regarded as the first sample and which one is regarded as the second sample -- if you swap them, the sum o...
Different ways of performing the Wilcoxon rank sum test and the interpretation of the resulting W-st
The Mann-Whitney U statistic counts 1 every time an observation in one sample is less than an observation in the other sample, across all cross-sample pairs of observations. However, it's arbitrary wh
Different ways of performing the Wilcoxon rank sum test and the interpretation of the resulting W-statistic The Mann-Whitney U statistic counts 1 every time an observation in one sample is less than an observation in the other sample, across all cross-sample pairs of observations. However, it's arbitrary which sample i...
Different ways of performing the Wilcoxon rank sum test and the interpretation of the resulting W-st The Mann-Whitney U statistic counts 1 every time an observation in one sample is less than an observation in the other sample, across all cross-sample pairs of observations. However, it's arbitrary wh