idx int64 1 56k | question stringlengths 15 155 | answer stringlengths 2 29.2k ⌀ | question_cut stringlengths 15 100 | answer_cut stringlengths 2 200 ⌀ | conversation stringlengths 47 29.3k | conversation_cut stringlengths 47 301 |
|---|---|---|---|---|---|---|
50,201 | How to detect changes in amplitude? | Changes in variance occur quite often in time series.We employ a search process based upon R. Tsay's innovative work to find the point in time that the variance of the errors has changed. This leads directly to Generalized Least Squares or otherwise known as Weighted Least Squares. His work appeared in the Journal of ... | How to detect changes in amplitude? | Changes in variance occur quite often in time series.We employ a search process based upon R. Tsay's innovative work to find the point in time that the variance of the errors has changed. This leads | How to detect changes in amplitude?
Changes in variance occur quite often in time series.We employ a search process based upon R. Tsay's innovative work to find the point in time that the variance of the errors has changed. This leads directly to Generalized Least Squares or otherwise known as Weighted Least Squares. ... | How to detect changes in amplitude?
Changes in variance occur quite often in time series.We employ a search process based upon R. Tsay's innovative work to find the point in time that the variance of the errors has changed. This leads |
50,202 | How to get a confidence interval on parameters that were fitted using multiple functions and datasets at once? | The easiest thing for you to do would be to string the data sets together (rbind() in R jargon, append in Stata jargon), so that you have variables $x_1,y_1$ coming from the first data set, $x_2,y_2$ coming from the second data set, and an identifier $I$ that takes the value of 0 for the first data set and the value of... | How to get a confidence interval on parameters that were fitted using multiple functions and dataset | The easiest thing for you to do would be to string the data sets together (rbind() in R jargon, append in Stata jargon), so that you have variables $x_1,y_1$ coming from the first data set, $x_2,y_2$ | How to get a confidence interval on parameters that were fitted using multiple functions and datasets at once?
The easiest thing for you to do would be to string the data sets together (rbind() in R jargon, append in Stata jargon), so that you have variables $x_1,y_1$ coming from the first data set, $x_2,y_2$ coming fr... | How to get a confidence interval on parameters that were fitted using multiple functions and dataset
The easiest thing for you to do would be to string the data sets together (rbind() in R jargon, append in Stata jargon), so that you have variables $x_1,y_1$ coming from the first data set, $x_2,y_2$ |
50,203 | Books about incremental data clustering | In a field that is this actively researched, a book will be quickly out of date. Just as with regular clustering: most books still discuss just hierarchical clustering, k-means and EM.
There is a book by C.C.Aggarwal, "Data streams: models and algorithms". Chapter 2 is on clustering.
It is better to check for recent pu... | Books about incremental data clustering | In a field that is this actively researched, a book will be quickly out of date. Just as with regular clustering: most books still discuss just hierarchical clustering, k-means and EM.
There is a book | Books about incremental data clustering
In a field that is this actively researched, a book will be quickly out of date. Just as with regular clustering: most books still discuss just hierarchical clustering, k-means and EM.
There is a book by C.C.Aggarwal, "Data streams: models and algorithms". Chapter 2 is on cluster... | Books about incremental data clustering
In a field that is this actively researched, a book will be quickly out of date. Just as with regular clustering: most books still discuss just hierarchical clustering, k-means and EM.
There is a book |
50,204 | Books about incremental data clustering | Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
Not a book, but a paper on this area.
Ailon, Nir, Rag... | Books about incremental data clustering | Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
| Books about incremental data clustering
Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
Not a book, bu... | Books about incremental data clustering
Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
|
50,205 | Books about incremental data clustering | I recommend you chapter 8 from "Cluster Analysis: Basic Concepts and Algorithms"
It provides a very definite overview on clustering, including Agglomerative Clustering, that we can count as incremental clustering. | Books about incremental data clustering | I recommend you chapter 8 from "Cluster Analysis: Basic Concepts and Algorithms"
It provides a very definite overview on clustering, including Agglomerative Clustering, that we can count as increment | Books about incremental data clustering
I recommend you chapter 8 from "Cluster Analysis: Basic Concepts and Algorithms"
It provides a very definite overview on clustering, including Agglomerative Clustering, that we can count as incremental clustering. | Books about incremental data clustering
I recommend you chapter 8 from "Cluster Analysis: Basic Concepts and Algorithms"
It provides a very definite overview on clustering, including Agglomerative Clustering, that we can count as increment |
50,206 | How should I use prop.test function? | If what you mean to test is whether more people reported an increase than the combined number who reported a decrease or no difference (which is what I think you mean) then your first version is closer to the correct one. Your null hypothesis in that case is that people choose 50-50 between "increase" and "no increase... | How should I use prop.test function? | If what you mean to test is whether more people reported an increase than the combined number who reported a decrease or no difference (which is what I think you mean) then your first version is close | How should I use prop.test function?
If what you mean to test is whether more people reported an increase than the combined number who reported a decrease or no difference (which is what I think you mean) then your first version is closer to the correct one. Your null hypothesis in that case is that people choose 50-5... | How should I use prop.test function?
If what you mean to test is whether more people reported an increase than the combined number who reported a decrease or no difference (which is what I think you mean) then your first version is close |
50,207 | Counting distinct elements when memory is limited | Fellow CVer @rrenaud cited this paper as a key reference for number of unique value estimation. He suggested also to check out the Good Turing frequency estimator, which was developed to estimate the proportion of elements that occur n times, including the case where n = 1 (i.e., unique values).
Here is a link to @rren... | Counting distinct elements when memory is limited | Fellow CVer @rrenaud cited this paper as a key reference for number of unique value estimation. He suggested also to check out the Good Turing frequency estimator, which was developed to estimate the | Counting distinct elements when memory is limited
Fellow CVer @rrenaud cited this paper as a key reference for number of unique value estimation. He suggested also to check out the Good Turing frequency estimator, which was developed to estimate the proportion of elements that occur n times, including the case where n ... | Counting distinct elements when memory is limited
Fellow CVer @rrenaud cited this paper as a key reference for number of unique value estimation. He suggested also to check out the Good Turing frequency estimator, which was developed to estimate the |
50,208 | How to predict a binary outcome with unbalanced repeated measures data? | The question seems to centre mainly on the concern that the data are not normally distributed.
There is no requirement, condition, or assumption that any of the data be normally distributed | How to predict a binary outcome with unbalanced repeated measures data? | The question seems to centre mainly on the concern that the data are not normally distributed.
There is no requirement, condition, or assumption that any of the data be normally distributed | How to predict a binary outcome with unbalanced repeated measures data?
The question seems to centre mainly on the concern that the data are not normally distributed.
There is no requirement, condition, or assumption that any of the data be normally distributed | How to predict a binary outcome with unbalanced repeated measures data?
The question seems to centre mainly on the concern that the data are not normally distributed.
There is no requirement, condition, or assumption that any of the data be normally distributed |
50,209 | How can I sample from a log transformed distribution using uniform distribution? | As I understand it, you've generally discretized to create a set of $n$ points, $x_1, \dots, x_n$, with probability $p_1, \dots, p_n$, and you then calculate the cumulative probabilities, say $c_i = \sum_{j=1}^i p_j$. So you can draw $U \sim Uniform(0,1)$ and then take $X = x_{i^*}$ where $i^* = \min_i \{i:c_i \ge U\}... | How can I sample from a log transformed distribution using uniform distribution? | As I understand it, you've generally discretized to create a set of $n$ points, $x_1, \dots, x_n$, with probability $p_1, \dots, p_n$, and you then calculate the cumulative probabilities, say $c_i = \ | How can I sample from a log transformed distribution using uniform distribution?
As I understand it, you've generally discretized to create a set of $n$ points, $x_1, \dots, x_n$, with probability $p_1, \dots, p_n$, and you then calculate the cumulative probabilities, say $c_i = \sum_{j=1}^i p_j$. So you can draw $U \... | How can I sample from a log transformed distribution using uniform distribution?
As I understand it, you've generally discretized to create a set of $n$ points, $x_1, \dots, x_n$, with probability $p_1, \dots, p_n$, and you then calculate the cumulative probabilities, say $c_i = \ |
50,210 | Correlating time stamps | I'm presuming the rows in the way you've presented data don't necessarily mean anything ie there is no necessary link between the third yawn, third whisper, and third stretch. What you are interested in with the third yawn is "how close is this in time to any whisper - not just the third whisper".
For each yawn I woul... | Correlating time stamps | I'm presuming the rows in the way you've presented data don't necessarily mean anything ie there is no necessary link between the third yawn, third whisper, and third stretch. What you are interested | Correlating time stamps
I'm presuming the rows in the way you've presented data don't necessarily mean anything ie there is no necessary link between the third yawn, third whisper, and third stretch. What you are interested in with the third yawn is "how close is this in time to any whisper - not just the third whispe... | Correlating time stamps
I'm presuming the rows in the way you've presented data don't necessarily mean anything ie there is no necessary link between the third yawn, third whisper, and third stretch. What you are interested |
50,211 | Correlating time stamps | I've similar problem my solution was naive - create new variables representing each minute of the day if given activite took place then mark that minute by 1 :
yawning -> yawning
... ...
2:21-2:22 2:21 1
3:42-3:45 2:22 1
9:20-925 2:23 0
14:45-14:32 .
. 3:42 1
. ... | Correlating time stamps | I've similar problem my solution was naive - create new variables representing each minute of the day if given activite took place then mark that minute by 1 :
yawning -> yawning
... | Correlating time stamps
I've similar problem my solution was naive - create new variables representing each minute of the day if given activite took place then mark that minute by 1 :
yawning -> yawning
... ...
2:21-2:22 2:21 1
3:42-3:45 2:22 1
9:20-925 2:23 0
14:45-14:32 .
. ... | Correlating time stamps
I've similar problem my solution was naive - create new variables representing each minute of the day if given activite took place then mark that minute by 1 :
yawning -> yawning
... |
50,212 | Inconsistency in calculating the Calinski-Harabasz index for a given clustering in R | There is one method of calculating Caliński & Harabasz (1974) index for the same distance matrix, so if two R functions show different results one of them is wrong. Hence your question is off-topic.
Look how Caliński & Harabasz index is calculated, in their original paper [1] or e.g. here.
Then check the source code o... | Inconsistency in calculating the Calinski-Harabasz index for a given clustering in R | There is one method of calculating Caliński & Harabasz (1974) index for the same distance matrix, so if two R functions show different results one of them is wrong. Hence your question is off-topic.
| Inconsistency in calculating the Calinski-Harabasz index for a given clustering in R
There is one method of calculating Caliński & Harabasz (1974) index for the same distance matrix, so if two R functions show different results one of them is wrong. Hence your question is off-topic.
Look how Caliński & Harabasz index ... | Inconsistency in calculating the Calinski-Harabasz index for a given clustering in R
There is one method of calculating Caliński & Harabasz (1974) index for the same distance matrix, so if two R functions show different results one of them is wrong. Hence your question is off-topic.
|
50,213 | Inconsistency in calculating the Calinski-Harabasz index for a given clustering in R | Using a synthetic, two dimensional dataset of 200 points, euclidean distance and complete linkage I am not able to reproduce the discrepancies which you encountered.
Also the clusterCrit package and another implementation return the same values
> # fpc
> ch1 <- calinhara(X, pc, cn=max(pc))
> # clusterSim
> ch2 <- index... | Inconsistency in calculating the Calinski-Harabasz index for a given clustering in R | Using a synthetic, two dimensional dataset of 200 points, euclidean distance and complete linkage I am not able to reproduce the discrepancies which you encountered.
Also the clusterCrit package and a | Inconsistency in calculating the Calinski-Harabasz index for a given clustering in R
Using a synthetic, two dimensional dataset of 200 points, euclidean distance and complete linkage I am not able to reproduce the discrepancies which you encountered.
Also the clusterCrit package and another implementation return the sa... | Inconsistency in calculating the Calinski-Harabasz index for a given clustering in R
Using a synthetic, two dimensional dataset of 200 points, euclidean distance and complete linkage I am not able to reproduce the discrepancies which you encountered.
Also the clusterCrit package and a |
50,214 | Discriminating periodic signals from aperiodic ones | I think that this is actually a difficult research question. As mentioned by @cardinal, the FT suffers from major drawbacks.
If I recall, the distribution of the square module of the coefficients is a scaled $\chi^2$ with 1 degree of freedom. This might be used to test that your signal is a white noise, but rejection w... | Discriminating periodic signals from aperiodic ones | I think that this is actually a difficult research question. As mentioned by @cardinal, the FT suffers from major drawbacks.
If I recall, the distribution of the square module of the coefficients is a | Discriminating periodic signals from aperiodic ones
I think that this is actually a difficult research question. As mentioned by @cardinal, the FT suffers from major drawbacks.
If I recall, the distribution of the square module of the coefficients is a scaled $\chi^2$ with 1 degree of freedom. This might be used to tes... | Discriminating periodic signals from aperiodic ones
I think that this is actually a difficult research question. As mentioned by @cardinal, the FT suffers from major drawbacks.
If I recall, the distribution of the square module of the coefficients is a |
50,215 | Discriminating periodic signals from aperiodic ones | Generally it is best to detect periodicity in the frequency domain. However, if for example there is a twelve month period and the time step is one month then at lag 12 and multiples of it you should see high correlations. If there is no periodic components there would be no peaks at a particular lag and its multiple... | Discriminating periodic signals from aperiodic ones | Generally it is best to detect periodicity in the frequency domain. However, if for example there is a twelve month period and the time step is one month then at lag 12 and multiples of it you should | Discriminating periodic signals from aperiodic ones
Generally it is best to detect periodicity in the frequency domain. However, if for example there is a twelve month period and the time step is one month then at lag 12 and multiples of it you should see high correlations. If there is no periodic components there wo... | Discriminating periodic signals from aperiodic ones
Generally it is best to detect periodicity in the frequency domain. However, if for example there is a twelve month period and the time step is one month then at lag 12 and multiples of it you should |
50,216 | Convergence proof for perceptron algorithm with margin | Well, the answer depends upon exactly which algorithm you have in mind. I would take a look in Brian Ripley's 1996 book, Pattern Recognition and Neural Networks, page 116. Here is a (very simple) proof of the convergence of Rosenblatt's perceptron learning algorithm if that is the algorithm you have in mind. This is re... | Convergence proof for perceptron algorithm with margin | Well, the answer depends upon exactly which algorithm you have in mind. I would take a look in Brian Ripley's 1996 book, Pattern Recognition and Neural Networks, page 116. Here is a (very simple) proo | Convergence proof for perceptron algorithm with margin
Well, the answer depends upon exactly which algorithm you have in mind. I would take a look in Brian Ripley's 1996 book, Pattern Recognition and Neural Networks, page 116. Here is a (very simple) proof of the convergence of Rosenblatt's perceptron learning algorith... | Convergence proof for perceptron algorithm with margin
Well, the answer depends upon exactly which algorithm you have in mind. I would take a look in Brian Ripley's 1996 book, Pattern Recognition and Neural Networks, page 116. Here is a (very simple) proo |
50,217 | Is there a package for R that allows smoothing splines in GEE? [closed] | The splines package has natural splines (ns), B-splines (bs), and a few other types. You can just use them as transformations for the predictors in the model:
geese(y ~ ns(x, 3) + z, ...) | Is there a package for R that allows smoothing splines in GEE? [closed] | The splines package has natural splines (ns), B-splines (bs), and a few other types. You can just use them as transformations for the predictors in the model:
geese(y ~ ns(x, 3) + z, ...) | Is there a package for R that allows smoothing splines in GEE? [closed]
The splines package has natural splines (ns), B-splines (bs), and a few other types. You can just use them as transformations for the predictors in the model:
geese(y ~ ns(x, 3) + z, ...) | Is there a package for R that allows smoothing splines in GEE? [closed]
The splines package has natural splines (ns), B-splines (bs), and a few other types. You can just use them as transformations for the predictors in the model:
geese(y ~ ns(x, 3) + z, ...) |
50,218 | Testing if data comes from a normal distribution with mean 0 and unknown variance in Matlab | You can use Spiegelhalter's test (1983, not the 'omnibus test' from 1977):
function pval = spiegel_test(x)
% compute pvalue under null of x normally distributed;
% x should be a vector;
% D. J. Spiegelhalter, 'Diagnostic tests of distributional shape,'
% Biometrika, 1983
xm = mean(x);
xs = std(x);
xz = (x - xm) ./ xs;... | Testing if data comes from a normal distribution with mean 0 and unknown variance in Matlab | You can use Spiegelhalter's test (1983, not the 'omnibus test' from 1977):
function pval = spiegel_test(x)
% compute pvalue under null of x normally distributed;
% x should be a vector;
% D. J. Spiege | Testing if data comes from a normal distribution with mean 0 and unknown variance in Matlab
You can use Spiegelhalter's test (1983, not the 'omnibus test' from 1977):
function pval = spiegel_test(x)
% compute pvalue under null of x normally distributed;
% x should be a vector;
% D. J. Spiegelhalter, 'Diagnostic tests o... | Testing if data comes from a normal distribution with mean 0 and unknown variance in Matlab
You can use Spiegelhalter's test (1983, not the 'omnibus test' from 1977):
function pval = spiegel_test(x)
% compute pvalue under null of x normally distributed;
% x should be a vector;
% D. J. Spiege |
50,219 | Testing if data comes from a normal distribution with mean 0 and unknown variance in Matlab | See https://www.mathworks.com/matlabcentral/fileexchange/60147-normality-test-package.
This package automatically runs 10 goodness of fit tests:
normalitytest(X)
Make sure X is a row vector.
This function provides ten Normality tests that are not altogether available under one compact routine as a compiled Matlab fu... | Testing if data comes from a normal distribution with mean 0 and unknown variance in Matlab | See https://www.mathworks.com/matlabcentral/fileexchange/60147-normality-test-package.
This package automatically runs 10 goodness of fit tests:
normalitytest(X)
Make sure X is a row vector.
This f | Testing if data comes from a normal distribution with mean 0 and unknown variance in Matlab
See https://www.mathworks.com/matlabcentral/fileexchange/60147-normality-test-package.
This package automatically runs 10 goodness of fit tests:
normalitytest(X)
Make sure X is a row vector.
This function provides ten Normali... | Testing if data comes from a normal distribution with mean 0 and unknown variance in Matlab
See https://www.mathworks.com/matlabcentral/fileexchange/60147-normality-test-package.
This package automatically runs 10 goodness of fit tests:
normalitytest(X)
Make sure X is a row vector.
This f |
50,220 | False discovery rate & q-values: how are q-values to be interpreted when rank of p-values is altered? | The method described in the Benjami-Hochberg paper does not have multiple q-values. What do you mean by "one can then 'order' the q-values?'. One fixes at the onset, a q-value (say 0.05). This means we want to control the FDR at the level q. That is, the expected ratio of incorrectly rejected to rejected hypothesis wil... | False discovery rate & q-values: how are q-values to be interpreted when rank of p-values is altered | The method described in the Benjami-Hochberg paper does not have multiple q-values. What do you mean by "one can then 'order' the q-values?'. One fixes at the onset, a q-value (say 0.05). This means w | False discovery rate & q-values: how are q-values to be interpreted when rank of p-values is altered?
The method described in the Benjami-Hochberg paper does not have multiple q-values. What do you mean by "one can then 'order' the q-values?'. One fixes at the onset, a q-value (say 0.05). This means we want to control ... | False discovery rate & q-values: how are q-values to be interpreted when rank of p-values is altered
The method described in the Benjami-Hochberg paper does not have multiple q-values. What do you mean by "one can then 'order' the q-values?'. One fixes at the onset, a q-value (say 0.05). This means w |
50,221 | False discovery rate & q-values: how are q-values to be interpreted when rank of p-values is altered? | The reason for the change in order is that the q-value measures a fundamentally different thing than the p-value. q-value is the false detection rate (FDR) at a given level of statistical significance. Let's say your 5th lowest observed p-value was 0.02 and by using some statistical method you estimated that you would ... | False discovery rate & q-values: how are q-values to be interpreted when rank of p-values is altered | The reason for the change in order is that the q-value measures a fundamentally different thing than the p-value. q-value is the false detection rate (FDR) at a given level of statistical significance | False discovery rate & q-values: how are q-values to be interpreted when rank of p-values is altered?
The reason for the change in order is that the q-value measures a fundamentally different thing than the p-value. q-value is the false detection rate (FDR) at a given level of statistical significance. Let's say your 5... | False discovery rate & q-values: how are q-values to be interpreted when rank of p-values is altered
The reason for the change in order is that the q-value measures a fundamentally different thing than the p-value. q-value is the false detection rate (FDR) at a given level of statistical significance |
50,222 | Analysing ratios of variables | For question 1 you can use standard methods if the denominator is bounded away from 0 (as mentioned in the comments). If your sample size is not large relative to the potential skewness in the ratios then you probably do not want to use normal based methods (t-tests, anova), but resampling methods (bootstrap, permutat... | Analysing ratios of variables | For question 1 you can use standard methods if the denominator is bounded away from 0 (as mentioned in the comments). If your sample size is not large relative to the potential skewness in the ratios | Analysing ratios of variables
For question 1 you can use standard methods if the denominator is bounded away from 0 (as mentioned in the comments). If your sample size is not large relative to the potential skewness in the ratios then you probably do not want to use normal based methods (t-tests, anova), but resamplin... | Analysing ratios of variables
For question 1 you can use standard methods if the denominator is bounded away from 0 (as mentioned in the comments). If your sample size is not large relative to the potential skewness in the ratios |
50,223 | Sampling from a marginal when full density is given | You are right!
The formulation you describe has a general name to it. The theorem is called de-Finetti's theorem for exchangeable sequences, and is the fundamental theorem behind the Bayesian philosophy and ideas. Specifically, $x_i$s are exchangeable, that is, any permutation of $x_i$s will have the same distribution ... | Sampling from a marginal when full density is given | You are right!
The formulation you describe has a general name to it. The theorem is called de-Finetti's theorem for exchangeable sequences, and is the fundamental theorem behind the Bayesian philosop | Sampling from a marginal when full density is given
You are right!
The formulation you describe has a general name to it. The theorem is called de-Finetti's theorem for exchangeable sequences, and is the fundamental theorem behind the Bayesian philosophy and ideas. Specifically, $x_i$s are exchangeable, that is, any pe... | Sampling from a marginal when full density is given
You are right!
The formulation you describe has a general name to it. The theorem is called de-Finetti's theorem for exchangeable sequences, and is the fundamental theorem behind the Bayesian philosop |
50,224 | Probabilities in case-controlled studies | Your proposal makes sense in this context. The Naive Bayes formulation (using the same language as Wikipedia) is:
$P(C|F_1,\ldots,F_n) \propto P(C) \prod_{i=1}^n P(F_i|C)$
The $P(F_i|C)$ terms are estimated from the data, but instead of estimating $P(C)$ from the data (study prevalence), you use a different measure (po... | Probabilities in case-controlled studies | Your proposal makes sense in this context. The Naive Bayes formulation (using the same language as Wikipedia) is:
$P(C|F_1,\ldots,F_n) \propto P(C) \prod_{i=1}^n P(F_i|C)$
The $P(F_i|C)$ terms are est | Probabilities in case-controlled studies
Your proposal makes sense in this context. The Naive Bayes formulation (using the same language as Wikipedia) is:
$P(C|F_1,\ldots,F_n) \propto P(C) \prod_{i=1}^n P(F_i|C)$
The $P(F_i|C)$ terms are estimated from the data, but instead of estimating $P(C)$ from the data (study pre... | Probabilities in case-controlled studies
Your proposal makes sense in this context. The Naive Bayes formulation (using the same language as Wikipedia) is:
$P(C|F_1,\ldots,F_n) \propto P(C) \prod_{i=1}^n P(F_i|C)$
The $P(F_i|C)$ terms are est |
50,225 | Probabilities in case-controlled studies | After a few days, I decided it may be best to use an alternative method. What I did was sample the data such that it reflected the reported distributions in the population. I repeated this a number of times, each time randomly sampling in appropriate proportions, and took the average performance on the classifier.
I co... | Probabilities in case-controlled studies | After a few days, I decided it may be best to use an alternative method. What I did was sample the data such that it reflected the reported distributions in the population. I repeated this a number of | Probabilities in case-controlled studies
After a few days, I decided it may be best to use an alternative method. What I did was sample the data such that it reflected the reported distributions in the population. I repeated this a number of times, each time randomly sampling in appropriate proportions, and took the av... | Probabilities in case-controlled studies
After a few days, I decided it may be best to use an alternative method. What I did was sample the data such that it reflected the reported distributions in the population. I repeated this a number of |
50,226 | Predicted probabilities from a multinomial regression model using zelig and R | The first thing to do is to construct the "linear predictors" or "logits" for each category for each prediction. So you have your model equation:
$$\eta_{ir}=\sum_{j=1}^{p}X_{ij}\hat{\beta}_{jr}\;\; (i=1,\dots,m\;\; r=1,\dots,R)$$
Where for notational convenience, the above is to be understood to have $\hat{\beta}_{jR... | Predicted probabilities from a multinomial regression model using zelig and R | The first thing to do is to construct the "linear predictors" or "logits" for each category for each prediction. So you have your model equation:
$$\eta_{ir}=\sum_{j=1}^{p}X_{ij}\hat{\beta}_{jr}\;\; | Predicted probabilities from a multinomial regression model using zelig and R
The first thing to do is to construct the "linear predictors" or "logits" for each category for each prediction. So you have your model equation:
$$\eta_{ir}=\sum_{j=1}^{p}X_{ij}\hat{\beta}_{jr}\;\; (i=1,\dots,m\;\; r=1,\dots,R)$$
Where for ... | Predicted probabilities from a multinomial regression model using zelig and R
The first thing to do is to construct the "linear predictors" or "logits" for each category for each prediction. So you have your model equation:
$$\eta_{ir}=\sum_{j=1}^{p}X_{ij}\hat{\beta}_{jr}\;\; |
50,227 | Comparing numbers of p-values from many linear models | Prior to answering your question - is the distribution of effect of the genes justify using a linear model (e.g: are they distributed more or less normally?)
Now to your question - I might offer to go a different way about it. It sounds like what you are asking for is to measure the correlation (e.g similarity of beh... | Comparing numbers of p-values from many linear models | Prior to answering your question - is the distribution of effect of the genes justify using a linear model (e.g: are they distributed more or less normally?)
Now to your question - I might offer to g | Comparing numbers of p-values from many linear models
Prior to answering your question - is the distribution of effect of the genes justify using a linear model (e.g: are they distributed more or less normally?)
Now to your question - I might offer to go a different way about it. It sounds like what you are asking fo... | Comparing numbers of p-values from many linear models
Prior to answering your question - is the distribution of effect of the genes justify using a linear model (e.g: are they distributed more or less normally?)
Now to your question - I might offer to g |
50,228 | Median of medians as robust mean of means? | If all the samples come from the same distribution, then yes the median of the sample medians is a fairly robust estimate of the median of the underlying distribution (though this need not be the same as the mean), since the median of a sample from a continuous distribution has probability 0.5 of being below (or above)... | Median of medians as robust mean of means? | If all the samples come from the same distribution, then yes the median of the sample medians is a fairly robust estimate of the median of the underlying distribution (though this need not be the same | Median of medians as robust mean of means?
If all the samples come from the same distribution, then yes the median of the sample medians is a fairly robust estimate of the median of the underlying distribution (though this need not be the same as the mean), since the median of a sample from a continuous distribution ha... | Median of medians as robust mean of means?
If all the samples come from the same distribution, then yes the median of the sample medians is a fairly robust estimate of the median of the underlying distribution (though this need not be the same |
50,229 | Data collection and storage for time series analysis | Option #2 is much more flexible than #1, particularly if you plan on using Excel pivot tables and/or R packages such as Hadley Wickham's excellent reshape package. I would store the data so that each row contains measured (event-level and contender-level) variables and any variables necessary to uniquely identify an i... | Data collection and storage for time series analysis | Option #2 is much more flexible than #1, particularly if you plan on using Excel pivot tables and/or R packages such as Hadley Wickham's excellent reshape package. I would store the data so that each | Data collection and storage for time series analysis
Option #2 is much more flexible than #1, particularly if you plan on using Excel pivot tables and/or R packages such as Hadley Wickham's excellent reshape package. I would store the data so that each row contains measured (event-level and contender-level) variables ... | Data collection and storage for time series analysis
Option #2 is much more flexible than #1, particularly if you plan on using Excel pivot tables and/or R packages such as Hadley Wickham's excellent reshape package. I would store the data so that each |
50,230 | Data collection and storage for time series analysis | In my experience, #1 is the better option. If you store the data in any flatfile setup (as you're suggesting) and don't put the rows as your time variable, it becomes that much harder to import into selected programs.
For example, I work primarily in Fortran/C, with secondary applications occasionally done in R or MATL... | Data collection and storage for time series analysis | In my experience, #1 is the better option. If you store the data in any flatfile setup (as you're suggesting) and don't put the rows as your time variable, it becomes that much harder to import into s | Data collection and storage for time series analysis
In my experience, #1 is the better option. If you store the data in any flatfile setup (as you're suggesting) and don't put the rows as your time variable, it becomes that much harder to import into selected programs.
For example, I work primarily in Fortran/C, with ... | Data collection and storage for time series analysis
In my experience, #1 is the better option. If you store the data in any flatfile setup (as you're suggesting) and don't put the rows as your time variable, it becomes that much harder to import into s |
50,231 | Weighted discrete measurements of a value changing over time | Sounds like you might want to look at (Weighted) Moving Average. | Weighted discrete measurements of a value changing over time | Sounds like you might want to look at (Weighted) Moving Average. | Weighted discrete measurements of a value changing over time
Sounds like you might want to look at (Weighted) Moving Average. | Weighted discrete measurements of a value changing over time
Sounds like you might want to look at (Weighted) Moving Average. |
50,232 | Weighted discrete measurements of a value changing over time | There might be a bit of confusion here with some imprecise statistical jargon. If you have data points that have been measured/reported with different precision/reliability/variability then one turns naturally to Generalized Least Squares where one transforms/weights the data by adjusting for the relative variability .... | Weighted discrete measurements of a value changing over time | There might be a bit of confusion here with some imprecise statistical jargon. If you have data points that have been measured/reported with different precision/reliability/variability then one turns | Weighted discrete measurements of a value changing over time
There might be a bit of confusion here with some imprecise statistical jargon. If you have data points that have been measured/reported with different precision/reliability/variability then one turns naturally to Generalized Least Squares where one transforms... | Weighted discrete measurements of a value changing over time
There might be a bit of confusion here with some imprecise statistical jargon. If you have data points that have been measured/reported with different precision/reliability/variability then one turns |
50,233 | Colinearity and scaling when using k-means | As CHL has already explained the use of center and scale to obtain standardized variables, I'll address collinearity:
There is good reason to reduce collinear variables when clustering.
Curse of Dimensionality
The more dimensions you use, the more likely you are to fall victim of Bellman's 'curse of dimensionality'. I... | Colinearity and scaling when using k-means | As CHL has already explained the use of center and scale to obtain standardized variables, I'll address collinearity:
There is good reason to reduce collinear variables when clustering.
Curse of Dimen | Colinearity and scaling when using k-means
As CHL has already explained the use of center and scale to obtain standardized variables, I'll address collinearity:
There is good reason to reduce collinear variables when clustering.
Curse of Dimensionality
The more dimensions you use, the more likely you are to fall victim... | Colinearity and scaling when using k-means
As CHL has already explained the use of center and scale to obtain standardized variables, I'll address collinearity:
There is good reason to reduce collinear variables when clustering.
Curse of Dimen |
50,234 | Gaussian kernel estimator as Nadaraya-Watson estimator? | The conditional mean is defined by:
$$E(Y|X)\equiv\int y f(y|x) dy$$
Where $f(Y|X)$ is the conditional density. Using the product rule, you can show:
$$f(y|x)=\frac{f(y,x)}{f(x)}$$
Substituting this back into the integral you get
$$E(Y|X)\equiv\frac{\int y f(y,x) dy}{f(x)}$$
Which is of the form you seek, if you use t... | Gaussian kernel estimator as Nadaraya-Watson estimator? | The conditional mean is defined by:
$$E(Y|X)\equiv\int y f(y|x) dy$$
Where $f(Y|X)$ is the conditional density. Using the product rule, you can show:
$$f(y|x)=\frac{f(y,x)}{f(x)}$$
Substituting this | Gaussian kernel estimator as Nadaraya-Watson estimator?
The conditional mean is defined by:
$$E(Y|X)\equiv\int y f(y|x) dy$$
Where $f(Y|X)$ is the conditional density. Using the product rule, you can show:
$$f(y|x)=\frac{f(y,x)}{f(x)}$$
Substituting this back into the integral you get
$$E(Y|X)\equiv\frac{\int y f(y,x)... | Gaussian kernel estimator as Nadaraya-Watson estimator?
The conditional mean is defined by:
$$E(Y|X)\equiv\int y f(y|x) dy$$
Where $f(Y|X)$ is the conditional density. Using the product rule, you can show:
$$f(y|x)=\frac{f(y,x)}{f(x)}$$
Substituting this |
50,235 | Computing confidence intervals for prevalence for several types of infection | So you have a population each of whom can have zero or more conditions. To answer the question: How many hospital patients have A? It seems to me that the best you can do is take your favourite proportion estimator and offer it up with your favourite confidence interval. There are lots of choices, which will make a ... | Computing confidence intervals for prevalence for several types of infection | So you have a population each of whom can have zero or more conditions. To answer the question: How many hospital patients have A? It seems to me that the best you can do is take your favourite prop | Computing confidence intervals for prevalence for several types of infection
So you have a population each of whom can have zero or more conditions. To answer the question: How many hospital patients have A? It seems to me that the best you can do is take your favourite proportion estimator and offer it up with your ... | Computing confidence intervals for prevalence for several types of infection
So you have a population each of whom can have zero or more conditions. To answer the question: How many hospital patients have A? It seems to me that the best you can do is take your favourite prop |
50,236 | Computing confidence intervals for prevalence for several types of infection | A few thoughts:
As people have mentioned, if you have the entire hospital population worth of data, and all the questions you have are restricted to that hospital, you can dispense with a confidence interval entirely. However, assuming that's not the case, and you either have a subsample of the hospital, or want to ta... | Computing confidence intervals for prevalence for several types of infection | A few thoughts:
As people have mentioned, if you have the entire hospital population worth of data, and all the questions you have are restricted to that hospital, you can dispense with a confidence | Computing confidence intervals for prevalence for several types of infection
A few thoughts:
As people have mentioned, if you have the entire hospital population worth of data, and all the questions you have are restricted to that hospital, you can dispense with a confidence interval entirely. However, assuming that's... | Computing confidence intervals for prevalence for several types of infection
A few thoughts:
As people have mentioned, if you have the entire hospital population worth of data, and all the questions you have are restricted to that hospital, you can dispense with a confidence |
50,237 | Which non-parametric test can I use to identify significant interactions of independent variables? | Non-parametric tests are likely to be less powerful than parametric tests and thus require a larger sample size. This is annoying because if you had a large sample size, sample means would be approximately normally distributed by the central limit theorem, and you thus wouldn't need non-parametric tests.
Look at genera... | Which non-parametric test can I use to identify significant interactions of independent variables? | Non-parametric tests are likely to be less powerful than parametric tests and thus require a larger sample size. This is annoying because if you had a large sample size, sample means would be approxim | Which non-parametric test can I use to identify significant interactions of independent variables?
Non-parametric tests are likely to be less powerful than parametric tests and thus require a larger sample size. This is annoying because if you had a large sample size, sample means would be approximately normally distri... | Which non-parametric test can I use to identify significant interactions of independent variables?
Non-parametric tests are likely to be less powerful than parametric tests and thus require a larger sample size. This is annoying because if you had a large sample size, sample means would be approxim |
50,238 | Which non-parametric test can I use to identify significant interactions of independent variables? | I had the same questions and made some research.
I came across some texts that seem to offer solutions but I ahve to admit that I did not seriously apply them until now.
Feller, A., Holmes, C.C., 2009. Beyond toplines: Heterogeneous treatment effects in random-ized experiments.
Leys, C., Schumann, S., 2010. A nonpara... | Which non-parametric test can I use to identify significant interactions of independent variables? | I had the same questions and made some research.
I came across some texts that seem to offer solutions but I ahve to admit that I did not seriously apply them until now.
Feller, A., Holmes, C.C., 20 | Which non-parametric test can I use to identify significant interactions of independent variables?
I had the same questions and made some research.
I came across some texts that seem to offer solutions but I ahve to admit that I did not seriously apply them until now.
Feller, A., Holmes, C.C., 2009. Beyond toplines: ... | Which non-parametric test can I use to identify significant interactions of independent variables?
I had the same questions and made some research.
I came across some texts that seem to offer solutions but I ahve to admit that I did not seriously apply them until now.
Feller, A., Holmes, C.C., 20 |
50,239 | How to make a combination (aggregation) of quantile forecast? | Short answer.
The problem you mention is well studied by Granger C.W.J. with co-authors, and known as the forecasts combination (or pooling) problem. The general idea is to choose the loss function criterion and the parameters (may be time dependent) that minimize the latter. Below I put some references that may be use... | How to make a combination (aggregation) of quantile forecast? | Short answer.
The problem you mention is well studied by Granger C.W.J. with co-authors, and known as the forecasts combination (or pooling) problem. The general idea is to choose the loss function cr | How to make a combination (aggregation) of quantile forecast?
Short answer.
The problem you mention is well studied by Granger C.W.J. with co-authors, and known as the forecasts combination (or pooling) problem. The general idea is to choose the loss function criterion and the parameters (may be time dependent) that mi... | How to make a combination (aggregation) of quantile forecast?
Short answer.
The problem you mention is well studied by Granger C.W.J. with co-authors, and known as the forecasts combination (or pooling) problem. The general idea is to choose the loss function cr |
50,240 | Multistage sampling in R | Yeah, the sampling package handles this, you can do cluster sampling or stratified or a few others: http://cran.r-project.org/web/packages/sampling/sampling.pdf
It can then also handle a lot of the special variance estimation techniques you'll have to do for any metric you calculate from the complex design. However, I... | Multistage sampling in R | Yeah, the sampling package handles this, you can do cluster sampling or stratified or a few others: http://cran.r-project.org/web/packages/sampling/sampling.pdf
It can then also handle a lot of the s | Multistage sampling in R
Yeah, the sampling package handles this, you can do cluster sampling or stratified or a few others: http://cran.r-project.org/web/packages/sampling/sampling.pdf
It can then also handle a lot of the special variance estimation techniques you'll have to do for any metric you calculate from the c... | Multistage sampling in R
Yeah, the sampling package handles this, you can do cluster sampling or stratified or a few others: http://cran.r-project.org/web/packages/sampling/sampling.pdf
It can then also handle a lot of the s |
50,241 | Multistage sampling in R | I think no extra package is needed for the task, just use the basic sample function, e.g.:
Get sample from the first group:
sample <- sample(data[data$"Care Type" == "Acute Care",], size = 25)
Get the choosen IDs out of the orig. dataset (making a backup could be a good idea before that):
data <- data[setdiff(data$pat... | Multistage sampling in R | I think no extra package is needed for the task, just use the basic sample function, e.g.:
Get sample from the first group:
sample <- sample(data[data$"Care Type" == "Acute Care",], size = 25)
Get th | Multistage sampling in R
I think no extra package is needed for the task, just use the basic sample function, e.g.:
Get sample from the first group:
sample <- sample(data[data$"Care Type" == "Acute Care",], size = 25)
Get the choosen IDs out of the orig. dataset (making a backup could be a good idea before that):
data... | Multistage sampling in R
I think no extra package is needed for the task, just use the basic sample function, e.g.:
Get sample from the first group:
sample <- sample(data[data$"Care Type" == "Acute Care",], size = 25)
Get th |
50,242 | Multistage sampling in R | What I would do is provide prob argument with weights for each data point based on number of levels in your variable. Example:
df <- data.frame(oks = sample(100),
grp = c(rep("trt1", times = 30), rep("trt2", times = 70)))
> head(df)
oks grp
1 40 trt1
2 29 trt1
3 12 trt1
4 25 trt1
5 19 trt1
6 45 trt1
... | Multistage sampling in R | What I would do is provide prob argument with weights for each data point based on number of levels in your variable. Example:
df <- data.frame(oks = sample(100),
grp = c(rep("trt1", times = 3 | Multistage sampling in R
What I would do is provide prob argument with weights for each data point based on number of levels in your variable. Example:
df <- data.frame(oks = sample(100),
grp = c(rep("trt1", times = 30), rep("trt2", times = 70)))
> head(df)
oks grp
1 40 trt1
2 29 trt1
3 12 trt1
4 25 trt... | Multistage sampling in R
What I would do is provide prob argument with weights for each data point based on number of levels in your variable. Example:
df <- data.frame(oks = sample(100),
grp = c(rep("trt1", times = 3 |
50,243 | Interaction between ordinal and categorical factor | I'd stick with logistic or probit regression, enter both factors as covariates, but enter the ordinal factor as if it was continuous. To test for interaction, do a likelihood-ratio test comparing models with and without an interaction between the two factors. This test will have a single degree of freedom and therefore... | Interaction between ordinal and categorical factor | I'd stick with logistic or probit regression, enter both factors as covariates, but enter the ordinal factor as if it was continuous. To test for interaction, do a likelihood-ratio test comparing mode | Interaction between ordinal and categorical factor
I'd stick with logistic or probit regression, enter both factors as covariates, but enter the ordinal factor as if it was continuous. To test for interaction, do a likelihood-ratio test comparing models with and without an interaction between the two factors. This test... | Interaction between ordinal and categorical factor
I'd stick with logistic or probit regression, enter both factors as covariates, but enter the ordinal factor as if it was continuous. To test for interaction, do a likelihood-ratio test comparing mode |
50,244 | How to setup a laboratory experiment in Ecological Research under high natural variability | Whether you have a reasonable chance of obtaining (i.e. power to obtain) reliable conclusions depends on how big the effects are you wish to be able to detect. With such small numbers they'll have to be very large. Clearly having fewer treatments and more replications per treatment will give you at least a bit more pow... | How to setup a laboratory experiment in Ecological Research under high natural variability | Whether you have a reasonable chance of obtaining (i.e. power to obtain) reliable conclusions depends on how big the effects are you wish to be able to detect. With such small numbers they'll have to | How to setup a laboratory experiment in Ecological Research under high natural variability
Whether you have a reasonable chance of obtaining (i.e. power to obtain) reliable conclusions depends on how big the effects are you wish to be able to detect. With such small numbers they'll have to be very large. Clearly having... | How to setup a laboratory experiment in Ecological Research under high natural variability
Whether you have a reasonable chance of obtaining (i.e. power to obtain) reliable conclusions depends on how big the effects are you wish to be able to detect. With such small numbers they'll have to |
50,245 | Automatic test measuring dissimilarity between two time series | There are many different distance measures. For starters, there's always the correlation.
You can look at the mean square error. In R, you can see the algorithm for time series in Rob Hyndman's ftsa package (see the error function).
See Liao (2005) for a nice short survey of time series similarity measures, includi... | Automatic test measuring dissimilarity between two time series | There are many different distance measures. For starters, there's always the correlation.
You can look at the mean square error. In R, you can see the algorithm for time series in Rob Hyndman's ft | Automatic test measuring dissimilarity between two time series
There are many different distance measures. For starters, there's always the correlation.
You can look at the mean square error. In R, you can see the algorithm for time series in Rob Hyndman's ftsa package (see the error function).
See Liao (2005) for ... | Automatic test measuring dissimilarity between two time series
There are many different distance measures. For starters, there's always the correlation.
You can look at the mean square error. In R, you can see the algorithm for time series in Rob Hyndman's ft |
50,246 | Can I use Synthetic Control Method for Comparative Case Studies with survey data? | [Caveat: I have not read the paper so the below may be nonsense for all I know ...]
Based on the summary of the R package I would venture to guess that you could use the proposed methodology for the survey data provided the following conditions are met:
You have survey data from control groups during pre-intervention ... | Can I use Synthetic Control Method for Comparative Case Studies with survey data? | [Caveat: I have not read the paper so the below may be nonsense for all I know ...]
Based on the summary of the R package I would venture to guess that you could use the proposed methodology for the s | Can I use Synthetic Control Method for Comparative Case Studies with survey data?
[Caveat: I have not read the paper so the below may be nonsense for all I know ...]
Based on the summary of the R package I would venture to guess that you could use the proposed methodology for the survey data provided the following cond... | Can I use Synthetic Control Method for Comparative Case Studies with survey data?
[Caveat: I have not read the paper so the below may be nonsense for all I know ...]
Based on the summary of the R package I would venture to guess that you could use the proposed methodology for the s |
50,247 | Dependent variable selection for loglinear segmented regression in time-series analysis of rare events | I think you're right to conclude that there's little hope of finding a 'statistically significant' result from 4 wards over 12 months. Of course, that doesn't mean the control measures don't work — just that your sample size is far too small (and the variability too large) to have much chance of finding evidence that i... | Dependent variable selection for loglinear segmented regression in time-series analysis of rare even | I think you're right to conclude that there's little hope of finding a 'statistically significant' result from 4 wards over 12 months. Of course, that doesn't mean the control measures don't work — ju | Dependent variable selection for loglinear segmented regression in time-series analysis of rare events
I think you're right to conclude that there's little hope of finding a 'statistically significant' result from 4 wards over 12 months. Of course, that doesn't mean the control measures don't work — just that your samp... | Dependent variable selection for loglinear segmented regression in time-series analysis of rare even
I think you're right to conclude that there's little hope of finding a 'statistically significant' result from 4 wards over 12 months. Of course, that doesn't mean the control measures don't work — ju |
50,248 | How can you approximate the number of trials to success given a particular Pr(Success)? | If I understand your question correctly you want to compute the quantiles for the "No of failures before the first success" given that $p=\frac{1}{104}$.
The distribution you should be looking at is the negative binomial distribution. The wiki discusses the negative binomial as:
In probability theory and statistics, t... | How can you approximate the number of trials to success given a particular Pr(Success)? | If I understand your question correctly you want to compute the quantiles for the "No of failures before the first success" given that $p=\frac{1}{104}$.
The distribution you should be looking at is t | How can you approximate the number of trials to success given a particular Pr(Success)?
If I understand your question correctly you want to compute the quantiles for the "No of failures before the first success" given that $p=\frac{1}{104}$.
The distribution you should be looking at is the negative binomial distributio... | How can you approximate the number of trials to success given a particular Pr(Success)?
If I understand your question correctly you want to compute the quantiles for the "No of failures before the first success" given that $p=\frac{1}{104}$.
The distribution you should be looking at is t |
50,249 | How to limit my input data for Jaccard item-item similarity calculation? | I'm confused: shouldn't you only need the 7900^2 item similarities, for which you use ratings from all users, which is still quite sparse?
UPDATE
I still think there's a more efficient way to do this, but maybe I'm just being dense. Specifically, consider item A and item B. For item A, generate a U-dimensional vector o... | How to limit my input data for Jaccard item-item similarity calculation? | I'm confused: shouldn't you only need the 7900^2 item similarities, for which you use ratings from all users, which is still quite sparse?
UPDATE
I still think there's a more efficient way to do this, | How to limit my input data for Jaccard item-item similarity calculation?
I'm confused: shouldn't you only need the 7900^2 item similarities, for which you use ratings from all users, which is still quite sparse?
UPDATE
I still think there's a more efficient way to do this, but maybe I'm just being dense. Specifically, ... | How to limit my input data for Jaccard item-item similarity calculation?
I'm confused: shouldn't you only need the 7900^2 item similarities, for which you use ratings from all users, which is still quite sparse?
UPDATE
I still think there's a more efficient way to do this, |
50,250 | How to limit my input data for Jaccard item-item similarity calculation? | I've solved similar problem with MinHash which is specifically designed to approximate Jaccard distance.
Idea is simple using MinHash probabilistic features you group your data into smaller groups (with same hash(s)) and then evalaute pairwise distance inside group (kind of block structure of matrix). The final answer ... | How to limit my input data for Jaccard item-item similarity calculation? | I've solved similar problem with MinHash which is specifically designed to approximate Jaccard distance.
Idea is simple using MinHash probabilistic features you group your data into smaller groups (wi | How to limit my input data for Jaccard item-item similarity calculation?
I've solved similar problem with MinHash which is specifically designed to approximate Jaccard distance.
Idea is simple using MinHash probabilistic features you group your data into smaller groups (with same hash(s)) and then evalaute pairwise dis... | How to limit my input data for Jaccard item-item similarity calculation?
I've solved similar problem with MinHash which is specifically designed to approximate Jaccard distance.
Idea is simple using MinHash probabilistic features you group your data into smaller groups (wi |
50,251 | Birthday paradox for non-uniform probabilities | Approach 1
Let's assume that $d$ is large and the distribution of birthdays on day $i$ can be modelled/approximated as independent Poisson variables.
Each day has a frequency of $\lambda_i = n \cdot p_i$ birthdays and the probability of no double birthday on day $i$.
$$P(X_i \leq 1) = e^{-\lambda_i}(1+ \lambda_i)$$
The... | Birthday paradox for non-uniform probabilities | Approach 1
Let's assume that $d$ is large and the distribution of birthdays on day $i$ can be modelled/approximated as independent Poisson variables.
Each day has a frequency of $\lambda_i = n \cdot p | Birthday paradox for non-uniform probabilities
Approach 1
Let's assume that $d$ is large and the distribution of birthdays on day $i$ can be modelled/approximated as independent Poisson variables.
Each day has a frequency of $\lambda_i = n \cdot p_i$ birthdays and the probability of no double birthday on day $i$.
$$P(X... | Birthday paradox for non-uniform probabilities
Approach 1
Let's assume that $d$ is large and the distribution of birthdays on day $i$ can be modelled/approximated as independent Poisson variables.
Each day has a frequency of $\lambda_i = n \cdot p |
50,252 | Who created the "soup analogy" for sampling | This saying is credit to George Gallup. It dates from before 1941, though I've not been able to find a primary source. It seems likely that he used the analogy multiple times.
For example the Ottawa Citizen writes:
When a cook want to taste the soup to see how it is coming he doesn't have to drink the whole boilerfu... | Who created the "soup analogy" for sampling | This saying is credit to George Gallup. It dates from before 1941, though I've not been able to find a primary source. It seems likely that he used the analogy multiple times.
For example the Ottawa | Who created the "soup analogy" for sampling
This saying is credit to George Gallup. It dates from before 1941, though I've not been able to find a primary source. It seems likely that he used the analogy multiple times.
For example the Ottawa Citizen writes:
When a cook want to taste the soup to see how it is coming... | Who created the "soup analogy" for sampling
This saying is credit to George Gallup. It dates from before 1941, though I've not been able to find a primary source. It seems likely that he used the analogy multiple times.
For example the Ottawa |
50,253 | How to formalize the following intutive reasoning (Bayes' rule)? | The issue here is that your prior does not fully specify the joint distribution of the relevant events at issue. If you let $\mathscr{E}_1,\mathscr{E}_2,\mathscr{E}_3,\mathscr{E}_4$ denote the individual events that each of the four respective compartments contain the laptop (only one of which can be true at most), th... | How to formalize the following intutive reasoning (Bayes' rule)? | The issue here is that your prior does not fully specify the joint distribution of the relevant events at issue. If you let $\mathscr{E}_1,\mathscr{E}_2,\mathscr{E}_3,\mathscr{E}_4$ denote the indivi | How to formalize the following intutive reasoning (Bayes' rule)?
The issue here is that your prior does not fully specify the joint distribution of the relevant events at issue. If you let $\mathscr{E}_1,\mathscr{E}_2,\mathscr{E}_3,\mathscr{E}_4$ denote the individual events that each of the four respective compartmen... | How to formalize the following intutive reasoning (Bayes' rule)?
The issue here is that your prior does not fully specify the joint distribution of the relevant events at issue. If you let $\mathscr{E}_1,\mathscr{E}_2,\mathscr{E}_3,\mathscr{E}_4$ denote the indivi |
50,254 | Interpretation of causal effect from instrumental variables | I think in an A/B test like this, where you have an encouragement design with one-sided non-compliance, you can make some progress under reasonable assumptions.
Using the notation from here, where C stands for compliers and not clickers, the LATE identified by IV is
$$\Delta_{IV} =\frac{E(Y_1 \vert C) \cdot Pr(C)−E(Y_... | Interpretation of causal effect from instrumental variables | I think in an A/B test like this, where you have an encouragement design with one-sided non-compliance, you can make some progress under reasonable assumptions.
Using the notation from here, where C s | Interpretation of causal effect from instrumental variables
I think in an A/B test like this, where you have an encouragement design with one-sided non-compliance, you can make some progress under reasonable assumptions.
Using the notation from here, where C stands for compliers and not clickers, the LATE identified by... | Interpretation of causal effect from instrumental variables
I think in an A/B test like this, where you have an encouragement design with one-sided non-compliance, you can make some progress under reasonable assumptions.
Using the notation from here, where C s |
50,255 | Why use the particular test statistic for hypothesis testing linear regression coefficients | Why use that particular test statistic and not another?
The t-test coincidences with the likelihood-ratio test and therefore has as good properties in being a powerful test.
Is it uniformly most powerful (UMP) among its peers?
Yes if you consider an alternative point hypothesis (think of Neyman's and Pearson's theor... | Why use the particular test statistic for hypothesis testing linear regression coefficients | Why use that particular test statistic and not another?
The t-test coincidences with the likelihood-ratio test and therefore has as good properties in being a powerful test.
Is it uniformly most pow | Why use the particular test statistic for hypothesis testing linear regression coefficients
Why use that particular test statistic and not another?
The t-test coincidences with the likelihood-ratio test and therefore has as good properties in being a powerful test.
Is it uniformly most powerful (UMP) among its peers?... | Why use the particular test statistic for hypothesis testing linear regression coefficients
Why use that particular test statistic and not another?
The t-test coincidences with the likelihood-ratio test and therefore has as good properties in being a powerful test.
Is it uniformly most pow |
50,256 | Show that $\min_{a \in \mathbb{R}} E \left[ \max \left( (1-a) V, a Z \right) \right]$ is minimized by $a$ such that $0<a<1$ | It is not possible to limit the solution to the open interval $a \in (0,1),$ because when for instance $(X,Y)$ is standard Normal, the global minimum is attained on the entire closed interval $[0,1]$ and, as you can readily compute, when $X$ is a nontrivial mixture of two Normals the global minimum is attained only at ... | Show that $\min_{a \in \mathbb{R}} E \left[ \max \left( (1-a) V, a Z \right) \right]$ is minimized b | It is not possible to limit the solution to the open interval $a \in (0,1),$ because when for instance $(X,Y)$ is standard Normal, the global minimum is attained on the entire closed interval $[0,1]$ | Show that $\min_{a \in \mathbb{R}} E \left[ \max \left( (1-a) V, a Z \right) \right]$ is minimized by $a$ such that $0<a<1$
It is not possible to limit the solution to the open interval $a \in (0,1),$ because when for instance $(X,Y)$ is standard Normal, the global minimum is attained on the entire closed interval $[0,... | Show that $\min_{a \in \mathbb{R}} E \left[ \max \left( (1-a) V, a Z \right) \right]$ is minimized b
It is not possible to limit the solution to the open interval $a \in (0,1),$ because when for instance $(X,Y)$ is standard Normal, the global minimum is attained on the entire closed interval $[0,1]$ |
50,257 | How do I model a paired experiment with pre-post data? | The model proposed is not incorrect but it can be improved.
I would suggest including the matching variables in the regression model too. The main reasons are two: 1. we might have a poor balance on the other covariates, even if the exact matching variables are perfectly balanced and 2. some of the covariates might ha... | How do I model a paired experiment with pre-post data? | The model proposed is not incorrect but it can be improved.
I would suggest including the matching variables in the regression model too. The main reasons are two: 1. we might have a poor balance on t | How do I model a paired experiment with pre-post data?
The model proposed is not incorrect but it can be improved.
I would suggest including the matching variables in the regression model too. The main reasons are two: 1. we might have a poor balance on the other covariates, even if the exact matching variables are per... | How do I model a paired experiment with pre-post data?
The model proposed is not incorrect but it can be improved.
I would suggest including the matching variables in the regression model too. The main reasons are two: 1. we might have a poor balance on t |
50,258 | Distribution of positive and negative values in a Brownian bridge | The distribution is uniform.
A more well known related relationship is Lévy's arcsine law: the distribution of time that a random walk is positive follows an arcsine distribution (or Beta 1/2,1/2).
On mathematics the same question was asked and answered Distribution of time spent above 0 by a Brownian Bridge.
We should... | Distribution of positive and negative values in a Brownian bridge | The distribution is uniform.
A more well known related relationship is Lévy's arcsine law: the distribution of time that a random walk is positive follows an arcsine distribution (or Beta 1/2,1/2).
On | Distribution of positive and negative values in a Brownian bridge
The distribution is uniform.
A more well known related relationship is Lévy's arcsine law: the distribution of time that a random walk is positive follows an arcsine distribution (or Beta 1/2,1/2).
On mathematics the same question was asked and answered ... | Distribution of positive and negative values in a Brownian bridge
The distribution is uniform.
A more well known related relationship is Lévy's arcsine law: the distribution of time that a random walk is positive follows an arcsine distribution (or Beta 1/2,1/2).
On |
50,259 | How can I test whether one empirical CDF is to the left or right of another? | Maybe you're interested in whether sample y
stochastically dominates sample x. If so,
you might want to look directly at ECDF plots, and do some formal tests.
Here are summaries and ECDF plots of two samples.
summary(x); length(x); sd(x)
Min. 1st Qu. Median Mean 3rd Qu. Max.
5.067 14.628 21.012 21.807 ... | How can I test whether one empirical CDF is to the left or right of another? | Maybe you're interested in whether sample y
stochastically dominates sample x. If so,
you might want to look directly at ECDF plots, and do some formal tests.
Here are summaries and ECDF plots of two | How can I test whether one empirical CDF is to the left or right of another?
Maybe you're interested in whether sample y
stochastically dominates sample x. If so,
you might want to look directly at ECDF plots, and do some formal tests.
Here are summaries and ECDF plots of two samples.
summary(x); length(x); sd(x)
M... | How can I test whether one empirical CDF is to the left or right of another?
Maybe you're interested in whether sample y
stochastically dominates sample x. If so,
you might want to look directly at ECDF plots, and do some formal tests.
Here are summaries and ECDF plots of two |
50,260 | Does bias eventually increase with model complexity? | I was wondering the same thing. What you have to realize though, is that the bias is defined for the expectation of the prediction for a given x over all possible training sets (D):
\begin{equation}
D=\left\{\left(x_{1}, y_{1}\right) \ldots,\left(x_{n}, y_{n}\right)\right\}
\end{equation}
\begin{equation}
\operatorname... | Does bias eventually increase with model complexity? | I was wondering the same thing. What you have to realize though, is that the bias is defined for the expectation of the prediction for a given x over all possible training sets (D):
\begin{equation}
D | Does bias eventually increase with model complexity?
I was wondering the same thing. What you have to realize though, is that the bias is defined for the expectation of the prediction for a given x over all possible training sets (D):
\begin{equation}
D=\left\{\left(x_{1}, y_{1}\right) \ldots,\left(x_{n}, y_{n}\right)\... | Does bias eventually increase with model complexity?
I was wondering the same thing. What you have to realize though, is that the bias is defined for the expectation of the prediction for a given x over all possible training sets (D):
\begin{equation}
D |
50,261 | Does bias eventually increase with model complexity? | It depends, is all I can say.
As a rule of thumb, bias decreases in a model as you add more parameters, but there can be some weird exceptions. | Does bias eventually increase with model complexity? | It depends, is all I can say.
As a rule of thumb, bias decreases in a model as you add more parameters, but there can be some weird exceptions. | Does bias eventually increase with model complexity?
It depends, is all I can say.
As a rule of thumb, bias decreases in a model as you add more parameters, but there can be some weird exceptions. | Does bias eventually increase with model complexity?
It depends, is all I can say.
As a rule of thumb, bias decreases in a model as you add more parameters, but there can be some weird exceptions. |
50,262 | Why use the EM Algorithm and not just marginalise the complete likelihood? | So I've discussed this with a colleague.
Consider the marginalisation
$$p(\mathbf{X} \mid \boldsymbol{\theta}) = \int p(\mathbf{X} \mid \mathbf{Z}, \boldsymbol{\theta})p(\mathbf{Z} \mid \boldsymbol{\theta}) d\mathbf{Z}.$$
This can be rewritten as the expectation
$$\mathbb{E}_{\mathbf{Z} \mid \boldsymbol{\theta}}\left[ ... | Why use the EM Algorithm and not just marginalise the complete likelihood? | So I've discussed this with a colleague.
Consider the marginalisation
$$p(\mathbf{X} \mid \boldsymbol{\theta}) = \int p(\mathbf{X} \mid \mathbf{Z}, \boldsymbol{\theta})p(\mathbf{Z} \mid \boldsymbol{\t | Why use the EM Algorithm and not just marginalise the complete likelihood?
So I've discussed this with a colleague.
Consider the marginalisation
$$p(\mathbf{X} \mid \boldsymbol{\theta}) = \int p(\mathbf{X} \mid \mathbf{Z}, \boldsymbol{\theta})p(\mathbf{Z} \mid \boldsymbol{\theta}) d\mathbf{Z}.$$
This can be rewritten a... | Why use the EM Algorithm and not just marginalise the complete likelihood?
So I've discussed this with a colleague.
Consider the marginalisation
$$p(\mathbf{X} \mid \boldsymbol{\theta}) = \int p(\mathbf{X} \mid \mathbf{Z}, \boldsymbol{\theta})p(\mathbf{Z} \mid \boldsymbol{\t |
50,263 | Why transformer in deep learning is called transformer? | Transformer, becuase it uses the attention mechanism with softmax transformation after that using the feedforward with nonlinear transformation. In short it uses different transformations(activation functions) to transform the input from intial representation into final representation if we would explain that in ver... | Why transformer in deep learning is called transformer? | Transformer, becuase it uses the attention mechanism with softmax transformation after that using the feedforward with nonlinear transformation. In short it uses different transformations(activation | Why transformer in deep learning is called transformer?
Transformer, becuase it uses the attention mechanism with softmax transformation after that using the feedforward with nonlinear transformation. In short it uses different transformations(activation functions) to transform the input from intial representation in... | Why transformer in deep learning is called transformer?
Transformer, becuase it uses the attention mechanism with softmax transformation after that using the feedforward with nonlinear transformation. In short it uses different transformations(activation |
50,264 | Parameter uncertainity in least squares optimization: rescaling Hessian | One can possibly look at the approximation for the covariance as a "rescaled inverse Hessian", but it kind of hides the simple deduction.
In principle it's the stopped series expansion of the term that is minimized. So, if one stops at the second term, the expression is in the simple 1D case around the (minimum) value ... | Parameter uncertainity in least squares optimization: rescaling Hessian | One can possibly look at the approximation for the covariance as a "rescaled inverse Hessian", but it kind of hides the simple deduction.
In principle it's the stopped series expansion of the term tha | Parameter uncertainity in least squares optimization: rescaling Hessian
One can possibly look at the approximation for the covariance as a "rescaled inverse Hessian", but it kind of hides the simple deduction.
In principle it's the stopped series expansion of the term that is minimized. So, if one stops at the second t... | Parameter uncertainity in least squares optimization: rescaling Hessian
One can possibly look at the approximation for the covariance as a "rescaled inverse Hessian", but it kind of hides the simple deduction.
In principle it's the stopped series expansion of the term tha |
50,265 | Probability that a device turns off in n seconds | The question concerns a sample space $\Omega$ of sequences of observations of the light made at times $1, 2, 3, \ldots.$ For it to be answerable, we have to suppose that the switch can be flipped no more than once in any interval $(n-1,n]$ (for otherwise the observations do not determine when the switch is flipped).
$\... | Probability that a device turns off in n seconds | The question concerns a sample space $\Omega$ of sequences of observations of the light made at times $1, 2, 3, \ldots.$ For it to be answerable, we have to suppose that the switch can be flipped no m | Probability that a device turns off in n seconds
The question concerns a sample space $\Omega$ of sequences of observations of the light made at times $1, 2, 3, \ldots.$ For it to be answerable, we have to suppose that the switch can be flipped no more than once in any interval $(n-1,n]$ (for otherwise the observations... | Probability that a device turns off in n seconds
The question concerns a sample space $\Omega$ of sequences of observations of the light made at times $1, 2, 3, \ldots.$ For it to be answerable, we have to suppose that the switch can be flipped no m |
50,266 | Probability that a device turns off in n seconds | To answer this question and confirm the answer that is proposed in the OP, I would first propose to slightly rephrase it as :
"A device is initially on and experiences a sequence of events $1, 2, \ldots$ which can turn it off. Each binary trigger event occurs with an independent probability given by a Bernouilli trial... | Probability that a device turns off in n seconds | To answer this question and confirm the answer that is proposed in the OP, I would first propose to slightly rephrase it as :
"A device is initially on and experiences a sequence of events $1, 2, \ld | Probability that a device turns off in n seconds
To answer this question and confirm the answer that is proposed in the OP, I would first propose to slightly rephrase it as :
"A device is initially on and experiences a sequence of events $1, 2, \ldots$ which can turn it off. Each binary trigger event occurs with an in... | Probability that a device turns off in n seconds
To answer this question and confirm the answer that is proposed in the OP, I would first propose to slightly rephrase it as :
"A device is initially on and experiences a sequence of events $1, 2, \ld |
50,267 | Distribution of $\langle x^2,a\rangle$ where $x$ is is a random direction? | Let's start with $m(a)=E[\langle x^2,a\rangle]$ and $s(a)=E[\langle x^2,a\rangle^2]$, so that the variance will be $s(a)-m(a)^2$.
We will calculate these by expressing $x$ in $n$-dimensional spherical coordinates, so
\begin{align}
& & x_1&=\cos(\phi_1)\\
0\ \le\ &\phi_1\le\pi & x_2&=\sin(\phi_1)\cos(\phi_2)\\
0\ \le\ &... | Distribution of $\langle x^2,a\rangle$ where $x$ is is a random direction? | Let's start with $m(a)=E[\langle x^2,a\rangle]$ and $s(a)=E[\langle x^2,a\rangle^2]$, so that the variance will be $s(a)-m(a)^2$.
We will calculate these by expressing $x$ in $n$-dimensional spherical | Distribution of $\langle x^2,a\rangle$ where $x$ is is a random direction?
Let's start with $m(a)=E[\langle x^2,a\rangle]$ and $s(a)=E[\langle x^2,a\rangle^2]$, so that the variance will be $s(a)-m(a)^2$.
We will calculate these by expressing $x$ in $n$-dimensional spherical coordinates, so
\begin{align}
& & x_1&=\cos(... | Distribution of $\langle x^2,a\rangle$ where $x$ is is a random direction?
Let's start with $m(a)=E[\langle x^2,a\rangle]$ and $s(a)=E[\langle x^2,a\rangle^2]$, so that the variance will be $s(a)-m(a)^2$.
We will calculate these by expressing $x$ in $n$-dimensional spherical |
50,268 | Internal validation steps | Expanding on the comment from gung - Reinstate Monica:
You have correctly grasped the major point. With this bootstrap approach you are validating your model-building process.
Under the bootstrap principle, resampling from your data set is akin to taking data sets from the underlying population. Thus, if repeating your... | Internal validation steps | Expanding on the comment from gung - Reinstate Monica:
You have correctly grasped the major point. With this bootstrap approach you are validating your model-building process.
Under the bootstrap prin | Internal validation steps
Expanding on the comment from gung - Reinstate Monica:
You have correctly grasped the major point. With this bootstrap approach you are validating your model-building process.
Under the bootstrap principle, resampling from your data set is akin to taking data sets from the underlying populatio... | Internal validation steps
Expanding on the comment from gung - Reinstate Monica:
You have correctly grasped the major point. With this bootstrap approach you are validating your model-building process.
Under the bootstrap prin |
50,269 | Internal validation steps | Normally, validation sets or validation is done inside of the cross validation function for hyperparameter testing. So, you actually don't care about a specific validation set. You normally end up with train, test.
What I believe is or could be the meaning of the part you quoted, without having access to the text of yo... | Internal validation steps | Normally, validation sets or validation is done inside of the cross validation function for hyperparameter testing. So, you actually don't care about a specific validation set. You normally end up wit | Internal validation steps
Normally, validation sets or validation is done inside of the cross validation function for hyperparameter testing. So, you actually don't care about a specific validation set. You normally end up with train, test.
What I believe is or could be the meaning of the part you quoted, without havin... | Internal validation steps
Normally, validation sets or validation is done inside of the cross validation function for hyperparameter testing. So, you actually don't care about a specific validation set. You normally end up wit |
50,270 | Linear Mixed Models non-independent data with unbalanced design and non-independent data? | You say that
the factor Year nested in Plant
If Year is nested within Plant. In that case, the moel should be
lmer(Productivity~Temperature +(1|Plant/Year),data = data)
or eqivalently:
lmer(Productivity~Temperature +(1|Plant) + (1|Plant:Year),data = data)
So, just to clarify, this means that each Year belongs to on... | Linear Mixed Models non-independent data with unbalanced design and non-independent data? | You say that
the factor Year nested in Plant
If Year is nested within Plant. In that case, the moel should be
lmer(Productivity~Temperature +(1|Plant/Year),data = data)
or eqivalently:
lmer(Product | Linear Mixed Models non-independent data with unbalanced design and non-independent data?
You say that
the factor Year nested in Plant
If Year is nested within Plant. In that case, the moel should be
lmer(Productivity~Temperature +(1|Plant/Year),data = data)
or eqivalently:
lmer(Productivity~Temperature +(1|Plant) +... | Linear Mixed Models non-independent data with unbalanced design and non-independent data?
You say that
the factor Year nested in Plant
If Year is nested within Plant. In that case, the moel should be
lmer(Productivity~Temperature +(1|Plant/Year),data = data)
or eqivalently:
lmer(Product |
50,271 | Intuitive understanding of least squares slope formula | Define "obvious"! What's obvious to you doesn't need be obvious to someone else. So I can only offer my perspective.
First, I hope it's obvious that the slope $\beta$ of the line doesn't change if we shift the data around; only the intercept changes. So, to simplify the formulas, we can, without loss of generality, ass... | Intuitive understanding of least squares slope formula | Define "obvious"! What's obvious to you doesn't need be obvious to someone else. So I can only offer my perspective.
First, I hope it's obvious that the slope $\beta$ of the line doesn't change if we | Intuitive understanding of least squares slope formula
Define "obvious"! What's obvious to you doesn't need be obvious to someone else. So I can only offer my perspective.
First, I hope it's obvious that the slope $\beta$ of the line doesn't change if we shift the data around; only the intercept changes. So, to simplif... | Intuitive understanding of least squares slope formula
Define "obvious"! What's obvious to you doesn't need be obvious to someone else. So I can only offer my perspective.
First, I hope it's obvious that the slope $\beta$ of the line doesn't change if we |
50,272 | Intuitive understanding of least squares slope formula | Answer 1 (simple):
The simple answer is that $\beta$ is the sensitivity of the response $y$ with respect to the regressor $X$ (assuming the relationship is true). Its estimator $\hat \beta$, which is what you quoted, is the result (the slope) of "forcing" (i.e. fitting) a line (consisting of an intercept and slope) to ... | Intuitive understanding of least squares slope formula | Answer 1 (simple):
The simple answer is that $\beta$ is the sensitivity of the response $y$ with respect to the regressor $X$ (assuming the relationship is true). Its estimator $\hat \beta$, which is | Intuitive understanding of least squares slope formula
Answer 1 (simple):
The simple answer is that $\beta$ is the sensitivity of the response $y$ with respect to the regressor $X$ (assuming the relationship is true). Its estimator $\hat \beta$, which is what you quoted, is the result (the slope) of "forcing" (i.e. fit... | Intuitive understanding of least squares slope formula
Answer 1 (simple):
The simple answer is that $\beta$ is the sensitivity of the response $y$ with respect to the regressor $X$ (assuming the relationship is true). Its estimator $\hat \beta$, which is |
50,273 | Regression with variance as outcome | For your example, for some parameter values $c,d$ the variance will be negative ... for that reason, in such models often is used a log link function for the variance. But such models (and many others) can be fitted with extensions of generalized linear models (glm's), also introducing link functions and linear predict... | Regression with variance as outcome | For your example, for some parameter values $c,d$ the variance will be negative ... for that reason, in such models often is used a log link function for the variance. But such models (and many others | Regression with variance as outcome
For your example, for some parameter values $c,d$ the variance will be negative ... for that reason, in such models often is used a log link function for the variance. But such models (and many others) can be fitted with extensions of generalized linear models (glm's), also introduci... | Regression with variance as outcome
For your example, for some parameter values $c,d$ the variance will be negative ... for that reason, in such models often is used a log link function for the variance. But such models (and many others |
50,274 | Identifying the correlation between a slope and a level | A slightly different characterisation of the problem
Instead of these separate variations/errors in $\alpha$ and $\beta$ you could describe the variance of $Y_i$ directly.
A common way (which you see a lot on this site) is to describe a linear function like $$y_i = a+bx_i + \epsilon \quad \text{where} \quad \epsilon \s... | Identifying the correlation between a slope and a level | A slightly different characterisation of the problem
Instead of these separate variations/errors in $\alpha$ and $\beta$ you could describe the variance of $Y_i$ directly.
A common way (which you see | Identifying the correlation between a slope and a level
A slightly different characterisation of the problem
Instead of these separate variations/errors in $\alpha$ and $\beta$ you could describe the variance of $Y_i$ directly.
A common way (which you see a lot on this site) is to describe a linear function like $$y_i ... | Identifying the correlation between a slope and a level
A slightly different characterisation of the problem
Instead of these separate variations/errors in $\alpha$ and $\beta$ you could describe the variance of $Y_i$ directly.
A common way (which you see |
50,275 | Identifying the correlation between a slope and a level | Let me not answer the question exactly as I posed it, but to answer a very related question (and in fact, the question I am interested in in the first place). Suppose we have potential outcomes $Y_1(X), Y_0(X)$, and suppose that for each individual has a default level of $X$, say $X_d$, in the absence of experimental i... | Identifying the correlation between a slope and a level | Let me not answer the question exactly as I posed it, but to answer a very related question (and in fact, the question I am interested in in the first place). Suppose we have potential outcomes $Y_1(X | Identifying the correlation between a slope and a level
Let me not answer the question exactly as I posed it, but to answer a very related question (and in fact, the question I am interested in in the first place). Suppose we have potential outcomes $Y_1(X), Y_0(X)$, and suppose that for each individual has a default l... | Identifying the correlation between a slope and a level
Let me not answer the question exactly as I posed it, but to answer a very related question (and in fact, the question I am interested in in the first place). Suppose we have potential outcomes $Y_1(X |
50,276 | Why are we checking the difference between q(z|x), and p(z|x) in variational encoders? | The first thing to appreciate about VAEs is that they are not just some magical deep generative model but that they are a special case of the Auto-Encoding Variational Bayes algorithm for doing variational Bayesian inference in generative models.
What that means is we consider a setup with a dataset $\mathcal{D} =\{x_i... | Why are we checking the difference between q(z|x), and p(z|x) in variational encoders? | The first thing to appreciate about VAEs is that they are not just some magical deep generative model but that they are a special case of the Auto-Encoding Variational Bayes algorithm for doing variat | Why are we checking the difference between q(z|x), and p(z|x) in variational encoders?
The first thing to appreciate about VAEs is that they are not just some magical deep generative model but that they are a special case of the Auto-Encoding Variational Bayes algorithm for doing variational Bayesian inference in gener... | Why are we checking the difference between q(z|x), and p(z|x) in variational encoders?
The first thing to appreciate about VAEs is that they are not just some magical deep generative model but that they are a special case of the Auto-Encoding Variational Bayes algorithm for doing variat |
50,277 | Law of the norm of the empirical mean of uniforms on the sphere? | Densities of short uniform walks in higher dimensions could be relevant. It discusses random walks with $n$ steps each of length $1$ in $\mathbb{R}^d$, where each step is taken in a uniformly random direction.
Theorem 2.1 states that the probability density function of the
distance to the origin in $d \ge 2$ dimensions... | Law of the norm of the empirical mean of uniforms on the sphere? | Densities of short uniform walks in higher dimensions could be relevant. It discusses random walks with $n$ steps each of length $1$ in $\mathbb{R}^d$, where each step is taken in a uniformly random d | Law of the norm of the empirical mean of uniforms on the sphere?
Densities of short uniform walks in higher dimensions could be relevant. It discusses random walks with $n$ steps each of length $1$ in $\mathbb{R}^d$, where each step is taken in a uniformly random direction.
Theorem 2.1 states that the probability densi... | Law of the norm of the empirical mean of uniforms on the sphere?
Densities of short uniform walks in higher dimensions could be relevant. It discusses random walks with $n$ steps each of length $1$ in $\mathbb{R}^d$, where each step is taken in a uniformly random d |
50,278 | Law of the norm of the empirical mean of uniforms on the sphere? | Similarity with a rubber band model.
This problem got me to think of the model for a 'rubber band' (See for instance wikipedia or section 3-7 in Herbert B. Callen's thermodynamics and an introduction to thermostatistics).
With a bit of hand waving:
consider the distribution of only one axis/component of the $U_i$ (th... | Law of the norm of the empirical mean of uniforms on the sphere? | Similarity with a rubber band model.
This problem got me to think of the model for a 'rubber band' (See for instance wikipedia or section 3-7 in Herbert B. Callen's thermodynamics and an introduction | Law of the norm of the empirical mean of uniforms on the sphere?
Similarity with a rubber band model.
This problem got me to think of the model for a 'rubber band' (See for instance wikipedia or section 3-7 in Herbert B. Callen's thermodynamics and an introduction to thermostatistics).
With a bit of hand waving:
cons... | Law of the norm of the empirical mean of uniforms on the sphere?
Similarity with a rubber band model.
This problem got me to think of the model for a 'rubber band' (See for instance wikipedia or section 3-7 in Herbert B. Callen's thermodynamics and an introduction |
50,279 | Law of the norm of the empirical mean of uniforms on the sphere? | I'm not sure if you want an upper or lower bound. You mention both in the question. The easiest way to get a very loose upper bound on this problem is to use Markov's inequality. Just in case it's been overlooked.
$$M=\left\|\frac{1}{n}\sum_{i=1}^n U_i \right\|.$$
$$P\left(M \ge \sqrt{\frac{\lambda}{n}}\right) \le \sqr... | Law of the norm of the empirical mean of uniforms on the sphere? | I'm not sure if you want an upper or lower bound. You mention both in the question. The easiest way to get a very loose upper bound on this problem is to use Markov's inequality. Just in case it's bee | Law of the norm of the empirical mean of uniforms on the sphere?
I'm not sure if you want an upper or lower bound. You mention both in the question. The easiest way to get a very loose upper bound on this problem is to use Markov's inequality. Just in case it's been overlooked.
$$M=\left\|\frac{1}{n}\sum_{i=1}^n U_i \r... | Law of the norm of the empirical mean of uniforms on the sphere?
I'm not sure if you want an upper or lower bound. You mention both in the question. The easiest way to get a very loose upper bound on this problem is to use Markov's inequality. Just in case it's bee |
50,280 | CDF*[1-CDF]/PDF --- name? integrable? | Logistic curve
One relationship might be with logistic growth which is based on the following differential equation:
$$f'(x) = f(x)(1-f(x))$$
But then for $F(x)$ and inhomogeneous (using some variable rate $g(x)$)
$$F'(x) = g(x) F(x)(1-F(x))$$
So if we express the CDF as a logistic curve
$$F(u) = \frac{1}{1+e^{-u}}$$
w... | CDF*[1-CDF]/PDF --- name? integrable? | Logistic curve
One relationship might be with logistic growth which is based on the following differential equation:
$$f'(x) = f(x)(1-f(x))$$
But then for $F(x)$ and inhomogeneous (using some variable | CDF*[1-CDF]/PDF --- name? integrable?
Logistic curve
One relationship might be with logistic growth which is based on the following differential equation:
$$f'(x) = f(x)(1-f(x))$$
But then for $F(x)$ and inhomogeneous (using some variable rate $g(x)$)
$$F'(x) = g(x) F(x)(1-F(x))$$
So if we express the CDF as a logistic... | CDF*[1-CDF]/PDF --- name? integrable?
Logistic curve
One relationship might be with logistic growth which is based on the following differential equation:
$$f'(x) = f(x)(1-f(x))$$
But then for $F(x)$ and inhomogeneous (using some variable |
50,281 | What is the application for using the `Boltzmann Machines`? | The idea behind the Boltzmann Machine is that it represents a closed system where an energy flows from one part to another, i.e. heat dissipation, and models the decrease in the entropy of a closed model - while the model starts with relatively low entropy (i.e. when there is a separation between 'hot' and 'cold' parts... | What is the application for using the `Boltzmann Machines`? | The idea behind the Boltzmann Machine is that it represents a closed system where an energy flows from one part to another, i.e. heat dissipation, and models the decrease in the entropy of a closed mo | What is the application for using the `Boltzmann Machines`?
The idea behind the Boltzmann Machine is that it represents a closed system where an energy flows from one part to another, i.e. heat dissipation, and models the decrease in the entropy of a closed model - while the model starts with relatively low entropy (i.... | What is the application for using the `Boltzmann Machines`?
The idea behind the Boltzmann Machine is that it represents a closed system where an energy flows from one part to another, i.e. heat dissipation, and models the decrease in the entropy of a closed mo |
50,282 | How can I show that two random variables are independent if their mutual information is 0? | Here's my take. In the discrete case,
$$ \operatorname{I}(X;Y) =
\sum_{y \in \mathcal Y} \sum_{x \in \mathcal X}
{p_{(X,Y)}(x,y) \log{ \left(\frac{p_{(X,Y)}(x,y)}{p_X(x)\,p_Y(y)} \right) }
} $$
So $I(X;Y)=0$ when, at all points, either:
${p_{(X,Y)}(x,y)} = 0$, or
$\log{ \left(\frac{p_{(X,Y)}(x,y)}{p_X(x)\,p_Y(y)}... | How can I show that two random variables are independent if their mutual information is 0? | Here's my take. In the discrete case,
$$ \operatorname{I}(X;Y) =
\sum_{y \in \mathcal Y} \sum_{x \in \mathcal X}
{p_{(X,Y)}(x,y) \log{ \left(\frac{p_{(X,Y)}(x,y)}{p_X(x)\,p_Y(y)} \right) }
} $$
S | How can I show that two random variables are independent if their mutual information is 0?
Here's my take. In the discrete case,
$$ \operatorname{I}(X;Y) =
\sum_{y \in \mathcal Y} \sum_{x \in \mathcal X}
{p_{(X,Y)}(x,y) \log{ \left(\frac{p_{(X,Y)}(x,y)}{p_X(x)\,p_Y(y)} \right) }
} $$
So $I(X;Y)=0$ when, at all poi... | How can I show that two random variables are independent if their mutual information is 0?
Here's my take. In the discrete case,
$$ \operatorname{I}(X;Y) =
\sum_{y \in \mathcal Y} \sum_{x \in \mathcal X}
{p_{(X,Y)}(x,y) \log{ \left(\frac{p_{(X,Y)}(x,y)}{p_X(x)\,p_Y(y)} \right) }
} $$
S |
50,283 | Likelihood ratio test for $H_0:(\mu_1,\mu_2)=(0,0)$ vs $H_1:(\mu_1,\mu_2) \neq (0,0)$ | Your solution seems to be correct. The strange shape of your parameter space (it's not a open subset of $\mathbb R^2$) creates this ambiguity in the final result: each combination of $(p,q,r)$ gives a different LRT. Some are more powered for $\mu_1=0$, some are more powered for $\mu_2=0$ and some are more powered for $... | Likelihood ratio test for $H_0:(\mu_1,\mu_2)=(0,0)$ vs $H_1:(\mu_1,\mu_2) \neq (0,0)$ | Your solution seems to be correct. The strange shape of your parameter space (it's not a open subset of $\mathbb R^2$) creates this ambiguity in the final result: each combination of $(p,q,r)$ gives a | Likelihood ratio test for $H_0:(\mu_1,\mu_2)=(0,0)$ vs $H_1:(\mu_1,\mu_2) \neq (0,0)$
Your solution seems to be correct. The strange shape of your parameter space (it's not a open subset of $\mathbb R^2$) creates this ambiguity in the final result: each combination of $(p,q,r)$ gives a different LRT. Some are more powe... | Likelihood ratio test for $H_0:(\mu_1,\mu_2)=(0,0)$ vs $H_1:(\mu_1,\mu_2) \neq (0,0)$
Your solution seems to be correct. The strange shape of your parameter space (it's not a open subset of $\mathbb R^2$) creates this ambiguity in the final result: each combination of $(p,q,r)$ gives a |
50,284 | Best method of quantifying probability of new datum belonging to either of two distanced normal distributions? | Your question is a bit vague and it seems the your figure does not quite match the rest of the problem. I think you may have put parts of two similar problems together in your Question.
I'll do my best to give most of the information you requested.
You say the means of the two normal populations are unknown with $\mu_A... | Best method of quantifying probability of new datum belonging to either of two distanced normal dist | Your question is a bit vague and it seems the your figure does not quite match the rest of the problem. I think you may have put parts of two similar problems together in your Question.
I'll do my bes | Best method of quantifying probability of new datum belonging to either of two distanced normal distributions?
Your question is a bit vague and it seems the your figure does not quite match the rest of the problem. I think you may have put parts of two similar problems together in your Question.
I'll do my best to give... | Best method of quantifying probability of new datum belonging to either of two distanced normal dist
Your question is a bit vague and it seems the your figure does not quite match the rest of the problem. I think you may have put parts of two similar problems together in your Question.
I'll do my bes |
50,285 | Best method of quantifying probability of new datum belonging to either of two distanced normal distributions? | Here the aim "is to find a threshold value between the two distributions such that a new datum can be assigned to $A$ if its value falls below this central point, and to $B$ if it lies above, with a certain level of accuracy".
Suppose we measure the accuracy as (probability of wrong assignment for data in $A$) + (proba... | Best method of quantifying probability of new datum belonging to either of two distanced normal dist | Here the aim "is to find a threshold value between the two distributions such that a new datum can be assigned to $A$ if its value falls below this central point, and to $B$ if it lies above, with a c | Best method of quantifying probability of new datum belonging to either of two distanced normal distributions?
Here the aim "is to find a threshold value between the two distributions such that a new datum can be assigned to $A$ if its value falls below this central point, and to $B$ if it lies above, with a certain le... | Best method of quantifying probability of new datum belonging to either of two distanced normal dist
Here the aim "is to find a threshold value between the two distributions such that a new datum can be assigned to $A$ if its value falls below this central point, and to $B$ if it lies above, with a c |
50,286 | Population vs. Data-Generating Process | In population approach, the model that you are fitting to a data can potentially be a reduced form of the true DGP. A crude example:
Say $X_t$ is a time-series that actually grows with time with white noise ($e_t$). Specifically, let DGP is
$X_t = a_0+a_1t+a_2t^2+e_t$
$\implies X_{t-1} = a_0+a_1(t-1)+a_2(t-1)^2 + e_{t-... | Population vs. Data-Generating Process | In population approach, the model that you are fitting to a data can potentially be a reduced form of the true DGP. A crude example:
Say $X_t$ is a time-series that actually grows with time with white | Population vs. Data-Generating Process
In population approach, the model that you are fitting to a data can potentially be a reduced form of the true DGP. A crude example:
Say $X_t$ is a time-series that actually grows with time with white noise ($e_t$). Specifically, let DGP is
$X_t = a_0+a_1t+a_2t^2+e_t$
$\implies X_... | Population vs. Data-Generating Process
In population approach, the model that you are fitting to a data can potentially be a reduced form of the true DGP. A crude example:
Say $X_t$ is a time-series that actually grows with time with white |
50,287 | $R^2$ of Logistic Regression Without Intercept? | (Throughout, I assume the labels are $0$ and $1$, not $\pm 1$.)
Let's look at what $R^2$ means in the setting where we use a linear regression with an intercept. While there are many equivalent definitions in this setting, the definition that I find to apply in the most generality is comparing the performance of our mo... | $R^2$ of Logistic Regression Without Intercept? | (Throughout, I assume the labels are $0$ and $1$, not $\pm 1$.)
Let's look at what $R^2$ means in the setting where we use a linear regression with an intercept. While there are many equivalent defini | $R^2$ of Logistic Regression Without Intercept?
(Throughout, I assume the labels are $0$ and $1$, not $\pm 1$.)
Let's look at what $R^2$ means in the setting where we use a linear regression with an intercept. While there are many equivalent definitions in this setting, the definition that I find to apply in the most g... | $R^2$ of Logistic Regression Without Intercept?
(Throughout, I assume the labels are $0$ and $1$, not $\pm 1$.)
Let's look at what $R^2$ means in the setting where we use a linear regression with an intercept. While there are many equivalent defini |
50,288 | Hypothesis test: numeric vs. ranked | Here are simulations comparing two samples of size 15 from
$\mathsf{Norm}(0,1)$ and $\mathsf{Norm}(1,1),$ respectively.
My simulation shows that the pooled t test has better power
than the two-sample Wilcoxon test, which is well-known, and
that neither test has power $0.8.$
set.seed(2020)
pv = replicate(10^4, t.test(rn... | Hypothesis test: numeric vs. ranked | Here are simulations comparing two samples of size 15 from
$\mathsf{Norm}(0,1)$ and $\mathsf{Norm}(1,1),$ respectively.
My simulation shows that the pooled t test has better power
than the two-sample | Hypothesis test: numeric vs. ranked
Here are simulations comparing two samples of size 15 from
$\mathsf{Norm}(0,1)$ and $\mathsf{Norm}(1,1),$ respectively.
My simulation shows that the pooled t test has better power
than the two-sample Wilcoxon test, which is well-known, and
that neither test has power $0.8.$
set.seed(... | Hypothesis test: numeric vs. ranked
Here are simulations comparing two samples of size 15 from
$\mathsf{Norm}(0,1)$ and $\mathsf{Norm}(1,1),$ respectively.
My simulation shows that the pooled t test has better power
than the two-sample |
50,289 | Why do we need to triangulate a convex polygon in order to sample uniformly from it? | The tldr answer is that in the square case, there are multiple ways to express a "deep" interior point as a convex combination of the vertices, but only one way for points that are nearer to the vertices.
To give a bit more detail and explain why this happens for squares and not triangles, let me first re-express your ... | Why do we need to triangulate a convex polygon in order to sample uniformly from it? | The tldr answer is that in the square case, there are multiple ways to express a "deep" interior point as a convex combination of the vertices, but only one way for points that are nearer to the verti | Why do we need to triangulate a convex polygon in order to sample uniformly from it?
The tldr answer is that in the square case, there are multiple ways to express a "deep" interior point as a convex combination of the vertices, but only one way for points that are nearer to the vertices.
To give a bit more detail and ... | Why do we need to triangulate a convex polygon in order to sample uniformly from it?
The tldr answer is that in the square case, there are multiple ways to express a "deep" interior point as a convex combination of the vertices, but only one way for points that are nearer to the verti |
50,290 | Random number generation for conjugate distribution of beta distribution | Here is an excerpt from our book, Introducing Monte Carlo methods with R, indirectly dealing with this case (by importance sampling). The graph of the target shows a smooth and regular shape for the conjugate, meaning a Normal or Student proposal could maybe be of used for accept-reject. An alternative is to use MCMC, ... | Random number generation for conjugate distribution of beta distribution | Here is an excerpt from our book, Introducing Monte Carlo methods with R, indirectly dealing with this case (by importance sampling). The graph of the target shows a smooth and regular shape for the c | Random number generation for conjugate distribution of beta distribution
Here is an excerpt from our book, Introducing Monte Carlo methods with R, indirectly dealing with this case (by importance sampling). The graph of the target shows a smooth and regular shape for the conjugate, meaning a Normal or Student proposal ... | Random number generation for conjugate distribution of beta distribution
Here is an excerpt from our book, Introducing Monte Carlo methods with R, indirectly dealing with this case (by importance sampling). The graph of the target shows a smooth and regular shape for the c |
50,291 | Why to downweight Precision in nominator of F-beta when I actually want to upweight Precision? | By setting $\beta$ to $0$ we are effectively getting "just Precision" that is because the Recall multipliers cancel each other out and we are left with Precision only. By using a lower $\beta$ we do not down-weight Precision in any way, we are up-weighting as we allow it to dominate the fraction's final value through t... | Why to downweight Precision in nominator of F-beta when I actually want to upweight Precision? | By setting $\beta$ to $0$ we are effectively getting "just Precision" that is because the Recall multipliers cancel each other out and we are left with Precision only. By using a lower $\beta$ we do n | Why to downweight Precision in nominator of F-beta when I actually want to upweight Precision?
By setting $\beta$ to $0$ we are effectively getting "just Precision" that is because the Recall multipliers cancel each other out and we are left with Precision only. By using a lower $\beta$ we do not down-weight Precision ... | Why to downweight Precision in nominator of F-beta when I actually want to upweight Precision?
By setting $\beta$ to $0$ we are effectively getting "just Precision" that is because the Recall multipliers cancel each other out and we are left with Precision only. By using a lower $\beta$ we do n |
50,292 | Mean-centering variables in glmer | When you use the scale function on a variable, this will apply to the whole variable.
That is not what you want here.
You need to try to disentangle the within-whale associations from the between-whale associations. One good way to do this is by mean-centering the variable(s) in question by group - that is, by whale in... | Mean-centering variables in glmer | When you use the scale function on a variable, this will apply to the whole variable.
That is not what you want here.
You need to try to disentangle the within-whale associations from the between-whal | Mean-centering variables in glmer
When you use the scale function on a variable, this will apply to the whole variable.
That is not what you want here.
You need to try to disentangle the within-whale associations from the between-whale associations. One good way to do this is by mean-centering the variable(s) in questi... | Mean-centering variables in glmer
When you use the scale function on a variable, this will apply to the whole variable.
That is not what you want here.
You need to try to disentangle the within-whale associations from the between-whal |
50,293 | Derivation of M step for Gaussian mixture model | TL;DR, we have that
$$\mu^*_k = \frac{\sum_{i=1}^n W_{ik}x_i}{\sum_{i=1}^n W_{ik}}$$
$$\Sigma^*_k = \frac{\sum_{i=1}^{n} W_{ik}(x_i -\mu^*_k)(x_i - \mu^*_k)'}{\sum_{i=1}^n W_{ik}}$$
In particular, this is the same as finding the MLE of a gaussian rv, but we weight by $W_{ik}$ for each $k$.
See below for the derivation,... | Derivation of M step for Gaussian mixture model | TL;DR, we have that
$$\mu^*_k = \frac{\sum_{i=1}^n W_{ik}x_i}{\sum_{i=1}^n W_{ik}}$$
$$\Sigma^*_k = \frac{\sum_{i=1}^{n} W_{ik}(x_i -\mu^*_k)(x_i - \mu^*_k)'}{\sum_{i=1}^n W_{ik}}$$
In particular, thi | Derivation of M step for Gaussian mixture model
TL;DR, we have that
$$\mu^*_k = \frac{\sum_{i=1}^n W_{ik}x_i}{\sum_{i=1}^n W_{ik}}$$
$$\Sigma^*_k = \frac{\sum_{i=1}^{n} W_{ik}(x_i -\mu^*_k)(x_i - \mu^*_k)'}{\sum_{i=1}^n W_{ik}}$$
In particular, this is the same as finding the MLE of a gaussian rv, but we weight by $W_{... | Derivation of M step for Gaussian mixture model
TL;DR, we have that
$$\mu^*_k = \frac{\sum_{i=1}^n W_{ik}x_i}{\sum_{i=1}^n W_{ik}}$$
$$\Sigma^*_k = \frac{\sum_{i=1}^{n} W_{ik}(x_i -\mu^*_k)(x_i - \mu^*_k)'}{\sum_{i=1}^n W_{ik}}$$
In particular, thi |
50,294 | When was Auto-Encoder used for anomaly detection for the first time? | A bit of reference chasing, combined with Google Scholar searches, suggests that the origin was
Japkowicz, N., Myers C., & Gluck M., (1995), “A Novelty Detection Approach to Classification”, in Mellish, C. (ed.) The International Joint Conference on Artificial Intelligence (IJCAI-95), Montreal, Canada. IJCAII & Morgan ... | When was Auto-Encoder used for anomaly detection for the first time? | A bit of reference chasing, combined with Google Scholar searches, suggests that the origin was
Japkowicz, N., Myers C., & Gluck M., (1995), “A Novelty Detection Approach to Classification”, in Mellis | When was Auto-Encoder used for anomaly detection for the first time?
A bit of reference chasing, combined with Google Scholar searches, suggests that the origin was
Japkowicz, N., Myers C., & Gluck M., (1995), “A Novelty Detection Approach to Classification”, in Mellish, C. (ed.) The International Joint Conference on A... | When was Auto-Encoder used for anomaly detection for the first time?
A bit of reference chasing, combined with Google Scholar searches, suggests that the origin was
Japkowicz, N., Myers C., & Gluck M., (1995), “A Novelty Detection Approach to Classification”, in Mellis |
50,295 | How to compute the Quantile Treatment Effect? | I think they will be the same in a setting like a homogenous treatment effect world ($Y1=Y0 + m$) or even an affine transformation ($Y1=k \cdot Y0 + m$) that preserves rank (i.e., $k>0$), but in general they will not overlap, so your concern is valid.
The second is definitely the more interesting counterfactual quantit... | How to compute the Quantile Treatment Effect? | I think they will be the same in a setting like a homogenous treatment effect world ($Y1=Y0 + m$) or even an affine transformation ($Y1=k \cdot Y0 + m$) that preserves rank (i.e., $k>0$), but in gener | How to compute the Quantile Treatment Effect?
I think they will be the same in a setting like a homogenous treatment effect world ($Y1=Y0 + m$) or even an affine transformation ($Y1=k \cdot Y0 + m$) that preserves rank (i.e., $k>0$), but in general they will not overlap, so your concern is valid.
The second is definite... | How to compute the Quantile Treatment Effect?
I think they will be the same in a setting like a homogenous treatment effect world ($Y1=Y0 + m$) or even an affine transformation ($Y1=k \cdot Y0 + m$) that preserves rank (i.e., $k>0$), but in gener |
50,296 | How do you calculate log likelihood p(x) for a VAE? | The IWAE ELBO provides a tighter bound to the true log-likelihood $\log p(x)$. This bound gets tighter as the number of importance weighted samples $k$ increases.
Therefore, the authors chose a large enough $k$, in this paper $k$=5000, to approximate the true likelihood of the test data as $\widehat{\log p(x)}$. As suc... | How do you calculate log likelihood p(x) for a VAE? | The IWAE ELBO provides a tighter bound to the true log-likelihood $\log p(x)$. This bound gets tighter as the number of importance weighted samples $k$ increases.
Therefore, the authors chose a large | How do you calculate log likelihood p(x) for a VAE?
The IWAE ELBO provides a tighter bound to the true log-likelihood $\log p(x)$. This bound gets tighter as the number of importance weighted samples $k$ increases.
Therefore, the authors chose a large enough $k$, in this paper $k$=5000, to approximate the true likeliho... | How do you calculate log likelihood p(x) for a VAE?
The IWAE ELBO provides a tighter bound to the true log-likelihood $\log p(x)$. This bound gets tighter as the number of importance weighted samples $k$ increases.
Therefore, the authors chose a large |
50,297 | PCA with polynomial kernel vs single layer autoencoder? | Assuming your polynomial kernel is given, for $d \in \mathbb I, c \in \mathbb R$ as
$$K(x,y) = \langle \phi(x)|\phi(y)\rangle = (\langle x|y\rangle + c)^d$$
The functional form of PCA is given by:
$$\phi(x')=WW^T\phi(x)\approx\phi(x)$$
In PCA $W$ is usually a rank-deficient matrix, so that the representation is lower d... | PCA with polynomial kernel vs single layer autoencoder? | Assuming your polynomial kernel is given, for $d \in \mathbb I, c \in \mathbb R$ as
$$K(x,y) = \langle \phi(x)|\phi(y)\rangle = (\langle x|y\rangle + c)^d$$
The functional form of PCA is given by:
$$\ | PCA with polynomial kernel vs single layer autoencoder?
Assuming your polynomial kernel is given, for $d \in \mathbb I, c \in \mathbb R$ as
$$K(x,y) = \langle \phi(x)|\phi(y)\rangle = (\langle x|y\rangle + c)^d$$
The functional form of PCA is given by:
$$\phi(x')=WW^T\phi(x)\approx\phi(x)$$
In PCA $W$ is usually a rank... | PCA with polynomial kernel vs single layer autoencoder?
Assuming your polynomial kernel is given, for $d \in \mathbb I, c \in \mathbb R$ as
$$K(x,y) = \langle \phi(x)|\phi(y)\rangle = (\langle x|y\rangle + c)^d$$
The functional form of PCA is given by:
$$\ |
50,298 | Asymptotic t test question- regression when you do not assume normality of errors | Since you are considering large sample theory where $n$ tends to infinity,
there are some additional assumptions you may need in order to make the assertion.
(1) $(\eta_1,\ldots,\eta_n)$ are uncorrelated with equal variance
(2) The design matrix $X$ grows as $n$ becomes larger. We should have something like: $\frac1n X... | Asymptotic t test question- regression when you do not assume normality of errors | Since you are considering large sample theory where $n$ tends to infinity,
there are some additional assumptions you may need in order to make the assertion.
(1) $(\eta_1,\ldots,\eta_n)$ are uncorrela | Asymptotic t test question- regression when you do not assume normality of errors
Since you are considering large sample theory where $n$ tends to infinity,
there are some additional assumptions you may need in order to make the assertion.
(1) $(\eta_1,\ldots,\eta_n)$ are uncorrelated with equal variance
(2) The design... | Asymptotic t test question- regression when you do not assume normality of errors
Since you are considering large sample theory where $n$ tends to infinity,
there are some additional assumptions you may need in order to make the assertion.
(1) $(\eta_1,\ldots,\eta_n)$ are uncorrela |
50,299 | Show that $\{X(t), = \cos(t+U)\}$, $U \sim \mathrm{Unif} (0, 2\pi)$ is a wide-sense stationary process | It is a typo, but you should get zero for the integral anyway. You should have:
$$\begin{aligned}
\mathbb{E}(\tfrac{1}{2} \cos(t_1+t_2 + 2U))
&= \int \limits_0^{2\pi} \frac{1}{4 \pi} \cdot \cos(t_1+t_2 + 2u) \ du \\[6pt]
&= \Bigg[ \frac{1}{8 \pi} \cdot \sin(t_1+t_2 + 2u) \Bigg]_{u=0}^{u=2\pi} \\[6pt]
&= \frac{1}{8 \p... | Show that $\{X(t), = \cos(t+U)\}$, $U \sim \mathrm{Unif} (0, 2\pi)$ is a wide-sense stationary proce | It is a typo, but you should get zero for the integral anyway. You should have:
$$\begin{aligned}
\mathbb{E}(\tfrac{1}{2} \cos(t_1+t_2 + 2U))
&= \int \limits_0^{2\pi} \frac{1}{4 \pi} \cdot \cos(t_1+t | Show that $\{X(t), = \cos(t+U)\}$, $U \sim \mathrm{Unif} (0, 2\pi)$ is a wide-sense stationary process
It is a typo, but you should get zero for the integral anyway. You should have:
$$\begin{aligned}
\mathbb{E}(\tfrac{1}{2} \cos(t_1+t_2 + 2U))
&= \int \limits_0^{2\pi} \frac{1}{4 \pi} \cdot \cos(t_1+t_2 + 2u) \ du \\[... | Show that $\{X(t), = \cos(t+U)\}$, $U \sim \mathrm{Unif} (0, 2\pi)$ is a wide-sense stationary proce
It is a typo, but you should get zero for the integral anyway. You should have:
$$\begin{aligned}
\mathbb{E}(\tfrac{1}{2} \cos(t_1+t_2 + 2U))
&= \int \limits_0^{2\pi} \frac{1}{4 \pi} \cdot \cos(t_1+t |
50,300 | Small dataset and optimal parameters for XGboost | It is difficult to give a smooth answer without having the data and all required subject knowledge at hand. Still I can throw in some comments.
Your validation strategy is fine as long as you decide everything by cross validation and not by the test data (but see 5.).
If your best solution picks value at the border, t... | Small dataset and optimal parameters for XGboost | It is difficult to give a smooth answer without having the data and all required subject knowledge at hand. Still I can throw in some comments.
Your validation strategy is fine as long as you decide | Small dataset and optimal parameters for XGboost
It is difficult to give a smooth answer without having the data and all required subject knowledge at hand. Still I can throw in some comments.
Your validation strategy is fine as long as you decide everything by cross validation and not by the test data (but see 5.).
I... | Small dataset and optimal parameters for XGboost
It is difficult to give a smooth answer without having the data and all required subject knowledge at hand. Still I can throw in some comments.
Your validation strategy is fine as long as you decide |
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