Understanding Variance-Covariance Matrix
$begingroup$
Suppose data set is expressed by the matrix $X inmathbb R^{n times d}$
where $n =$ Number of samples and $d =$ dimension/features of each sample
Then what does $operatorname{Cov}(X) inmathbb R^{d times d}$ (Variance-Covariance matrix of $X$) represent. Does below interpretation would be right
Variance-Covariance matrix of $X$ represents covariance between every pair of dimension/feature for all samples.
linear-algebra matrices covariance
edited Dec 31 '18 at 21:10
Davide Giraudo
128k17156268
asked Dec 31 '18 at 9:44
AtineshAtinesh
4694822
$endgroup$
add a comment |
$begingroup$
Suppose data set is expressed by the matrix $X inmathbb R^{n times d}$
where $n =$ Number of samples and $d =$ dimension/features of each sample
Then what does $operatorname{Cov}(X) inmathbb R^{d times d}$ (Variance-Covariance matrix of $X$) represent. Does below interpretation would be right
Variance-Covariance matrix of $X$ represents covariance between every pair of dimension/feature for all samples.
linear-algebra matrices covariance
edited Dec 31 '18 at 21:10
Davide Giraudo
128k17156268
asked Dec 31 '18 at 9:44
AtineshAtinesh
4694822
$endgroup$
add a comment |
$begingroup$
Suppose data set is expressed by the matrix $X inmathbb R^{n times d}$
where $n =$ Number of samples and $d =$ dimension/features of each sample
Then what does $operatorname{Cov}(X) inmathbb R^{d times d}$ (Variance-Covariance matrix of $X$) represent. Does below interpretation would be right
Variance-Covariance matrix of $X$ represents covariance between every pair of dimension/feature for all samples.
linear-algebra matrices covariance
edited Dec 31 '18 at 21:10
Davide Giraudo
128k17156268
asked Dec 31 '18 at 9:44
AtineshAtinesh
4694822
$endgroup$
Suppose data set is expressed by the matrix $X inmathbb R^{n times d}$
where $n =$ Number of samples and $d =$ dimension/features of each sample
Then what does $operatorname{Cov}(X) inmathbb R^{d times d}$ (Variance-Covariance matrix of $X$) represent. Does below interpretation would be right
Variance-Covariance matrix of $X$ represents covariance between every pair of dimension/feature for all samples.
linear-algebra matrices covariance
linear-algebra matrices covariance
edited Dec 31 '18 at 21:10
Davide Giraudo
128k17156268
asked Dec 31 '18 at 9:44
AtineshAtinesh
4694822
edited Dec 31 '18 at 21:10
Davide Giraudo
128k17156268
asked Dec 31 '18 at 9:44
AtineshAtinesh
4694822
edited Dec 31 '18 at 21:10
Davide Giraudo
128k17156268
edited Dec 31 '18 at 21:10
Davide Giraudo
128k17156268
edited Dec 31 '18 at 21:10
Davide Giraudo
128k17156268
128k17156268
asked Dec 31 '18 at 9:44
AtineshAtinesh
4694822
asked Dec 31 '18 at 9:44
AtineshAtinesh
4694822
asked Dec 31 '18 at 9:44
AtineshAtinesh
4694822
4694822
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let's review how the covariance matrix is computed in this context. Let $mu_j$ denote the mean of the $j$th column, and let $mu$ denote the row-vector $mu = (mu_1,mu_2,dots,mu_d)$. Then
$$
operatorname{cov}(X) = (X - mu 1_n)(X - mu 1_n)^T
$$
where $1_n$ denotes the column vector $(1,dots,1)^T$ of length $n$.
With this in mind, the $i,j$ entry of the covariance matrix is given by
$$
operatorname{cov}(X)[i,j] = sum_{k=1}^n (x_{ik} - mu_k)(x_{jk} - mu_k)
$$
So, $frac 1n operatorname{cov}(X)[i,j]$ is the covariance between the $i$th feature and $j$th feature, and $frac 1n operatorname{cov}(X)[i,i]$ is the variance of the $i$th feature.
answered Dec 31 '18 at 19:50
OmnomnomnomOmnomnomnom
129k794188
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3057551%2funderstanding-variance-covariance-matrix%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Let's review how the covariance matrix is computed in this context. Let $mu_j$ denote the mean of the $j$th column, and let $mu$ denote the row-vector $mu = (mu_1,mu_2,dots,mu_d)$. Then
$$
operatorname{cov}(X) = (X - mu 1_n)(X - mu 1_n)^T
$$
where $1_n$ denotes the column vector $(1,dots,1)^T$ of length $n$.
With this in mind, the $i,j$ entry of the covariance matrix is given by
$$
operatorname{cov}(X)[i,j] = sum_{k=1}^n (x_{ik} - mu_k)(x_{jk} - mu_k)
$$
So, $frac 1n operatorname{cov}(X)[i,j]$ is the covariance between the $i$th feature and $j$th feature, and $frac 1n operatorname{cov}(X)[i,i]$ is the variance of the $i$th feature.
answered Dec 31 '18 at 19:50
OmnomnomnomOmnomnomnom
129k794188
$endgroup$
add a comment |
$begingroup$
Let's review how the covariance matrix is computed in this context. Let $mu_j$ denote the mean of the $j$th column, and let $mu$ denote the row-vector $mu = (mu_1,mu_2,dots,mu_d)$. Then
$$
operatorname{cov}(X) = (X - mu 1_n)(X - mu 1_n)^T
$$
where $1_n$ denotes the column vector $(1,dots,1)^T$ of length $n$.
With this in mind, the $i,j$ entry of the covariance matrix is given by
$$
operatorname{cov}(X)[i,j] = sum_{k=1}^n (x_{ik} - mu_k)(x_{jk} - mu_k)
$$
So, $frac 1n operatorname{cov}(X)[i,j]$ is the covariance between the $i$th feature and $j$th feature, and $frac 1n operatorname{cov}(X)[i,i]$ is the variance of the $i$th feature.
answered Dec 31 '18 at 19:50
OmnomnomnomOmnomnomnom
129k794188
$endgroup$
add a comment |
$begingroup$
Let's review how the covariance matrix is computed in this context. Let $mu_j$ denote the mean of the $j$th column, and let $mu$ denote the row-vector $mu = (mu_1,mu_2,dots,mu_d)$. Then
$$
operatorname{cov}(X) = (X - mu 1_n)(X - mu 1_n)^T
$$
where $1_n$ denotes the column vector $(1,dots,1)^T$ of length $n$.
With this in mind, the $i,j$ entry of the covariance matrix is given by
$$
operatorname{cov}(X)[i,j] = sum_{k=1}^n (x_{ik} - mu_k)(x_{jk} - mu_k)
$$
So, $frac 1n operatorname{cov}(X)[i,j]$ is the covariance between the $i$th feature and $j$th feature, and $frac 1n operatorname{cov}(X)[i,i]$ is the variance of the $i$th feature.
answered Dec 31 '18 at 19:50
OmnomnomnomOmnomnomnom
129k794188
$endgroup$
Let's review how the covariance matrix is computed in this context. Let $mu_j$ denote the mean of the $j$th column, and let $mu$ denote the row-vector $mu = (mu_1,mu_2,dots,mu_d)$. Then
$$
operatorname{cov}(X) = (X - mu 1_n)(X - mu 1_n)^T
$$
where $1_n$ denotes the column vector $(1,dots,1)^T$ of length $n$.
With this in mind, the $i,j$ entry of the covariance matrix is given by
$$
operatorname{cov}(X)[i,j] = sum_{k=1}^n (x_{ik} - mu_k)(x_{jk} - mu_k)
$$
So, $frac 1n operatorname{cov}(X)[i,j]$ is the covariance between the $i$th feature and $j$th feature, and $frac 1n operatorname{cov}(X)[i,i]$ is the variance of the $i$th feature.
answered Dec 31 '18 at 19:50
OmnomnomnomOmnomnomnom
129k794188
answered Dec 31 '18 at 19:50
OmnomnomnomOmnomnomnom
129k794188
answered Dec 31 '18 at 19:50
OmnomnomnomOmnomnomnom
129k794188
answered Dec 31 '18 at 19:50
OmnomnomnomOmnomnomnom
129k794188
129k794188
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3057551%2funderstanding-variance-covariance-matrix%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
