Completing MDS manually in R
$begingroup$
Given a matrix A
, I want to complete Multidimensional Scaling by hand, instead of using any given R functions.
As such, I have calculated the centered matrix B
with the following code:
n<-nrow(A)
id<-diag(n)
e<-diag(id)
H <- id - (1/n)*e %*% etranspose
B <- (-1/2)* H %*% A %*% H
My question is: how can I use my B matrix to complete multidimensional scaling on my A matrix, without the cmdscale
function or anything along those lines?
machine-learning statistics dimensionality-reduction
$endgroup$
bumped to the homepage by Community♦ 4 mins ago
This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
add a comment |
$begingroup$
Given a matrix A
, I want to complete Multidimensional Scaling by hand, instead of using any given R functions.
As such, I have calculated the centered matrix B
with the following code:
n<-nrow(A)
id<-diag(n)
e<-diag(id)
H <- id - (1/n)*e %*% etranspose
B <- (-1/2)* H %*% A %*% H
My question is: how can I use my B matrix to complete multidimensional scaling on my A matrix, without the cmdscale
function or anything along those lines?
machine-learning statistics dimensionality-reduction
$endgroup$
bumped to the homepage by Community♦ 4 mins ago
This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
add a comment |
$begingroup$
Given a matrix A
, I want to complete Multidimensional Scaling by hand, instead of using any given R functions.
As such, I have calculated the centered matrix B
with the following code:
n<-nrow(A)
id<-diag(n)
e<-diag(id)
H <- id - (1/n)*e %*% etranspose
B <- (-1/2)* H %*% A %*% H
My question is: how can I use my B matrix to complete multidimensional scaling on my A matrix, without the cmdscale
function or anything along those lines?
machine-learning statistics dimensionality-reduction
$endgroup$
Given a matrix A
, I want to complete Multidimensional Scaling by hand, instead of using any given R functions.
As such, I have calculated the centered matrix B
with the following code:
n<-nrow(A)
id<-diag(n)
e<-diag(id)
H <- id - (1/n)*e %*% etranspose
B <- (-1/2)* H %*% A %*% H
My question is: how can I use my B matrix to complete multidimensional scaling on my A matrix, without the cmdscale
function or anything along those lines?
machine-learning statistics dimensionality-reduction
machine-learning statistics dimensionality-reduction
edited Dec 28 '18 at 0:50
Alex L
1478
1478
asked Nov 16 '15 at 1:53
potatosouppotatosoup
262
262
bumped to the homepage by Community♦ 4 mins ago
This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
bumped to the homepage by Community♦ 4 mins ago
This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If you're interested in the classic MDS algorithm, it is spelled out very nicely on the Wikipedia page:
Classical MDS uses the fact that the coordinate matrix can be derived
by eigenvalue decomposition from $B=XX'$. And the matrix $B$ can be
computed from proximity matrix $D$ by using double centering.
- Set up the squared proximity matrix $D^{(2)}=[d_{ij}^{2}]$
Apply double centering: $B=-{frac {1}{2}}JD^{(2)}J$ using the centering matrix $J=I-{frac {1}{n}}11'$, where $n$ is the number of
objects.
Determine the $m$ largest eigenvalues $lambda _{1},lambda _{2},...,lambda _{m}$ and corresponding eigenvectors $e_{1},e_{2},...,e_{m}$ of $B$ (where $m$ is the number of dimensions
desired for the output).
Now, $X=E_{m}Lambda _{m}^{1/2}$, where $E_{m}$ is the matrix of $m$ eigenvectors and $Lambda _{m}$ is the diagonal matrix of $m$
eigenvalues of $B$.
I can't really tell what your $A$ matrix is, but if each entry is a measure of the distance from the $i^{th}$ to $j^{th}$ entry, then that would be your proximity matrix.
Also, $11'$ is an n-by-n matrix of all 1's. This should be fairly self-explanatory, however most of these terms can be Googled if you're unsure. It really shouldn't be any more than about 10 lines of code.
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "557"
};
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: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
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
},
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%2fdatascience.stackexchange.com%2fquestions%2f8853%2fcompleting-mds-manually-in-r%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
$begingroup$
If you're interested in the classic MDS algorithm, it is spelled out very nicely on the Wikipedia page:
Classical MDS uses the fact that the coordinate matrix can be derived
by eigenvalue decomposition from $B=XX'$. And the matrix $B$ can be
computed from proximity matrix $D$ by using double centering.
- Set up the squared proximity matrix $D^{(2)}=[d_{ij}^{2}]$
Apply double centering: $B=-{frac {1}{2}}JD^{(2)}J$ using the centering matrix $J=I-{frac {1}{n}}11'$, where $n$ is the number of
objects.
Determine the $m$ largest eigenvalues $lambda _{1},lambda _{2},...,lambda _{m}$ and corresponding eigenvectors $e_{1},e_{2},...,e_{m}$ of $B$ (where $m$ is the number of dimensions
desired for the output).
Now, $X=E_{m}Lambda _{m}^{1/2}$, where $E_{m}$ is the matrix of $m$ eigenvectors and $Lambda _{m}$ is the diagonal matrix of $m$
eigenvalues of $B$.
I can't really tell what your $A$ matrix is, but if each entry is a measure of the distance from the $i^{th}$ to $j^{th}$ entry, then that would be your proximity matrix.
Also, $11'$ is an n-by-n matrix of all 1's. This should be fairly self-explanatory, however most of these terms can be Googled if you're unsure. It really shouldn't be any more than about 10 lines of code.
$endgroup$
add a comment |
$begingroup$
If you're interested in the classic MDS algorithm, it is spelled out very nicely on the Wikipedia page:
Classical MDS uses the fact that the coordinate matrix can be derived
by eigenvalue decomposition from $B=XX'$. And the matrix $B$ can be
computed from proximity matrix $D$ by using double centering.
- Set up the squared proximity matrix $D^{(2)}=[d_{ij}^{2}]$
Apply double centering: $B=-{frac {1}{2}}JD^{(2)}J$ using the centering matrix $J=I-{frac {1}{n}}11'$, where $n$ is the number of
objects.
Determine the $m$ largest eigenvalues $lambda _{1},lambda _{2},...,lambda _{m}$ and corresponding eigenvectors $e_{1},e_{2},...,e_{m}$ of $B$ (where $m$ is the number of dimensions
desired for the output).
Now, $X=E_{m}Lambda _{m}^{1/2}$, where $E_{m}$ is the matrix of $m$ eigenvectors and $Lambda _{m}$ is the diagonal matrix of $m$
eigenvalues of $B$.
I can't really tell what your $A$ matrix is, but if each entry is a measure of the distance from the $i^{th}$ to $j^{th}$ entry, then that would be your proximity matrix.
Also, $11'$ is an n-by-n matrix of all 1's. This should be fairly self-explanatory, however most of these terms can be Googled if you're unsure. It really shouldn't be any more than about 10 lines of code.
$endgroup$
add a comment |
$begingroup$
If you're interested in the classic MDS algorithm, it is spelled out very nicely on the Wikipedia page:
Classical MDS uses the fact that the coordinate matrix can be derived
by eigenvalue decomposition from $B=XX'$. And the matrix $B$ can be
computed from proximity matrix $D$ by using double centering.
- Set up the squared proximity matrix $D^{(2)}=[d_{ij}^{2}]$
Apply double centering: $B=-{frac {1}{2}}JD^{(2)}J$ using the centering matrix $J=I-{frac {1}{n}}11'$, where $n$ is the number of
objects.
Determine the $m$ largest eigenvalues $lambda _{1},lambda _{2},...,lambda _{m}$ and corresponding eigenvectors $e_{1},e_{2},...,e_{m}$ of $B$ (where $m$ is the number of dimensions
desired for the output).
Now, $X=E_{m}Lambda _{m}^{1/2}$, where $E_{m}$ is the matrix of $m$ eigenvectors and $Lambda _{m}$ is the diagonal matrix of $m$
eigenvalues of $B$.
I can't really tell what your $A$ matrix is, but if each entry is a measure of the distance from the $i^{th}$ to $j^{th}$ entry, then that would be your proximity matrix.
Also, $11'$ is an n-by-n matrix of all 1's. This should be fairly self-explanatory, however most of these terms can be Googled if you're unsure. It really shouldn't be any more than about 10 lines of code.
$endgroup$
If you're interested in the classic MDS algorithm, it is spelled out very nicely on the Wikipedia page:
Classical MDS uses the fact that the coordinate matrix can be derived
by eigenvalue decomposition from $B=XX'$. And the matrix $B$ can be
computed from proximity matrix $D$ by using double centering.
- Set up the squared proximity matrix $D^{(2)}=[d_{ij}^{2}]$
Apply double centering: $B=-{frac {1}{2}}JD^{(2)}J$ using the centering matrix $J=I-{frac {1}{n}}11'$, where $n$ is the number of
objects.
Determine the $m$ largest eigenvalues $lambda _{1},lambda _{2},...,lambda _{m}$ and corresponding eigenvectors $e_{1},e_{2},...,e_{m}$ of $B$ (where $m$ is the number of dimensions
desired for the output).
Now, $X=E_{m}Lambda _{m}^{1/2}$, where $E_{m}$ is the matrix of $m$ eigenvectors and $Lambda _{m}$ is the diagonal matrix of $m$
eigenvalues of $B$.
I can't really tell what your $A$ matrix is, but if each entry is a measure of the distance from the $i^{th}$ to $j^{th}$ entry, then that would be your proximity matrix.
Also, $11'$ is an n-by-n matrix of all 1's. This should be fairly self-explanatory, however most of these terms can be Googled if you're unsure. It really shouldn't be any more than about 10 lines of code.
answered Dec 27 '18 at 22:48
Alex LAlex L
1478
1478
add a comment |
add a comment |
Thanks for contributing an answer to Data Science 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%2fdatascience.stackexchange.com%2fquestions%2f8853%2fcompleting-mds-manually-in-r%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