what does eigenvalues expres in the covariance matrix?

Question

Mohamed Moawed 2013-4-23

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/73298-what-does-eigenvalues-expres-in-the-covariance-matrix

评论： TUSHAR MURATKAR 2020-2-18

is there a relationship between a covariance matrix and eigenvalues? like an example

Let us consider a 321 × 261 image dimention 321 × 261 = 83781. We have only 32 observations and 83781 unknowns then we have a matrix of (32 row X 83781 column)

then we will calculate the covariance matrix (32 X 32) so we get 32 eigenvalues the question is: does these eigenvalues express the 32 images? or there is no any relationship between eigenvalues and images

thanks for you,

8 个评论
显示 6更早的评论隐藏 6更早的评论

Mohamed Moawed 2013-4-23

what about image point of view??? when i want to select some eigenvectors.how can i say this eigenvector for this observation or for this image

Vincent Spruyt 2015-3-10

The eigenvalues in this case represent the magnitude of the spread in the direction of the principal components. If you data has a diagonal covariance matrix (covariances are zero), then the eigenvalues are equal to the variances:

If the covariance matrix is not diagonal, then the eigenvalues still define the variance of the data along the the principal components, whereas the covariance matrix operates along the axes:

Here is an article (and the source of the above images) that discusses this in more detail: http://www.visiondummy.com/2014/04/geometric-interpretation-covariance-matrix/

请先登录，再进行评论。

请先登录，再回答此问题。

Answer 1

Kye Taylor 2013-4-23

3
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/73298-what-does-eigenvalues-expres-in-the-covariance-matrix#answer_83253

编辑：Kye Taylor 2013-4-23

在 MATLAB Online 中打开

Long story short: The eigenvalues of the covariance matrix encode the variability of the data in an orthogonal basis that captures as much of the data's variability as possible in the first few basis functions (aka the principle component basis).

For example, this code creates an ellipse, whos major axis is the x-axis, and whos minor axis is the y-axis.

t = linspace(0,2*pi,256);
data = [cos(t);0.2*sin(t)];
plot(data(1,:),data(2,:),'.')
axis([-1,1,-1,1])

Now, compute the variance of the data's coordinates

% tranpose to get variance down each column
% computes variance of each coordinate of the data
v = var(data')

Finally, observe the eigenvalues of the covariance matrix are equal to the variance of the data's coordinates

% need to transpose since input to cov must have
% rows = observations
% columns = variables
l = eig(cov(data'))

In other words, v and l contain the same values. The order may be different because the eig function returns eigenvalues in no particular order.

Note that if the data is rotated so that the major and minor axes are no longer the x and y axes, then var(data') no longer computes the variance about the coordinates from their principle axes, but eig(cov(data')) does automatically since the eigenvectors are principle components.

For example

% rotate everything
r = [cos(pi/4),-sin(pi/4);sin(pi/4),cos(pi/4)]; % rotation matrix
rData = r*data;
hold on
plot(rData(1,:),rData(2,:),'r.') % plot rotated data
vr = var(rData') % different
lr = eig(cov(rData')) % should be same as l

4 个评论
显示 2更早的评论隐藏 2更早的评论

Kye Taylor 2013-4-24

编辑：Kye Taylor 2013-4-24

在 MATLAB Online 中打开

It sounds like you're asking a different question that what the main thread is, and what I answered above but here's a nice little example getting at what you might be interested in .. If you can, give me more insight into what you're trying to do?

load mri.mat % included w MATLAB
montage(D,map) % 27 images in mri.mat
% organize the data into a nImages-by-nTotalPixelsPerImage array
D = permute(D,[4,1,2,3]);
data = reshape(D,[size(D,1),size(D,2)*size(D,3)]);
% do something crazy, and compute the first 12 
% eigenvectors of this 2^14-dimensional matrix
[v,d] = eigs( cov(double(data)), 12,'LA');
d = diag(d);  % only the eigenvalues
maxD = max(d);

This figure contains the principle components of your data, scaled according to the variance along that component. I know, it's cool.

figure
for j = 1:12
    subplot(4,3,j)
    imagesc(reshape(v(:,j)*d(j)/maxD,siz(1),siz(2)))
    colorbar
    colormap gray
end

If you look at a plot of the eigenvalues, notice that more than half of the variance in the data is captured in the first three principle components

figure,plot( d/maxD, '.-')

Mohamed Moawed 2013-4-24

wow, that is nice, thanks for your kindly reply. i need more thing, in my work i have several fMRI brain images for Alzheimer Disease. and i get these images and compare between each part in these images to get the parts that are different between positive and negative images. how can i get these parts? i want to use PCA to get eigenvalues that they are the principle component of these images but how can i say this eigenvalue for this part in this image?

Kye Taylor 2013-4-24

Open up a new thread... and kindly accept my answer.

请先登录，再进行评论。

Answer 2

Shashank Prasanna 2013-4-23

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/73298-what-does-eigenvalues-expres-in-the-covariance-matrix#answer_83244

编辑：Shashank Prasanna 2013-4-23

Essentially what you are describing are the principal components of your data.

https://en.wikipedia.org/wiki/Principal_component_analysis#Computing_PCA_using_the_covariance_method

Its a popularly used dimensionality reduction technique, for example to make your image smaller such that it still retains most of its variance.

The PCA command in MATLAB does all this for you directly.

http://www.mathworks.com/help/stats/pca.html

2 个评论
显示无隐藏无

Mohamed Moawed 2013-4-23

yes i know i want to use PCA . but i dont understand how to use it with more than one image to get the principle component of these images

TUSHAR MURATKAR 2020-2-18

In my reference paper related to wireless communication the covariance matrix is made from vector comprising of channel coefficients. And non zero eigen values of the covariance matrix are calculated. What does this signify with reference to wireless communication.

请先登录，再进行评论。

what does eigenvalues expres in the covariance matrix?

8 个评论
显示 6更早的评论隐藏 6更早的评论

采纳的回答

4 个评论
显示 2更早的评论隐藏 2更早的评论

更多回答（1 个）

2 个评论
显示无隐藏无

另请参阅

类别

标签

Community Treasure Hunt

what does eigenvalues expres in the covariance matrix?

8 个评论 显示 6更早的评论隐藏 6更早的评论

采纳的回答

4 个评论 显示 2更早的评论隐藏 2更早的评论

更多回答（1 个）

2 个评论 显示 无隐藏 无

另请参阅

类别

标签

Community Treasure Hunt

8 个评论
显示 6更早的评论隐藏 6更早的评论

4 个评论
显示 2更早的评论隐藏 2更早的评论

2 个评论
显示无隐藏无