I am now studying PCA using an Eigen matrix. For doing PCA, obtaining covariance matrix is essential, but every time I obtain a covariance matrix by my hand and compare with my Matlab result, they are completely different.
Below is simple code that obtains a covariance matrix.
x=[-4 9 5;3 3 5;1 3 -1;8 1 7];
c=cov(x);
M=[2 4 4];
beforecov=x-repmat(M,4,1);
summat=zeros(3,3,4);
for i=1:4
summat(:,:,i)= beforecov(i,:)'*beforecov(i,:);
end
cov_onmyown=(summat(:,:,1)+summat(:,:,2)+summat(:,:,3)+summat(:,:,4))/4;
x is the matrix that has 4 samples with 3 features. The result is
c=[24.667 -16 6;
-16 12 0;
6 0 12]
Now I obtain the covariance matrix manually. What I tried is by using the definition of the covariance matrix which is written below:
COV[X]=E[(X-u)(X-u)']
The mean matrix is [2 4 4] so I did X-u for each sample (beforecov in code). Then I made a 3x3 matrix for each 4 samples and divided by 4 (the sample number).
But the results c and cov_onmyown in the code are totally different.
cov_onmyown=[18.5, -12, 4.5;
-12, 9, 0;
4.5, 0, 9]
Could anyone tell me what is wrong with my idea?
From the help text:
cov(X) or cov(X,Y) normalizes by (N-1) if N>1
So to get the same result as cov use
cov_onmyown=(summat(:,:,1)+summat(:,:,2)+summat(:,:,3)+summat(:,:,4))/(4 - 1);