Why I get two different covariance matrix?
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Hi all,
Prof. Andrew Ng in his ML class says that we can calculate covariance matrix as: 1/m*X'*X .
Where;
examples are in rows of X,
X' is transpose of X,
and, m is number of examples.
For example:
X=randi(12,[6,2]);
cov1=1/size(X,1)*X'*X
And, covariance with cov function is:
cov2=cov(X)
As you can see, cov1 is different from cov2 !!!
What is the reasan for that? Do you have any idea?
Thanks
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Paul
2022-7-14
Hi Ali,
Perhaps Prof. Ng has some additional assumptions about the data that aren't included in your question. To compute the covariance we have to subtract off the mean. As for whether or not the outer product should be scaled by 1/m or 1/(m-1) depends on assumptions about the underlying data. IIRC, if we know the data is drawn from a Normal distribution then divide by m (perhaps also for other distributions as well?), but typically we don't assume that and so divide by m-1 for unbiased estimation. As can be seen below, cov subtracts the mean and divides by m-1.
rng(100);
X=randi(12,[6,2])
cov1=1/(size(X,1)-1)*(X-mean(X))'*(X - mean(X))
cov(X)
As for the second question
sigma=[6 2;2 3]; % cov matrix
[a1,v] = eig(sigma)
[a2,s,~] = svd(sigma)
we see that eig and svd just have a different order for the results.
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Paul
2022-7-14
Which one of eig() or svd() to use? it would never occur to me to use svd() to get the eigenvectors of a symmetric matrix. Don't know if eig() or svd() is better for that special case. Asking that as a new question is more likely to get the attention of knowledgeable people that can answer.
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ali yaman
2022-7-14
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John D'Errico
2022-7-14
编辑:John D'Errico
2022-7-14
Since this is probably of some general interest, I'll actually post a question of my own, then answer it myself, discussing the relative issues between eig and svd.
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