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
cov1 = 2×2
92.5000 42.0000 42.0000 21.8333
And, covariance with cov function is:
cov2=cov(X)
cov2 = 2×2
2.7000 -0.9000 -0.9000 1.9000
As you can see, cov1 is different from cov2 !!!
What is the reasan for that? Do you have any idea?
Thanks

Accepted Answer

Paul
Paul on 14 Jul 2022
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])
X = 6×2
7 9 4 10 6 2 11 7 1 11 2 3
cov1=1/(size(X,1)-1)*(X-mean(X))'*(X - mean(X))
cov1 = 2×2
13.3667 -1.6000 -1.6000 14.0000
cov(X)
ans = 2×2
13.3667 -1.6000 -1.6000 14.0000
As for the second question
sigma=[6 2;2 3]; % cov matrix
[a1,v] = eig(sigma)
a1 = 2×2
0.4472 -0.8944 -0.8944 -0.4472
v = 2×2
2.0000 0 0 7.0000
[a2,s,~] = svd(sigma)
a2 = 2×2
-0.8944 -0.4472 -0.4472 0.8944
s = 2×2
7 0 0 2
we see that eig and svd just have a different order for the results.
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More Answers (1)

ali yaman
ali yaman on 14 Jul 2022
What about the second question!!
Just like covarince matrixe we can get two different eigenvectors.
For example;
sigma=[6 2;2 3]; % cov matrix
[a,~]=eig(sigma)
a = 2×2
0.4472 -0.8944 -0.8944 -0.4472
And, lets calculate eigenvectors with svd, singular value decomposition function:
[a,~,~]=svd(sigma)
a = 2×2
-0.8944 -0.4472 -0.4472 0.8944
As you can see we are getting two different eigenvectors value.
What is the reason?
Thanks, in advance.
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