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Jan 30, 2007, 1:17:37 PM1/30/07

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hi, all

I searched the group and found some message dealing with the singular

covariance matrix problem. One of which is using pseudo inverse based

on svd to generate pseudo inversed covariance matrix for the singular

matrix. My question, in order to calculate the Gaussian pdf, how

should I deal with the determinant of the singular matrix which is 0

and will give inf in gaussian pdf? Thanks for comments.

zl2k

I searched the group and found some message dealing with the singular

covariance matrix problem. One of which is using pseudo inverse based

on svd to generate pseudo inversed covariance matrix for the singular

matrix. My question, in order to calculate the Gaussian pdf, how

should I deal with the determinant of the singular matrix which is 0

and will give inf in gaussian pdf? Thanks for comments.

zl2k

Jan 30, 2007, 3:12:54 PM1/30/07

to

Consider first the bivariate normal case in which the variables are

perfectly correlated. Then the density is zero everywhere except on

some line, where it is univariate normal.

The n-variate case is analogous. If the covariance matrix has rank k

then the density is zero everywhere except in some k-dimensional

hyperplane, where it is k-variate normal.

Let x be such an n-vector, with mean vector m and covariance matrix S.

Let P be the n by k matrix of unit-length eigenvectors corresponding

to the nonzero eigenvalues of S, and let Q be any n by (n-k) matrix

whose columns span the null space of S. Let y = P'(x-m), and let z =

Q'(x-m). If z'z > 0 then x is not in the hyperplane, and the density

is zero. If z = 0 then x is in the hyperplane, where its conditional

density is that of y, which is k-variate normal with mean vector 0

and covariance matrix P'SP (which is diagonal; the variances are the

nonzero eigenvalues of S).

Feb 2, 2007, 8:53:00 AM2/2/07

to

It's the product of the non-zero singular values. Ray gave a good

geometric explanation of why this works. It's the determinant of the

full support basis vectors of the covariance matrix.

Feb 3, 2007, 6:05:14 AM2/3/07

to

Go back to that thread and find the sentence

Now replace inv(C) with C# and det(C) with the product s1*s2*...sr.

Hope this helps.

Greg

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