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Re: Multivariate pdf of a normal distribution
From: |
Mike Miller |
Subject: |
Re: Multivariate pdf of a normal distribution |
Date: |
Sat, 5 Nov 2005 17:38:37 -0600 (CST) |
On Sat, 5 Nov 2005, Paul Kienzle wrote:
Of course, the density goes to infinity when Sigma is singular. Is your
use of chol() just meant to check that the matrix is PD?
The wikipedia entry on cholesky decomposition
(http://en.wikipedia.org/wiki/Choleskey_decomposition) suggests it is faster
and more stable than the lu decomposition which would be used to compute the
inverse. The speed doesn't matter in this case, but accuracy is always a
concern. The side effect of checking positive definiteness of sigma is a
bonus.
Some numerical tests with e.g.,
n=11; x = prolate(n); cn=cond(x), d=norm(x*inv(x)-eye(n)), r=chol(x);
c=norm(x*inv(r)*inv(r)'-eye(n)),
shows that it chol() is indeed a little better than inv() for ill-conditioned
positive definite matrices. The function prolate() is from higham's test
matrix toolbox.
Similarly for hilb(), though norm(inv(x) - invhilb(n)) and
norm(inv(r)*inv(r)' - invhilb(n)) are both pretty bad.
Interesting. Thanks for sharing this information.
Mike
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- Multivariate pdf of a normal distribution, Gorazd Brumen, 2005/11/05
- Re: Multivariate pdf of a normal distribution, Paul Kienzle, 2005/11/05
- Re: Multivariate pdf of a normal distribution, Mike Miller, 2005/11/05
- Re: Multivariate pdf of a normal distribution, Paul Kienzle, 2005/11/05
- Re: Multivariate pdf of a normal distribution,
Mike Miller <=
- Re: Multivariate pdf of a normal distribution, Prasenjit Kapat, 2005/11/05
- Re: Multivariate pdf of a normal distribution, Mike Miller, 2005/11/05
- Re: Multivariate pdf of a normal distribution, Prasenjit Kapat, 2005/11/05
- Re: Multivariate pdf of a normal distribution, Paul Kienzle, 2005/11/05
- Re: Multivariate pdf of a normal distribution, Mike Miller, 2005/11/06
Re: Multivariate pdf of a normal distribution, Michael Creel, 2005/11/07