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st: RE: Finding the largest positive definite submatrix


From   "Feiveson, Alan H. (JSC-SK311)" <alan.h.feiveson@nasa.gov>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: Finding the largest positive definite submatrix
Date   Wed, 7 Jun 2006 13:15:25 -0500

Is the matrix not positive definite because it has less than full rank?
If so, you could add a small increment say eps to the diagonal and use
-corr again to get a new (positive definite) correlation matrix. If eps
is small, the results of the factor analysis should give you the about
the same information you would get as if the matrix were PD to begin
with.

On the other hand if the matrix is not PD because it was estimated
pairwise, or by some other non-optimal method,this method would not
work. In this case, I would try re-estimating the matrix using imputed
values or a similar method. 

I did some work along these lines some years ago, addressing the problem
of finding the best regression subset of a given size when the original
matrix was not of full rank. The algorithm depended on finding principal
sub-matrices of a given size that were (or were not) positive definite.
The "were or were not" depends on whether an elimination or direct
approach was used. See the paper below:

Feiveson, A. H. (1994). "Finding the Best Regression Subset by Reduction
in Non-Full-Rank Cases", SIAM (Society for Industrial and Applied
Mathematics) Journal of Matrix Analysis and Applications, Vol. 15, No.
1, 194-204.


Al F

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Joseph
Coveney
Sent: Wednesday, June 07, 2006 10:55 AM
To: Statalist
Subject: st: Finding the largest positive definite submatrix

I'm using Stas Kolenikov's -polychoric- on data from a 70-item
questionnaire
(102 respondents with complete responses) to obtain the matrix of
polychoric correlation coefficients to feed to Stata's -factormat-.

Alas, -factormat- complains that the correlation matrix is not positive
definite.  I would like to retain a subset of as many of the
questionnaire items as possible in the factor analysis.  What is the
best way to return the largest positive-definite submatrix from the
correlation matrix?

Joseph Coveney

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