Hi.I tried to run a principal components analysis in SPSS and I received the message 'correlation matrix is not positive definite'. However, the output generates a rotated component matrix in spite of the error message. How serious is this problem (i.e., should I ignore the output until I correct the problem)? Does anybody have any suggestions on how to address this issue?Thank you.

Hi msr,If correlation matirx is not SPD, you will likely have negative eigenvalues. So. your PCA analisys is not valid.You could try to transform (proxy) your correlation matrix in a SPD one.

we should have a forum solely dedicated to non-positve definite correlation matrices...

QuoteOriginally posted by: mjwe should have a forum solely dedicated to non-positve definite correlation matrices...yes indeed, I agree.....

Well, for full generality, a forum on using 'search' in the Wilmott website.

Thank you for the feedback. It appears from the other responses that this topic has been an issue for others as well. Is there any reading you could suggest that explains how to transform the matrix (I'm guessing it is too involved for you to explain, although by all means, anything you can add to point me in the right direction is much appreciated!). Thank youMSR

http://www.wilmott.com/messageview.cfm? ... eadid=9910

I recommend the paper by A.Kreinin and Alex Levin"Robust Monte Carlo Simulation for Approximate Covariance Matrices and VaR Analyses", in “Probablistic Constrained Optimization: Theory and Applications”, 2000, Kluwer Academic Publ., p. 59 - 71,

Actually the idea is very simple: you change negative eigenvalues to max( abs(negative)). This works nicely in the situation you have smallnegative after rounding.

I recommend the paper by F. Rapisarda, D. Brigo and F. Mercurio (2002):"Parameterizing correlations: a geometric interpretation". Thanks

In my line of work, most people run pca on a sample correlation matrix. Usually they have more stocks than time periods. In that case the sample correlation matrix of returns is not positive definite. But there's nothing wrong with that (as long as you don't use the eigenvectors that have negative eigenvalues). Let say you keep the 5 eignvectors with the 5 largest eignevalues (i.e., the eigenvectors that explain the largest fraction of variance). The correlation matrix constructed with 5 eigenvectors will be positive definite.Of course, keep in mind, that in my line of work I often misunderstand the question.