Re: inversion of covariance sub-matrices

john.tramuta@xxxxxxxxxxxxxx wrote:
Hi,

I am having the following problem and was wondering whether anybody
could help.

I have an n x p data matrix X containing n observations on p
variables, which I use to compute the p x p covariance matrix X ' X
and its inverse.

Now, call D ' D the q x q sub-matrix of X ' X obtained by taking q
columns and q rows of X ' X.

I need to efficiently compute the inverse of D ' D for all q=1,
2, ..., p-1, sequentially (i.e. starting with only one variable, and
adding one variable at a time).

Can this be done recursively, or efficiently using the inverse of X '
X?


Are you saying that you are only interested in the combinations (X1,X2), (X1,X2,X3), (X1,X2,X3,X4), ...? So you're not considering, say, (X1,X3,X4)? Otherwise, you don't have p covariance matrices to worry about, you have 2^p matrices.

Either way, you could use the old-fashioned method of minors to computer the inverses (http://en.wikipedia.org/wiki/Adjugate). If you are in fact doing all 2^p combinations, you work your way up in dimension (do all single variables, then all pairs, all triples, ...). At each step, you compute the new minors recursively from the ones you already have.

/Paul
.