ABSTRACT
This shows that wj depends on two terms, where u·Qj - 1u represents the spread sj and
(1-P)as 'N 2 represents the noise variance. The quantity P represents the relative
weight we give to each quantity. Exercise 12-1: Derive a similar result when N is not diagonal. There is a point that I have glossed over: remember, we required that the matrix
Qj be nonsingular, so that equation (17) will make sense. But suppose, after we have determined the retrieval vector rj for a given yj, we decide to add another measurement at some xm that is very close to one of the xj’s that we have already
used. This added measurement would make Q j singular, or nearly so: would this invalidate the method? Certainly, we would expect that adding an additional measurement, even if it is redundant, would still improve the retrieval a little bit, since it would reduce the effect of noise, at least. As we have seen, matrices really span a continuum, from the truly singular, to the grossly ill-conditioned, to moderately ill-conditioned, and finally to well conditioned matrices. Given a matrix based on some physical measurements, it would be surprising to find it to be truly singular, in the strict mathematical sense-even though, for all computational purposes, it might as well be. So: does an additional measurement make Qj more illconditioned and make the retrieval worse?
