Yale University
Department of Statistics

Monday, March 4, 1996

Jonathan Buckheit

Department of Statistics
Stanford University

Alternatives to Karhunen-Loeve Expansions Using a Library of Orthogonal Bases

Karhunen-Loeve expansion is often described as an ``optimal representation'' of stochastic processes, even when the assumptions necessary for such optimality do not apply. In searching for more satisfying notions of optimal representations we are hamstrung by the fact that it is of little use to discuss representations which cannot be found constructively by modern computing methods. In this talk we describe a way to define an ``optimal orthogonal basis'' for representing a stochastic process which has both an effective algorithm and an appealing intuitive interpretation. The idea is to search over a finite list of orthogonal bases for that basis which optimizes a numerical measure of the quality of representation in that coordinate system. We use recently-developed libraries of time/frequency bases such as wavelet packets and cosine packets for our searches. We quantify optimality using measures of departure from the Gaussian distribution, such as standardized cumulants and goodness-of-fit against Gaussian models. For such libraries and such measures we have available a fast algorithm for finding an optimal representation, the Coifman-Wickerhauser best-basis algorithm, which was originally developed in a data compression context. We use our technique to study several interesting examples. In each case we uncover representations that are much more intuitively satisfying than the Karhunen-Loeve expansion. These examples have interesting interpretations; for example, they show that in several examples for which Fourier analysis provides the K-L transform, more general types of time/frequency analysis are somehow intrinsically ``better,'' and representations such as wavelets and Gabor functions can be found as explicit solutions of an optimization problem. We will sketch many interesting questions arising from this work -- dealing with searching through massive collections of bases in finite sample sizes, relations to work of Mallat, Papanicolaou and Zhang, and possible applied connections with projection pursuit, discrimination, and deconvolution.

Seminar to be held in Room 107, 24 Hillhouse @ 4:15 pm