### Yale University

Department of Statistics

Seminar

#### 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