STANFORD UNIVERSITY
PROBABILITY & STOCHASTIC PROCESSES SEMINAR
4:15 p.m., Monday, April 21, 2003
Sequoia Hall, Room 200
Cookies at 4:00 p.m., 1st Floor Lounge
Jonathan Taylor
Statistics Department
Stanford University
Validity of the expected Euler characteristic heuristic
Abstract.
We consider the well-studied problem of approximating the distribution of
the maximum of a smooth Gaussian process (X(t), t in T) where T is
a nice subset of Euclidean space, or, more generally an abstract
manifold, with or without boundary.
The expected Euler characteristic (EC) is a heuristic approximation to
this distribution that has a relatively simple closed form when the mean
and variance are constant. However, as the argument is heuristic, there
are relatively few results known about the accuracy of the EC
approximation.
Previous results relate the EC approximation to two different
approximations, corresponding to two special cases: the volume of tubes
approach for finite Karhunen-Loeve processes, and the ``double sum''
approach, for ``almost isotropic'' processes.
In this talk, we describe an exact form for the error in the EC
approximation. The expression for the error is based on a point process
representation of the EC approximation based on Morse theory, and a
simple point process representation of the maximum of (X(t), t in T).
The argument does not use the assumption that X is a Gaussian process,
nor does it assume constant mean and variance, and is applicable to all
suitably regular processes, though computations are simplest in the
Gaussian case, with constant variance.
Based on this form for the error, we give a proof that the error, in the
constant mean and variance Gaussian case, is exponentially smaller than
any term in the EC approximation. We also characterize the exponential
rate of decay, which strengthens previous results for ``almost
isotropic'' processes, even in 1 dimension.
This is based on joint work with Akimichi Takemura, Robert Adler and
Satoshi Kuriki.
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