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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, January 30, 2001}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Jonathan Taylor}
\centerline{\sl Mc Gill University}
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\centerline{\bf Random fields on surfaces and volumes without
stationarity: Applications to brain imaging}
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An important tool in signal detection in brain imaging is the distribution
of the maximum of a random field on some Euclidean space, which is taken to
be the parameter space of the random field. One approach that has proved
successful in approximating the tail of this distribution is the so-called
EC or (expected) Euler characteristic approach. This approach uses
techniques from integral geometry to approximate the distribution of the
maximum of the random field and provides an approximation of the tail in
terms of certain integral invariants of S, known in integral geometry as
the intrinsic volumes of S.

A fundamental assumption of this approach was that the random field was
isotropic, i.e. invariant under the group of Euclidean rigid motions.
However, when studying random fields on manifolds such as the cortical
surface, there is no notion of isotropy because of the irregularity of the
manifold. Further, the distribution of the maximum of the random field is
invariant under diffeomorphisms of the parameter space, so that our
inference should not be based on how we visualize the data, i.e.,
flattening the cortical surface to view data in the sulci of the cortical
surface should not affect the inference about the presence of a signal or not.

These ideas point to the fact that the relevant geometry in the study of
such random fields is one that is intrinsic to the random field itself. In
other words, the relevant geometry is derived from the Riemannian structure
that the random field induces on the manifold and not (necessarily) from
the space in which we visualize the data. In the case of an isotropic field
on a Euclidean space, the assumption of isotropy restricts the geometry to
be  the geometry of the Euclidean space (modulo a constant).

We describe the EC approach and how it can be extended smoothly to random
fields on manifolds as well as how to estimate the relevant geometric
quantities that appear in the EC approximation. We illustrate the method on
some fMRI (functional magnetic resonance imaging) data, restricted to the
cortical surface.

If time permits, we will describe a version of the classical Kinematic
Fundamental Formula from integral geometry for smooth vector-valued
Gaussian random fields. This has applications for the EC approach in a
non-Gaussian framework, i.e. for random fields built from i.i.d. Gaussian
fields.

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