ABSTRACTS
DISTINGUISHED SPEAKERS
Prof. Rudy
Beran
Title:
Better Estimation of
Multivariate Means
Abstract:
We consider practical strategies for low risk estimation of
multivariate means:
First formulation. Estimate the rows of an unknown constant matrix M
that is observed with additive, mean zero, random error. Constraining
the rank of the matrix estimator can reduce its quadratic risk. An
adaptive total least squares estimator has smallest asymptotic risk
among all reduced rank total least squares fits to the data matrix. The
asymptotic risk of a more flexible adaptive total shrinkage estimator
is smaller still. Both adaptive estimators are easily computed
functions of the singular value decomposition of the data matrix. In
the asymptotics, the row dimension of M tends to infinity while the
column dimension stays fixed.
Second formulation. Estimate the mean multivariate responses in a
multi-way MANOVA layout. We combine the previous rank shrinkage
strategy with the strategy of shrinking the fitted multivariate means
towards MANOVA submodels. Adaptive dual shrinkage yields an estimator
superior in asymptotic risk to that obtainable from either shrinkage
strategy alone.
Prof.
Rabi Bhattacharya
Title:
Nonparametric Inference on Manifolds, with
Applications to Shapes
Abstract:
The
talk begins with a survey of recent results on (1) general consistency
for the
nonparametric estimation of extrinsic and intrinsic means on manifolds,
(2) asymptotic distribution
theory for such estimates leading to confidence regions and tests, and
(3)
applications of these results to the statistical analysis of “shapes”
(as
defined by Kendall and others). We then
discuss a nonparametric approach to multivariate analysis of shapes,
especially
on Reimannian manifolds.<>
Prof. Ronald Butler
Title:
Bootstrapping in the Transform Domain: Double Bootstrap Confidence Bands for
Survival Times in Semi-Markov Models by using Saddlepoint Approximation
Abstract:
Finite state semi-Markov
processes are often used to model the transitions in states for a
patient with a degenerative disorder with death occurring upon passage
to a fatal state. If patients are followed over time, the records of
their sojourns through the semi-Markov process provide the data that
are used for estimating the dynamic parameters of the semi-Markov
process including transition probabilities of the jump chain and
empirical CDFs for the various holding time distributions in system
states. From the estimated process, various properties such as the
first passage distribution to the fatal state can be estimated by
resampling first passage transitions routed through the estimated
semi-Markov. The resulting estimator would be considered a "single
bootstrap" estimate for the survival time distribution of a patient.
This talk shows how the single bootstrap estimator may be computed
analytically from a saddlepoint approximation without the need to
actually resample sojourns through the estimated process.
A confidence band for this survival
time distribution requires the "double bootstrap" and entails a double
layer of resampling of sojourn times through the estimated process. In
theory, double bootstrap resampling should provide such a confidence
band, however the simulation effort required for its implementation is
generally beyond our current computing capabilities. This talk shows
how such double bootstrap confidence bands can be computed quickly and
practically if the inner layer of resampling is replaced with
analytical saddlepoint approximation; thus resampling is only retained
in the outer resampling. Such combined use of outer resampling and
inner analytical approximation leads to confidence bands that are
reasonably quick to compute and which attain coverage levels that are
extremely close to the nominal coverage.
If time permits, the talk will
address the additional complication in which patients can be randomly
censored from the various states of the semi-Markov process. For such
settings, saddlepoint approximations lead to survival distribution
estimates and confidence bands and thus provide an elegant solution for
the Fix-Neyman problem in multistate survival analysis.
Prof. Dipak Dey
Title:
Shape Classification procedures with Application to Schizophrenia Diagnosis
Abstract:
We discuss classification procedures in a shape analysis context. We
derive discriminants in shape space, while considering a complex Watson
shape distribution for the data, as well as in a tangent space to shape
space. Both frequentist and Bayesian approaches are considered. Using
MAP (Maximum A Posteriori) estimates of parameters involved, we derive
discriminant rules, and calculate missclassification probabilities
using Monte Carlo methods. The methods are exemplified through an
example, where we are interested in classifying patients into the
normal or schizophrenic groups, based on shapes created by MRI's
(Magnetic Resonance Images) of
their brain. Hence, the methods provide us with a new way of diagnosis
of this medical condition while controlling the error of misallocation.
Prof. John Kent
Title:
Shapes and images
Abstract:
The statistical theory of shape is a subject that has seen tremendous
development over the past twenty years. Two objects are said to
have the same shape if they are identical up to a similarity
transformation. The simplest version of the subject deals with
(finite) labelled configurations of landmarks, and changes in shape
correspond to the relative movement of the landmarks. From this
point of view, shape analysis becomes a variant of multivariate
analysis. A richer theory of shape regards an object as a solid
body and changes in shape are represented by deformations of the
underlying space. From this point of view, shape analysis is
closely related to functional data analysis. A commnon method of
obtaining geometric information about objects is by taking
images. Image modalities in 2 and 3 dimensions include
photographs, X-rays, CT, MRI and laser scans. Applications of shape
analysis include cross-sectional and longitudinal (growth) studies of
shape, and the identification and tracking of objects in images.
Prof. Peter Kim
Title:
Sharp Adaptation for Statistical Inverse Problems on Manifolds with
Application to Medical Imaging
Abstract:
This talk will examine the estimation of an indirect signal
embedded in white noise over a compact manifold. It will be
shown that the sharp minimax bound is determined by the degree to
which the indirect signal is embedded in the linear operator. Thus when
the linear operator has polynomial decay, recovery of the signal is
polynomial, whereas if the linear operator has exponential decay,
recovery of the signal is logarithmic. The constants are
determined for both of these classes and adaptive sharp estimation is
also carried out. In the polynomial case a blockwise shrinkage
estimator is needed while in the exponential case, a straight
projection estimator will suffice. Some of the results depend on
aspects of spectral geometry and in particular, the asymptotic
eigenvalue calculations associated with H. Weyl.
The framework of this analysis include applications to medical imaging
where the manifold is taken to be the two-dimensional sphere. The first
application, cone-beam image reconstruction, is associated with single
photon emission computed tomography (SPECT). The statistical
formulation of the problem involves Compton scattering through the
Klein-Nishina distribution.
A second application, and also dealing with medical imaging, is that
associated with diffusion magnetic resonance imaging (MRI). The
data comes as three-dimensional euclidean Fourier transforms, however,
of particular interest is the angular (directional) portion since
medically, this can clinically reveal certain anomalies in the brain
arising from trauma for example. This angular composition can
also be structured as a statistical inverse problem on the sphere.
Prof. Madan Puri
Title:
A Tribute to Frit's Ruymgaart
Abstract:
TBA
OUTSTANDING EARLY-CAREER SPEAKERS
Dr.Stephan Huckemann
Title:
Principal Component Analysis based on Geodesics for Shape Spaces
Abstract:
Kendall's landmark based shape spaces are so called pre-shape spheres
modulo a rotation group. Currently, principal component analysis on
shape spaces is performed in the tangent space of the pre-shape sphere
taken at a point in the fibre of a mean shape.
The tangent space is equipped with an euclidean metric stemming from
embedding the pre-shape sphere in euclidean space. In our approach we
fit geodesics in the shape space directly to the data. This leads to
the concept of principal component geodesics. As a consequence
different definitions of means appear yielding different concepts of
data variation.
In the work presented we apply these concepts and derive an algorithmic
approach.
Dr. Kathryn Leonard
Title:
Deterministic Model Selection for 2D
Shape
Abstract:
In the rapidly changing field of
image analysis, a shape model is only as good as the latest algorithm
that uses it. We wish to establish a coherent theory of shape modeling
outside the realm of image analysis tasks. To do so, we derive an
intrinsic, quantitative measure of the suitablity of a model for a
particular shape. Our work builds on the work of Shannon and Rissanen
in that it uses efficiency of representation as the suitability
criterion, but does so in a deterministic setting. We compare two shape
models, the boundary curve and Blum's medial axis, and classify
databases of shapes based on the best-suited model. Along the way, we
estimate the epsilon-entropy of two compact classes of curves and
construct two explicit adaptive encodings for non-compact classes of
shapes, one using the boundary curve and the other using the medial
axis.
Prof. Anuj Srivastava
Title:
Path-Straightening Flows for Constructing Geodesics on Shape Spaces of
Closed Curves
Abstract:
Curves play an important role in statistical analysis of shapes.For
instance, objects in 2D images can be characterized by shapes of their
boundaries, or shapes of surfaces of 3D objects can be studied through
shapes of certain level curves on these surfaces.
A fundamental tool in analyzing shapes of closed curves is the
construction of geodesics between any two such curves. In past we have
used a shooting method for constructing geodesics on spaces of planar,
closed curves and have studied the resulting shape statistics. In this
talk, we describe a path-straightening approach to finding geodesics
between closed curves in any Euclidean space. The basic idea is to
connect the given two curves on an appropriate manifold using any path,
and to iteratively straighten this path until it becomes a geodesic. We
illustrate this approach using examples from image analysis.
CVIAL INVITED SPEAKER
Dr. Hilary Thompson
Title:
Statistical Analysis of Hyperspectral Biomedical Images
Abstract:
TBA.