Prof. Rudy Beran

Better Estimation of Multivariate Means

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

Nonparametric Inference on Manifolds, with Applications to Shapes

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

Bootstrapping in the Transform Domain: 
Double Bootstrap Confidence Bands for Survival Times in Semi-Markov Models by using Saddlepoint Approximation

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

Shape Classification procedures with Application to Schizophrenia

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

Shapes and images

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
Sharp Adaptation for Statistical Inverse Problems on Manifolds with Application to Medical Imaging

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

A Tribute to Frit's Ruymgaart



Dr.Stephan Huckemann

Principal Component Analysis based on Geodesics for Shape Spaces

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

Deterministic Model Selection for 2D Shape

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

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.


Dr. Hilary Thompson

Statistical Analysis of Hyperspectral Biomedical Images