Nonparametric Statistics on Manifolds with Applications to Shape Spaces

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ProQuest, 2008 - 152 pages
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This thesis presents certain recent methodologies and some new results for the statistical analysis of probability distributions on non-Euclidean manifolds. The notions of Frechet mean and variation as measures of center and spread are introduced and their properties are discussed. The sample estimates from a random sample are shown to be consistent under fairly broad conditions. Depending on the choice of distance on the manifold, intrinsic and extrinsic statistical analyses are carried out. In both cases, sufficient conditions are derived for the uniqueness of the population means and for the asymptotic normality of the sample estimates. Analytic expressions for the parameters in the asymptotic distributions are derived. The manifolds of particular interest in this thesis are the shape spaces of k-ads. The statistical analysis tools developed on general manifolds are applied to the spaces of direct similarity shapes, planar shapes, reflection similarity shapes, affine shapes and projective shapes. Two-sample nonparametric tests are constructed to compare the mean shapes and variation in shapes for two random samples. The samples in consideration can be either independent of each other or be the outcome of a matched pair experiment. The testing procedures are based on the asymptotic distribution of the test statistics, or on nonparametric bootstrap methods suitably constructed. Real life examples are included to illustrate the theory.
 

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Contents

Table of ContentsContinued
11
FRECHET ANALYSIS ON METRIC SPACES
25
INTRINSIC ANALYSIS ON MANIFOLDS
36
EXTRINSIC ANALYSIS ON MANIFOLDS
51
INTRODUCTION TO SHAPE SPACES
66
References 149
67
Table of ContentsContinued
70
THE PLANAR SHAPE SPACE
76
REFLECTION SIMILARITY SHAPE SPACES RΣkm
101
AFFINE SHAPE SPACES AΣkm
123
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