Nonparametric Statistics on Manifolds with Applications to Shape Spaces
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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Table of ContentsContinued
FRECHET ANALYSIS ON METRIC SPACES
INTRINSIC ANALYSIS ON MANIFOLDS
EXTRINSIC ANALYSIS ON MANIFOLDS
INTRODUCTION TO SHAPE SPACES
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THE PLANAR SHAPE SPACE
REFLECTION SIMILARITY SHAPE SPACES RΣkm
AFFINE SHAPE SPACES AΣkm
asymptotic distribution asymptotic level asymptotic normality asymptotic p-value Bhattacharya 2008a Bhattacharya and Bhattacharya Bhattacharya and Patrangenaru bootstrap methods Chapter compact conﬁdence region Consider Corollary deﬁned deﬁnition denote diﬀerent diﬀerential eigenvalues eigenvectors embedding equation Euclidean space exponential map extrinsic analysis extrinsic mean set extrinsic mean shapes ﬁrst Fn(p follows FrŽechet function FrŽechet mean FrŽechet variation geodesic distance identiﬁed iid sample injectivity radius k-ad landmarks linear projection matrix mean of Q means and variations metric space metric tensor nonfocal point oP(l orbit orthonormal basis p-value Patrangenaru 2005 pivotal bootstrap planar shape space probability distribution Procrustes Procrustes coordinates projective shape Proof Proposition 2.2.2 Rd+1 reﬂection similarity reject H0 respectively Riemannian manifold sample estimates sample extrinsic mean sample extrinsic variations sample from Q sample mean sectional curvature shape analysis similarity shape space SO(m supp(Q tangent space test statistic Theorem Tp^M unique intrinsic mean variation of Q vector