Nonparametric Inference on Manifolds

by Bhattacharya, Abhishek; Bhattacharya, Rabi

• ISBN13: 9781107019584
• ISBN10: 1107019583
• Format: Hardcover
• Copyright: 2012-05-28
• Publisher: Cambridge Univ Pr

Summary

This book introduces in a systematic manner a general nonparametric theory of statistics on manifolds, with emphasis on manifolds of shapes. The theory has important and varied applications in medical diagnostics, image analysis, and machine vision. An early chapter of examples establishes the effectiveness of the new methods and demonstrates how they outperform their parametric counterparts. Inference is developed for both intrinsic and extrinsic Fréchet means of probability distributions on manifolds, then applied to shape spaces defined as orbits of landmarks under a Lie group of transformations - in particular, similarity, reflection similarity, affine and projective transformations. In addition, nonparametric Bayesian theory is adapted and extended to manifolds for the purposes of density estimation, regression and classification. Ideal for statisticians who analyze manifold data and wish to develop their own methodology, this book is also of interest to probabilists, mathematicians, computer scientists and morphometricians with mathematical training.

Table of Contents

Introduction

Examples

Location and spread on metric spaces

Extrinsic analysis on manifolds

Intrinsic analysis on manifolds

Landmark-based shape spaces

Kendall's similarity shape spaces ¿km

The planar shape space ¿k2

Reflection similarity shape spaces R¿km

Stiefel manifolds

Affine shape spaces A¿km

Real projective spaces and projective shape spaces

Nonparametric Bayes inference

Regression, classification and testing

Differentiable manifolds

Riemannian manifolds

Dirichlet processes

Parametric models on Sd and ¿k2

References

Subject index

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