Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis

By Victor Patrangenaru, Leif Ellingson

© 2015 – CRC Press

542 pages | 111 B/W Illus.

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pub: 2015-09-08
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About the Book

A New Way of Analyzing Object Data from a Nonparametric Viewpoint

Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis provides one of the first thorough treatments of the theory and methodology for analyzing data on manifolds. It also presents in-depth applications to practical problems arising in a variety of fields, including statistics, medical imaging, computer vision, pattern recognition, and bioinformatics.

The book begins with a survey of illustrative examples of object data before moving to a review of concepts from mathematical statistics, differential geometry, and topology. The authors next describe theory and methods for working on various manifolds, giving a historical perspective of concepts from mathematics and statistics. They then present problems from a wide variety of areas, including diffusion tensor imaging, similarity shape analysis, directional data analysis, and projective shape analysis for machine vision. The book concludes with a discussion of current related research and graduate-level teaching topics as well as considerations related to computational statistics.

Researchers in diverse fields must combine statistical methodology with concepts from projective geometry, differential geometry, and topology to analyze data objects arising from non-Euclidean object spaces. An expert-driven guide to this approach, this book covers the general nonparametric theory for analyzing data on manifolds, methods for working with specific spaces, and extensive applications to practical research problems. These problems show how object data analysis opens a formidable door to the realm of big data analysis.

Reviews

"… the first extensive book on [this subject] … This book succeeds in unifying the field by bringing in disparate topics, already available in several papers, but not easy to understand, under one roof. … a brilliant and a bold idea by an active researcher, who is now joined in coauthorship by an enthusiastic, hardworking, and talented younger peer. … it exceeds all expectations, in particular regarding the extent to which complex differential geometric notions permeate statistics."

—From the Foreword by Victor Pambuccian, Professor of Mathematics, Arizona State University

Table of Contents

Nonparametric Statistics on Manifolds

Data on Manifolds

Directional and Axial Data

Similarity Shape Data and Size and Shape Data

Digital Camera Images

Stereo Imaging Data of the Eye Fundus

CT Scan Data

DTI Data

Data Tables

Basic Nonparametric Multivariate Inference

Basic Probability Theory

Integration on Euclidean Spaces

Random Vectors

Sampling Distributions of Estimators

Consistency and Asymptotic Distributions of Estimators

The Multivariate Normal Distribution

Convergence in Distribution

Limit Theorems

Elementary Inference

Comparison of Two Mean Vectors

Principal Components Analysis (PCA)

Multidimensional Scaling

Nonparametric Bootstrap and Edgeworth Expansion

Nonparametric Function Estimation

Data Analysis on Hilbert Spaces

Exercises

Geometry and Topology of Manifolds

Manifolds, Submanifolds, Embeddings, Lie Groups

Riemannian Structures, Curvature, Geodesics

The Laplace-Beltrami Operator

Topology of Manifolds

Manifolds as Spaces of Objects in Data Analysis

Exercises

Consistency of Fréchet Moments on Manifolds

Introduction

Fréchet Means and Cartan Means

Exercises

Nonparametric Distributions of Fréchet Means

Introduction

Fréchet Total Sample Variance-Nonparametrics

Elementary CLT for Extrinsic Means

CLT and Bootstrap for Fréchet Means

CLT for Extrinsic Sample Means

Exercises

Inference for Two Samples on Manifolds

Introduction

Two-Sample Test for Total Extrinsic Variances

Bhattacharya’s Two-Sample Test for Means

Test for Mean Change in Matched Pairs on Lie Groups

Two-Sample Test for Simply Transitive Group Actions

Nonparametric Bootstrap for Two-Sample Tests

Exercises

Function Estimation on Manifolds

Introduction

Statistical Inverse Estimation

Proofs of Main results

Kernel Density Estimation

Asymptotic Theory and Nonparametric Bootstrap on Special Manifolds

Statistics on Homogeneous Hadamard Manifolds

Introduction

Considerations for Two-Sample Tests

Intrinsic Means on Hadamard Manifolds

Two-Sample Tests for Intrinsic Means

Analysis on Stiefel Manifolds

Stiefel Manifolds

Special Orthogonal Groups

Intrinsic Analysis on Spheres

Asymptotic Distributions on Projective Spaces

Total Variance of Projective Shape Asymptotics

Asymptotic Distributions of VW-Means

Asymptotic Distributions of VW-Means of k-ads

Inference for Projective Shapes of k-ads

Two-Sample Tests for Mean Projective Shapes

Nonparametric Statistics on Hilbert Manifolds

Introduction

Hilbert Manifolds

Extrinsic Analysis of Means on Hilbert Manifolds

A One-Sample Test of the Neighborhood Hypothesis

Analysis on Spaces of Congruences of k-ads

Introduction

Equivariant Embeddings of SSk2 and RSSkm,0

Extrinsic Means and Their Estimators

Asymptotic Distribution of Extrinsic Sample Mean

Mean Size-and-Shape of Protein Binding Sites

Similarity Shape Analysis

Introduction

Equivariant Embeddings of Sk2 and RSkm,0

Extrinsic Mean Planar Shapes and Their Estimators

Asymptotic Distribution of Mean Shapes

A Data-Driven Example

Statistics on Grassmannians

Equivariant Embeddings of Grassmann Manifolds

Dimitric Mean of a Random Object on a Grassmannian

Extrinsic Sample Covariance Matrix on a Grassmannian

Applications in Object Data Analysis on Manifolds

DTI Data Analysis

Introduction

Tests for Equality of Generalized Frobenius Means

Application to Diffusion Tensor Imaging Data

Application of Directional Data Analysis

Introduction

The Pluto Controversy

The Solar Nebula Theory

Distributions for the Mean Direction

Implementation of the Nonparametric Approach

Direct Similarity Shape Analysis in Medical Imaging

Introduction

University School X-Ray Data Analysis

LEGS Data Analysis

Similarity Shape Analysis of Planar Contours

Introduction

Similarity Shape Space of Planar Contours

The Extrinsic Mean Direct Similarity Shape

Asymptotic Distribution of the Sample Mean

The Neighborhood Hypothesis Test for Mean Shape

Application of the One Sample Test

Bootstrap Confidence Regions for the Sample Mean

Approximation of Planar Contours

Application to Einstein’s Corpus Callosum

Estimating Mean Skull Size and Shape from CT Scans

Introduction

CT Scans

Bone Surface Segmentation

Skull Reconstruction

Landmark-Based Size-and-Shape Analysis

Affine Shape and Linear Shape Applications

Introduction

The Affine Shape Space in Computer Vision

Extrinsic Means of Affine Shapes

Analysis of Gel Electrophoresis (2DGE)

Projective Shape Analysis of Planar Contours

Introduction

Hilbert Space Representations of Projective Shapes

The One-Sample Problem for Mean Projective Shapes

3D Projective Shape Analysis of Camera Images

Introduction

Test for Coplanarity

Projective Geometry for Pinhole Camera Imaging

3D Reconstruction and Projective Shape

Applications

Two-Sample Tests for Mean Projective Shapes

Projective Shape Analysis Examples in 1D and 2D

Test for VW Means of 3D Projective Shapes

Mean Glaucomatous Shape Change Detection

Introduction

Glaucoma and LEGS Stereo Eye Fundus Data

Shape-Based Glaucoma Index

Reconstruction of 3D Eye Fundus Configurations

Application of Density Estimation on Manifolds

Introduction

Pelletier Density Estimators on Homogeneous Spaces

Density Estimation on Symmetric Spaces

An Example of Projective Shape Density Estimation

Additional Topics

Persistent Homology

Introduction

Nonparametric Regression on Manifolds

Main Results

Discussion

Proofs

Further Directions in Statistics on Manifolds

Introduction

Additional Topics

Computational Issues

Summary

About the Authors

Victor Patrangenaru is a professor of statistics at Florida State University. He received his PhD from the University of Haifa; his dissertation on locally homogeneous Riemannian and pseudo-Riemannian manifolds was conferred the Morris Pulver award. He has also been a recipient of the Rothrock Mathematics Faculty Teaching Award from Indiana University. Dr. Patrangenaru’s research interests include applications of statistics on stratified spaces to medical imaging, astronomy, proteomics, phylogenetics, and pattern recognition. He is a leading researcher in nonparametric statistics on manifolds and object data analysis, with his research consistently funded by the U.S. National Security Agency and U.S. National Science Foundation.

Leif Ellingson is an assistant professor at Texas Tech University. He received his PhD in statistics from Florida State University; his dissertation "Statistical Shape Analysis on Manifolds with Applications to Planar Contours and Structural Proteomics" received the Ralph A. Bradley award. He has also been a recipient of the New Faculty Award from the Texas Tech Alumni Association. His current research interests include nonparametric statistics on manifolds, shape analysis, computational methods in statistics, and utilizing statistical methods in structural proteomics.

Subject Categories

BISAC Subject Codes/Headings:
MAT029000
MATHEMATICS / Probability & Statistics / General
TEC015000
TECHNOLOGY & ENGINEERING / Imaging Systems