1st Edition

Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis




ISBN 9781439820506
Published September 25, 2015 by CRC Press
517 Pages 111 B/W Illustrations

USD $115.00

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Book Description

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.

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

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Author(s)

Biography

Victor Patrangenaru is a professor of statistics at Florida State University. He received his first PhD from the University of Haifa; his differential geometry dissertation on locally homogeneous Riemannian and pseudo-Riemannian manifolds was conferred the Morris Pulver award. His second PhD was conferred at Indiana University for his dissertation on asymptotic statistics on manifolds and their applications. He has been a recipient of the Rothrock Mathematics Teaching Award from Indiana University.

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.

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