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Nonparametric statistics on manifolds and their applications to object data analysis

By: Patrangenaru, Victor.
Contributor(s): Ellingson, Leif.
Material type: materialTypeLabelBookPublisher: Boca Raton CRC Press 2016Description: xxiv, 517 p.ISBN: 9781439820506.Subject(s): Spatial analysis - Statistics | Geography - Statistical methods | Nonparametric - Statistics | Manifolds - MathematicsDDC classification: 519.535 Summary: 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. (https://www.crcpress.com/Nonparametric-Statistics-on-Manifolds-and-Their-Applications-to-Object/Patrangenaru-Ellingson/9781439820506)
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Table of Contents


I. NONPARAMETRIC STATISTICS ON MANIFOLDS


1. 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

2. 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

3. 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

4. Consistency of Fréchet Moments on Manifolds

Introduction
Fréchet Means and Cartan Means
Exercises

5. 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

6. 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

7. Function Estimation on Manifolds

Introduction
Statistical Inverse Estimation
Proofs of Main results
Kernel Density Estimation


II. ASYMPTOTIC THEORY AND NONPARAMETRIC BOOTSTRAP ON SPECIAL MANIFOLDS


8. Statistics on Homogeneous Hadamard Manifolds

Introduction
Considerations for Two-Sample Tests
Intrinsic Means on Hadamard Manifolds
Two-Sample Tests for Intrinsic Means

9. Analysis on Stiefel Manifolds

Stiefel Manifolds
Special Orthogonal Groups
Intrinsic Analysis on Spheres

10. 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

11. Nonparametric Statistics on Hilbert Manifolds

Introduction
Hilbert Manifolds
Extrinsic Analysis of Means on Hilbert Manifolds
A One-Sample Test of the Neighborhood Hypothesis

12. 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

13. 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

14. Statistics on Grassmannians

Equivariant Embeddings of Grassmann Manifolds
Dimitric Mean of a Random Object on a Grassmannian
Extrinsic Sample Covariance Matrix on a Grassmannian


III. APPLICATIONS IN OBJECT DATA ANALYSIS ON MANIFOLDS


15. DTI Data Analysis

Introduction
Tests for Equality of Generalized Frobenius Means
Application to Diffusion Tensor Imaging Data

16. Application of Directional Data Analysis

Introduction
The Pluto Controversy
The Solar Nebula Theory
Distributions for the Mean Direction
Implementation of the Nonparametric Approach

17. Direct Similarity Shape Analysis in Medical Imaging

Introduction
University School X-Ray Data Analysis
LEGS Data Analysis

18. 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

19. Estimating Mean Skull Size and Shape from CT Scans

Introduction
CT Scans
Bone Surface Segmentation
Skull Reconstruction
Landmark-Based Size-and-Shape Analysis

20. Affine Shape and Linear Shape Applications

Introduction
The Affine Shape Space in Computer Vision
Extrinsic Means of Affine Shapes
Analysis of Gel Electrophoresis (2DGE)
21. Projective Shape Analysis of Planar Contours

Introduction
Hilbert Space Representations of Projective Shapes
The One-Sample Problem for Mean Projective Shapes

22. 3D Projective Shape Analysis of Camera Images

Introduction
Test for Coplanarity
Projective Geometry for Pinhole Camera Imaging
3D Reconstruction and Projective Shape
Applications

23. Two-Sample Tests for Mean Projective Shapes

Projective Shape Analysis Examples in 1D and 2D
Test for VW Means of 3D Projective Shapes

24. Mean Glaucomatous Shape Change Detection

Introduction
Glaucoma and LEGS Stereo Eye Fundus Data
Shape-Based Glaucoma Index
Reconstruction of 3D Eye Fundus Configurations

25. 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



IV. ADDITIONAL TOPICS



26. Persistent Homology

Introduction
Nonparametric Regression on Manifolds
Main Results
Discussion
Proofs

27. Further Directions in Statistics on Manifolds

Introduction
Additional Topics
Computational Issues
Summary

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.

(https://www.crcpress.com/Nonparametric-Statistics-on-Manifolds-and-Their-Applications-to-Object/Patrangenaru-Ellingson/9781439820506)

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