# Nonparametric statistics on manifolds and their applications to object data analysis

##### By: Patrangenaru, Victor.

##### Contributor(s): Ellingson, Leif.

Material type: BookPublisher: 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)Item type | Current location | Collection | Call number | Status | Date due | Barcode |
---|---|---|---|---|---|---|

Books | Vikram Sarabhai Library General Stacks | Non-fiction | 519.535 P2N6 (Browse shelf) | Checked out | 09/01/2020 | 190946 |

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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