Opis: Nonparametric Statistics on Manifolds and Their Applications - Lief Ellingson, Victor Patrangenaru

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. "... 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 UniversityNonparametric 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 Frechet Moments on Manifolds Introduction Frechet Means and Cartan Means Exercises Nonparametric Distributions of Frechet Means Introduction Frechet Total Sample Variance-Nonparametrics Elementary CLT for Extrinsic Means CLT and Bootstrap for Frechet 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

Szczegóły: Nonparametric Statistics on Manifolds and Their Applications - Lief Ellingson, Victor Patrangenaru

Tytuł: Nonparametric Statistics on Manifolds and Their Applications
Autor: Lief Ellingson, Victor Patrangenaru
Producent: CRC Press Inc.
ISBN: 9781439820506
Rok produkcji: 2012
Ilość stron: 541
Oprawa: Twarda

Recenzje: Nonparametric Statistics on Manifolds and Their Applications - Lief Ellingson, Victor Patrangenaru