Statistics on Special Manifolds / Edition 1

Statistics on Special Manifolds / Edition 1

by Yasuko Chikuse, Y. Chikuse
     
 

ISBN-10: 0387001603

ISBN-13: 9780387001609

Pub. Date: 02/06/2003

Publisher: Springer New York

This book is concerned with statistical analysis on the two special manifolds, the Stiefel manifold and the Grassmann manifold, treated as statistical sample spaces consisting of matrices. The former is represented by the set of m x k matrices whose columns are mutually orthogonal k-variate vectors of unit length, and the latter by the set of m x m orthogonal

Overview

This book is concerned with statistical analysis on the two special manifolds, the Stiefel manifold and the Grassmann manifold, treated as statistical sample spaces consisting of matrices. The former is represented by the set of m x k matrices whose columns are mutually orthogonal k-variate vectors of unit length, and the latter by the set of m x m orthogonal projection matrices idempotent of rank k. The observations for the special case k=1 are regarded as directed vectors on a unit hypersphere and as axes or lines undirected, respectively. Statistical analysis on these manifolds is required, especially for low dimensions in practical applications, in the earth (or geological) sciences, astronomy, medicine, biology, meteorology, animal behavior and many other fields. The Grassmann manifold is a rather new subject treated as a statistical sample space, and the development of statistical analysis on the manifold must make some contributions to the related sciences. The reader may already know the usual theory of multivariate analysis on the real Euclidean space and intend to deeper or broaden the research area to statistics on special manifolds, which is not treated in general textbooks of multivariate analysis. The author rather concentrates on the topics to which a considerable amount of personal effort has been devoted. Starting with fundamental material of the special manifolds and some knowledge in multivariate analysis, the book discusses population distributions (especially the matrix Langevin distributions that are used for the most of the statistical analyses in this book), decompositions of the special manifolds, sampling distributions, and statistical inference on the parameters (estimation and tests for hypotheses). Asymptotic theory in sampling distributions and statistical inference is developed for large sample size, for large concentration and for high dimension. Further investigated are Procrustes methods applied

Product Details

ISBN-13:
9780387001609
Publisher:
Springer New York
Publication date:
02/06/2003
Series:
Lecture Notes in Statistics Series, #174
Edition description:
2003
Pages:
403
Product dimensions:
6.10(w) x 9.25(h) x 0.36(d)

Table of Contents

The Special Manifolds and Related Multivariate Topics
• Distributions on the Special Manifolds
• Decompositions of the Special Manifolds
• Distributional Problems in the Decomposition Theorems and the Sampling Theory
• The Inference on the Parameters of the Matrix Langevin Distributions
• Large Sample Asymptotic Theorems in Connection with Tests for Uniformity
• Asymptotic Theorems for Concentrated Matrix Langevin Distributions
• High Dimensional Asymptotic Theorems
• Procrustes Analysis on the Special Manifolds
• Density Estimation on the Special Manifolds
• Measures of Orthogonal Association on the Special Manifolds

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