Statistics on special manifolds
type: Item
instance of: scholarly article
title: Statistics on special manifolds
zbMATH Open document ID: 1026.62051
author: Yasuko Chikuse
published in: Lecture Notes in Statistics
publication date: 2003-04-24T00:00:00Z
review text: The special manifolds of interest in this book are the Stiefel manifold and the Grassmann manifold. Formally, the Stiefel manifold \(V_{k,m}\) is the space of \(k\)-frames in the \(m\)-dimensional real Euclidean space \(\mathbb R^m\), represented by the set of \(m\times k\) matrices \(X\) such that \(X'X = I_k\), where \(I_k\) is the \(k\times k\) identity matrix, and the Grassmann manifold \(G_{k,m-k}\) is the space of \(k\)-planes (\(k\)-dimensional hyperplanes) in \(\mathbb R^m\). The manifold \(P_{k,m-k}\) of \(m\times m\) orthogonal projection matrices idempotent of rank \(k\) corresponds uniquely to \(G_{k,m-k}\). This book is concerned with statistical analysis on the manifolds \(V_{k,m}\) and \(P_{k,m-k}\) as statistical sample spaces consisting of matrices. The discussion is carried out on the real spaces so that scalars, vectors, and matrices treated in this book are all real, unless explicitly stated otherwise. For the special case \(k= 1\), the observations from \(V_{l,m}\) and \(G_{1.m-1}\) are regarded as directed vectors on a unit sphere and as undirected axes or lines, respectively. There exists a large literature of applications of directional statistics and its statistical analysis, mostly occurring for \(m = 2\) or 3 in practice, in the Earth (or Geological) Sciences, Astrophysics, Medicine, Biology, Meteorology, Animal Behavior, and many other fields. Examples of observations on the general Grassmann manifold \(G_{k,m-k}\) arise in signal processing of radars with \(m\) elements observing \(k\) targets. The Grassmann manifold is a rather new subject treated as a statistical sample space, and the development of statistical analysis on Grassmann manifolds, which is one of the purposes of this book, must make some contributions to the study in the related sciences. Ch. 1, The Special Manifolds and Related Multivariate Topics, presents fundamental material which may be helpful for reading the main part of the book: the backgrounds of the special mani…
Mathematics Subject Classification ID: 62Hxx
Mathematics Subject Classification ID: 62-02
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 33C90
Mathematics Subject Classification ID: 53A99
Mathematics Subject Classification ID: 53B99
zbMATH DE Number: 1900778
zbMATH Keywords: Stiefel manifolds
zbMATH Keywords: Grassmann manifolds
zbMATH Keywords: orientation statistics
zbMATH Keywords: multivariate distributions
zbMATH Keywords: distributions of orientations of random matrices
zbMATH Keywords: simulation methods for generating pseudo-random matrices
zbMATH Keywords: decompositions of special manifolds
zbMATH Keywords: sampling
zbMATH Keywords: canonical correlation coefficients of general dimension
zbMATH Keywords: inference on parameters of matrix Langevin distribudistributions
zbMATH Keywords: Fisher scoring methods
zbMATH Keywords: maximum likelihood estimators
zbMATH Keywords: profile likelihood method
zbMATH Keywords: inference on orientation parameters
zbMATH Keywords: Bayes estimators
zbMATH Keywords: large sample asymptotic theorems
zbMATH Keywords: tests for uniformity
zbMATH Keywords: asymptotic expansions for sample mean matrices
zbMATH Keywords: testing problems
zbMATH Keywords: Rayleigh-style tests
zbMATH Keywords: likelihood ratio test
zbMATH Keywords: Rao score test
zbMATH Keywords: profile score functions and tests
zbMATH Keywords: locally best invariant tests
zbMATH Keywords: concentrated matrices
zbMATH Keywords: large concentration parameters
zbMATH Keywords: generalized Stam limit theorems
zbMATH Keywords: Procrustes analysis
zbMATH Keywords: perturbation theory
zbMATH Keywords: embeddings
zbMATH Keywords: density estimation
zbMATH Keywords: measures of orthogonal association
zbMATH Keywords: invariant polynomials with matrix arguments
zbMATH Keywords: hypergeometric functions with matrix arguments
zbMATH Keywords: generalized Hermite and Laguerre polynomials with matrix arguments
zbMATH Keywords: Edgeworth expansions for multiple random symmetric metric matrices
zbMATH Keywords: random rectangular matrices
zbMATH Keywords: generalized multivariate Meixner classes of invariant distributions of multiple random matrices
zbMATH Keywords: Edgeworth and saddle-point expansions for random matrices
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1900778