By Yasuko Chikuse
This publication is anxious with statistical research at the designated manifolds, the Stiefel manifold and the Grassmann manifold, taken care of as statistical pattern areas which includes matrices. the previous is represented through the set of m x okay matrices whose columns are at the same time orthogonal k-variate vectors of unit size, and the latter by way of the set of m x m orthogonal projection matrices idempotent of rank okay. The observations for the detailed case k=1 are considered as directed vectors on a unit hypersphere and as axes or strains undirected, respectively. Statistical research on those manifolds is needed, specifically for low dimensions in sensible functions, within the earth (or geological) sciences, astronomy, medication, biology, meteorology, animal habit and lots of different fields. The Grassmann manifold is a slightly new topic handled as a statistical pattern house, and the improvement of statistical research at the manifold needs to make a few contributions to the comparable sciences. The reader may perhaps already be aware of the standard idea of multivariate research at the actual Euclidean area and intend to deeper or expand the examine quarter to stats on targeted manifolds, which isn't taken care of usually textbooks of multivariate research.
The writer particularly concentrates at the themes to which a large amount of own attempt has been dedicated. beginning with basic fabric of the particular manifolds and a few wisdom in multivariate research, the e-book discusses inhabitants distributions (especially the matrix Langevin distributions which are used for the main of the statistical analyses during this book), decompositions of the targeted manifolds, sampling distributions, and statistical inference at the parameters (estimation and exams for hypotheses). Asymptotic thought in sampling distributions and statistical inference is constructed for big pattern measurement, for giant focus and for prime measurement. additional investigated are Procrustes tools utilized at the particular manifolds, density estimation, and size of orthogonal organization.
This ebook is designed as a reference publication for either theoretical and utilized statisticians. The publication can be used as a textbook for a graduate direction in multivariate research. it can be assumed that the reader knows the standard thought of univariate facts and an intensive history in arithmetic, particularly, wisdom of multivariate calculation innovations. To make the ebook self-contained, a short evaluate of a few of these features and comparable themes is given.
Yasuko Chikuse is Professor of records and arithmetic at Kagawa collage, Japan. She earned a Ph.D. in records from Yale collage and Sc.D. in arithmetic from Kyushu collage, Japan. She is a member of the overseas Statistical Institute, the Institute of Mathematical records, the yankee Statistical organization, the Japan Statistical Society, and the Mathematical Society of Japan. She has held traveling examine and/or instructing appointments on the CSIRO, the college of Pittsburgh, the collage of California at Santa Barbara, York college, McGill college, and the college of St Andrews.
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Additional info for Statistics on Special Manifolds
2, the Hermite polynomials Him) (S), A I1 = 0,1,2, ... , associated with the normal N mm (0, 1m) distribution and, in general, the generalized Hermite polynomials Hi~L)(S[q]; A[r])' 4J E Al ... Ar , with q m x m symmetric matrix arguments SI, ... ,Sq (= S[q]) and r h x m constant matrices AI' ... ' Ar (= A[r]) (q ~ r), associated with the joint distribution of q independent m x m symmetric matrix-variate standard normal Nmm(O, 1m) distributions. 3. Z'Z). 13) That is, the elements of the m x k random matrix Z are independent and identically distributed as normal N(O, 1).
2. 3. 1. 3 (ii). 3. 2. 1 (iii)' given earlier. 4. 10)]. 26)]. , for k = 1 and m=2). 3. 1. Non-uniform Distributions on V k,m The Matrix Langevin Distribution A random matrix X on Vk,m is said to have the matrix Langevin (or von MisesFisher) distribution, denoted by L(m, kj F), if its density function is given by [Downs (1972)] F (! 1. 1) where F is an m x k matrix. 3) of X on Rm,k with F = ME-I and the condition X' X = 1k imposed, and has been used most commonly as an exponential population distribution on Vk ,m in the literature.
3) by generalizing the orientationally rotational symmetry to the rotational symmetry around the subspace V. 3. Non-uniform Distributions 35 The matrix generalized Langevin distribution [g-L(m, k; q, V; F)], with F being an m x k matrix, has the density function F (! 7) where Pv denotes the orthogonal projection matrix onto V. 4, for k = 1, suggested by Scheiddegger (1965) and Watson (1965). 3), respectively. Here M(r) denotes the subspace generated by the columns of r. These distributions can be further generalized to those having density functions of the form f(PvX) for a suitable function f(·).