# Statistics on Special Manifolds by Yasuko Chikuse

By Yasuko Chikuse

This booklet is anxious with statistical research at the certain manifolds, the Stiefel manifold and the Grassmann manifold, handled as statistical pattern areas such as matrices. the previous is represented by means of the set of m x ok matrices whose columns are together orthogonal k-variate vectors of unit size, and the latter by means of the set of m x m orthogonal projection matrices idempotent of rank ok. The observations for the distinct 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, particularly for low dimensions in functional purposes, within the earth (or geological) sciences, astronomy, drugs, biology, meteorology, animal habit and plenty of different fields. The Grassmann manifold is a slightly new topic handled as a statistical pattern area, and the improvement of statistical research at the manifold needs to make a few contributions to the similar sciences. The reader might already comprehend the standard thought of multivariate research at the actual Euclidean house and intend to deeper or expand the study quarter to statistical data on designated manifolds, which isn't handled normally textbooks of multivariate research.

The writer particularly concentrates at the subject matters to which a large amount of own attempt has been dedicated. beginning with basic fabric of the distinct 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 distinct manifolds, sampling distributions, and statistical inference at the parameters (estimation and checks for hypotheses). Asymptotic conception in sampling distributions and statistical inference is built for giant pattern measurement, for big focus and for prime measurement. additional investigated are Procrustes tools utilized at the particular manifolds, density estimation, and size of orthogonal organization.

This publication is designed as a reference publication for either theoretical and utilized statisticians. The ebook may also be used as a textbook for a graduate direction in multivariate research. it can be assumed that the reader knows the standard idea of univariate statistics and an intensive historical past in arithmetic, particularly, wisdom of multivariate calculation suggestions. To make the publication self-contained, a short overview of a few of these features and comparable subject matters is given.

Yasuko Chikuse is Professor of data and arithmetic at Kagawa collage, Japan. She earned a Ph.D. in facts from Yale collage and Sc.D. in arithmetic from Kyushu college, Japan. She is a member of the overseas Statistical Institute, the Institute of Mathematical statistics, the yankee Statistical organization, the Japan Statistical Society, and the Mathematical Society of Japan. She has held vacationing learn and/or instructing appointments on the CSIRO, the collage of Pittsburgh, the collage of California at Santa Barbara, York college, McGill collage, and the collage of St Andrews.

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Extra info for Statistics on Special Manifolds

Example text

2: am 2: 0, then we define A 1/ 2 = HAY2 H " h were A 1/2 d 1/2) . 3) of a random matrix Z will be frequently used in later chapters. 4) where H E Vk,m and T = (t ij ) is a k x k upper-triangular matrix with positive diagonal elements; so that the unique upper-triangular matrix T is chosen such that T'T = Z'Z and H is then given by H = ZT-1. 1. 2. 7) > i{m - 1), where the integral is taken over the space of all m x m positive definite matrices. 4), we have (dZ) A: . A: = II t~-' /\ 1 dtij{dH). 14].

The density function of Y may be written as The symmetric matrix-variate normal Nmm(O, 1m) distribution is a typical orthogonally invariant distribution of an m x m symmetric matrix variate S, which is invariant under the orthogonal transformation S -t HSH' for H E Oem). In general, let S be an m x m orthogonally invariant symmetric matrix variate which has the density function f(S) with the invariance property f(S) = f(HSH') for HE Oem). 5. Jacobians and Some Related Distributions 23 with S being an m x m orthogonally invariant symmetric matrix variate with the density function I(S) and for an m x m symmetric matrix M and an m x m nonsingular matrix A.

We will frequently use it in later chapters. 4. 8) where H2 E O{k), and . . Here VA: m denotes the 2-A:th part of VA: m consisting of matrices Xl E VA: m whose elements of the first row are positive with the normalized invariant measure [dXl ] = 2A:{dXl )fv{k,m). 5. 3. 9), we have m (dS) = II~(li -li) /\ dli(dH). 2 (i). 1 could be considered in more general terms as (random) analytic matrix-valued functions defined on some analytic manifold. We have given most of the results we need concerning Jacobians without proofs.