By Yasuko Chikuse
This booklet is worried with statistical research at the precise manifolds, the Stiefel manifold and the Grassmann manifold, handled as statistical pattern areas together with matrices. the previous is represented through the set of m x ok matrices whose columns are collectively 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 okay. The observations for the targeted case k=1 are considered as directed vectors on a unit hypersphere and as axes or traces undirected, respectively. Statistical research on those manifolds is needed, particularly for low dimensions in useful functions, within the earth (or geological) sciences, astronomy, medication, 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 may possibly already recognize the standard concept of multivariate research at the actual Euclidean area and intend to deeper or increase the study sector to statistical data on exact manifolds, which isn't taken care of quite often textbooks of multivariate research.
The writer fairly concentrates at the themes to which a large amount of own attempt has been committed. beginning with primary fabric of the distinctive 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 certain manifolds, sampling distributions, and statistical inference at the parameters (estimation and exams for hypotheses). Asymptotic idea in sampling distributions and statistical inference is built for giant pattern dimension, for giant focus and for top measurement. extra investigated are Procrustes equipment utilized at the targeted manifolds, density estimation, and size of orthogonal organization.
This booklet is designed as a reference booklet for either theoretical and utilized statisticians. The booklet can also be used as a textbook for a graduate path in multivariate research. it can be assumed that the reader knows the standard idea of univariate records and an intensive history in arithmetic, specifically, wisdom of multivariate calculation suggestions. To make the booklet self-contained, a quick evaluation of a few of these elements and similar subject matters is given.
Yasuko Chikuse is Professor of facts and arithmetic at Kagawa college, Japan. She earned a Ph.D. in data 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 information, 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 college of Pittsburgh, the collage of California at Santa Barbara, York college, McGill collage, and the college of St Andrews.
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Extra info for Statistics on Special Manifolds
In view of 2. 1), we obtain the density function of Hz, which is proportional to p+1Fq [a 1, ... m; b1, ... ,bq;H~BHz(H~AHz)-1]IH~AHzl-m/2. 4), and the condition A = 1m gives the density function proportional to PHFq(a1, ... m; b1, ... , bq; H'zBH z ). 10) l=O >'I-! (B) are zonal polynomials (see Appendix A), and we can put d(O) = 1 without loss of generality. k; b1, . m; B) x pFq(a 1, ... , ap; b1, . k\ . B) etr(X' BX), 1 2 ' 2m, 1 (1 I ) F. 13) 1 (1 . I ) F (! k;X' BX). 3. 4, respectively. 2.
15)(P). 13)(P), having the density function 1 1 F (lk' 1 . 22) is a slight modification of the Downs (1972) distribution on the Stiefel manifold, and may be called the matrix Langevin distribution on Pk,m-k' which is denoted by L
X' BrX) = :E: >'[r];q, = 1, ... -C~[r] (X' Bl X, . [r] Il1i! 19) Here we use the notation in the theory of invariant polynomials j that is, C~[r] (B l , ... , Br) with the matrix arguments B l , ... , B r , and the d~[r are suitable coefficients. 2. 9) on V,. 21) 38 2. 11). 17)(P). 15)(P). 13)(P), having the density function 1 1 F (lk' 1 . 22) is a slight modification of the Downs (1972) distribution on the Stiefel manifold, and may be called the matrix Langevin distribution on Pk,m-k' which is denoted by L
Statistics on Special Manifolds by Yasuko Chikuse