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By Yasuko Chikuse

This booklet is anxious with statistical research at the specified manifolds, the Stiefel manifold and the Grassmann manifold, handled as statistical pattern areas which includes matrices. the previous is represented via the set of m x okay matrices whose columns are together 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 ok. The observations for the exact 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, specially for low dimensions in sensible purposes, 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 taken care of as a statistical pattern area, and the advance of statistical research at the manifold needs to make a few contributions to the comparable sciences. The reader could already understand the standard thought of multivariate research at the actual Euclidean house and intend to deeper or develop the study sector to stats on distinctive manifolds, which isn't taken care of typically textbooks of multivariate research.

The writer quite concentrates at the themes to which a large amount of own attempt has been committed. beginning with basic fabric of the particular manifolds and a few wisdom in multivariate research, the ebook 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 checks for hypotheses). Asymptotic concept in sampling distributions and statistical inference is built for big pattern measurement, for big focus and for top measurement. extra investigated are Procrustes tools utilized at the distinctive manifolds, density estimation, and dimension of orthogonal organization.

This e-book is designed as a reference ebook for either theoretical and utilized statisticians. The ebook can be used as a textbook for a graduate path in multivariate research. it can be assumed that the reader understands the standard concept of univariate records and a radical historical past in arithmetic, specifically, wisdom of multivariate calculation innovations. To make the e-book self-contained, a short evaluation of a few of these features 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 college 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 examine and/or educating appointments on the CSIRO, the college of Pittsburgh, the collage of California at Santa Barbara, York college, McGill collage, and the collage of St Andrews.

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a lengthy and intricate historical past. In 1938-39, Nielsen gave a sequence of lectures on
discontinuous teams of motions within the non-euclidean aircraft, and this led him - in the course of
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Statistics on Special Manifolds

This ebook is anxious with statistical research at the distinctive manifolds, the Stiefel manifold and the Grassmann manifold, handled as statistical pattern areas along with matrices. the previous is represented by means of the set of m x ok matrices whose columns are jointly orthogonal k-variate vectors of unit size, and the latter via the set of m x m orthogonal projection matrices idempotent of rank ok.

Extra resources for Statistics on Special Manifolds

Sample text

The matrix Langevin L(m, kj F) and L(P) (m, kj B) distributions are expon~n­ tial and modal distributions on Vk,m and Pk,m-k' respectively, and possess desirable properties and tractability corresponding to those possessed by the multivariate normal distributions on the real spaces Rm, Rm,k' and Sm. The special cases F = 0 and B = 0 yield the uniform distributions on Vk,m and Pk,m-k' respectively. Most of the statistical analyses on the manifolds Vk,m and Pk,m-k treated in later chapters are carried out for these matrix Langevin distributions.

The orientation of the orbit can be specified by the matrix X = (3: 1 3:2 ) in V;,3. Jupp (1995) gave applications of directional statistics to some astronomical problems on the orbits of comets using Marsden's catalogue of cometary orbits [see Marsden and Williams (1993)]; the same article considered directional problems on the orbits of visual binary stars and spin axes of asteroids as well. See Mardia (1975), Jupp and Mardia (1979), Waston (1983a), and Fisher, Lewis, and Embleton (1987) for analysis of related data.

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

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