Statistics on Special Manifolds

This book is concerned with statistical analysis on the two special manifolds, the Stiefel manifold and the Grassmann manifold, treated as statistical sample spaces consisting of matrices. The former is represented by the set of m x k matrices whose columns are mutually orthogonal k-variate vectors of unit length, and the latter by the set of m x m orthogonal projection matrices idempotent of rank k. The observations for the special case k=3D1 are regarded as directed vectors on a unit hypersphere and as axes or lines undirected, respectively.

Statistical analysis on these manifolds is required, especially for low dimensions in practical applications, in the earth (or geological) sciences, astronomy, medicine, biology, meteorology, animal behavior and many other fields. The Grassmann manifold is a rather new subject treated as a statistical sample space, and the development of statistical analysis on the manifold must make some contributions to the related sciences. The reader may already know the usual theory of multivariate analysis on the real Euclidean space and intend to deeper or broaden the research area to statistics on special manifolds, which is not treated in general textbooks of multivariate analysis.

The author rather concentrates on the topics to which a considerable amount of personal effort has been devoted. Starting with fundamental material of the special manifolds and some knowledge in multivariate analysis, the book discusses population distributions (especially the matrix Langevin distributions that are used for the most of the statistical analyses in this book), decompositions of the special manifolds, sampling distributions, and statistical inference on the parameters (estimation and tests for hypotheses). Asymptotic theory in sampling distributions and statistical inference is developed for large sample size, for large concentration and for high dimension.