Approximating the matrix Fisher and Bingham distributions: applications to spherical regression and Procrustes analysis

We obtain approximations to the distribution of the exponent in the matrix Fisher distributions on SO(p) and on O(p) whose density with respect to Haar measure is proportional to exp(Tr GX0tX). Similar approximations are found for the distribution of the exponent in the Bingham distribution, with density proportional to exp(xtGx), on the unit sphere Sp-1 in Euclidean p-dimensional space. The matrix Fisher distribution arises as the exact conditional distribution of the maximum likelihood estmate of the unknown orthogonal matrix in the spherical regression model on Sp-1 with Fisher distributed errors. It also arises as the exact conditional distribution of the maximum likelihood estimate of the unknown orthogonal matrix in a model of Procrustes analysis in which location and orientation, but not scale, changes are allowed. These methods allow determination of a confidence region for the unknown rotation for moderate sample sizes with moderate error concentrations when the error concentration parameter is known.

[1]  J. Gower Generalized procrustes analysis , 1975 .

[2]  J. Mackenzie,et al.  The estimation of an orientation relationship , 1957 .

[3]  Martin S. Hanna,et al.  On graphically representing the confidence region for an unknown rotation in three dimensions , 1990 .

[4]  J. Stuelpnagel,et al.  A Least Squares Estimate of Satellite Attitude (Grace Wahba) , 1966 .

[5]  Robb J. Muirhead,et al.  Latent Roots and Matrix Variates: A Review of Some Asymptotic Results , 1978 .

[6]  M. J. Prentice Orientation Statistics Without Parametric Assumptions , 1986 .

[7]  C. Goodall Procrustes methods in the statistical analysis of shape , 1991 .

[8]  A. James Distributions of Matrix Variates and Latent Roots Derived from Normal Samples , 1964 .

[9]  John T. Kent,et al.  Asymptotic expansions for the Bingham distribution , 1987 .

[10]  Ted Chang Spherical Regression and the Statistics of Tectonic Plate Reconstructions , 1993 .

[11]  G. A. Anderson An Asymptotic Expansion for the Noncentral Wishart Distribution , 1970 .

[12]  O. Barndorff-Nielsen,et al.  Exponential transformation models , 1982, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[13]  I. Olkin,et al.  Sampling theory of surveys, with applications , 1955 .

[14]  Louis-Paul Rivest,et al.  Spherical regression for concentrated Fisher-von Mises distributions , 1989 .

[15]  K. Mardia,et al.  The von Mises–Fisher Matrix Distribution in Orientation Statistics , 1977 .

[16]  Donald St. P. Richards,et al.  Hypergeometric functions on complex matrix space , 1991 .

[17]  Christopher Bingham An Antipodally Symmetric Distribution on the Sphere , 1974 .

[18]  R. Muirhead Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[19]  G. S. Watson Statistics on Spheres , 1983 .