Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and procrustes analysis

Christopher Bingham, Ted Chang, Donald Richards

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)314-337
Number of pages24
JournalJournal of Multivariate Analysis
Volume41
Issue number2
DOIs
StatePublished - Jan 1 1992

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Procrustes Analysis
Regression Analysis
Orthogonal matrix
Maximum likelihood
Exact Distribution
Conditional Distribution
Unknown
Directly proportional
Exponent
Spherical Model
Haar Measure
Confidence Region
Approximation
Unit Sphere
Maximum Likelihood Estimate
Maximum Likelihood
Euclidean
Regression Model
Sample Size

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty

Cite this

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Approximating the matrix Fisher and Bingham distributions : Applications to spherical regression and procrustes analysis. / Bingham, Christopher; Chang, Ted; Richards, Donald.

In: Journal of Multivariate Analysis, Vol. 41, No. 2, 01.01.1992, p. 314-337.

Research output: Contribution to journalArticle

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