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August 2005 Statistical analysis on high-dimensional spheres and shape spaces
Ian L. Dryden
Ann. Statist. 33(4): 1643-1665 (August 2005). DOI: 10.1214/009053605000000264

Abstract

We consider the statistical analysis of data on high-dimensional spheres and shape spaces. The work is of particular relevance to applications where high-dimensional data are available—a commonly encountered situation in many disciplines. First the uniform measure on the infinite-dimensional sphere is reviewed, together with connections with Wiener measure. We then discuss densities of Gaussian measures with respect to Wiener measure. Some nonuniform distributions on infinite-dimensional spheres and shape spaces are introduced, and special cases which have important practical consequences are considered. We focus on the high-dimensional real and complex Bingham, uniform, von Mises–Fisher, Fisher–Bingham and the real and complex Watson distributions. Asymptotic distributions in the cases where dimension and sample size are large are discussed. Approximations for practical maximum likelihood based inference are considered, and in particular we discuss an application to brain shape modeling.

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Ian L. Dryden. "Statistical analysis on high-dimensional spheres and shape spaces." Ann. Statist. 33 (4) 1643 - 1665, August 2005. https://doi.org/10.1214/009053605000000264

Information

Published: August 2005
First available in Project Euclid: 5 August 2005

zbMATH: 1078.62058
MathSciNet: MR2166558
Digital Object Identifier: 10.1214/009053605000000264

Subjects:
Primary: 60G15 , 62H11

Keywords: Bingham distribution , complex Bingham , complex Watson , directional data , Functional data analysis , infinite-dimensional sphere , shape , sphere , von Mises–Fisher distribution , Watson distribution , Wiener measure , Wiener process

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 4 • August 2005
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