The Annals of Statistics

Statistical analysis on high-dimensional spheres and shape spaces

Ian L. Dryden

Full-text: Open access

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.

Article information

Source
Ann. Statist., Volume 33, Number 4 (2005), 1643-1665.

Dates
First available in Project Euclid: 5 August 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1123250225

Digital Object Identifier
doi:10.1214/009053605000000264

Mathematical Reviews number (MathSciNet)
MR2166558

Zentralblatt MATH identifier
1078.62058

Subjects
Primary: 62H11: Directional data; spatial statistics 60G15: Gaussian processes

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

Citation

Dryden, Ian L. Statistical analysis on high-dimensional spheres and shape spaces. Ann. Statist. 33 (2005), no. 4, 1643--1665. doi:10.1214/009053605000000264. https://projecteuclid.org/euclid.aos/1123250225


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