The Annals of Statistics

Statistical analysis on high-dimensional spheres and shape spaces

Ian L. Dryden

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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

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

First available in Project Euclid: 5 August 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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