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Statistical analysis on high-dimensional spheres and shape spaces. (English) Zbl 1078.62058

Summary: 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 the Wiener measure. We then discuss densities of Gaussian measures with respect to the 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.

MSC:

62H11 Directional data; spatial statistics
62H10 Multivariate distribution of statistics
62M99 Inference from stochastic processes
60G15 Gaussian processes
62H05 Characterization and structure theory for multivariate probability distributions; copulas

Software:

fda (R)

References:

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