Statistical analysis on high-dimensional spheres and shape spaces

• 成果类型: Article
• 署名作者: Dryden, IL
• 署名单位: University of Nottingham
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053605000000264
• 发表日期: 2005
• 页码: 1643-1665
• 关键词: limit-theorems; latent roots; MANIFOLDS; bingham

摘要

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.