THE RIEMANNIAN STRUCTURE OF EUCLIDEAN SHAPE SPACES - A NOVEL ENVIRONMENT FOR STATISTICS

• 成果类型: Article
• 署名作者: LE, HL; KENDALL, DG
• 署名单位: University of Cambridge
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176349259
• 发表日期: 1993
• 页码: 1225-1271
• 关键词: random spherical triangles; distributions; densities; sets; diffusion; points

摘要

The Riemannian metric structure of the shape space SIGMA(m)k for k labelled points in R(m) was given by Kendall for the atypically simple situations in which m = 1 or 2 and k greater-than-or-equal-to 2. Here we deal with the general case (m greater-than-or-equal-to 1, k greater-than-or-equal-to 2) by using the properties of Riemannian submersions and warped products as studied by O'Neill. The approach is via the associated size-and-shape space that is the warped product of the shape space and the half-line R+ (carrying size), the warping function being equal to the square of the size. When combined with parallel studies by Le of the corresponding global geodesic geometry, the results obtained here determine the environment in which shape-statistical calculations have to be acted out. Finally three different applications are discussed that illustrate the theory and its use in practice.