Title: Statistical analysis on manifolds: A nonparametric approach for inference on shape spaces

Author(s): Abhishek Bhattacharya
Issue: Volume 70 Series A Part 2 Year 2008
Pages: 223 -- 266
This article concerns nonparametric statistics on manifolds with special emphasis on ${\it landmark \ based \ shape}$ spaces in which a $k$-ad, i.e., a set of $k$ points or landmarks on an object or a scene, is observed in 2D or 3D, for purposes of identification, discrimination, or diagnostics. Two different notions of shape are considered: ${\it reflection \ shape}$ invariant under all translations, scaling and orthogonal transformations, and ${\it affine \ shape}$ invariant under all affine transformations. A computation of the ${\it extrinsic \ mean}$ reflection shape, which has remained unresolved in earlier works, is given in arbitrary dimensions, enabling one to extend nonparametric inference on Kendall type shape manifolds from 2D to higher dimensions. For both reflection and affine shapes, two sample test statistics are constructed based on appropriate choice of orthonormal frames on the tangent bundle and computations of differentials of projection maps with respect to them at the sample extrinsic mean. The samples in consideration can be either independent or be the outcome of a matched pair experiment. Examples are included to illustrate the theory.
AMS (2000) subject classification. Primary 62H35; Secondary 62G20, 62F40.
Keywords and phrases: Manifold, equivariant embedding, shape space of $k$ - ads, Fréchet function, extrinsic mean and variation, nonparametric analysis.