Extrinsic Regression and Anti-Regression on Projective Shape Manifolds

Abstract

Necessary and sufficient conditions for the existence of the extrinsic mean and extrinsic antimean of a random object (r.o.) X on a compact metric space $\mathcal M,$ lead to considerations of extrinsic regression and antiregression functions on manifolds. One derives asymptotic distributions of kernel based estimators for antiregression functions with a numerical predictor, and use these in deriving confidence tubes for antiregression functions. In particular one considers VW-regression and VW-antiregression, for 3D projective shapes depending on a number of covariates. As an example, using 3D projective shape data extracted from multiple digital cameras images of a species of clamshells, one estimates the age dependent VW-regression and VW-antiregression for their 3D projective shapes, where the proxy for age is the number of seasonal ridges marks on shells, and the response is the 3D projective shape of landmark configurations of seven corresponding points marked on a shell’s surface.