Extrinsic Regression and Anti-Regression on Projective Shape Manifolds

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 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 3 D projective shape of landmark configurations of seven corresponding points marked on a shell’s surface.