Anti-regression on manifolds with an application to 3D projective shape analysis

Given a random object X on a compact metric space M, provided with a “chord” distance induced by the Euclidean distance on the numerical space where M is embedded, one considers the Fréchet function, expected square of the chord distance from the random object X to a point on M. This function attains its maximum at a set of points, called the extrinsic antimean set. In case the extrinsic antimean set has one point only, that point is called extrinsic antimean of X. Given a pair of random objects (Y,X) on a product of object spaces N ×M, where M is compact and is embedded into a numerical space, the value of the anti-regression function at a point y on N as the conditional extrinsic antimean of X given Y = y. Here one gives necessary and sufficient conditions that insure that the antiregression function is well defined, and as an application one gives an example of extrinsic anti-regression in projective shape analysis for a clamshells species found in the Florida panhandle, where the predictor is age and the response is 3D projective shape. M.S.C. 2010: 62G08, 62H35.