Spherical Regression Models Using Projective Linear Transformations

This article studies the problem of modeling relationship between two spherical (or directional) random variables in a regression setup. Here the predictor and the response variables are constrained to be on a unit sphere and, due to this nonlinear condition, the standard Euclidean regression models do not apply. Several past papers have studied this problem, termed spherical regression, by modeling the response variable with a von Mises-Fisher (VMF) density with the mean given by a rotation of the predictor variable. The few papers that go beyond rigid rotations are limited to one- or two-dimensional spheres. This article extends the mean transformations to a larger group--the projective linear group of transformations--on unit spheres of arbitrary dimensions, while keeping the VMF density to model the noise. It develops a Newton-Raphson algorithm on the special linear group for estimating the MLE of regression parameter and establishes its asymptotic properties when the sample-size becomes large. Through a variety of experiments, using data taken from projective shape analysis, cloud tracking, etc., and some simulations, this article demonstrates improvements in the prediction and modeling performance of the proposed framework over previously used models. Supplementary materials for this article are available online.