Spherical regression models with general covariates and anisotropic errors

Existing parametric regression models in the literature for response data on the unit sphere assume that the covariates have particularly simple structure, for example that they are either scalar or are themselves on the unit sphere, and/or that the error distribution is isotropic. In many practical situations, such models are too inflexible. Here, we develop richer parametric spherical regression models in which the covariates can have quite general structure (for example, they may be on the unit sphere, in Euclidean space, categorical or some combination of these) and in which the errors are anisotropic. We consider two anisotropic error distributions—the Kent distribution and the elliptically symmetric angular Gaussian distribution—and two parametrisations of each which enable distinct ways to model how the response depends on the covariates. Various hypotheses of interest, such as the significance of particular covariates, or anisotropy of the errors, are easy to test, for example by classical likelihood ratio tests. We also introduce new model-based residuals for evaluating the fitted models. In the examples we consider, the hypothesis tests indicate strong evidence to favour the novel models over simpler existing ones.