Abstract
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.
Highlights
Spherical data are observations that lie on the unit sphere Sp−1 = y ∈ Rp : y y = 1
Recent work in regression modelling on general Riemannian manifolds, for which the unit sphere is a special case, includes the nonparametric approach of Lin et al (2017), who develop local regression models assuming Euclidean covariates, and the semi-parametric approach of Cornea et al (2017), who use parametric link functions mapping from a general covariate space to the manifold, with a nonparametric error distribution; though in neither is the possibility of anisotropic errors explicitly considered
The regression models we have introduced are rather more general than existing regression models in the literature, allowing covariates with general structure, and errors that are nonisotropic
Summary
The second parametrisation we consider, generalising (2), is in terms of a pair of vectors, μ ∈ R3 and γ ∈ R2, in which, as in (2), μ controls the mean direction and concentration; γ ∈ R2 controls eccentricity and orientation of the elliptical contours These two parametrisations lend themselves to different ways of modelling how the response variable depends on covariates. Presnell et al (1998) considers regression on S1 for a general covariate xi , assuming IAG errors We mention this model in particular because it is a close analogue on S1 of our ESAG2 model on S2 in the isotropic case (which corresponds to γ = 0), as discussed later. Code for fitting the models in this paper is available on the second author’s web page
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