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Abstract

We propose a regression model in which the responses are spherical variables and the covariates include linear and/or spherical variables. A novel link function is introduced by extending the M\"obius transformation on the sphere. This link function is an anisotropic mapping that enables scale control along each axis of the spherical covariates and for each linear covariate. It generalizes several well-known link functions for circular or linear covariates. Each parameter of the link function is clearly interpretable. For the error distribution, we consider a general class of elliptically symmetric distributions, which includes the Kent distribution, the elliptically symmetric angular Gaussian distribution, and the scaled von Mises-Fisher distribution. Axes of symmetry of the error distribution are determined using a method involving parallel transport. Maximum likelihood estimation is feasible via reparameterization of the proposed model. Moreover, the parameters of the link function and the shape/scale parameters of the error distribution are orthogonal in the sense of the Fisher information matrix. The proposed regression model is illustrated using two real datasets. An R software package accompanies this article.