🤖 AI Summary
This paper addresses regression modeling with spherical responses and mixed covariates—both linear and spherical. We propose an anisotropic link function based on an extended Möbius transformation, which enables flexible, interpretable directional scaling and generalizes conventional spherical regression links. Crucially, the link ensures orthogonality—in the Fisher information sense—between regression parameters and shape parameters of the error distribution. Coupled with elliptically symmetric error distributions (e.g., Kent and scaled von Mises–Fisher distributions), we employ parallel transport to identify the symmetry axis and reparameterize the model for numerically stable maximum likelihood estimation. The methodology is validated on real-world data, and an accompanying R package facilitates practical implementation. Our approach substantially enhances the flexibility, interpretability, and statistical robustness of spherical regression models.
📝 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 Möbius 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.