Nonparametric Regression for Spherical Data

Abstract

We are grateful to an Associate Editor and two anonymous Referees for a careful reading of the manuscript and helpful comments, which led to an improved version of this article. We also thank Isabella Fabbri for having read and discussed some parts of the manuscript. 2 We develop nonparametric smoothing for regression when both the predictor and the response variables are defined on a sphere of whatever dimension. A local polynomial fitting approach is pursued, which retains all the advantages in terms of rate optimality, interpretability and ease of implementation widely observed in the standard setting. Our estimates have a multi-output nature, meaning that each coordinate is separately estimated, within a scheme of a regression with a linear response. The main properties include linearity and rotational equivariance. This research has been motivated by the fact that very few models describe this kind of regression. Such current methods are surely not widely employable since they have a parametric nature, and also require the same dimensionality for prediction and response spaces, along with non-random design. Our approach does not suffer these limitations. Real data case studies and simulation experiments are used to illustrate the effectiveness of the method.