Optimal designs for regression with spherical data

TL;DR

This paper develops optimal experimental designs for regression models with spherical predictors, crucial for material science applications involving hyperspherical harmonic series estimation.

Contribution

It explicitly determines optimal designs for spherical regression problems, including uniform and discrete designs, with applications to crystallography.

Findings

Uniform distribution on the sphere is optimal.

Constructed discrete designs match continuous optimal information matrices.

Designs improve series estimation of hyperspherical harmonics.

Abstract

In this paper optimal designs for regression problems with spherical predictors of arbitrary dimension are considered. Our work is motivated by applications in material sciences, where crystallographic textures such as the missorientation distribution or the grain boundary distribution (depending on a four dimensional spherical predictor) are represented by series of hyperspherical harmonics, which are estimated from experimental or simulated data. For this type of estimation problems we explicitly determine optimal designs with respect to Kiefers $\Phi_p$-criteria and a class of orthogonally invariant information criteria recently introduced in the literature. In particular, we show that the uniform distribution on the $m$-dimensional sphere is optimal and construct discrete and implementable designs with the same information matrices as the continuous optimal designs. Finally, we illustrate the advantages of the new designs for series estimation by hyperspherical harmonics, which are symmetric with respect to the first and second crystallographic point group.