Electronic Journal of Statistics

Optimal designs for regression with spherical data

Holger Dette, Maria Konstantinou, Kirsten Schorning, and Josua Gösmann

Full-text: Open access

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 misorientation 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 the $\Phi _{p}$-criteria introduced by Kiefer (1974) 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.

Article information

Source
Electron. J. Statist., Volume 13, Number 1 (2019), 361-390.

Dates
Received: April 2018
First available in Project Euclid: 9 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1549681241

Digital Object Identifier
doi:10.1214/18-EJS1524

Mathematical Reviews number (MathSciNet)
MR3910487

Zentralblatt MATH identifier
1407.62285

Subjects
Primary: 62K05: Optimal designs
Secondary: 33C55: Spherical harmonics

Keywords
Optimal design hyperspherical harmonics series estimation $\Phi _{p}$-optimality

Rights
Creative Commons Attribution 4.0 International License.

Citation

Dette, Holger; Konstantinou, Maria; Schorning, Kirsten; Gösmann, Josua. Optimal designs for regression with spherical data. Electron. J. Statist. 13 (2019), no. 1, 361--390. doi:10.1214/18-EJS1524. https://projecteuclid.org/euclid.ejs/1549681241


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