Vol. 11, No. 2, 2020

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Algebraic analysis of rotation data

Michael F. Adamer, András C. Lőrincz, Anna-Laura Sattelberger and Bernd Sturmfels

Vol. 11 (2020), No. 2, 189–211
Abstract

We develop algebraic tools for statistical inference from samples of rotation matrices. This rests on the theory of D-modules in algebraic analysis. Noncommutative Gröbner bases are used to design numerical algorithms for maximum likelihood estimation, building on the holonomic gradient method of Sei, Shibata, Takemura, Ohara, and Takayama. We study the Fisher model for sampling from rotation matrices, and we apply our algorithms to data from the applied sciences. On the theoretical side, we generalize the underlying equivariant D-modules from SO(3) to arbitrary Lie groups. For compact groups, our D-ideals encode the normalizing constant of the Fisher model.

Keywords
algebraic analysis, Fisher model, maximum likelihood estimation, directional statistics, holonomic gradient method, Fourier transform
Mathematical Subject Classification
Primary: 14F10, 33F10, 62F10, 62H11, 62R01, 90C90
Milestones
Received: 3 December 2019
Revised: 9 April 2020
Accepted: 30 April 2020
Published: 28 December 2020
Authors
Michael F. Adamer
D-BSSE
ETH Zürich
4058 Basel
Switzerland
András C. Lőrincz
Humboldt-Universität zu Berlin
Institut für Mathematik
10099 Berlin
Germany
Anna-Laura Sattelberger
Max Planck Institute for Mathematics in the Sciences
04103 Leipzig
Germany
Bernd Sturmfels
University of California
Berkeley, CA 94720
United States
Max Planck Institute for Mathematics in the Sciences
04103 Leipzig
Germany