#### Vol. 11, No. 2, 2020

 Recent Issues Volume 11, Issue 2 Volume 11, Issue 1
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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 $SO\left(3\right)$ 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