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Abstract
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We develop algebraic tools for statistical inference from samples of rotation matrices. This rests on the
theory of
-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
-modules
from
to arbitrary Lie groups. For compact groups, our
-ideals
encode the normalizing constant of the Fisher model.
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Keywords
algebraic analysis, Fisher model, maximum likelihood
estimation, directional statistics, holonomic gradient
method, Fourier transform
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Mathematical Subject Classification
Primary: 14F10, 33F10, 62F10, 62H11, 62R01, 90C90
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Milestones
Received: 3 December 2019
Revised: 9 April 2020
Accepted: 30 April 2020
Published: 28 December 2020
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