Bayesian Inference for a New Class of Distributions on Equivalence Classes of Three-Dimensional Orientations With Applications to Materials Science

Abstract

Experiments in materials science investigating cubic crystalline structures often collect data which are in truth equivalence classes of crystallographically symmetric orientations. These intend to represent how lattice structures of particles are orientated relative to a reference coordinate system. Motivated by a materials science application, we formulate parametric probability models for “unlabeled orientation data.” This amounts to developing models on equivalence classes of three-dimensional rotations. We use a flexible existing model class for random rotations (called uniform-axis-random-spin models) to induce probability distributions on the equivalence classes of rotations. We develop one-sample Bayesian inference for the parameters in these models, and compare this methodology to some likelihood-based approaches. We also contrast the new parametric analysis of unlabeled orientation data with other analyses that proceed as if the data have been preprocessed into honest orientation data. Supplementary materials for this article are available online.

• Cone-shaped region
• Rotation matrix
• Texture analysis
• Uniform-axis-random-spin

ACKNOWLEDGMENTS

The authors wish to thank Editor Hugh Chipman, an associate editor, and three reviewers for their careful reading and thoughtful comments, which significantly improved the article.