One-sample Bayes inference for symmetric distributions of 3-D rotations

A variety of existing symmetric parametric models for 3-D rotations found in both statistical and materials science literatures are considered from the point of view of the “uniform-axis-random-spin” (UARS) construction. One-sample Bayes methods for non-informative priors are provided for all of these models and attractive frequentist properties for corresponding Bayes inference on the model parameters are confirmed. Taken together with earlier work, the broad efficacy of non-informative Bayes inference for symmetric distributions on 3-D rotations is conclusively demonstrated

A Wrapped Trivariate Normal Distribution and Bayes Inference for 3-D Rotations

For modeling orientation data represented as 3 × 3 rotation matrices, we develop a wrapped trivariate normal distribution (wTND) under which random rotations have simple geometric construction as symmetric errors about a mean. While of interest in its own right, the wTND also provides simple and effective approximations to the isotropic Gaussian distribution on rotations, with some advantages over approximations based on other commonly used models. We develop non-informative Bayes inference for the wTND via Markov Chain Monte Carlo methods that allow straightforward computations in a model where maximum likelihood is undefined. Credible regions for model parameters (including a fixed 3 × 3 mean rotation) are shown to possess good frequentist coverage properties. We illustrate the model and inference method with orientation data collected in texture analysis from materials science

Isotropic Distributions for 3-Dimension Rotations and One-Sample Bayes Inference

This dissertation discusses models for 3-dimensional orientations. A new Uniform-Axis- Random-Spin (UARS) family is developed and the corresponding Bayes inferences for all UARS distributions are studied. The effectiveness of one-sample non-informative Bayes methods are demonstrated which indicates the broad use in material science. Then an R package has been developed which is useful for practical modelers

Likelihood and Bayes inference for a class of distributions on orientations in 3 dimensions

This dissertation concerns variation in 3-dimensional orientations (represented by 3 x 3 orthogonal matrices with positive determinant). The Uniform-Axis-Random-Spin (UARS) class of distributions for 3-dimensional orientations is identified and likelihood and Bayes inference for the class are developed, resulting in new practical statistical methods for modeling orientation data. Two members of the UARS class, the von Mises version and the symmetric matrix von Mises-Fisher distribution, are considered in detail. The methodology developed is used in materials science and human kinematics applications

Finite-sample investigation of likelihood and Bayes inference for the symmetric von Mises-Fisher distribution

We consider likelihood and Bayes analyses for the symmetric matrix von Mises-Fisher (matrix Fisher) distribution, which is a common model for three-dimensional orientations (represented by 3x3 orthogonal matrices with a positive determinant). One important characteristic of this model is a 3x3 rotation matrix representing the modal rotation, and an important challenge is to establish accurate confidence regions for it with an interpretable geometry for practical implementation. While we provide some extensions of one-sample likelihood theory (e.g., Euler angle parametrizations of modal rotation), our main contribution is the development of MCMC-based Bayes inference through non-informative priors. In one-sample problems, the Bayes methods allow the construction of inference regions with transparent geometry and accurate frequentist coverages in a way that standard likelihood inference cannot. Simulation is used to evaluate the performance of Bayes and likelihood inference regions. Furthermore, we illustrate how the Bayes framework extends inference from one-sample problems to more complicated one-way random effects models based on the symmetric matrix Fisher model in a computationally straightforward manner. The inference methods are then applied to a human kinematics example for illustration.

Modeling, inference and clustering for equivalence classes of 3-D orientations

Investigating cubic crystalline structures of specimens is an important way to study properties of materials in text analysis. Crystals in metal specimens have internally homogeneous orientations relative to a pre-chosen reference coordinate system. Clusters of crystals in the metal with locally similar orientations constitute so-called grains. The nature of these grains (shape, size, etc.) affects physical properties (e.g., hardness, conductivity, etc.) of the material. Electron backscatter diffraction (EBSD) machines are often use to measure orientations of crystals in metal specimens. However, orientations reported by EBSD machines are in truth equivalence classes of crystallographically symmetric orientations. Motivated by the materials science applications, we formulate parametric probability models for unlabeled orientation data. This amounts to developing models on equivalence classes of 3-D rotations. A Bayesian method is developed for inferencing parameters in the models, which is generally superior to large-sample methods based on likelihood estimation. We also proposed an algorithms for clustering equivalence classes of 3-D orientations. As we continue to work on this area, we found and studied an interesting class of Markov chains with state spaces partitions of a finite set. These Markov chains have some properties that make them attractive in their own right, and they are potentially helpful in Bayesian model-based clustering