Melissa A. Bingham

Modeling and Inference for Measured Crystal Orientations and a Tractable Class of Symmetric Distributions for Rotations in Three Dimensions

Electron backscatter diffraction (EBSD) is a technique used in materials science to study the microtexture of metals, producing data that measure the orientations of crystals in a specimen. We examine the precision of such data based on a useful class of distributions on orientations in three dimensions (as represented by 3×3 orthogonal matrices with positive determinants). Although such modeling has received attention in the statistical literature, the approach taken typically has been based on general “special manifold” considerations, and the resulting methodology may not be easily accessible to nonspecialists. We take a more direct modeling approach, beginning from a simple, intuitively appealing mechanism for generating random orientations specifically in three-dimensional space. The resulting class of distributions has many desirable properties, including directly interpretable parameters and relatively simple theory. We investigate the basic properties of the entire class and one-sample quasi-likelihood–based inference for one member of the model class, producing a new statistical methodology that is practically useful in the analysis of EBSD data. This article has supplementary material online.

Bayes one-sample and one-way random effects analyses for 3-D orientations with application to materials science

We consider Bayes inference for a class of distributions on orien- tations in 3 dimensions described by 3 3 rotation matrices. Non-informative priors are identied and Metropolis-Hastings within Gibbs algorithms are used to generate samples from posterior distributions in one-sample and one-way random eects models. A simulation study investigates the performance of Bayes analyses based on non-informative priors in the one-sample case, making comparisons to quasi-likelihood inference. A second simulation study investigates the behavior of posteriors for some informative priors. Bayes one-way random eect analyses of orientation matrix data are then developed and the Bayes methods are illustrated in a materials science application.

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.

Uniformly Hyper-Efficient Bayes Inference in a Class of Nonregular Problems

We present a tractable class of nonregular continuous statistical models where 1) likelihoods have multiple singularities and ordinary maximum likelihood is intrinsically unavailable, but 2) Bayes procedures achieve convergence rates better than n−1 across the whole parameter space. In fact, for every p>1, there is a member of the class for which the posterior distribution is consistent at rate n−p uniformly in the parameter.

Using Enrollment Data to Predict Retention Rate

First- to second-year retention rates are one metric reported by colleges and universities to convey institutional success to a variety of external constituents. But how much of a retention rate is institutional inputs, and how much can be understood by examining student inputs? The authors utilize multi-year, multi-institutional data to examine differences in predicted and observed retention rates at a mid-size, selective, comprehensive public university in the Midwest.

A permutation test for the spread of three-dimensional rotation data

ABSTRACT The permutation test is a nonparametric test that can be used to compare measures of spread for two data sets, but is yet to be explored in the context of three-dimensional rotation data. A permutation test for such data is developed and the statistical power of this test is considered under various conditions. The test is then used in a brief application comparing movement around the calcaneocuboid joint for a human, chimpanzee, and baboon.