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

Bingham, Melissa
Vardeman, Stephen
Nordman, Daniel
Vardeman, Stephen
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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.

<p>This is an Accepted Manuscript of an article published by Taylor & Francis in Journal of the American Statistical Association in 2009, available online: <a href="" target="_blank"></a></p>
Directional data, Electron backscatter diffraction, Euler angle, Haar measure, Orthogonal matrix, Quasi-likelihood ratio test, Uniform axis–random spin distribution, von Mises distribution, Wald test