Statistica Sinica 24 (2014), 897-917

A WRAPPED TRIVARIATE NORMAL DISTRIBUTION

AND BAYES INFERENCE FOR 3-D ROTATIONS

Yu Qiu, Daniel J. Nordman and Stephen B. Vardeman

Iowa State University

Abstract: 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.

Key words and phrases: CLT, credible set of cones, isotropic Gaussian distribution, MCMC, UARS model.