Speaker:  Eva Biswas, PhD Candidate, Department of Statistics, Iowa State University

Title: A Bootstrap-based test of rotational symmetry in orientation data. 

Abstract: Orientation data arise in a variety of studies, including human kinematics and materials science, where each observation is a 3x3 rotation matrix representing an object's position in a three-dimensional reference frame. In many applications, rotationally symmetric distributions are commonly assumed for modeling purposes, which views an orientation as a random isotropic perturbation of an underlying location parameter, as a type of location model for random matrices with symmetric errors.  However, despite the popularity of these models for orientations, no formal assessment has existed to adequately assess the appropriateness of such symmetry assumptions.

This work provides a valid test statistic for diagnosing rotational symmetry in orientation data.  The test characterizes distributional symmetry in orientation through the joint distributional form of three random variables extracted from the orientation, which can be examined for data using empirical characteristic functions.  We also develop a formal and practical bootstrap procedure for approximating a reference distribution for the test statistic.  The bootstrap serves to ''re-create" data under a null hypothesis of rotational symmetry, which induces good properties in the test, and bootstrap is further valuable as test statistics have complicated limit distributions that are impractical for direct use. Numerical studies indicate that the proposed test maintains size while exhibiting high power in a variety of departures from symmetry.  The testing approach is further illustrated with orientations collected in texture analysis from materials science.