Point Estimation of the Central Orientation of Random Rotations

Abstract

Data as three-dimensional rotations have application in computer science, kinematics, and materials sciences, among other areas. Estimating the central orientation from a sample of such data is an important problem, which is complicated by the fact that several different approaches exist for this, motivated by various geometrical and decision-theoretical considerations. However, little is known about how such estimators compare, especially on common distributions for location models with random rotations. We examine four location estimators, three of which are commonly found in different literatures and the fourth estimator (a projected median) is newly introduced. Our study unifies existing literature and provides a detailed numerical investigation of location estimators for three commonly used rotation distributions in statistics and materials science. While the data-generating model influences the best choice of an estimator, the proposed projected median emerges as an overall good performer, which can be suggested without particular distributional assumptions. We illustrate the estimators and our findings with data from a materials science study by approximating the central orientation of cubic crystals on the microsurface of a metal. Accompanying supplementary materials are available online.

rotations: An R Package for SO(3) Data

“…where r ∈ [−π, π) satisfies tr (R) = 1 + 2 cos r and tr(·) denotes the trace of a matrix. For more on the correspondence between SO(3) and skew-symmetric matrices see Stanfill et al (2013).…”

“…For this reason, the Riemannian distance is often considered the more natural metric on SO (3). As demonstrated in Stanfill et al (2013), the Euclidean and Riemannian distances are related by…”

“…The last nine columns, denoted v1-v9, represent the elements of the rotation matrix at that location in vector form. See Bingham et al (2009Bingham et al ( , 2010 and Stanfill et al (2013) for more details. In the example below we estimate the central orientation at location one.…”

Statistical methods for random rotations

“…Chang and Rivest 2001), where the data are then interpreted as random perturbations of an underlying central orientation S. In the following, we consider the projected arithmetic mean S n , which is perhaps the most common and popular estimator of the location S parameter for rotation data (cf. Moakher 2002;Fletcher et al 2003;Bhattacharya and Patrangenaru 2003;Bachmann et al 2010); see Stanfill et al (2013) for a discussion of point estimators for the central orientation.…”

“…We demonstrate these results with several common distributions for SO(3) data, involving the Cayley, circularvon Mises and matrix Fisher distributions (cf. Stanfill et al 2013). In Section 3.6, we apply our bootstrap to a real data set from materials science involving rotation data from electron backscatter diffraction.…”