Robust statistical methods for the rotation group

Electron backscatter diffraction (EBSD) is a technique used in material sciences to determine the orientation of crystals on the surface of metals. Because crystal orientation plays a key role in determining material strength, accurate identification of a crystal's orientation is a crucial problem. To achieve a high level of accuracy repeated EBSD sessions are aggregated, but material degradation and misalignment can occur between sessions resulting in spurious crystal orientations. Currently the projected mean is used to estimate the true crystal orientation from repeated EBSD sessions, but it has been shown to perform poorly when extreme observations are present. The projected and geometric medians have been proposed as robust alternatives, but a rigorous treatment of their behavior has not been undertaken. In this manuscript we evaluate the behavior of the projected estimators as well as the geometric median when extreme observations are present. We also quantify the effect of extreme observations on the projected estimators by deriving their influence functions. Then we propose a statistic that can be used to identify extreme observations and propose a novel mean-type estimator that has improved robustness properties over the projected mean estimator. The investigated estimators are illustrated with a EBSD data set.