On Optimal Tests for Rotational Symmetry Against New Classes of Hyperspherical Distributions

Discrete mixture representations of spherical distributions

We obtain discrete mixture representations for parametric families of probability distributions on Euclidean spheres, such as the von Mises–Fisher, the Watson and the angular Gaussian families. In addition to several special results we present a general approach to isotropic distribution families that is based on density expansions in terms of special surface harmonics. We discuss the connections to stochastic processes on spheres, in particular random walks, discrete mixture representations derived from spherical diffusions, and the use of Markov representations for the mixing base to obtain representations for families of spherical distributions.

Using directional statistics to test hypotheses regarding rigid body attitude: Comparison to univariate and multivariate Cardan angle tests

Rigid body attitude and single-joint kinematics are typically expressed using three Cardan angles which represent rotations in anatomical planes. It was recently shown in the Biomechanics literature that Cardan angles inaccurately estimate true mean attitude due to an important mathematical inadequacy: attitude under-representation; at least four quantities are needed to unambiguously specify attitude. Directional statistics, which is the multivariate generalization of (univariate) circular statistics, solves this problem using four-dimensional unit vectors and the mathematics of hyperspherical geometry. The purpose of this study was to compare the results of directional analysis to the results of uni- and multi-variate Cardan analysis for representative joint kinematic data during gait. We analyzed hip, knee and pelvis data from three open datasets and report exemplary results for knee kinematics in v-cut vs. side shuffle tasks. We also conducted Monte Carlo simulations, using synthetic data with precisely controlled true angular effects, to systematically compare directional and Cardan analyses. Results show that directional analysis yielded considerably smaller p values (p<0.03) than Cardan analysis (p>0.055) for the exemplary dataset. Simulation results confirmed that directional analysis is considerably more powerful (i.e., much more able to detect true angular effects) than both uni- and multi-variate Cardan analysis. These results suggest that directional statistics should be used to analyse attitude, including 3D joint kinematics, to avoid false negatives.