Small-sphere distributions for directional data with application to medical imaging

We propose novel parametric concentric multi-unimodal small-subsphere families of densities for p − 1 ≥ 2-dimensional spherical data. Their parameters describe a common axis for K small hypersubspheres, an array of K directional modes, one mode for each subsphere, and K pairs of concentrations parameters, each pair governing horizontal (within the subsphere) and vertical (orthogonal to the subsphere) concentrations. We introduce two kinds of distributions. In its one-subsphere version, the first kind coincides with a special case of the Fisher–Bingham distribution, and the second kind is a novel adaption that models independent horizontal and vertical variations. In its multisubsphere version, the second kind allows for a correlation of horizontal variation over different subspheres. In medical imaging, the situation of p − 1 = 2 occurs precisely in modeling the variation of a skeletally represented organ shape due to rotation, twisting, and bending. For both kinds, we provide new computationally feasible algorithms for simulation and estimation and propose several tests. To the best knowledge of the authors, our proposed models are the first to treat the variation of directional data along several concentric small hypersubspheres, concentrated near modes on each subsphere, let alone horizontal dependence. Using several simulations, we show that our methods are more powerful than a recent nonparametric method and ad hoc methods. Using data from medical imaging, we demonstrate the advantage of our method and infer on the dominating axis of rotation of the human knee joint at different walking phases.