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

Abstract

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.

Recent advances in directional statistics

“…Kent et al (2016) introduced a fiveparameter special case of the Fisher-Bingham model for use with data patterns that are unimodal and concentrated near a great circle. More recently, Kim et al (2019) proposed two kinds of small-sphere distributions, one of which is a member of the Fisher-Bingham family. Previously, Oualkacha and Rivest (2009) had developed an alternative to the Bingham distribution for modelling symmetric axial data, with a simple closed-form normalising constant.…”

“…Related regression problems for a S 2 -valued response include the fitting of small circles to spherical data (Rivest 1999) and the analysis of rotational deformations through fitting small circles on the sphere nonparametrically (Schulz et al 2015) and parametrically (Kim et al 2019).…”

Exploratory Factor Analysis of Data on a Sphere

“…However, this approach is not practical in high dimensions: indeed, in their application, [27] use a priori dimension reduction (into the first three principal components) on the raw data as a preprocessing step before modeling and analysis. Other distributions such as the complex Bingham [28], Fisher-Bingham [29] and the real/complex Watson distributions [30], [31] also suffer from intractability with dimensions greater than three.…”

Geodesic projection of the von Mises–Fisher distribution for projection pursuit of directional data

“…We consider the one-dimensional projection of directional data onto geodesics, a notion closest to the orthogonal projection onto a vector in Euclidean space. Projections onto geodesics and the evaluation of the projection scores and residuals are fundamental ingredients of modern directional statistics and, in general, statistics on manifolds; applications include dimension reduction (Fletcher et al, 2004;Jung, Dryden and Marron, 2012), regression (Fletcher, 2013;Cornea et al, 2017), classification (Pizer and Marron, 2017) and developments of parametric models (Schulz et al, 2015;Kim et al, 2019). Here, the projection onto a geodesic γ is defined intrinsically.…”