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

Abstract

We investigate geodesic projections of von Mises–Fisher (vMF) distributed directional data. The vMF distribution for random directions on the $(p-1)$-dimensional unit hypersphere ${\mathbb{S}}^{p-1}\subset {\mathbb{R}}^p$ plays the role of multivariate normal distribution in directional statistics. For one-dimensional circle ${\mathbb{S}}^1$, the vMF distribution is called von Mises (vM) distribution. Projections onto geodesics are one of main ingredients of modeling and exploring directional data. We show that the projection of vMF distributed random directions onto any geodesic is approximately vM-distributed, albeit not exactly the same. In particular, the distribution of the geodesic-projected score is an infinite scale mixture of vM distributions. Approximations by vM distributions are given along various asymptotic scenarios including large and small concentrations $(\mathrm{\kappa }\to \mathrm{\infty },\mathrm{\kappa }\to 0$), high-dimensions $(p\to \mathrm{\infty })$, and two important cases of double-asymptotics $(p,\mathrm{\kappa }\to \mathrm{\infty }$, $\mathrm{\kappa }∕p\to c$ or $\mathrm{\kappa }∕\sqrt{p}\to \mathrm{\lambda }$), to support our claim: geodesic projections of the vMF are approximately vM. As one of potential applications of the result, we contemplate a projection pursuit exploration of high-dimensional directional data. We show that in a high dimensional model almost all geodesic-projections of directional data are nearly vM, thus measures of non-vM-ness are a viable candidate for projection index.