Parameter estimation and model-based clustering with spherical normal distribution on the unit hypersphere

Abstract

In directional statistics, the von Mises-Fisher (vMF) distribution is one of the most basic and popular probability distributions for data on the unit hypersphere. Recently, the spherical normal (SN) distribution was proposed as an intrinsic counterpart to the vMF distribution by replacing the standard Euclidean norm with the great-circle distance, which is length of the shortest path joining two points on the unit sphere. Focusing on an isotropic version of SN distribution, it is shown that maximum likelihood estimators uniquely exist under mild support conditions. Since no analytic formula are available for the estimation, efficient numerical routines are proposed for parameter estimation. The estimation is considered in a general setting where non-negative weights are assigned to observations. This leads to a more interesting contribution for model-based clustering on the unit hypersphere by finite mixture model with SN distributions. Efficiency of optimization-based estimation procedures and effectiveness of SN mixture model are validated using simulated and real data examples.