Abstract
This paper presents a Bayesian model for the clustering of non-ordered multivariate directional or circular data. The particular trait of our data is that each single observation is made up of $k \ge 2$ non-ordered points on the circle. We introduce a hierarchical model that combines a symmetrization technique, Projected Normal distributions and a Dirichlet Process. One parameter is introduced to model the non-ordered trait and another one to control the variability of the angles on the circle. An informative prior on the relative locations of the $k$ angles is also provided. The gain of the symmetrization is highlighted by a theoretical study. The parameters of the model are then inferred using a Metropolis-Hastings within Gibbs algorithm. Simulated datasets are analyzed to study the sensitivity to hyperparameters. Then, the benefits of our approach are illustrated by clustering real data made up of the positions of five separate radiotherapy x-ray beams on a circle.