Unsupervised clustering of multivariate circular data.

In this paper, we study an unsupervised clustering problem. The originality of this problem lies in the data, which consist of the positions of five separate X-ray beams on a circle. Radiation therapists positioned the five X-ray beam 'projectors' around each patient on a predefined circle. However, similarities exist in positioning for certain groups of patients, and we aim to describe these similarities with the goal of creating pre-adjustment settings that could help save time during X-ray positioning. We therefore performed unsupervised clustering of observed X-ray positions. Because the data for each patient consist of five angle measurements, Euclidean distances are not appropriated. Furthermore, we cannot perform k-means algorithm, usually used for minimizing corresponding distortion because we cannot calculate centers of clusters. We present here solutions to these problems. First, we define a suitable distance on the circle. Then, we adapt an algorithm based on simulated annealing to minimize distortion. This algorithm is shown to be theoretically convergent. Finally, we present simulations on simulated and real data.