Extended Box Clustering for Classification Problems

In this work we address a technique, based on elementary convex sets called box hulls, for effectively grouping finite point sets into non-convex objects, called box clusters. The proposed clustering approach is based on homogeneity conditions, not according to some distance measure, and it is situated inside the theoretical framework of Supervised clustering. This approach extends the so-called (convex) box clustering, originally developed in the context of the logical analysis of data, to non-convex geometry. We briefly discuss the topological properties of these clusters and introduce a family of hypergraphs, called incompatibility hypergraphs; the main aim for these hypergraphs is their role in clustering algorithms, even if they have strong theoretical properties as shown in other works in literature. We also discuss of supervised classification problems and generalized Voronoi diagrams are considered to define a classifier based on box clusters. Finally, computational experiments on real world data are used to show the efficacy of our methods both in terms of clustering and accuracy.