Predictive clustering

We show how to convert any clustering into a prediction set. This has the effect of converting the clustering into a (possibly overlapping) union of spheres or ellipsoids. The tuning parameters can be chosen to minimize the size of the prediction set. When applied to k-means clustering, this method solves several problems: the method tells us how to choose k, how to merge clusters and how to replace the Voronoi partition with more natural shapes. We show that the same reasoning can be applied to other clustering methods.