Multilevel clustering models and interval convexities

The k-weakly hierarchical, pyramidal and paired hierarchical models are alternative multilevel clustering models that extend hierarchical clustering. In this paper, we study these various multilevel clustering models in the framework of general convexity. We prove a characterization of the paired hierarchical model via a four-point condition on the segment operator, and examine the case of k-weakly hierarchical models for k≥3. We also prove sufficient conditions for an interval convexity to be either hierarchical, paired hierarchical, pyramidal, weakly hierarchical or k-weakly hierarchical. Moreover, we propose a general algorithm for computing the interval convexity induced by any given interval operator, and deduce a unified clustering scheme for capturing either of the considered multilevel clustering models. We illustrate our results with two interval operators that can be defined from any dissimilarity index and propose a parameterized definition of an adaptive interval operator for cluster analysis.