Some Aspects of Clustering Functions

Abstract

Janowitz’s concept of hierarchical clustering which includes the concepts of hierarchical clustering due to Jardine and Sibson and Matula is extended—following the main stream of the theory of partially ordered sets—to describe all connections between hierarchies and isotone functions which measure the homogeneity or compactness of sets of data. In particular a very general description of those hierarchies which correspond bijectively to Hubert’s k-clustering functions is presented. As a consequence an exact characterization of the discrepancies between the original concept of Jardine and Sibson—its generalization due to Janowitz—and Hubert’s concept of hierarchical clustering is possible.