Axiomatic hierarchical clustering given intervals of metric distances

Abstract

This paper examines metric spaces in which the distance between any pair of nodes is given by an interval. The goal is to investigate methods for hierarchical clustering, i.e., a family of nested partitions indexed by a connectivity parameter, deduced from the underlying distance intervals of the metric spaces. Our construction is based on designing admissible methods abiding to the axioms of value and transformation. Two admissible methods are constructed and are shown to provide upper and lower bounds in the space of all admissible methods. Practical implications are explored by clustering moving points via snapshots. The proposed clustering methods succeed in identifying underlying clustering structures via the maximum and minimum distances in all snapshots.