Hierarchies and Weak-hierarchies as Interval Convexities
Patrice Bertrand, Jean Diatta · HAL (Le Centre pour la Communication Scientifique Directe) · 2022
Hierarchies and weak-hierarchies as interval convexities P. Bertrand and J. Diatta There are several ways to characterize a hierarchy, one being a collection of nonempty subsets that are convex according to a type of interval function. This characterization in terms of interval convexity, extends to general classes of multilevel clusterings, thus providing a unifying heoretical framework [1, 2]. We expand this line of research, with a special attention to specifications allowing the capture of clusterings usually constructed in data mining practice, such as the Apresjan and the single-link hierarchies. We propose: (a) New characterizations of hierarchies and weak hierarchies as interval convexities, (b) Interval functions which induce known clustering schemes such as the Single Link hierarchy or the Apresjan hierarchy, (c) A sequence of nested families of interval convexities that is gradually increasing from the Apresjan hierarchy to the Single-Link hierarchy, which enables the detection of redundant clusters.