Clustering and isolation in the consensus problem for partitions

We examine the problem of aggregating several partitions of a finite set into a single consensus partition We note that the dual concepts of clustering and isolation are especially significant in this connection. The hypothesis that a consensus partition should respect unanimity with respect to either concept leads us to stress a consensus interval rather than a single partition. The extremes of this interval are characterized axiomatically. If a sufficient totality of traits has been measured, and if measurement errors are independent, then a true classifying partition can be expected to lie in the consensus interval. The structure of the partitions in the interval lends itself to partial solutions of the consensus problem Conditional entropy may be used to quantify the uncertainty inherent in the interval as a whole     

Preserving consensus hierarchies

In numerical taxonomy we often have the task of finding a consensus hierarchy for a given set of hierarchies. This consensus hierarchy should reflect the substructures which are common to all hierarchies of the set. Because there are several kinds of substructures in a hierarchy, the general axiom to preserve common substructures leads to different axioms for each kind of substructure. In this paper we consider the three substructurescluster, separation, andnesting, and we give several characterizations of hierarchies preserving these substructures. These characterizations facilitate interpretation of axioms for preserving substructures and the examination of properties of consensus methods. Finally some extensions concerning the preserving of qualified substructures are discussed.The author is grateful to the editor and the referees for their helpful suggestions and to H. J. Bandelt for his comments on an earlier version of this paper.     

On the use of ordered sets in problems of comparison and consensus of classifications

Ordered set theory provides efficient tools for the problems of comparison and consensus of classifications Here, an overview of results obtained by the ordinal approach is presented Latticial or semilatticial structures of the main sets of classification models are described Many results on partitions are adaptable to dendrograms; many results on n-trees hold in any median semilattice and thus have counterparts on ordered trees and Buneman (phylogenetic) trees For the comparison of classifications, the semimodularity of the ordinal structures involved yields computable least-move metrics based on weighted or unweighted elementary transformations In the unweighted case, these metrics have simple characteristic properties For the consensus of classifications, the constructive, axiomatic, and optimization approaches are considered Natural consensus rules (majoritary, oligarchic, ) have adequate ordinal formalizations A unified presentation of Arrow-like characterization results is given In the cases of n-trees, ordered trees and Buneman trees, the majority rule is a significant example where the three approaches convergeThe authors would like to thank the anonymous referees for helpful suggestions on the first draft of this paper, and W H E Day for his comments and his significant improvements of style