Consensus and fuzzy partition of dendrograms from a three-way dissimilarity array

In cluster analysis, a problem often addressed is finding a consensus on a set of hierarchical classifications of the same set of objects, named primary hierarchies (dendrograms). A unique consensus of the primary hierarchies, called a secondary hierarchy, is sufficient to synthesize the relevant clustering information only when the primary dendrograms are similar to one another. In contrast, when several differences are observed, more than one consensus secondary hierarchy is required to clearly synthesize the different primary hierarchies. Our methodology, PARoDENo3WD (PARtition of DENdrograms of a 3-Way Data array), aims to obtain a secondary fuzzy partition of the primary hierarchies, where hierarchies belonging to the same class are perceived as similar. Each class is associated with a consensus hierarchy. The fuzzy approach allows each primary hierarchy to contribute to the definition of all classes of the secondary partition, according to different membership degrees. In this way, the ‘clustering uncertainty’ is taken into account. Indeed, in several realistic applications, primary dendrograms belonging to different classes of the secondary partition may share some similar features. The performance of PARoDENo3WD is evaluated by an extended simulation study, generating 1800 three-way datasets. An application of PARoDENo3WD on a real data set is given