Principal classifications analysis: a method for generating consensus dendrograms and its application to three-way data

Abstract

A three-way data set is the result of the observation of data characterized by three modes: units, variables, and occasions. It is often useful to classify the elements of one mode on the basis of the other two. This is referred to as One Mode Classification (OMC) of a three-way data set and it can be seen as a synthesis of a set of hierarchical classifications, each one defined by applying a hierarchical algorithm to a two-mode matrix of the three-way data set (after standardizing variables if necessary). For example, the OMC of the units according to variables and occasions is a consensus of the set of hierarchical classifications defined by clustering the same units according to variables for each different occasion; i.e., the classification of the same multivariate units observed in different occasions. This last case will be considered in this paper. Many consensus methods can be used to achieve OMC of a three-way data set, however, often the set of hierarchical classifications is wide and some of these are dissimilar, hence a single synthesis, is generally not representative of the entire set. This might happen because in the observed three-way data set there are several groups of similar hierarchical classifications, i.e., formed by dendrograms that partially or completely repeat in different occasions without systematic differences. Therefore a consensus classification for each subset of similar classifications can be defined. Furthermore, when a single consensus is used for a set of classifications relative to different time situations, a loss of meaningful information on the evolution of the set of classifications is generally observed. To overcome these difficulties we propose the PRINcipal CLassifications Analysis of a three-way data set. The idea, in PRINCLA, is to find few non-observable hierarchical classifications, called principal classifications, each defined as a weighted mean hierarchical classification. No pairs of principal classifications are computed with the contribution of a same original hierarchical classification.