One-Mode Classification of a Three-Way Data Matrix

X is the automatic hierarchical classification of one mode (units or variables or occasions) of X on the basis of the other two. In this paper the case of OMC of units according to variables and occasions is discussed. OMC is the synthesis of a set of hierarchical classifications Delta obtained from X; e.g., the OMC of units is the consensus (synthesis) among the set of dendograms individually defined by clustering units on the basis of variables, separately for each given occasion of X. However, because Delta is often formed by a large number of classifications, it may be unrealistic that a single synthesis is representative of the entire set. In this case, subsets of similar (homegeneous) dendograms may be found in Delta so that a consensus representative of each subset may be identified. This paper proposes, PARtition and Least Squares Consensus cLassifications Analysis (PARLSCLA) of a set of r hierarchical classifications Delta. PARLSCLA identifies the best least-squares partition of Delta into m (1 <= m <= r) subsets of homogeneous dendograms and simultaneously detects the closest consensus classification (a median classification called Least Squares Consensus Dendogram (LSCD) for each subset. PARLSCLA is a generalization of the problem to find a least-squares consensus dendogram for Delta. PARLSCLA is formalized as a mixed-integer programming problem and solved with an iterative, two-step algorithm. The method proposed is applied to an empirical data set.