MACPLUS: A Mathematical Programming Approach to Fitting the ADCLUS Model

We present a new algorithm, MAPCLUS ( MA thematical P rogramming CLUS tering), for fitting the Shepard-Arabie ADCLUS (for AD ditive CLUS tering) model. MAPCLUS utilizes an alternating least squares method combined with a mathematical programming optimization procedure based on a penalty function approach, to impose discrete (0,1) constraints on parameters defining cluster membership. This procedure is supplemented by several other numerical techniques (notably a heuristically based combinatorial optimization procedure) to provide an efficient general-purpose computer implemented algorithm for obtaining ADCLUS representations. MAPCLUS is illustrated with an application to one of the examples given by Shepard and Arabie using the older ADCLUS procedure. The MAPCLUS solution uses half as many clusters to achieve nearly the same level of goodness-of-fit. Finally, we consider an extension of the present approach to fitting a three-way generalization of the ADCLUS model, called INDCLUS ( IN dividual D ifferences CLUS tering).