K-Means, Ward and Probabilistic Distance-Based Clustering Methods with Contiguity Constraint

Abstract

We analyze some possibilities of using contiguity (neighbourhood) matrix as a constraint in the clustering made by the k-means and Ward methods as well as by an approach based on distances and probabilistic assignments aimed at obtaining a solution of the multi-facility location problem (MFLP). That is, some special two-stage algorithms being the kinds of clustering with relational constraint are proposed. They optimize division of set of objects into clusters respecting the requirement that neighbours have to belong to the same cluster. In the case of the probabilistic d-clustering, relevant modification of its target function is suggested and studied. Versatile simulation study and empirical analysis verify the practical efficiency of these methods. The quality of clustering is assessed on the basis of indices of homogeneity, heterogeneity and correctness of clusters as well as the silhouette index. Using these tools and similarity indices (Rand, Peirce and Sokal and Sneath), it was shown that the probabilistic d-clustering can produce better results than Ward’s algorithm. In comparison with the k-means approach, the probabilistic d-clustering—although gives rather similar results—is more robust to creation of trivial (of which empty) clusters and produces less diversified (in replications, in terms of correctness) results than k-means approach, i.e. is more predictable from the point of view of the clustering quality.