Convexity-based clustering criteria: theory,algorithms, and applications in statistics

This paper deals with the construction of optimum partitions of for a clustering criterion which is based on a convex function of the class centroids as a generalization of the classical SSQ clustering criterion for n data points. We formulate a dual optimality problem involving two sets of variables and derive a maximum-support-plane (MSP) algorithm for constructing a (sub-)optimum partition as a generalized k-means algorithm. We present various modifications of the basic criterion and describe the corresponding MSP algorithm. It is shown that the method can also be used for solving optimality problems in classical statistics (maximizing Csiszárs -divergence) and for simultaneous classification of the rows and columns of a contingency table.     

Krishnaiah,P.R. and C.R. Rao (eds):Sampling. Handbook of Statistics,vol. 6

In a recent paper Yager (1988) introduced a special type of aggregation operators which he called OWA-operators and applied it to multicriteria decisionmaking. In our paper we give a more general definition of such operators and investigate in some detail the structural properties of them. In particular we introduce a well motivated ordering relation on weight systems which is different but compatible with Yager's degree of “orness”. Our results have immediate applications to the theory of generalized quantifiers as also discussed by Yager. Details concerning this topic will appear in a future paper.