Square-error Clustering Scheme and Clustering Networks P C R=1 K

The goal of cluster analysis is to separate a set of objects into constituent groups so that the members of any one group diier from one another as little as possible, according to a given criteria 6]. Pal et al. 5] proposed a generalized learning vector quantiza-tion (GLVQ) algorithm and compared it with learning vector quantization (LVQ) algorithm on clustering Anderson's IRIS data 3]. In this paper, performance of hard c-means (HCM), fuzzy c-means (FCM), LVQ and GLVQ algorithm are evaluated using \wine recognition data". 1 Square-error clustering Let < be the set of reals (feature space), < p the set of p tuples of reals and a nite set X < p , X = fx 1 ; x 2 ; ; x n g and an integer c (c partition), 2 c n. Every function u : X ! f0; 1g (hard membership) or 0; 1] (fuzzy membership) is said to assign its grade of membership to each x 2 X 4]. Let v be the c tuples, that is v i 2 < p , the cluster center or prototype of class i. Then, the objective functional J m is deened as J m (U; v) = P n k=1 P c i=1 (u ik) m (d ik) 2 where d 2 ik =k x k ? v i k 2 , m = 1 for HCM and 1 < m < 1 for FCM. The HCM and FCM algorithm via iterative optimization of J m produces a hard or fuzzy c partition of data set. 2 Clustering networks LVQ is not a clustering algorithm per se; rather it can be used to generate hard c-partitions of data sets un-labeled with the nearest prototype classiier designed with its terminal prototypes 5]. In LVQ one tries to discover cluster structure in unlabeled p-dimensional data. Let X = fx 1 ; x 2 ; ; x n g < p denote samples and use c to denote the number of nodes, clusters in X, in the competitive layer. The input layer of LVQ network is connected directly to the output layer. Each node of output layer has a prototype (or weight vector) attached to it. For 1 i c , the prototypes V = (v 1 ; v 2 ; ; v n) are a network array of unknown cluster centers v i 2 < p. In this context …