This correspondence describes extensions to the k-modes algorithm for clustering categorical data. By modifying a simple matching dissimilarity measure for categorical objects, a heuristic approach was developed in [4], [12] which allows the use of the k-modes paradigm to obtain a cluster with strong intrasimilarity and to efficiently cluster large categorical data sets. The main aim of this paper is to rigorously derive the updating formula of the k-modes clustering algorithm with the new dissimilarity measure and the convergence of the algorithm under the optimization framework.

Abstract-In this paper, we present an agglomerative fuzzy K-Means clustering algorithm for numerical data, an extension to the standard fuzzy K-Means algorithm by introducing a penalty term to the objective function to make the clustering process not sensitive to the initial cluster centers. The new algorithm can produce more consistent clustering results from different sets of initial clusters centers. Combined with cluster validation techniques, the new algorithm can determine the number of clusters in a data set, which is a well-known problem in K-Means clustering. Experimental results on synthetic data sets (2 to 5 dimensions, 500 to 5,000 objects and 3 to 7 clusters), the BIRCH two-dimensional data set of 20,000 objects and 100 cluster0and the WINE data set of 178 objects, 17 dimensions, and 3 clusters from UCI have demonstrated the effectiveness of the new algorithm in producing consistent clustering results and determining the correct number of clusters in different data sets, some with overlapping inherent clusters.