Contributions of Fuzzy Concepts to Data Clustering

This chapter tries to answer the fundamental question of what main contributions of fuzzy clustering to the theory of cluster analysis from theoretical viewpoints. While fuzzy clustering is thought to be clearly useful by users of this technique, others think that the concept of fuzziness is not needed in clustering. Thus the usefulness of fuzzy clustering is not trivial. The discussion here is divided into two: one is on fuzzy c-means which is best-known fuzzy method of clustering. However, there is another techniques, discussed by Zadeh, in hierarchical clustering which is equivalent to the old technique of the single linkage. This chapter overviews the both techniques, beginning from basic discussion of fuzzy c-means, and introducing the fundamental concept of fuzzy classifiers and its usefulness. A concept of inductive clustering is introduced which means that a result of clustering can be extended to a partition of the whole space. Moreover hierarchical fuzzy clustering is briefly discussed where the transitive closure gives a simple algebraic form of clusters.