Kazutaka Umayahara

Methods in Hard and Fuzzy Clustering

Clustering, also referred to as cluster analysis, is a class of unsupervised classification methods for data analysis. There have been numerous studies of clustering, which are both theoretical and applicational. Applications to scientific classifications, engineering problems, behavioral sciences. etc., have been investigated and usefulness of this technique has been appreciated.

Two Methods of Fuzzy c-Means and Classification Functions

A regularization method using an entropy function is studied and contrasted with the ordinary fuzzy c-means. The way in which two algorithms lead to similar formulas is discussed. Classification functions derived from the two methods, which are naturally obtained when the algorithm of clustering is convergent, are compared. Theoretical properties of the two classification functions are studied.

Generalization of discriminant analysis for possibility distributions

This paper aims to generalize the method of possibility discriminant analysis proposed by Tanaka et al. (1992) using t-norms. The two classes discussed in the possibility discriminant analysis by Tanaka are generalized into m classes and, moreover, the minimum operation is generalized into t-norm operations. The ordering of a set of the functions calculated from possibility distributions for given classes determines the class of an observation. In particular, the first two functions in the ordering are essential for the classification.

Four c-Regression Methods and Classification Functions

Four methods of c-regression are compared. Two of them are methods of fuzzy clustering: (a) the fuzzy c-regression methods, and (b) an entropy method proposed by the authors. Two others are probabilistic methods of (c) the deterministic annealing, and (d) the mixture distribution method using the EM algorithm. It is shown that the entropy method yields the same formula as that of the deterministic annealing. Clustering results as well as classification functions are compared. The classification functions for fuzzy clustering are fuzzy rules interpolating cluster memberships, while those for the latter two are probabilistic rules. Theoretical properties of the classification functions are studied. A numerical example is shown.