Towards Dynamical Systems Approach to Fuzzy Clustering

In many application areas, there is a need for clustering, and there is a need to take fuzzy uncertainty into account when clustering. Most existing fuzzy clustering techniques are based on the idea that an object belongs to a certain cluster if this object is close to a typical object from this cluster. In some application areas, however, this idea does not work well. One example of such application is clustering in education that is used to convert a detailed number grade into a letter grade. In such application, it is more appropriate to use clustering techniques which are based on a different idea: that an object tends to belong to the same cluster as its nearest neighbor. In this paper, we explain the relationship between this idea and dynamical systems, and we discuss how fuzzy uncertainty can be taken into account in this approach to clustering. 1 Formulation of the Problem Clustering is important. In many applications areas we have a large number of different objects, with different behavior. For example, in biology, we have many different plants, animals, bacteria, etc. In chemistry, we have many different substances. In astronomy, we have a large number of different objects (planets, stars, galaxies, etc.). In all these situations, the number of objects is so huge and the behavior of these objects is so different that it is not possible to study each object individually. In such cases, it is often possible to group objects into clusters in such a way that all the objects within the same cluster are similar to each other (in some reasonable sense). Clustering enables us to replace a practically impossible task of studying all individual objects with a more doable task of studying the behavior of typical objects from different clusters. Since all the objects within a cluster are similar to each other, the analysis of typical objects provides us with a good description of how all the objects behave. For example, in biology, instead of studying each of millions of animals individually, we classify them into species, subspecies, etc., and study typical