Numerical methods for fuzzy clustering

Authors:
Enrique H. Ruspini
Affiliations:
-
Venue:
Information Sciences: an International Journal
Year:
1970

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Abstract

In a previous paper ^[^1^] the use of the concept of fuzzy sets in clustering was proposed. The convenience of fuzzy clustering over conventional representation was then stressed. Assigning each point a degree of belongingness to each cluster provides a way of characterizing bridges, strays, and undetermined points. This is especially useful when considering scattered data. The classificatory process may be considered as the breakdown of the probability density function of the original set into the weighted sum of the component fuzzy set densities. Such decomposition should be performed so that the components really represent clusters. This is done by optimization of some functional defined over all possible fuzzy classifications of the data set. Several functionals were suggested in ^[^1^]. The bulk of this paper is concerned with numerical techniques useful in the solution of such problems. The first two formulas treated do not provide an acceptable fuzzy classification but yield good starting points for the minimization of a third functional. This last method obtains very good dichotomies and is characterized by slower convergence than the previous processes. Using that functional, a modification is suggested to obtain partitions in more than two sets. Numerous computational experiments are presented.