A New Criterion for Grouping
Hiroshi Omori · Journal of Japan Society for Fuzzy Theory and Intelligent Informatics · 1993
The grouping of n objects into g groups is equivalent to the assignment of the every membership α_ of ith object to jth group. When these α_ 's are un-known, a criterion to get optimal α_ 's from obtained date is proposed. Making use of the hypothesis decomposition for the contingency table, the change in the amount of information by grouping I_g is defined as I_g=n log n+Σ__iΣ__j α_ logα_ -Σ__j n_jlogn_j, 0≦α_ ≦1,Σ__j α__ij=1,where n_j is the number of members in jth group. On the other hand, we also have the parallel hypothesis decomposition with that for the contingency table, if the object data is assumed to be drawn from the normal population. Then from a geometrical viewpoint a new criterion FGC (Fuzzy Group Criterion) is proposed in order to search for an optimal grouping such as FGC=S_W/S_T+I_g/I_n, where S_T and S_W denote the total S.S. and the within S.S., respectively, and I_n=nlog n which denotes the change in the amount of information by grouping of n objects into n non-fuzzy groups. An optimal grouping is then selected to have the minimum FGC. This grouping strategy is intended to have the small within S.S. without a clear-cut grouping, that is, with small I_g. It may lead to a fuzzy grouping optimal. The application of this strategy to the fuzzy k-means method is described, where the optimal number of clusters and the parameter value which controls the degree of separation between clusters are searched for. An example to two artificial two-dimensional data sets is also given in which we get a result that the fuzzy clusters are optimal if the boundary between clusters seems obscure and that the non-fuzzy ones are optimal if it seems clear.