Hard n-Means Algorithm

The FNM procedure may be notified to detect hard cluster structure in the data set. In this case clusters will be described by classical (hard) subsets of X. In the iterative process the classes are modified according to the rule 

GFNM is a one-level (or horizontal) clustering procedure. It may be used to detect spherical or ellipsoidal clusters. The shape of the clusters detected by this procedure is influenced by the matrix M. If M = I il 

If the clusters are close it is possible that the hyperplane of equal membership will intersect the greater cluster. Some points of the greater cluster will be captured by its neighbor, as shown in Fig. 4. This represents a pathological situation that may be avoided by using a data-dependent (or adaptive) distance.With an adaptive metric the apparent sizes of clusters become equal. An adaptive distance may be induced by the radius or by the diameter of each fuzzy class A,. The diameter 6 of the fuzzy class Ai is defined as 

FIGURE 4 Two unequal clusters. The smaller cluster captures some points. 

The diameter induced adaptive distance with respect to is defined as 

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