The Generalized Fuzzy n-Means Algorithm

If the classes are compact clusters of spherical or ellipsoidal shape, they may be represented by the points (vectors) L . L in R If we let L = L, L, . L we may consider L as the representation of the fuzzy partition P. The inadequacy AL ) between the fuzzy class A and its prototype may be defined as 

Using d we may also define a distance with respect to fuzzy class A. This distance is denoted by d, and may be defined as 

If the prototype L is from X, then we may assume that the prototype has the greatest membership in A, that is, 

The inadequacy J(P, L) between the fuzzy partition P and its representation L may be defined as 

If we consider the distance d as generated by a scalar product induced norm we have 

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