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With these newly formed clusters, the new cluster centers are calculated. The process is continued until some convergence criterion is met, usually when assignments have remained unchanged. The algorithm is relatively efficient: O(tkn), where n is the number of objects, k the number of clusters, and t the number of iterations.

An (xn )) if f −1 (Y ) = ∅ 0 iff −1 (Y ) = ∅ A crucial concept in fuzzy set theory is that of fuzzy relations, which is a generalization of the conventional crisp relations. An n-ary fuzzy relation in X1 × X2 × · · · × Xn is, in fact, a fuzzy set on X1 × X2 × · · · × Xn . Fuzzy relations can be composed (and this composition is closely related to the extension principle shown above). For instance, if U is a relation from X1 to X2 (or, equivalently, a relation in X1 × X2 ), and V is a relation from X2 to X3 , then the composition of U and V is a fuzzy relation from X1 to X3 which is denoted by U ◦V and defined by μU ◦V (x1 , x3 ) = max min(μU (x1 , x2 ), μV (x2 , x3 )), x1 ∈ X1 , x3 ∈ X3 x2 ∈X2 A convenient way of representing a binary fuzzy relation is to use a matrix.

Each rule is weighted by the percentage of positive examples in the set of examples covered by it. The weights of rules of the same class are combined to a weight for the entire class and the class with the highest weight will be returned. 2 CN2 CN2 induces propositional classification rules from data [58]. The algorithm consists of two main procedures: the search procedure and the control procedure. The search procedure performs beam search to locate a single rule, employing the classification accuracy of the rule as a guiding heuristic.

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