8. Fuzzy Clustering Algorithms
Society for Industrial and Applied Mathematics eBooks · 2007
Hard (or crisp) clustering algorithms require that each data point of the data set belong to one and only one cluster. Fuzzy clustering extends this notion to associate each data point in the data set with every cluster using a membership function. Since the concept of fuzzy sets (Zadeh, 1965) was introduced, fuzzy clustering has been widely discussed, studied, and applied in various areas. Early work on applying fuzzy set theory in cluster analysis was proposed by Bellman et al. (1966) and Ruspini (1969). Let D be a data set with n objects, each of which is described by d attributes, and let c be an integer between one and n. Then a fuzzy c-partition is defined by a c × n matrix U = (uli) that satisfies uli ∈ [0,1], 1≤l≤c,1≤i≤n, 8.1a ∑ l=1 c uli =1,1≤i≤n, 8.1b ∑ i=1 n uli >0,1≤l≤c, 8.1c where uli denotes the degree of membership of the object i in the lth cluster. For each fuzzy c-partition, there is a corresponding hard c-partition. Let uli (l = 1, 2, …, c, i = 1, 2, …, n) be the membership of any fuzzy c-partition. Then the corresponding hard c-partition of uli can be defined as ωli as follows (Xie and Beni, 1991): ωli ={1if l=arg max 1≤j≤c uji ,0otherwise. 8.1 Fuzzy Sets The concept of the fuzzy set was first introduced by Zadeh (1965). A fuzzy set is defined to be a class of objects with a continuum of grades of membership.