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2.3. FUZZY CLUSTERINGAn intuitive approach to objective rule generation is based upon fuzzy clustering of input-output data. One simple and applicable idea, especially for systems with large numbers of input variables, was suggested by Sugeno and Yasukawa (1993). In this approach, one first clusters only the output space, which can be always considered as a single-dimensional space in fuzzy models. The fuzzy partition of the input space is specified at the next step by generating the projection of the output clusters into each input variable space separately. Using this method, the rule generation step could be separated from the input selection step, as will be discussed later. The idea of fuzzy clustering is to divide the output data into fuzzy clusters that overlap each other. Therefore, the assignment of each data point to each cluster is defined by a membership grade in [0,1]. Formally, clustering unlabeled data X = {x1, x2, ..., xN} where 0 [less than or equal to] uik [less than or equal to] 1, A is a positive definite matrix that specifies the shape of the cluster. The common selection for the matrix A is the identity matrix, leading to the definition of Euclidean distance, and consequently to spherical clusters. There are, however, investigations where the matrix A is taken as the covariance matrix that generates models with elliptic clusters (Kosko, 1996). In current literature, most fuzzy clustering studies are carried out by the Fuzzy C-Means (FCM) algorithm through an iterative optimization. However, there are three major difficulties with FCM clustering algorithm:
That is, different choices of initial V0 might lead to different local extrema. Therefore, it is desirable to modify the FCM algorithm.
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