Fuzzy identification with clustering methods of rules´ construction

A method is investigated to map functions given a finite training set and using fuzzy models that are based on pattern recognition criteria. The fundamental problem to be explored concerns the determination of the parameters, both dimensional and substantive, of a matrix of membership functions U each element of which represents the normalized sampled value of a basis function evoked by a discrete input training vector x(k). This paper proposes a new validity functional, to judge the fitness of candidate membership function matrices. The functional is sensitive to the separation, or classification, objectives characteristic in pattern recognition applications, and also to the topological specific traits of the desired mapping, traits that are embedded in the training data itself. Minimization of a metric resulting from the validity functional guides the selection of the parameters, including dimension, that constrain and define the matrix of basis functions. The methodology provides a significant reduction in computational complexity in the determination of consequent parameters (eigenvalues): Given an optimal or suboptimal U matrix obtained by a modified fuzzy-c-means algorithm prior to training, structures that behave as eigenvalues are obtained by backpropagation of error during the training phase of the model. The degree to which each rule fires is available in a known matrix, prior to training of the consequent parameters that provide a specific gain to each fuzzy basis function. Hence, each rule´s response, and the overall network, does not require during training repeated calculations of a minimum (fuzzy “and”) operation that typically entails a calculation of the membership values of subrules along each dimension of the training vector. Additionally, because the sum of the elements in any column of any column of U is a-priori and algorithmically constrained to equal one, the need for repeated normalization over all the rule membership values is eliminated from the outset