Cooperative-competitive genetic evolution of radial basis function centers and widths for time series prediction

In a radial basis function (RBF) network, the RBF centers and widths can be evolved by a cooperative-competitive genetic algorithm. The set of genetic strings in one generation of the algorithm represents one REP network, not a population of competing networks. This leads to moderate computation times for the algorithm as a whole. Selection operates on individual RBFs rather than on whole networks. Selection therefore requires a genetic fitness function that promotes competition among RBFs which are doing nearly the same job while at the same time promoting cooperation among RBFs which cover different parts of the domain of the function to be approximated. Niche creation resulting from a fitness function of the form |w_i|^β/E(|w_i´|^β), 1<β<2 can facilitate the desired cooperative-competitive behavior. The feasibility of the resulting algorithm to evolve networks of Gaussian, inverse multiquadric, and thin-plate spline RBFs is demonstrated by predicting the Mackey-Glass time series. For each type of RBF, and for networks of 25, 50, 75, 100, 125, and 150 RBF units, prediction errors for the evolved Gaussian RBF networks are 50-70% lower than RBF networks obtained by k-means clustering