Analysis of input-output clustering for determining centers of RBFN

The key point in design of radial basis function networks is to specify the number and the locations of the centers. Several heuristic hybrid learning methods, which apply a clustering algorithm for locating the centers and subsequently a linear least-squares method for the linear weights, have been previously suggested. These hybrid methods can be put into two groups, which will be called as input clustering (IC) and input-output clustering (IOC), depending on whether the output vector is also involved in the clustering process. The idea of concatenating the output vector to the input vector in the clustering process has independently been proposed by several papers in the literature although none of them presented a theoretical analysis on such procedures, but rather demonstrated their effectiveness in several applications. The main contribution of this paper is to present an approach for investigating the relationship between clustering process on input-output training samples and the mean squared output error in the context of a radial basis function network (RBFN). We may summarize our investigations in that matter as follows: (1) A weighted mean squared input-output quantization error, which is to be minimized by IOC, yields an upper bound to the mean squared output error. (2) This upper bound and consequently the output error can be made arbitrarily small (zero in the limit case) by decreasing the quantization error which can be accomplished through increasing the number of hidden units.