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 s ubsequently a linear least-squares method for the linear weights, have been previously suggested. These hybrid methods can be put into two groups, whi ch will be called as input clustering (IC) and input-output clustering (IOC ), depending on whether the output vector is also involved in the clusterin g 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 o n such procedures, but rather demonstrated their effectiveness in several a pplications. The main contribution of this paper is to present an approach for investigating the relationship between clustering process on input-outp ut training samples and the mean squared output error in the context of a r adial basis function network (RBFN), We may summarize our investigations in that matter as follows: 1) A weighted mean squared input-output quantizati on error, which is to be minimized by IOC, yields an upper bound to the mea n squared output error. 2) This upper bound and consequently the output err or 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.