Euclidean distance and second derivative based widths optimization of radial basis function neural networks

The design of radial basis function widths of Radial Basis Function Neural Network (RBFNN) is thoroughly studied in this paper. Firstly, the influence of the widths on performance of RBFNN is illustrated with three simple function approximation experiments. Based on the conclusions drawn from the experiments, we find that two key factors including the spatial distribution of the training data set and the nonlinearity of the function should be considered in the width design. We propose to use Euclidean distances between center nodes and the second derivative of function to measure these two factors respectively. Secondly, a two step method is proposed to design the widths based on the information about the aforementioned two key factors obtained from comprehensive analysis of the given training data set. In the first step the data set spatial distribution features are analyzed according to the Euclidean distances between the data points, and the second derivative of each center node is estimated with finite difference approximation method. Based on the analysis an initial design of the widths is given with a heuristic equation. In the second step optimization techniques are used to optimize the widths which can effectively find the optimum with the good initial baseline. Thirdly, one mathematical example is taken to verify the efficiency of the proposed method, and followed by conclusions.