Nonlinear prediction using radial basis function network incorporating coordinate transformation

Abstract

Nonlinear short-term prediction incorporating modeling technique is used in various fields such as stock prices, exchange, daily temperature and power demand, and is one of the crucial research topics. Radial basis function (RBF) network, which is one of the artificial neural networks (ANNs), is widely used for the modeling and prediction (He and Lapedes, 1993; Rosupal et al., 1998; Gan et al., 2012). Kondo used an ANN with four layers for modeling and prediction of Sulfur Dioxide (SO2) (Kondo, 1993), in which he reported that highly accurate prediction could be made by the ANN in comparison with a linear regression model and an auto-regressive (AR) model. Cowper et al. adopted the RBF network for nonlinear modeling and prediction (Cowper et al., 2002), in which they pointed out that the width of Gaussian kernel was a key factor for highly accurate modeling and prediction using the RBF network. They adopted a simple estimate for the width proposed by Haykin (1994), and the validation of normalization of Gaussian kernel was discussed. Du and Zhang also adopted the RBF network for modeling and prediction (Du and Zhang, 2008), in which genetic algorithm (GA) was used to optimize several parameters (the width and center of Gaussian kernel, the number of hidden layers) in the RBF network unlike Cowper et al. (Cowper et al., 2002). Manjunatha et al. adopted the RBF network for predicting diesel engine emissions, and concluded that the highly accurate prediction could be made in comparison with back propagation neural network (Manjunatha et al., 2012). Based on the above review, we developed a system for modeling and prediction using the RBF network and applied it to several benchmarks. Here, as an illustrative example, let us consider Mackey-Glass delay-differential equation given by Eq. (1).