Nonlinear modeling by radial basis function networks

Deterministic nonlinear prediction is applied to both artificial and real time series data in order to investigate orbital-instabilities, short-term predictabilities and long-term unpredictabilities, which are important characteristics of deterministic chaos. As an example of artificial data, bimodal maps of chaotic neuron models are approximated by radial basis Function networks, and the approximation abilities are evaluated by applying deterministic nonlinear prediction, estimating Lyapunov exponents acid reconstructing bifurcation diagrams of chaotic neuron models. The functional approximation is also applied to squid giant axon response as an example of real data. Two methods, the standard and smoothing interpolation, are adopted to construct radial basis function networks; while the former is the conventional method that reproduces data points strictly, the latter considers both faithfulness and smoothness of interpolation which is suitable under existence of noise. In order to take a balance between faithfulness and smoothness of interpolation, cross validation is applied to obtain an optimal one. As a result, it is confirmed that by the smoothing interpolation prediction performances are very high and estimated Lyapunov exponents are very similar to actual ones, even though in the case of periodic responses. Moreover, it is confirmed that reconstructed bifurcation diagrams are very similar to the original ones.