Chaotic time series prediction with residual analysis method using hybrid Elman–NARX neural networks

Residual analysis using hybrid Elman–NARX neural network along with embedding theorem is used to analyze and predict chaotic time series. Using embedding theorem, the embedding parameters are determined and the time series is reconstructed into proper phase space points. The embedded phase space points are fed into an Elman neural network and trained. The residual of predicted time series is analyzed, and it was observed that residuals demonstrate chaotic behaviour. The residuals are considered as a new chaotic time series and reconstructed according to embedding theorem. A new Elman neural network is trained to predict the future value of the residual time series. The residual analysis is repeated several times. Finally, a NARX network is used to capture the relationship among the predicted value of original time series and residuals and original time series. The method is applied to Mackey–Glass and Lorenz equations which produce chaotic time series, and to a real life chaotic time series, Sunspot time series, to evaluate the validity of the proposed technique. Numerical experimental results confirm that the proposed method can predict the chaotic time series more effectively and accurately when compared with the existing prediction methods.