Hybrid Enhanced Echo State Network for Nonlinear Prediction of Multivariate Chaotic Time Series

As a kind of special nonlinear phenomenon, chaos has obtained much attention due to its interesting characteristics, such as randomness, sensibility, and complexity. How to predict chaos effectively and accurately is a significant issue in the nonlinear area. In this paper, a hybrid enhanced echo state network (HEESN) is proposed for the nonlinear prediction of multivariate chaotic time series. The HEESN scheme is contributed by three interactional aspects: output weight regularization, initial parameter optimization, and chaotic signal reconstruction. First, to enhance noise robustness, a sparse regression based on L2 regularization is employed to finely learn the output weights of ESN. Second, vital reservoir parameters (i.e., global scaling factor, reservoir size, scaling coefficient, and sparsity degree) are learned by a linear-weighted particle swarm optimization (LW-PSO) to further improve prediction accuracy and reliability. Third, recommendations of key settings in the signal reconstruction stage (i.e., embedding dimension and time delay) are studied and given according to the temporal complexity and signal-to-noise ratio of the predicted time series. Extensive experiments about computational complexity and three evaluating metrics are carried out on one chaotic benchmark. The analyzed results indicate that the proposed HEESN performs promisingly on multivariate chaotic time series prediction.