Multistep-Ahead Chaotic Time Series Prediction Based on Hierarchical Echo State Network With Augmented Random Features

Multistep-ahead chaotic time series prediction is a kind of highly nonlinear problem, which puts forward higher requirements both for the dynamical memory and nonlinearity of the model. Echo state network (ESN) is frequently employed in the realm of chaotic time series modeling and prediction, but the basic ESN has been proved to have an antagonistic tradeoff between nonlinear transformation and memory capacity. To overcome this tradeoff, a new architecture named hierarchical ESN with augmented random features (HESN-ARF) is proposed. On the basis of the traditional linear random projection, the proposed HESN-ARF further leverages nonlinear kernel transformation to construct augmented random features, which can enable the linear and nonlinear properties to be fully represented. Moreover, the HESN-ARF utilizes low-rank kernel approximation to further reduce the computational cost, preserving the advantage of efficient modeling as much as possible while ensuring the capacities of nonlinear transformation and dynamical memory simultaneously. The proposed HESN-ARF can mine and learn the latent evolution patterns hidden in the dynamic system layer by layer through the hierarchical strategy, and achieves excellent performance in multistep-ahead chaotic time series prediction, as demonstrated by experimental findings on two synthetic chaotic systems and a real-world meteorological data set.