Chaotic time series prediction by noisy echo state network

Abstract

We have applied noisy echo state networks to the short-term forecasting of hyperchaotic and chaotic time series. The hyperchaotic time series were generated using the augmented Lorenz equations as a star network of Q nonidentical Lorenz systems and a four-dimensional Lorenz system. The echo state networks were used mainly in the recursive forecasting mode, wherein the output value of the network, i.e., the predicted value, at the current time step was recursively fed back to the input node at the next time step of prediction. The addition of external noise to the reservoir network has been found to considerably improve the fidelity of the geometrical structures of the chaotic attractors reconstructed from the predicted time series. We discuss these observations on the basis of Ueda's theory of chaos.