Analyzing the state space property of echo state networks for chaotic system prediction

For chaotic system prediction, ESNs (echo state networks) are realization of neural state reconstruction, in which the reconstructed state variable is from the internal neurons´ activation, rather than the delay vector obtained from delay coordinate reconstruction. In the framework of the neural state reconstruction, some quantitative analyses can be further made on the issues such as the network structure configuration and initial state determination. Based on the simulation study on chaotic data from Chua´s circuit, it is shown that the ESN is a non-minimum state space realization of the target time series, and the initial state can be freely chosen in the training process, and in the phase of prediction, ESN needs to know where the prediction begins by being set a proper initial state through a process of teacher forcing.