Forecasting chaotic time series using echo state networks: limits of predictability and relationship with entropy

This study demonstrates the feasibility of forecasting chaotic time series using an echo state network, with the Mackey–Glass system serving as a representative example. We investigated how prediction accuracy depends on the input sequence length, the degree of chaoticity in the time series, and the forecasting horizon. The analysis revealed the existence of threshold (optimal) input lengths for different prediction horizons, with the threshold remaining nearly constant for a fixed level of chaoticity. Furthermore, we observed that the optimal input length decreases as the chaoticity of the series increases. A strong linear correlation was also identified between the minimum prediction error and the sample entropy of the time series, indicating that prediction error can serve as an indirect measure of time series complexity.