Embedding dimension estimation of high dimensional chaotic time series using distributed time delay neural network

In the face of a practical chaotic system whose mathematical model is not available, because of unknown input factors and unavailable dynamical equations, using time series approach can be useful. Therefore, space state reconstruction of a chaotic system by using a scalar time series from its output observations is considered for obtaining information on this system from its one-dimensional signal. In this paper a method for estimation of an appropriate embedding dimension for phase space reconstruction of underlying high dimensional system from the observed chaotic time series by a Distributed Time Delay Neural Network (DTDNN) is proposed. Various methods for embedding dimension estimation have been previously studied from which False Nearest Neighbours (FNN) is the most conventional method, however the performance of this method for the high dimensional chaotic systems is not acceptable. The proposed method is applied to high dimensional chaotic systems such as approximated Mackey-Glass time series with dimensions 7, 13 and data set D-1 from the Santa Fe institute. Our method for embedding dimension estimation has been compared with the conventional estimation method, and their comparisons showed the effectiveness of the proposed methodology. The results show that this method is feasible and fit for the embedding dimension estimation of high-dimensional chaotic systems.