Selection of Embedding Dimension and Delay Time in Phase Space Reconstruction via Symbolic Dynamics

The modeling and prediction of chaotic time series require proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Thus, one must choose appropriate time delay tau* and embedding dimension p for phase space reconstruction. The value of tau* can be estimated from the Mutual Information, but this method is rather cumbersome computationally. Additionally, some researchers have recommended that tau* should be chosen to be dependent on the embedding dimension p by means of an appropriate value for the time delay tau w=(p-1)tau*, which is the optimal time delay for independence of the time series. The C-C method, based on Correlation Integral, is a method simpler than Mutual Information and has been proposed to select optimally tau w and tau*. In this paper, we suggest a simple method for estimating tau* and tau w based on symbolic analysis and symbolic entropy. As in the C-C method, tau* is estimated as the first local optimal time delay and tau w as the time delay for independence of the time series. The method is applied to several chaotic time series that are the base of comparison for several techniques. The numerical simulations for these systems verify that the proposed symbolic-based method is useful for practitioners and, according to the studied models, has a better performance than the C-C method for the choice of the time delay and embedding dimension. In addition, the method is applied to EEG data in order to study and compare some dynamic characteristics of brain activity under epileptic episodes

Automated summary

Modeling and predicting chaotic time series require proper reconstruction of the state space by choosing appropriate time delay and embedding dimension. This paper proposes a simple method based on symbolic analysis and entropy to estimate time delays, showing better performance than traditional methods in numerical simulations. The method is validated on various chaotic time series, demonstrating its effectiveness for practitioners.