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 τ* and embedding dimension p for phase space reconstruction. The value of τ* can be estimated from the Mutual Information, but this method is rather cumbersome computationally. Additionally, some researchers have recommended that τ* should be chosen to be dependent on the embedding dimension p by means of an appropriate value for the time delay τw=(p−1)τ*, 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 τw and τ*. In this paper, we suggest a simple method for estimating τ* and τw based on symbolic analysis and symbolic entropy. As in the C-C method, τ* is estimated as the first local optimal time delay and τw 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

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In this article we introduce a test for independence between two processes {X_t} and {Y_t}. To this end we rely on symbolic dynamics and permutation entropy as a measure of dependence. As a result, a nonparametric (model-free) test for either linear or nonlinear processes is presented. The test is consistent for a broad range of dependent alternatives. Empirical simulations indicate and highlight the general utility of the test for time-series analysts. Copyright Copyright 2010 Blackwell Publishing Ltd

This paper proposes two new nonparametric tests for independence between time series. Both tests are based on symbolic analysis, specifically on symbolic correlation integral, in order to be robust to potential unknown nonlinearities. The first test is developed for a scenario in which each considered time series is independent and therefore the interest is to ascertain if two internally independent time series share a relationship of an unknown form. This is especially relevant as the test is nuisance parameter free, as proved in the paper. The second proposed statistic tests for independence among variables, allowing these time series to exhibit within-dependence. Monte Carlo experiments are conducted to show the empirical properties of the tests