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

Abstract

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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“…The values of the time delay and embedding dimensions reveal [13,14] the average logarithmic divergence code. Figure 8 shows the comparison of the average logarithmic divergence of bending deflection against time due to critical buckling load at fiber volume fraction (80%) and aspect ratio (2.5).…”

Optimal state space reconstruction via Monte Carlo decision tree search

“…However, by setting τ j , j > 2 to multiples of τ 2 , one ignores the fact that this "measure" of independence strictly holds only for the first two components of reconstruction vectors (m = 2) [51,52], even though in practice it works fine for most cases. More sophisticated ideas, like high-dimensional conditional mutual information [53,54] and other statistics [54][55][56][57][58][59], some of which include nonuniform delays and the extension to multivariate input data [30,38,39,53,54,60,[60][61][62][63][64][65]65], have been presented. (2) A statistic, we call it Γ throughout this paper, which serves as an objective function and quantifies the goodness of a reconstruction, given that delays τ j have been estimated.…”