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

Mariano Matilla-García,Manuel Ruiz Marín,Isidro Morales,Jose Miguel Rodríguez

doi:10.3390/e23020221

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

Highlights

IntroductionIs the base for data-driven analysis and prediction of chaotic systems
For the theory of state space reconstruction suggested by Packard, Takens et al [1,2]is the base for data-driven analysis and prediction of chaotic systems
We propose a new method for selection p and τ ∗ based on symbolic dynamics and Information Theory

Summary

Introduction

Is the base for data-driven analysis and prediction of chaotic systems. It can be proved through Taken’s theorem [2] that the strange attractor of the chaotic systems could be properly recovered from only one projection of the dynamic system. The fundamental theorem of reconstruction of Takens establishes a sufficient condition (but not necessary) given by p ≥ 2d + 1, where d is the fractal dimension of the underlying chaotic attractor, and p stands for the embedding dimension used for phase space reconstruction. A popular method for state space reconstruction is the method of delays It consists of embedding the observed scalar time series { Xt }t∈ I in one p-dimensional space X τp (t) =

Objectives

Methods

Conclusion