Design of a Network Permutation Entropy and Its Applications for Chaotic Time Series and EEG Signals

Abstract

Measuring the complexity of time series provides an important indicator for characteristic analysis of nonlinear systems. The permutation entropy (PE) is widely used, but it still needs to be modified. In this paper, the PE algorithm is improved by introducing the concept of the network, and the network PE (NPE) is proposed. The connections are established based on both the patterns and weights of the reconstructed vectors. The complexity of different chaotic systems is analyzed. As with the PE algorithm, the NPE algorithm-based analysis results are also reliable for chaotic systems. Finally, the NPE is applied to estimate the complexity of EEG signals of normal healthy persons and epileptic patients. It is shown that the normal healthy persons have the largest NPE values, while the EEG signals of epileptic patients are lower during both seizure-free intervals and seizure activity. Hence, NPE could be used as an alternative to PE for the nonlinear characteristics of chaotic systems and EEG signal-based physiological and biomedical analysis.Entropy 2019, 21, 849 2 of 18 algorithm; thus, complexity can be analyzed in a multiple scale sense. After this, many different multiscale complexity algorithms such as the multiscale PE (MPE) algorithm [27], the multiscale SampEn algorithm [28], the multiscale permutation Rényi entropy [29], and the multivariate multiscale entropy [30] were designed. Among those methods, the PE algorithm and MPE algorithm have a fast estimation speed and are gaining more and more academic attention. Therefore, this article will concentrate on the PE algorithm.In 2002, Bandt and Pompe [23] proposed the PE algorithm based on the patterns deduced from constructed vectors. However, it was indicated in many literature works that the PE algorithm has some drawbacks that cannot always measure complexity effectively. For instance, Zunino et al. [31] figured out that the Bandt-Pompe method for processing equal values could lead to erroneous conclusions. Until now, how to improve the PE algorithm is still an open project. At present, several improved PE algorithms have been proposed. Bian et al. [32] modified the PE algorithm by mapping the equal value onto the same symbol (rank), and the modified PE (mPE) algorithm was designed. Fadlallah et al. [33] proposed the weighted PE (WPE) algorithm, and EEG signals were analyzed to verify its effectiveness. Azami et al. [34] indicated that PE does not consider the average of the amplitude values and equal amplitude values and designed the amplitude-aware permutation entropy (AAPE). Chen et al. [35]proposed the improved PE (IPE) algorithm by introducing a symbolic process and combining some advantages of previous modifications of PE. As a result, IPE has more patterns and better robustness for noise-polluted time series. Those algorithms have better measuring results when comparing with the original PE algorithm. However, they do not consider the relationship between different patterns since they actually treat the vectors as independent units. Currently, appro...

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