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

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.

[1]  K. Diethelm.  AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 1997 .

[2]  Ganesh R. Naik,et al.  Low-complexity hardware design methodology for reliable and automated removal of ocular and muscular artifact from EEG , 2018, Comput. Methods Programs Biomed..

[3]  Kehui Sun,et al.  Multiscale permutation Rényi entropy and its application for EEG signals , 2018, PloS one.

[4]  K Lehnertz,et al.  Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: dependence on recording region and brain state. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Abraham Lempel,et al.  On the Complexity of Finite Sequences , 1976, IEEE Trans. Inf. Theory.

[6]  Petr Klimes,et al.  Intracerebral EEG Artifact Identification Using Convolutional Neural Networks , 2018, Neuroinformatics.

[7]  Jie Sun,et al.  Convergence of C0 Complexity , 2009, Int. J. Bifurc. Chaos.

[8]  Jing Yu,et al.  Improved Permutation Entropy for Measuring Complexity of Time Series under Noisy Condition , 2019, Complex..

[9]  E. Basar,et al.  Wavelet entropy: a new tool for analysis of short duration brain electrical signals , 2001, Journal of Neuroscience Methods.

[10]  Shuen-De Wu,et al.  Modified multiscale entropy for short-term time series analysis , 2013 .

[11]  Jing Li,et al.  Using Permutation Entropy to Measure the Changes in EEG Signals During Absence Seizures , 2014, Entropy.

[12]  Adem Kiliçman,et al.  Dynamics and Complexity of a New 4D Chaotic Laser System , 2019, Entropy.

[13]  Atilla Kilicarslan,et al.  Characterization and real-time removal of motion artifacts from EEG signals , 2019, Journal of neural engineering.

[14]  Yuqin Li,et al.  Chaos and Complexity Analysis of a Discrete Permanent-Magnet Synchronous Motor System , 2018, Complex..

[15]  Paul M. B. Vitányi,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 1993, Graduate Texts in Computer Science.

[16]  Qianli D. Y. Ma,et al.  Modified permutation-entropy analysis of heartbeat dynamics. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  J. Richman,et al.  Physiological time-series analysis using approximate entropy and sample entropy. , 2000, American journal of physiology. Heart and circulatory physiology.

[18]  Viet-Thanh Pham,et al.  On fractional–order discrete–time systems: Chaos, stabilization and synchronization , 2019, Chaos, Solitons & Fractals.

[19]  Luciano Zunino,et al.  Permutation entropy based time series analysis: Equalities in the input signal can lead to false conclusions , 2017 .

[20]  M. Stokić,et al.  Memory load effect in auditory–verbal short-term memory task: EEG fractal and spectral analysis , 2015, Experimental Brain Research.

[21]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[22]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[23]  Kehui Sun,et al.  Complexity Analysis and DSP Implementation of the Fractional-Order Lorenz Hyperchaotic System , 2015, Entropy.

[24]  Ming Zhu,et al.  Nonlinear Dynamic Analysis of Air Traffic Flow at Different Temporal Scales: Nonlinear Analysis Approach versus Complex Networks Approach , 2018, 2018 IEEE International Conference on Software Quality, Reliability and Security Companion (QRS-C).

[25]  Phillip P. A. Staniczenko,et al.  Rapidly detecting disorder in rhythmic biological signals: a spectral entropy measure to identify cardiac arrhythmias. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Carreño Carreño,et al.  Evaluación de la diversidad taxonómica y funcional de la comunidad microbiana relacionada con el ciclo del nitrógeno en suelos de cultivo de arroz con diferentes manejos del tamo , 2020 .

[27]  Hamed Azami,et al.  Amplitude-aware permutation entropy: Illustration in spike detection and signal segmentation , 2016, Comput. Methods Programs Biomed..

[28]  Xiangyang Wang,et al.  Complexity Analysis of Electroencephalogram Dynamics in Patients with Parkinson's Disease , 2017, Parkinson's disease.

[29]  Wen-Jie Xie,et al.  Tetradic motif profiles of horizontal visibility graphs , 2018, Commun. Nonlinear Sci. Numer. Simul..

[30]  Ales Procházka,et al.  Permutation Entropy and Signal Energy Increase the Accuracy of Neuropathic Change Detection in Needle EMG , 2018, Comput. Intell. Neurosci..

[31]  Ganesh R. Naik,et al.  Online and automated reliable system design to remove blink and muscle artefact in EEG , 2015, 2015 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC).

[32]  Xinmin Wang,et al.  EEG-Based Spatio–Temporal Convolutional Neural Network for Driver Fatigue Evaluation , 2019, IEEE Transactions on Neural Networks and Learning Systems.

[33]  Min Zhou,et al.  ECG Classification Using Wavelet Packet Entropy and Random Forests , 2016, Entropy.

[34]  Kehui Sun,et al.  Fractional fuzzy entropy algorithm and the complexity analysis for nonlinear time series , 2018, The European Physical Journal Special Topics.

[35]  Yudong Zhang,et al.  Tea Category Identification Using a Novel Fractional Fourier Entropy and Jaya Algorithm , 2016, Entropy.

[36]  Badong Chen,et al.  Weighted-permutation entropy: a complexity measure for time series incorporating amplitude information. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Kehui Sun,et al.  Detecting chaos in fractional-order nonlinear systems using the smaller alignment index , 2019, Physics Letters A.

[38]  Weiting Chen,et al.  Measuring complexity using FuzzyEn, ApEn, and SampEn. , 2009, Medical engineering & physics.

[39]  Shuen-De Wu,et al.  Refined Composite Multiscale Permutation Entropy to Overcome Multiscale Permutation Entropy Length Dependence , 2015, IEEE Signal Processing Letters.

[40]  Kehui Sun,et al.  Dynamics of a Simplified Lorenz System , 2009, Int. J. Bifurc. Chaos.

[41]  Asymmetry in Stock Comovements: An Entropy Approach , 2017 .

[42]  Kehui Sun,et al.  Multivariate permutation entropy and its application for complexity analysis of chaotic systems , 2016 .

[43]  Rifai Chai,et al.  Driver Fatigue Classification With Independent Component by Entropy Rate Bound Minimization Analysis in an EEG-Based System , 2017, IEEE Journal of Biomedical and Health Informatics.

[44]  Francesco Carlo Morabito,et al.  Entropic Measures of EEG Complexity in Alzheimer's Disease Through a Multivariate Multiscale Approach , 2013, IEEE Sensors Journal.

[45]  Ram Bilas Pachori,et al.  Automated diagnosis of atrial fibrillation ECG signals using entropy features extracted from flexible analytic wavelet transform , 2018 .

[46]  Yang Li,et al.  A New Epileptic Seizure Detection Method Based on Fusion Feature of Weighted Complex Network , 2018, ISNN.

[47]  Kehui Sun,et al.  Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems , 2018, Entropy.

[48]  Ganesh R. Naik,et al.  Dynamically identifying relevant EEG channels by utilizing channels classification behaviour , 2017, Expert Syst. Appl..

[49]  Yali Zhang,et al.  The complexity–entropy causality plane based on multivariate multiscale distribution entropy of traffic time series , 2018, Nonlinear Dynamics.

[50]  Hasan Ocak,et al.  Automatic detection of epileptic seizures in EEG using discrete wavelet transform and approximate entropy , 2009, Expert Syst. Appl..

[51]  J. Kurths,et al.  Complex network approaches to nonlinear time series analysis , 2019, Physics Reports.

[52]  S. Pincus.  Approximate entropy (ApEn) as a complexity measure. , 1995, Chaos.

[53]  Ricardo López-Ruiz,et al.  A Statistical Measure of Complexity , 1995, ArXiv.

[54]  B. Pompe,et al.  Permutation entropy: a natural complexity measure for time series. , 2002, Physical review letters.

[55]  Wei-Dong Dang,et al.  Multiscale limited penetrable horizontal visibility graph for analyzing nonlinear time series , 2016, Scientific Reports.

[56]  Madalena Costa,et al.  Multiscale entropy analysis of biological signals. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[57]  Rytis Maskeliūnas,et al.  Removal of Movement Artefact for Mobile EEG Analysis in Sports Exercises , 2019, IEEE Access.