Special Issue "Permutation Entropy: Theory and Applications"

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Multidisciplinary Applications".

Deadline for manuscript submissions: 30 November 2019.

Special Issue Editor

Guest Editor
Dr. Massimiliano Zanin Website E-Mail
Centro de Tecnología Biomédica, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón,Spain
Interests: complex systems; complex networks; network science; data mining

Special Issue Information

Dear Colleagues,

Introduced 16 years ago in a seminal paper by C. Bandt and B. Pompe, the concept of permutation entropy (PE) has attracted much attention from researchers from a plethora of fields. Undeniably, part of this success originates from the elegance and simplicity of PE. It is based on comparing neighbouring values in a time series, for then finding the order patterns that result in sorted (ascending) sequences, and finally on studying the corresponding probability distribution. PE thus allows us to synthesise, in a simple metric, the temporal causality of the time series, and to do this in a computationally efficient and almost parameter-free way. If originally introduced to assess the complexity of a time series, it was soon modified to, e.g., discriminate between chaotic and random dynamics, to quantify the irreversibility of a time series, or to test the presence of causality relationships.

The theoretical advantages offered by PE have fostered its application to heterogeneous real-world problems. The best examples can be found in the biomedical field, where it has been used to characterise the dynamics of different entities in healthy and pathological conditions—including individual neurons, brain, heart or gait. However, further examples can also be found in economics, laser physics, mechanical engineering, or climate analysis, to name a few. In synthesis, whenever it is required to characterise the complexity of the mechanisms behind a time series, PE offers an efficient and effective option.

In spite of this, many questions remain open, and the community has only scratched the surface of some theoretical problems. To illustrate, if PE allows characterising the temporal causality in one time series, few options are available to move to multivariate measurement. Additionally, the community still has to reach a consensus on how to handle equal values, or on how to optimise parameters like the embedding delay or the order of permutation.

This Special Issue seeks contributions from researchers working on this active topic, from both theoretical and applied perspectives, in the form of both original research and review papers. The application of PE to real-world problems, and the adaptation of PE to tackle new theoretical problems will especially be welcome.

Dr. Massimiliano Zanin
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Permutation entropy
  • Forbidden patterns
  • Complexity
  • Time series analysis
  • Causality

Published Papers (4 papers)

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Research

Open AccessArticle
Permutation Entropy and Irreversibility in Gait Kinematic Time Series from Patients with Mild Cognitive Decline and Early Alzheimer’s Dementia
Entropy 2019, 21(9), 868; https://doi.org/10.3390/e21090868 - 06 Sep 2019
Abstract
Gait is a basic cognitive purposeful action that has been shown to be altered in late stages of neurodegenerative dementias. Nevertheless, alterations are less clear in mild forms of dementia, and the potential use of gait analysis as a biomarker of initial cognitive [...] Read more.
Gait is a basic cognitive purposeful action that has been shown to be altered in late stages of neurodegenerative dementias. Nevertheless, alterations are less clear in mild forms of dementia, and the potential use of gait analysis as a biomarker of initial cognitive decline has hitherto mostly been neglected. Herein, we report the results of a study of gait kinematic time series for two groups of patients (mild cognitive impairment and mild Alzheimer’s disease) and a group of matched control subjects. Two metrics based on permutation patterns are considered, respectively measuring the complexity and irreversibility of the time series. Results indicate that kinematic disorganisation is present in early phases of cognitive impairment; in addition, they depict a rich scenario, in which some joint movements display an increased complexity and irreversibility, while others a marked decrease. Beyond their potential use as biomarkers, complexity and irreversibility metrics can open a new door to the understanding of the role of the nervous system in gait, as well as its adaptation and compensatory mechanisms. Full article
(This article belongs to the Special Issue Permutation Entropy: Theory and Applications)
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Open AccessArticle
Fault Diagnosis for Rail Vehicle Axle-Box Bearings Based on Energy Feature Reconstruction and Composite Multiscale Permutation Entropy
Entropy 2019, 21(9), 865; https://doi.org/10.3390/e21090865 - 05 Sep 2019
Abstract
The fault response signals of an axle-box bearing of a rail vehicle have strongly non-linear and non-stationary characteristics, which can reflect the operating state of the running gears. This paper proposes a novel method for bearing fault diagnosis based on frequency-domain energy feature [...] Read more.
The fault response signals of an axle-box bearing of a rail vehicle have strongly non-linear and non-stationary characteristics, which can reflect the operating state of the running gears. This paper proposes a novel method for bearing fault diagnosis based on frequency-domain energy feature reconstruction (EFR) and composite multiscale permutation entropy (CMPE). First, a wavelet packet transform (WPT) is applied to decompose the vibration signals into multiple frequency bands. Then, considering that the bearing-localized defects cause the axle-box bearing system to resonate at a high frequency, which will lead to uneven energy distribution of the signal in the frequency domain, the energy factors of each frequency band are calculated by an energy feature extraction algorithm, from which the frequency band with maximum energy factor (which contains abundant fault information) is reconstructed to the time-domain signal. Next, the complexity of the reconstructed signals is calculated by CMPE as fault feature vectors. Finally, the feature vectors are input into a medium Gaussian support vector machine (MG-SVM) for bearing condition classification. The proposed method is validated by a public bearing data set and a wheelset-bearing system test bench data set. The experimental results indicate that the proposed method can effectively extract bearing fault features and provides a new solution for condition monitoring and fault diagnosis of rail vehicle axle-box bearings. Full article
(This article belongs to the Special Issue Permutation Entropy: Theory and Applications)
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Open AccessArticle
Design of a Network Permutation Entropy and Its Applications for Chaotic Time Series and EEG Signals
Entropy 2019, 21(9), 849; https://doi.org/10.3390/e21090849 - 30 Aug 2019
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Permutation Entropy: Theory and Applications)
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Open AccessArticle
Time-Shift Multi-scale Weighted Permutation Entropy and GWO-SVM Based Fault Diagnosis Approach for Rolling Bearing
Entropy 2019, 21(6), 621; https://doi.org/10.3390/e21060621 - 25 Jun 2019
Abstract
Multi-scale permutation entropy (MPE) is an effective nonlinear dynamic approach for complexity measurement of time series and it has been widely applied to fault feature representation of rolling bearing. However, the coarse-grained time series in MPE becomes shorter and shorter with the increase [...] Read more.
Multi-scale permutation entropy (MPE) is an effective nonlinear dynamic approach for complexity measurement of time series and it has been widely applied to fault feature representation of rolling bearing. However, the coarse-grained time series in MPE becomes shorter and shorter with the increase of the scale factor, which causes an imprecise estimation of permutation entropy. In addition, the different amplitudes of the same patterns are not considered by the permutation entropy used in MPE. To solve these issues, the time-shift multi-scale weighted permutation entropy (TSMWPE) approach is proposed in this paper. The inadequate process of coarse-grained time series in MPE was optimized by using a time shift time series and the process of probability calculation that cannot fully consider the symbol mode is solved by introducing a weighting operation. The parameter selections of TSMWPE were studied by analyzing two different noise signals. The stability and robustness were also studied by comparing TSMWPE with TSMPE and MPE. Based on the advantages of TSMWPE, an intelligent fault diagnosis method for rolling bearing is proposed by combining it with gray wolf optimized support vector machine for fault classification. The proposed fault diagnostic method was applied to two cases of experimental data analysis of rolling bearing and the results show that it can diagnose the fault category and severity of rolling bearing accurately and the corresponding recognition rate is higher than the rate provided by the existing comparison methods. Full article
(This article belongs to the Special Issue Permutation Entropy: Theory and Applications)
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