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Entropy
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Ordinal Pattern-Based Entropies: New Ideas and Challenges
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Ordinal Pattern-Based Entropies: New Ideas and Challenges

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

Deadline for manuscript submissions: closed (31 January 2025) | Viewed by 7033

Special Issue Editor

Dr. Meryem Jabloun

E-Mail Website
Guest Editor
Laboratoire Pluridisciplinaire de Recherche en Ingénierie des Systèmes, Mécanique, Énergétique (PRISME), University of Orleans, 45100 Orleans, France
Interests: multiscale permutation entropy; time series analysis and forecasting; Bayesian inference; biomedical signal processing; financial time series

Special Issue Information

Dear Colleagues,

Ordinal pattern-based entropies have received significant attention in recent decades as complexity measures for time series. Their utility extends across numerous domains, including biomedical signal analysis, image processing, bearing fault diagnosis, financial analytics, and more. Recently, they found applications in multidimensional time series and encrypted data analysis.

This Special Issue’s aim is twofold. Firstly, it seeks to address theoretical investigations to enrich our understanding of the applicability of ordinal pattern-based entropies. Secondly, it aims to explore new and promising areas as well as novel concepts.

We invite original, unpublished papers and comprehensive reviews exploring the following research areas:

  • Advancements and development of innovative concepts in ordinal pattern-based entropies and methodologies.
  • Theoretical investigations to enhance the interpretability and applicability of permutation entropy.
  • Investigation of linear and nonlinear preprocessing of multiscale permutation entropy on processes involving forbidden patterns.
  • Investigation of the potential of permutation entropy as features in machine learning algorithms, particularly in the context of large and complex datasets.
  • Mathematical modelling and engineering problem-solving using the ordinal pattern-based entropies.
  • Analysis of nonlinear dynamical systems and nonlinear phenomena from the perspective of ordinal patterns.
  • Practical applications of permutation entropy in real-world problems.

Dr. Meryem Jabloun
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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 2600 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
  • multiscale permutation entropy and its variants
  • Tsallis permutation entropy
  • ordinal pattern-based complexity measure of time series
  • real data application
  • multidimensional time series
  • encrypted data
  • theoretical understanding for appropriate application
  • ordinal pattern based-entropy in machine learning

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Published Papers (5 papers)

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Research

17 pages, 5532 KiB  
Article
Permutation Entropy: An Ordinal Pattern-Based Resilience Indicator for Industrial Equipment
by Christian Salas, Orlando Durán, José Ignacio Vergara and Adolfo Arata
Entropy 2024, 26(11), 961; https://doi.org/10.3390/e26110961 - 8 Nov 2024
Cited by 1 | Viewed by 835
Abstract
In a highly dynamic and complex environment where risks and uncertainties are inevitable, the ability of a system to quickly recover from disturbances and maintain optimal performance is crucial for ensuring operational continuity and efficiency. In this context, resilience has become an increasingly [...] Read more.
In a highly dynamic and complex environment where risks and uncertainties are inevitable, the ability of a system to quickly recover from disturbances and maintain optimal performance is crucial for ensuring operational continuity and efficiency. In this context, resilience has become an increasingly important topic in the field of engineering and the management of productive systems. However, there is no single quantitative indicator of resilience that allows for the measurement of this characteristic in a productive system. This study proposes the use of permutation entropy of ordinal patterns in time series as an indicator of resilience in industrial equipment and systems. Based on the definition of resilience, the developed method enables precise and efficient assessment of a system’s ability to withstand and recover from disturbances. The methodology includes the identification of ordinal patterns and their analysis through the calculation of a permutation entropy indicator to characterize the dynamics of industrial systems. Case studies are presented and the results are compared with other resilience models existing in the literature, aiming to demonstrate the effectiveness of the proposed approach. The results are promising and highlight a highly applicable and simple indicator for resilience in industrial systems. Full article
(This article belongs to the Special Issue Ordinal Pattern-Based Entropies: New Ideas and Challenges)
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23 pages, 5276 KiB  
Article
Generalized Gaussian Distribution Improved Permutation Entropy: A New Measure for Complex Time Series Analysis
by Kun Zheng, Hong-Seng Gan, Jun Kit Chaw, Sze-Hong Teh and Zhe Chen
Entropy 2024, 26(11), 960; https://doi.org/10.3390/e26110960 - 7 Nov 2024
Viewed by 1187
Abstract
To enhance the performance of entropy algorithms in analyzing complex time series, generalized Gaussian distribution improved permutation entropy (GGDIPE) and its multiscale variant (MGGDIPE) are proposed in this paper. First, the generalized Gaussian distribution cumulative distribution function is employed for data normalization to [...] Read more.
To enhance the performance of entropy algorithms in analyzing complex time series, generalized Gaussian distribution improved permutation entropy (GGDIPE) and its multiscale variant (MGGDIPE) are proposed in this paper. First, the generalized Gaussian distribution cumulative distribution function is employed for data normalization to enhance the algorithm’s applicability across time series with diverse distributions. The algorithm further processes the normalized data using improved permutation entropy, which maintains both the absolute magnitude and temporal correlations of the signals, overcoming the equal value issue found in traditional permutation entropy (PE). Simulation results indicate that GGDIPE is less sensitive to parameter variations, exhibits strong noise resistance, accurately reveals the dynamic behavior of chaotic systems, and operates significantly faster than PE. Real-world data analysis shows that MGGDIPE provides markedly better separability for RR interval signals, EEG signals, bearing fault signals, and underwater acoustic signals compared to multiscale PE (MPE) and multiscale dispersion entropy (MDE). Notably, in underwater target recognition tasks, MGGDIPE achieves a classification accuracy of 97.5% across four types of acoustic signals, substantially surpassing the performance of MDE (70.5%) and MPE (62.5%). Thus, the proposed method demonstrates exceptional capability in processing complex time series. Full article
(This article belongs to the Special Issue Ordinal Pattern-Based Entropies: New Ideas and Challenges)
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17 pages, 924 KiB  
Article
Legendre Polynomial Fitting-Based Permutation Entropy Offers New Insights into the Influence of Fatigue on Surface Electromyography (sEMG) Signal Complexity
by Meryem Jabloun, Olivier Buttelli and Philippe Ravier
Entropy 2024, 26(10), 831; https://doi.org/10.3390/e26100831 - 30 Sep 2024
Viewed by 862
Abstract
In a recently published work, we introduced local Legendre polynomial fitting-based permutation entropy (LPPE) as a new complexity measure for quantifying disorder or randomness in time series. LPPE benefits from the ordinal pattern (OP) concept and incorporates a natural, aliasing-free multiscaling effect by [...] Read more.
In a recently published work, we introduced local Legendre polynomial fitting-based permutation entropy (LPPE) as a new complexity measure for quantifying disorder or randomness in time series. LPPE benefits from the ordinal pattern (OP) concept and incorporates a natural, aliasing-free multiscaling effect by design. The current work extends our previous study by investigating LPPE’s capability to assess fatigue levels using both synthetic and real surface electromyography (sEMG) signals. Real sEMG signals were recorded during biceps brachii fatiguing exercise maintained at 70% of maximal voluntary contraction (MVC) until exhaustion and were divided into four consecutive temporal segments reflecting sequential stages of exhaustion. As fatigue levels rise, LPPE values can increase or decrease significantly depending on the selection of embedding dimensions. Our analysis reveals two key insights. First, using LPPE with limited embedding dimensions shows consistency with the literature. Specifically, fatigue induces a decrease in sEMG complexity measures. This observation is supported by a comparison with the existing multiscale permutation entropy (MPE) variant, that is, the refined composite downsampling (rcDPE). Second, given a fixed OP length, higher embedding dimensions increase LPPE’s sensitivity to low-frequency components, which are notably present under fatigue conditions. Consequently, specific higher embedding dimensions appear to enhance the discrimination of fatigue levels. Thus, LPPE, as the only MPE variant that allows a practical exploration of higher embedding dimensions, offers a new perspective on fatigue’s impact on sEMG complexity, complementing existing MPE approaches. Full article
(This article belongs to the Special Issue Ordinal Pattern-Based Entropies: New Ideas and Challenges)
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15 pages, 3678 KiB  
Article
Tsallis Entropy-Based Complexity-IPE Casualty Plane: A Novel Method for Complex Time Series Analysis
by Zhe Chen, Changling Wu, Junyi Wang and Hongbing Qiu
Entropy 2024, 26(6), 521; https://doi.org/10.3390/e26060521 - 17 Jun 2024
Cited by 2 | Viewed by 1401
Abstract
Due to its capacity to unveil the dynamic characteristics of time series data, entropy has attracted growing interest. However, traditional entropy feature extraction methods, such as permutation entropy, fall short in concurrently considering both the absolute amplitude information of signals and the temporal [...] Read more.
Due to its capacity to unveil the dynamic characteristics of time series data, entropy has attracted growing interest. However, traditional entropy feature extraction methods, such as permutation entropy, fall short in concurrently considering both the absolute amplitude information of signals and the temporal correlation between sample points. Consequently, this limitation leads to inadequate differentiation among different time series and susceptibility to noise interference. In order to augment the discriminative power and noise robustness of entropy features in time series analysis, this paper introduces a novel method called Tsallis entropy-based complexity-improved permutation entropy casualty plane (TC-IPE-CP). TC-IPE-CP adopts a novel symbolization approach that preserves both absolute amplitude information and inter-point correlations within sequences, thereby enhancing feature separability and noise resilience. Additionally, by incorporating Tsallis entropy and weighting the probability distribution with parameter q, it integrates with statistical complexity to establish a feature plane of complexity and entropy, further enriching signal features. Through the integration of multiscale algorithms, a multiscale Tsallis-improved permutation entropy algorithm is also developed. The simulation results indicate that TC-IPE-CP requires a small amount of data, exhibits strong noise resistance, and possesses high separability for signals. When applied to the analysis of heart rate signals, fault diagnosis, and underwater acoustic signal recognition, experimental findings demonstrate that TC-IPE-CP can accurately differentiate between electrocardiographic signals of elderly and young subjects, achieve precise bearing fault diagnosis, and identify four types of underwater targets. Particularly in underwater acoustic signal recognition experiments, TC-IPE-CP achieves a recognition rate of 96.67%, surpassing the well-known multi-scale dispersion entropy and multi-scale permutation entropy by 7.34% and 19.17%, respectively. This suggests that TC-IPE-CP is highly suitable for the analysis of complex time series. Full article
(This article belongs to the Special Issue Ordinal Pattern-Based Entropies: New Ideas and Challenges)
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27 pages, 2852 KiB  
Article
Benefits of Zero-Phase or Linear Phase Filters to Design Multiscale Entropy: Theory and Application
by Eric Grivel, Bastien Berthelot, Gaetan Colin, Pierrick Legrand and Vincent Ibanez
Entropy 2024, 26(4), 332; https://doi.org/10.3390/e26040332 - 14 Apr 2024
Cited by 3 | Viewed by 1874
Abstract
In various applications, multiscale entropy (MSE) is often used as a feature to characterize the complexity of the signals in order to classify them. It consists of estimating the sample entropies (SEs) of the signal under study and its coarse-grained (CG) versions, where [...] Read more.
In various applications, multiscale entropy (MSE) is often used as a feature to characterize the complexity of the signals in order to classify them. It consists of estimating the sample entropies (SEs) of the signal under study and its coarse-grained (CG) versions, where the CG process amounts to (1) filtering the signal with an average filter whose order is the scale and (2) decimating the filter output by a factor equal to the scale. In this paper, we propose to derive a new variant of the MSE. Its novelty stands in the way to get the sequences at different scales by avoiding distortions during the decimation step. To this end, a linear-phase or null-phase low-pass filter whose cutoff frequency is well suited to the scale is used. Interpretations on how the MSE behaves and illustrations with a sum of sinusoids, as well as white and pink noises, are given. Then, an application to detect attentional tunneling is presented. It shows the benefit of the new approach in terms of p value when one aims at differentiating the set of MSEs obtained in the attentional tunneling state from the set of MSEs obtained in the nominal state. It should be noted that CG versions can be replaced not only for the MSE but also for other variants. Full article
(This article belongs to the Special Issue Ordinal Pattern-Based Entropies: New Ideas and Challenges)
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