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Entropy
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Ordinal Patterns-Based Tools and Their Applications
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Ordinal Patterns-Based Tools and Their Applications

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

Deadline for manuscript submissions: 15 August 2025 | Viewed by 3225

Special Issue Editor

Dr. Luciano Zunino

E-Mail Website
Guest Editor
Centro de Investigaciones Ópticas (CONICET La Plata—CIC—UNLP), La Plata, Buenos Aires, Argentina
Interests: time series analysis; nonlinear dynamics; complex systems; data analysis; permutation entropy; ordinal patterns; chaos; long-range correlations; fractality; multifractality
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

It is clear that in the current era of Big Data, there is a growing interest in finding efficient and reliable methods to deal with the concomitant data deluge. Simplicity, low computational cost, wide applicability, and less susceptibility to outliers and artifacts are highly valued properties for these tools. Ordinal symbolic quantifiers satisfy all these requirements and, consequently, they seem to be particularly suited to meet the challenge. Actually, even when their utility within this context has already been largely proven, there is room for more progress. In this Special Issue, novel ordinal pattern-based tools and/or strategies that help to support this claim are sought after. This also includes the possibility of combining them with algorithms and techniques from other fields, such as machine learning, in order to enhance their performance. Furthermore, successful interdisciplinary implementations that illustrate their potential in real-world applications are also greatly welcomed.

The characterization of time series from complex systems, the identification of intrinsic temporal scales, discrimination between stochastic and chaotic dynamics, time series classification, time series segmentation, and time series irreversibility, by using ordinal pattern-based tools, are just some of the topics of interest for this Special Issue. Original works and comprehensive reviews from both theoretical and applied perspectives will also be considered. Researchers and practitioners in the field are encouraged to make their contributions.

Dr. Luciano Zunino
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
  • ordinal pattern-based quantifiers
  • multiscale ordinal pattern-based quantifiers
  • interdisciplinary applications
  • complexity measures
  • classification and discrimination tasks
  • multivariate time series analysis
  • ordinal pattern-based methodologies and machine learning
  • ordinal symbolic mapping

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Published Papers (5 papers)

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Research

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15 pages, 419 KiB  
Article
Ordinal Random Processes
by Christoph Bandt
Entropy 2025, 27(6), 610; https://doi.org/10.3390/e27060610 - 7 Jun 2025
Viewed by 201
Abstract
Ordinal patterns have proven to be a valuable tool in many fields. Here, we address the need for theoretical models. A paradigmatic example shows that a model for frequencies of ordinal patterns can be determined without any numerical values. We specify the important [...] Read more.
Ordinal patterns have proven to be a valuable tool in many fields. Here, we address the need for theoretical models. A paradigmatic example shows that a model for frequencies of ordinal patterns can be determined without any numerical values. We specify the important concept of stationary order and the fundamental problems to be solved in order to establish a genuine statistical methodology for ordinal time series. Full article
(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)
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10 pages, 8363 KiB  
Article
Improved Reconstruction of Chaotic Signals from Ordinal Networks
by Antonio Politi and Leonardo Ricci
Entropy 2025, 27(5), 499; https://doi.org/10.3390/e27050499 - 6 May 2025
Viewed by 374
Abstract
Permutation entropy is customarily implemented to quantify the intrinsic indeterminacy of complex time series, under the assumption that determinism manifests itself by lowering the (permutation) entropy of the resulting symbolic sequence. We expect this to be roughly true, but, in general, it is [...] Read more.
Permutation entropy is customarily implemented to quantify the intrinsic indeterminacy of complex time series, under the assumption that determinism manifests itself by lowering the (permutation) entropy of the resulting symbolic sequence. We expect this to be roughly true, but, in general, it is not clear to what extent a given ordinal pattern indeed provides a faithful reconstruction of the original signal. Here, we address this question by attempting the reconstruction of the original time series by invoking an ergodic Markov approximation of the symbolic dynamics, thereby inverting the encoding procedure. Using the Hénon map as a testbed, we show that a meaningful reconstruction can also be made in the presence of a small observational noise. Full article
(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)
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16 pages, 1374 KiB  
Article
Quantifying Deviations from Gaussianity with Application to Flight Delay Distributions
by Felipe Olivares and Massimiliano Zanin
Entropy 2025, 27(4), 354; https://doi.org/10.3390/e27040354 - 28 Mar 2025
Viewed by 330
Abstract
We propose a novel approach for quantifying deviations from Gaussianity by leveraging the Jensen–Shannon distance. Using stable distributions as a flexible framework, we analyze the effects of skewness and heavy tails in synthetic sequences. We employ phase-randomized surrogates as Gaussian references to systematically [...] Read more.
We propose a novel approach for quantifying deviations from Gaussianity by leveraging the Jensen–Shannon distance. Using stable distributions as a flexible framework, we analyze the effects of skewness and heavy tails in synthetic sequences. We employ phase-randomized surrogates as Gaussian references to systematically evaluate the statistical distance between this reference and stable distributions. Our methodology is validated using real flight delay datasets from major airports in Europe and the United States, revealing significant deviations from Gaussianity, particularly at high-traffic airports. These results highlight systematic air traffic management strategy differences between the two geographic regions. Full article
(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)
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13 pages, 4232 KiB  
Article
Universality of Dynamical Symmetries in Chaotic Maps
by Marcos Acero, Sean Lyons, Andrés Aragoneses and Arjendu K. Pattanayak
Entropy 2024, 26(11), 969; https://doi.org/10.3390/e26110969 - 12 Nov 2024
Cited by 1 | Viewed by 1171
Abstract
Identifying signs of regularity and uncovering dynamical symmetries in complex and chaotic systems is crucial both for practical applications and for enhancing our understanding of complex dynamics. Recent approaches have quantified temporal correlations in time series, revealing hidden, approximate dynamical symmetries that provide [...] Read more.
Identifying signs of regularity and uncovering dynamical symmetries in complex and chaotic systems is crucial both for practical applications and for enhancing our understanding of complex dynamics. Recent approaches have quantified temporal correlations in time series, revealing hidden, approximate dynamical symmetries that provide insight into the systems under study. In this paper, we explore universality patterns in the dynamics of chaotic maps using combinations of complexity quantifiers. We also apply a recently introduced technique that projects dynamical symmetries into a “symmetry space”, providing an intuitive and visual depiction of these symmetries. Our approach unifies and extends previous results and, more importantly, offers a meaningful interpretation of universality by linking it with dynamical symmetries and their transitions. Full article
(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)
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Review

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14 pages, 470 KiB  
Review
Permutation Entropy and Its Niche in Hydrology: A Review
by Dragutin T. Mihailović
Entropy 2025, 27(6), 598; https://doi.org/10.3390/e27060598 - 3 Jun 2025
Viewed by 273
Abstract
One effective method for analyzing complexity involves applying information measures to time series derived from observational data. Permutation entropy (PE) is one such measure designed to quantify the degree of disorder or complexity within a time series by examining the order relations among [...] Read more.
One effective method for analyzing complexity involves applying information measures to time series derived from observational data. Permutation entropy (PE) is one such measure designed to quantify the degree of disorder or complexity within a time series by examining the order relations among its values. PE is distinguished by its simplicity, robustness, and exceptionally low computational cost, making it a benchmark tool for complexity analysis. This text reviews the advantages and limitations of PE while exploring its diverse applications in hydrology from 2002 to 2025. Specifically, it categorizes the uses of PE across various subfields, including runoff prediction, streamflow analysis, water level forecasting, assessment of hydrological changes, and evaluating the impact of infrastructure on hydrological systems. By leveraging PE’s ability to capture the intricate dynamics of hydrological processes, researchers can enhance predictive models and improve our understanding of water-related phenomena. Full article
(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)
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