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Entropy
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Theoretical Developments and Applications of Entropy and Ordinal Patterns
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Theoretical Developments and Applications of Entropy and Ordinal Patterns

A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (15 December 2019) | Viewed by 24883

Special Issue Editors

Prof. Dr. José María Amigó

E-Mail Website
Guest Editor
Centro de Investigación Operativa, Universidad Miguel Hernández, Avenida de la Universidad s/n, 03202 Elche, Spain
Interests: dynamical systems; nonlinear time-series analysis; ergodic theory; mathematical physics
Special Issues, Collections and Topics in MDPI journals
Prof. Yoshito Hirata

E-Mail Website
Guest Editor
Mathematics and Informatics Center, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Interests: nonlinear time-series analysis; time-series prediction; point processes; mathematical medicine

Special Issue Information

Dear Colleagues,

The concept of entropy (whether as a measure of disorder, uncertainty, randomness, or complexity) is ubiquitous in applied mathematics. This is due both to its exceptional mathematical properties, such as invariance under relevant transformations, and, especially, to its generality, which causes other similar quantifiers to be related to it. In this context, one of the scopes of this Special Issue is to develop new theoretical insights and practical applications with the concept of entropy, in any of its different  materializations, as a leitmotif.

At the same, we are also interested in papers devoted to the study of ordinal patterns. Permutation entropy, an entropy of ordinal patterns originally introduced by Bandt and Pompe (2002), has led to a paradigm shift in nonlinear time-series analysis, because we do not have to estimate a generating partition for rigorously analysing a given time series by preserving the information for the underlying dynamics. Now, we can estimate metric and topological entropies much more easily. In addition, there are lots of emerging applications of ordinal patterns such as change-point detections, time-series predictions, detection of determinism, directional coupling, and surrogate data. Thus, this Special Issue aims as well at accelerating theoretical developments of ordinal patterns, and expanding their applications in science, engineering, medicine, and society. Both theoretical and/or application-oriented papers will be considered for the publication in this Special Issue of Entropy.

Prof. José María Amigó
Prof. Yoshito Hirata
Guest Editors

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

  • entropy and entropy-like quantities
  • complexity quantified with entropy and/or ordinal patterns
  • determinism and stochasticity using entropy and/or ordinal patterns
  • causality/directional coupling using entropy and/or ordinal patterns
  • surrogate data using entropy and/or ordinal patterns
  • early warning signals/change-point detection using entropy and/or ordinal patterns
  • time-series analysis/time-series prediction using entropy and/or ordinal patterns
  • other emerging applications using entropy and/or ordinal patterns
  • theoretical justification for analysis using entropy and/or ordinal patterns

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (7 papers)

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Research

26 pages, 418 KiB  
Article
Ordinal Pattern Based Entropies and the Kolmogorov–Sinai Entropy: An Update
by Tim Gutjahr and Karsten Keller
Entropy 2020, 22(1), 63; https://doi.org/10.3390/e22010063 - 2 Jan 2020
Cited by 8 | Viewed by 3381
Abstract
Different authors have shown strong relationships between ordinal pattern based entropies and the Kolmogorov–Sinai entropy, including equality of the latter one and the permutation entropy, the whole picture is however far from being complete. This paper is updating the picture by summarizing some [...] Read more.
Different authors have shown strong relationships between ordinal pattern based entropies and the Kolmogorov–Sinai entropy, including equality of the latter one and the permutation entropy, the whole picture is however far from being complete. This paper is updating the picture by summarizing some results and discussing some mainly combinatorial aspects behind the dependence of Kolmogorov–Sinai entropy from ordinal pattern distributions on a theoretical level. The paper is more than a review paper. A new statement concerning the conditional permutation entropy will be given as well as a new proof for the fact that the permutation entropy is an upper bound for the Kolmogorov–Sinai entropy. As a main result, general conditions for the permutation entropy being a lower bound for the Kolmogorov–Sinai entropy will be stated. Additionally, a previously introduced method to analyze the relationship between permutation and Kolmogorov–Sinai entropies as well as its limitations will be investigated. Full article
(This article belongs to the Special Issue Theoretical Developments and Applications of Entropy and Ordinal Patterns)
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19 pages, 1167 KiB  
Article
Permutation Entropy: Enhancing Discriminating Power by Using Relative Frequencies Vector of Ordinal Patterns Instead of Their Shannon Entropy
by David Cuesta-Frau, Antonio Molina-Picó, Borja Vargas and Paula González
Entropy 2019, 21(10), 1013; https://doi.org/10.3390/e21101013 - 18 Oct 2019
Cited by 11 | Viewed by 3009
Abstract
Many measures to quantify the nonlinear dynamics of a time series are based on estimating the probability of certain features from their relative frequencies. Once a normalised histogram of events is computed, a single result is usually derived. This process can be broadly [...] Read more.
Many measures to quantify the nonlinear dynamics of a time series are based on estimating the probability of certain features from their relative frequencies. Once a normalised histogram of events is computed, a single result is usually derived. This process can be broadly viewed as a nonlinear I R n mapping into I R , where n is the number of bins in the histogram. However, this mapping might entail a loss of information that could be critical for time series classification purposes. In this respect, the present study assessed such impact using permutation entropy (PE) and a diverse set of time series. We first devised a method of generating synthetic sequences of ordinal patterns using hidden Markov models. This way, it was possible to control the histogram distribution and quantify its influence on classification results. Next, real body temperature records are also used to illustrate the same phenomenon. The experiments results confirmed the improved classification accuracy achieved using raw histogram data instead of the PE final values. Thus, this study can provide a very valuable guidance for the improvement of the discriminating capability not only of PE, but of many similar histogram-based measures. Full article
(This article belongs to the Special Issue Theoretical Developments and Applications of Entropy and Ordinal Patterns)
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20 pages, 334 KiB  
Article
Two Tests for Dependence (of Unknown Form) between Time Series
by M. Victoria Caballero-Pintado, Mariano Matilla-García, Jose M. Rodríguez and Manuel Ruiz Marín
Entropy 2019, 21(9), 878; https://doi.org/10.3390/e21090878 - 9 Sep 2019
Cited by 2 | Viewed by 2627
Abstract
This paper proposes two new nonparametric tests for independence between time series. Both tests are based on symbolic analysis, specifically on symbolic correlation integral, in order to be robust to potential unknown nonlinearities. The first test is developed for a scenario in which [...] Read more.
This paper proposes two new nonparametric tests for independence between time series. Both tests are based on symbolic analysis, specifically on symbolic correlation integral, in order to be robust to potential unknown nonlinearities. The first test is developed for a scenario in which each considered time series is independent and therefore the interest is to ascertain if two internally independent time series share a relationship of an unknown form. This is especially relevant as the test is nuisance parameter free, as proved in the paper. The second proposed statistic tests for independence among variables, allowing these time series to exhibit within-dependence. Monte Carlo experiments are conducted to show the empirical properties of the tests. Full article
(This article belongs to the Special Issue Theoretical Developments and Applications of Entropy and Ordinal Patterns)
16 pages, 1105 KiB  
Article
Surrogate Data Preserving All the Properties of Ordinal Patterns up to a Certain Length
by Yoshito Hirata, Masanori Shiro and José M. Amigó
Entropy 2019, 21(7), 713; https://doi.org/10.3390/e21070713 - 22 Jul 2019
Cited by 8 | Viewed by 3427
Abstract
We propose a method for generating surrogate data that preserves all the properties of ordinal patterns up to a certain length, such as the numbers of allowed/forbidden ordinal patterns and transition likelihoods from ordinal patterns into others. The null hypothesis is that the [...] Read more.
We propose a method for generating surrogate data that preserves all the properties of ordinal patterns up to a certain length, such as the numbers of allowed/forbidden ordinal patterns and transition likelihoods from ordinal patterns into others. The null hypothesis is that the details of the underlying dynamics do not matter beyond the refinements of ordinal patterns finer than a predefined length. The proposed surrogate data help construct a test of determinism that is free from the common linearity assumption for a null-hypothesis. Full article
(This article belongs to the Special Issue Theoretical Developments and Applications of Entropy and Ordinal Patterns)
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22 pages, 3032 KiB  
Article
Small Order Patterns in Big Time Series: A Practical Guide
by Christoph Bandt
Entropy 2019, 21(6), 613; https://doi.org/10.3390/e21060613 - 21 Jun 2019
Cited by 30 | Viewed by 4645
Abstract
The study of order patterns of three equally-spaced values x t , x t + d , x t + 2 d in a time series is a powerful tool. The lag d is changed in a wide range so that the differences [...] Read more.
The study of order patterns of three equally-spaced values x t , x t + d , x t + 2 d in a time series is a powerful tool. The lag d is changed in a wide range so that the differences of the frequencies of order patterns become autocorrelation functions. Similar to a spectrogram in speech analysis, four ordinal autocorrelation functions are used to visualize big data series, as for instance heart and brain activity over many hours. The method applies to real data without preprocessing, and outliers and missing data do not matter. On the theoretical side, we study the properties of order correlation functions and show that the four autocorrelation functions are orthogonal in a certain sense. An analysis of variance of a modified permutation entropy can be performed with four variance components associated with the functions. Full article
(This article belongs to the Special Issue Theoretical Developments and Applications of Entropy and Ordinal Patterns)
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19 pages, 2243 KiB  
Article
Ordinal Patterns in Heartbeat Time Series: An Approach Using Multiscale Analysis
by María Muñoz-Guillermo
Entropy 2019, 21(6), 583; https://doi.org/10.3390/e21060583 - 12 Jun 2019
Cited by 3 | Viewed by 3033
Abstract
In this paper, we simultaneously use two different scales in the analysis of ordinal patterns to measure the complexity of the dynamics of heartbeat time series. Rényi entropy and weighted Rényi entropy are the entropy-like measures proposed in the multiscale analysis in which, [...] Read more.
In this paper, we simultaneously use two different scales in the analysis of ordinal patterns to measure the complexity of the dynamics of heartbeat time series. Rényi entropy and weighted Rényi entropy are the entropy-like measures proposed in the multiscale analysis in which, with the new scheme, four parameters are involved. First, the influence of the variation of the new parameters in the entropy values is analyzed when different groups of subjects (with cardiac diseases or healthy) are considered. Secondly, we exploit the introduction of multiscale analysis in order to detect differences between the groups. Full article
(This article belongs to the Special Issue Theoretical Developments and Applications of Entropy and Ordinal Patterns)
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14 pages, 446 KiB  
Article
Using Permutations for Hierarchical Clustering of Time Series
by Jose S. Cánovas, Antonio Guillamón and María Carmen Ruiz-Abellón
Entropy 2019, 21(3), 306; https://doi.org/10.3390/e21030306 - 21 Mar 2019
Cited by 3 | Viewed by 3917
Abstract
Two distances based on permutations are considered to measure the similarity of two time series according to their strength of dependency. The distance measures are used together with different linkages to get hierarchical clustering methods of time series by dependency. We apply these [...] Read more.
Two distances based on permutations are considered to measure the similarity of two time series according to their strength of dependency. The distance measures are used together with different linkages to get hierarchical clustering methods of time series by dependency. We apply these distances to both simulated theoretical and real data series. For simulated time series the distances show good clustering results, both in the case of linear and non-linear dependencies. The effect of the embedding dimension and the linkage method are also analyzed. Finally, several real data series are properly clustered using the proposed method. Full article
(This article belongs to the Special Issue Theoretical Developments and Applications of Entropy and Ordinal Patterns)
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