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Time-series analysis in terms of ordinal patterns is revisited by introducing a generalized permutation entropy $H_p(w,L)$, which depends on two different window lengths: $w$, implicitly defining the resolution of the underlying partition;…

Chaotic Dynamics · Physics 2019-02-20 Stuart J Watt , Antonio Politi

Ordinal Patterns are a time-series data analysis tool used as a preliminary step to construct the Permutation Entropy which itself allows the same characterization of dynamics as chaotic or regular as more theoretical constructs such as the…

Adaptation and Self-Organizing Systems · Physics 2021-02-24 I. Gunther , Arjendu K. Pattanayak , Andrés Aragoneses

In 2002, in a seminal article, Christoph Bandt and Bernd Pompe proposed a new methodology for the analysis of complex time series, now known as Ordinal Analysis. The ordinal methodology is based on the computation of symbols (known as…

Data Analysis, Statistics and Probability · Physics 2022-06-07 Inmaculada Leyva , Johann Martinez , Cristina Masoller , Osvaldo A. Rosso , Massimiliano Zanin

We address the excess entropy, which is a measure of complexity for stationary time series, from the ordinal point of view. We show that the permutation excess entropy is equal to the mutual information between two adjacent semi-infinite…

Chaotic Dynamics · Physics 2011-12-13 Taichi Haruna , Kohei Nakajima

We propose to examine the predictability and the complexity characteristics of the Standard&Poor500 dynamics behaviors in a coarse-grained way using the symbolic dynamics method and under the prism of the Information theory through the…

Statistical Finance · Quantitative Finance 2021-05-11 Geoffrey Ducournau

Kolmogorov complexity of a finite binary word reflects both algorithmic structure and the empirical distribution of symbols appearing in the word. Words with symbol frequencies far from one half have smaller combinatorial richness and…

Computation · Statistics 2025-12-25 Brani Vidakovic

Shannon Entropy is the preeminent tool for measuring the level of uncertainty (and conversely, information content) in a random variable. In the field of communications, entropy can be used to express the information content of given…

Information Theory · Computer Science 2024-11-06 Bill Kay , Audun Myers , Thad Boydston , Emily Ellwein , Cameron Mackenzie , Iliana Alvarez , Erik Lentz

The Random Permutation Set (RPS) is a new type of set proposed recently, which can be regarded as the generalization of evidence theory. To measure the uncertainty of RPS, the entropy of RPS and its corresponding maximum entropy have been…

Information Theory · Computer Science 2024-03-12 Jiefeng Zhou , Zhen Li , Kang Hao Cheong , Yong Deng

In this work we introduce a method for estimating entropy rate and entropy production rate from finite symbolic time series. From the point of view of statistics, estimating entropy from a finite series can be interpreted as a problem of…

Statistical Mechanics · Physics 2021-02-24 Raul Salgado-Garcia , Cesar Maldonado

We derive the asymptotic distribution of ordinal-pattern frequencies under weak dependence conditions and investigate the long-run covariance matrix not only analytically for moving-average, Gaussian, and the novel generalized coin-tossing…

Statistics Theory · Mathematics 2025-07-24 Angelika Silbernagel , Christian Weiß

Based on the data gained from a full-scale experiment, the order/disorder characteristics of the compartment fire temperatures are analyzed. Among the known permutation/encoding type entropies used to analyze time series, we look for those…

Data Analysis, Statistics and Probability · Physics 2019-08-14 Flavia-Corina Mitroi-Symeonidis , Ion Anghel , Octavian Lalu , Constantin Popa

For discrete-time stochastic processes, there is a close connection between return/waiting times and entropy. Such a connection cannot be straightforwardly extended to the continuous-time setting. Contrarily to the discrete-time case one…

Probability · Mathematics 2007-05-23 Jean-Rene Chazottes , Cristian Giardina , Frank Redig

Since Bandt et al. have shown that the permutation entropy and the Kolmogorov-Sinai entropy coincide for piecewise monotone interval maps, the relationship of both entropies for time-discrete dynamical systems is of a certain interest. The…

Chaotic Dynamics · Physics 2014-07-25 Karsten Keller , Anton M. Unakafov , Valentina A. Unakafova

In a genetic algorithm, fluctuations of the entropy of a genome over time are interpreted as fluctuations of the information that the genome's organism is storing about its environment, being this reflected in more complex organisms. The…

Computational Engineering, Finance, and Science · Computer Science 2010-11-04 Manuel Cebrian , Manuel Alfonseca , Alfonso Ortega

In [Haruna, T. and Nakajima, K., 2011. Physica D 240, 1370-1377], the authors introduced the duality between values (words) and orderings (permutations) as a basis to discuss the relationship between information theoretic measures for…

Chaotic Dynamics · Physics 2015-06-04 Taichi Haruna , Kohei Nakajima

Entropy has been a common index to quantify the complexity of time series in a variety of fields. Here, we introduce increment entropy to measure the complexity of time series in which each increment is mapped into a word of two letters,…

Data Analysis, Statistics and Probability · Physics 2016-01-20 Xiaofeng Liu , Aimin Jiang , Ning Xu , Jianru Xue

Transfer entropy is a widely used measure for quantifying directed information flows in complex systems. While the challenges of estimating transfer entropy for continuous data are well known, it has two major shortcomings for data of…

Data Analysis, Statistics and Probability · Physics 2025-11-27 Alec Kirkley

The paper makes the observation that all orders of information entropy are equal in signals composed of repeating units of distinct symbols where the units can be classified as a member of a symmetry group. This leads to an improved metric…

Information Theory · Computer Science 2010-07-14 Reginald D. Smith

Subshifts of deterministic substitutions are ubiquitous objects in dynamical systems and aperiodic order (the mathematical theory of quasicrystals). Two of their most striking features are that they have low complexity (zero topological…

Dynamical Systems · Mathematics 2026-01-14 Philipp Gohlke , Andrew Mitchell , Dan Rust , Tony Samuel

Permutation entropy and its associated frameworks are remarkable examples of physics-inspired techniques adept at processing complex and extensive datasets. Despite substantial progress in developing and applying these tools, their use has…

Data Analysis, Statistics and Probability · Physics 2024-08-14 Leonardo G. J. M. Voltarelli , Arthur A. B. Pessa , Luciano Zunino , Rafael S. Zola , Ervin K. Lenzi , Matjaz Perc , Haroldo V. Ribeiro