Article Permutation Entropy and Its Main Biomedical and Econophysics Applications: A Review
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paper_title: Distinguishing noise from chaos.

paper_content:

Chaotic systems share with stochastic processes several properties that make them almost undistinguishable. In this communication we introduce a representation space, to be called the complexity-entropy causality plane. Its horizontal and vertical axis are suitable functionals of the pertinent probability distribution, namely, the entropy of the system and an appropriate statistical complexity measure, respectively. These two functionals are evaluated using the Bandt-Pompe recipe to assign a probability distribution function to the time series generated by the system. Several well-known model-generated time series, usually regarded as being of either stochastic or chaotic nature, are analyzed so as to illustrate the approach. The main achievement of this communication is the possibility of clearly distinguishing between them in our representation space, something that is rather difficult otherwise.
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paper_title: Causality and the Entropy-Complexity Plane: Robustness and Missing Ordinal Patterns

paper_content:

We deal here with the issue of determinism versus randomness in time series. One wishes to identify their relative weights in a given time series. Two different tools have been advanced in the literature to such effect, namely, (i) the “causal” entropy–complexity plane [O.A. Rosso, H.A. Larrondo, M.T. Martin, A. Plastino, M.A. Fuentes, Distinguishing noise from chaos, Phys. Rev. Lett. 99 (2007) 154102] and (ii) the estimation of the decay rate of missing ordinal patterns [J.M. Amigo, S. Zambrano, M.A.F. Sanjuan, True and false forbidden patterns in deterministic and random dynamics, Europhys. Lett. 79 (2007) 50001; L.C. Carpi, P.M. Saco, O.A. Rosso, Missing ordinal patterns in correlated noises. Physica A 389 (2010) 2020–2029]. In this work we extend the use of these techniques to address the analysis of deterministic finite time series contaminated with additive noises of different degree of correlation. The chaotic series studied here was via the logistic map (r=4) to which we added correlated noise (colored noise with f−k Power Spectrum, 0≤k≤2) of varying amplitudes. In such a fashion important insights pertaining to the deterministic component of the original time series can be gained. We find that in the entropy–complexity plane this goal can be achieved without additional computations.
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paper_title: Wavelet entropy: a new tool for analysis of short duration brain electrical signals

paper_content:

Since traditional electrical brain signal analysis is mostly qualitative, the development of new quantitative methods is crucial for restricting the subjectivity in the study of brain signals. These methods are particularly fruitful when they are strongly correlated with intuitive physical concepts that allow a better understanding of brain dynamics. Here, new method based on orthogonal discrete wavelet transform (ODWT) is applied. It takes as a basic element the ODWT of the EEG signal, and defines the relative wavelet energy, the wavelet entropy (WE) and the relative wavelet entropy (RWE). The relative wavelet energy provides information about the relative energy associated with different frequency bands present in the EEG and their corresponding degree of importance. The WE carries information about the degree of order:disorder associated with a multi-frequency signal response, and the RWE measures the degree of similarity between different segments of the signal. In addition, the time evolution of the WE is calculated to give information about the dynamics in the EEG records. Within this framework, the major objective of the present work was to characterize in a quantitative way functional dynamics of order:disorder microstates in short duration EEG signals. For that aim, spontaneous EEG signals under different physiological conditions were analyzed. Further, specific quantifiers were derived to characterize how stimulus affects electrical events in terms of frequency synchronization (tuning) in the event related potentials. © 2001 Published by Elsevier Science B.V.
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paper_title: Permutation Entropy: A Natural Complexity Measure for Time Series

paper_content:

We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.
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paper_title: A spectral entropy method for distinguishing regular and irregular motion of Hamiltonian systems

paper_content:

Regular and irregular motions of bounded conservative Hamiltonian systems of N degrees of freedom can be distinguished by the structure of the frequency spectrum of a single trajectory. The spectral entropy S is introduced which provides a measure of the distribution of the frequency components. Numerical calculations on the model Henon and Heiles system and a realistic molecular model are performed. Power spectra are obtained from numerical solutions to Hamilton's equations using fast Fourier transforms and the Hanning method. For regular trajectories S is found to stabilise after a finite time of integration, while for irregular cases S increases erratically. Estimates of the relative volume of regular regions of phase space as a function of energy are given for the two systems.
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paper_title: GENERALIZED STATISTICAL COMPLEXITY MEASURE

paper_content:

A generalized Statistical Complexity Measure (SCM) is a functional that characterizes the probability distribution P associated to the time series generated by a given dynamical system. It quantifies not only randomness but also the presence of correlational structures. We review here several fundamental issues in such a respect, namely, (a) the selection of the information measure ; (b) the choice of the probability metric space and associated distance ; (c) the question of defining the so-called generalized disequilibrium ; (d) the adequate way of picking up the probability distribution P associated to a dynamical system or time series under study, which is indeed a fundamental problem. In this communication we show (point d) that sensible improvements in the final results can be expected if the underlying probability distribution is "extracted" via appropriate consideration regarding causal effects in the system's dynamics.
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paper_title: Generalized statistical complexity measures: Geometrical and analytical properties

paper_content:

We discuss bounds on the values adopted by the generalized statistical complexity measures [M.T. Martin et al., Phys. Lett. A 311 (2003) 126; P.W. Lamberti et al., Physica A 334 (2004) 119] introduced by Lopez Ruiz et al. [Phys. Lett. A 209 (1995) 321] and Shiner et al. [Phys. Rev. E 59 (1999) 1459]. Several new theorems are proved and illustrated with reference to the celebrated logistic map.
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paper_title: Shakespeare and other English Renaissance authors as characterized by Information Theory complexity quantifiers

paper_content:

We introduce novel Information Theory quantifiers in a computational linguistic study that involves a large corpus of English Renaissance literature. The 185 texts studied (136 plays and 49 poems in total), with first editions that range from 1580 to 1640, form a representative set of its period. Our data set includes 30 texts unquestionably attributed to Shakespeare; in addition we also included A Lover’s Complaint, a poem which generally appears in Shakespeare collected editions but whose authorship is currently in dispute. Our statistical complexity quantifiers combine the power of Jensen–Shannon’s divergence with the entropy variations as computed from a probability distribution function of the observed word use frequencies. Our results show, among other things, that for a given entropy poems display higher complexity than plays, that Shakespeare’s work falls into two distinct clusters in entropy, and that his work is remarkable for its homogeneity and for its closeness to overall means.
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paper_title: Entropy and Information Theory

paper_content:

This book is an updated version of the information theory classic, first published in 1990. About one-third of the book is devoted to Shannon source and channel coding theorems; the remainder addresses sources, channels, and codes and on information and distortion measures and their properties. New in this edition:Expanded treatment of stationary or sliding-block codes and their relations to traditional block codesExpanded discussion of results from ergodic theory relevant to information theoryExpanded treatment of B-processes -- processes formed by stationary coding memoryless sourcesNew material on trading off information and distortion, including the Marton inequalityNew material on the properties of optimal and asymptotically optimal source codesNew material on the relationships of source coding and rate-constrained simulation or modeling of random processesSignificant material not covered in other information theory texts includes stationary/sliding-block codes, a geometric view of information theory provided by process distance measures, and general Shannon coding theorems for asymptotic mean stationary sources, which may be neither ergodic nor stationary, and d-bar continuous channels.
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paper_title: Construction of Symbolic Dynamics from Experimental Time Series

paper_content:

Symbolic dynamics play a central role in the description of the evolution of nonlinear systems. Yet there are few methods for determining symbolic dynamics of chaotic data. One difficulty is that the data contains random fluctuations associated with the experimental process. Using data obtained from a magnetoelastic ribbon experiment we show how a topological approach that allows for experimental error and bounded noise can be used to obtain a description of the dynamics in terms of subshift dynamics on a finite set of symbols.
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paper_title: Randomizing nonlinear maps via symbolic dynamics

paper_content:

Pseudo Random Number Generators (PRNG) have attracted intense attention due to their obvious importance for many branches of science and technology. A randomizing technique is a procedure designed to improve the PRNG randomness degree according the specific requirements. It is obviously important to quantify its effectiveness. In order to classify randomizing techniques based on a symbolic dynamics’ approach, we advance a novel, physically motivated representation based on the statistical properties of chaotic systems. Recourse is made to a plane that has as coordinates (i) the Shannon entropy and (ii) a form of the statistical complexity measure. Each statistical quantifier incorporates a different probability distribution function, generating thus a representation that (i) sheds insight into just how each randomizing technique operates and also (ii) quantifies its effectiveness. Using the Logistic Map and the Three Way Bernoulli Map as typical examples of chaotic dynamics it is shown that our methodology allows for choosing the more convenient randomizing technique in each instance. Comparison with measures of complexity based on diagonal lines on the recurrence plots [N. Marwan, M.C. Romano, M. Thiel, J. Kurths, Phys. Rep. 438 (2007) 237] support the main conclusions of this paper.
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paper_title: Measures of Statistical Complexity: Why?

paper_content:

We review several statistical complexity measures proposed over the last decade and a half as general indicators of structure or correlation. Recently, L\`opez-Ruiz, Mancini, and Calbet [Phys. Lett. A 209 (1995) 321] introduced another measure of statistical complexity $C_{\rm LMC}$ that, like others, satisfies the ``boundary conditions'' of vanishing in the extreme ordered and disordered limits. We examine some properties of $C_{\rm LMC}$ and find that it is neither an intensive nor an extensive thermodynamic variable. It depends nonlinearly on system size and vanishes exponentially in the thermodynamic limit for all one-dimensional finite-range spin systems. We propose a simple alteration of $C_{\rm LMC}$ that renders it extensive. however, this remedy results in a quantity that is a trivial function of the entropy density and hence of no use as a measure of structure or memory. We conclude by suggesting that a useful ``statistical complexity'' must not only obey the ordered-random bounary conditions of vanishing, it must also be defined in a setting that gives a clear interpretation to what structures are quantified.
---
paper_title: Distinguishing noise from chaos.

paper_content:

Chaotic systems share with stochastic processes several properties that make them almost undistinguishable. In this communication we introduce a representation space, to be called the complexity-entropy causality plane. Its horizontal and vertical axis are suitable functionals of the pertinent probability distribution, namely, the entropy of the system and an appropriate statistical complexity measure, respectively. These two functionals are evaluated using the Bandt-Pompe recipe to assign a probability distribution function to the time series generated by the system. Several well-known model-generated time series, usually regarded as being of either stochastic or chaotic nature, are analyzed so as to illustrate the approach. The main achievement of this communication is the possibility of clearly distinguishing between them in our representation space, something that is rather difficult otherwise.
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paper_title: Permutation entropy improves fetal behavioural state classification based on heart rate analysis from biomagnetic recordings in near term fetuses

paper_content:

The relevance of the complexity of fetal heart rate fluctuations with regard to the classification of fetal behavioural states has not been satisfyingly clarified so far. Because of the short behavioural states, the permutation entropy provides an advantageous complexity estimation leading to the Kullback–Leibler entropy (KLE). We test the hypothesis that parameters derived from KLE can improve the classification of fetal behaviour states based on classical heart rate fluctuation parameters (SDNN, RMSSD, ln(LF), ln(HF)). From measured heartbeat sequences (35 healthy fetuses at a gestational age between 35 and 40 completed weeks) representative intervals of 256 heartbeats were visually preclassified into fetal behavioural states. Employing discriminant analysis to separate the states 1F, 2F and 4F, the best classification result by classical parameters was 80.0% (SDNN). After additionally considering KLE parameters it was improved significantly (p<0.0005) to 94.3% (ln(LF), KLE_Mean). It could be confirmed that KLE can improve the state classification. This might reflect the consideration of different physiological aspects by classical and complexity measures.
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paper_title: Permutation Entropy: A Natural Complexity Measure for Time Series

paper_content:

We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.
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paper_title: Ordinal time series analysis

paper_content:

Abstract We discuss robust methods of time series analysis which use only comparisons of values and not their actual size. Local and global order structure are defined as matrices or by rank numbers. Local ranks, autocorrelation by Kendall’s tau, and permutation entropy as complexity measure are introduced in such a way that they contain a scale parameter which allows to study time series on different scales.
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paper_title: Permutation-information-theory approach to unveil delay dynamics from time-series analysis

paper_content:

In this paper an approach to identify delay phenomena from time series is developed. We show that it is possible to perform a reliable time delay identification by using quantifiers derived from information theory, more precisely, permutation entropy and permutation statistical complexity. These quantifiers show clear extrema when the embedding delay τ of the symbolic reconstruction matches the characteristic time delay τ(S) of the system. Numerical data originating from a time delay system based on the well-known Mackey-Glass equations operating in the chaotic regime were used as test beds. We show that our method is straightforward to apply and robust to additive observational and dynamical noise. Moreover, we find that the identification of the time delay is even more efficient in a noise environment. Our permutation approach is also able to recover the time delay in systems with low feedback rate or high nonlinearity.
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paper_title: Entropy and Information Theory

paper_content:

This book is an updated version of the information theory classic, first published in 1990. About one-third of the book is devoted to Shannon source and channel coding theorems; the remainder addresses sources, channels, and codes and on information and distortion measures and their properties. New in this edition:Expanded treatment of stationary or sliding-block codes and their relations to traditional block codesExpanded discussion of results from ergodic theory relevant to information theoryExpanded treatment of B-processes -- processes formed by stationary coding memoryless sourcesNew material on trading off information and distortion, including the Marton inequalityNew material on the properties of optimal and asymptotically optimal source codesNew material on the relationships of source coding and rate-constrained simulation or modeling of random processesSignificant material not covered in other information theory texts includes stationary/sliding-block codes, a geometric view of information theory provided by process distance measures, and general Shannon coding theorems for asymptotic mean stationary sources, which may be neither ergodic nor stationary, and d-bar continuous channels.
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paper_title: Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator

paper_content:

In the dynamics of optical systems, one commonly needs to cope with the problem of coexisting deterministic and stochastic components. The separation of these components is an important, although difficult, task. Often the time scales at which determinism and noise dominate the system's dynamics differ. In this Letter we propose to use information-theory-derived quantifiers, more precisely, permutation entropy and statistical complexity, to distinguish between the two behaviors. Based on experiments of a paradigmatic opto-electronic oscillator, we demonstrate that the time scales at which deterministic or noisy behavior dominate can be identified. Supporting numerical simulations prove the accuracy of this identification.
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paper_title: Time Scales of a Chaotic Semiconductor Laser With Optical Feedback Under the Lens of a Permutation Information Analysis

paper_content:

We analyze the intrinsic time scales of the chaotic dynamics of a semiconductor laser subject to optical feedback by estimating quantifiers derived from a permutation information approach. Based on numerically and experimentally obtained times series, we find that permutation entropy and permutation statistical complexity allow the extraction of important characteristics of the dynamics of the system. We provide evidence that permutation statistical complexity is complementary to permutation entropy, giving valuable insights into the role of the different time scales involved in the chaotic regime of the semiconductor laser dynamics subject to delay optical feedback. The results obtained confirm that this novel approach is a conceptually simple and computationally efficient method to identify the characteristic time scales of this relevant physical system.
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paper_title: Classifying cardiac biosignals using ordinal pattern statistics and symbolic dynamics

paper_content:

The performance of (bio-)signal classification strongly depends on the choice of suitable features (also called parameters or biomarkers). In this article we evaluate the discriminative power of ordinal pattern statistics and symbolic dynamics in comparison with established heart rate variability parameters applied to beat-to-beat intervals. As an illustrative example we distinguish patients suffering from congestive heart failure from a (healthy) control group using beat-to-beat time series. We assess the discriminative power of individual features as well as pairs of features. These comparisons show that ordinal patterns sampled with an additional time lag are promising features for efficient classification.
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paper_title: Causality and the Entropy-Complexity Plane: Robustness and Missing Ordinal Patterns

paper_content:

We deal here with the issue of determinism versus randomness in time series. One wishes to identify their relative weights in a given time series. Two different tools have been advanced in the literature to such effect, namely, (i) the “causal” entropy–complexity plane [O.A. Rosso, H.A. Larrondo, M.T. Martin, A. Plastino, M.A. Fuentes, Distinguishing noise from chaos, Phys. Rev. Lett. 99 (2007) 154102] and (ii) the estimation of the decay rate of missing ordinal patterns [J.M. Amigo, S. Zambrano, M.A.F. Sanjuan, True and false forbidden patterns in deterministic and random dynamics, Europhys. Lett. 79 (2007) 50001; L.C. Carpi, P.M. Saco, O.A. Rosso, Missing ordinal patterns in correlated noises. Physica A 389 (2010) 2020–2029]. In this work we extend the use of these techniques to address the analysis of deterministic finite time series contaminated with additive noises of different degree of correlation. The chaotic series studied here was via the logistic map (r=4) to which we added correlated noise (colored noise with f−k Power Spectrum, 0≤k≤2) of varying amplitudes. In such a fashion important insights pertaining to the deterministic component of the original time series can be gained. We find that in the entropy–complexity plane this goal can be achieved without additional computations.
---
paper_title: Permutation Entropy: A Natural Complexity Measure for Time Series

paper_content:

We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.
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paper_title: Order Patterns and Chaos

paper_content:

Chaotic maps can mimic random behavior in a quite impressive way. In particular, those possessing a generating partition can produce any symbolic sequence by properly choosing the initial state. We study in this Letter the ability of chaotic maps to generate order patterns and come to the conclusion that their performance in this respect falls short of expectations. This result reveals some basic limitation of a deterministic dynamic as compared to a random one. This being the case, we propose a non-statistical test based on ‘forbidden’ order patterns to discriminate chaotic from truly random time series with, in principle, arbitrarily high probability. Some relations with discrete chaos and chaotic cryptography are also discussed.
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paper_title: Order Patterns in Time Series

paper_content:

Recent use of order patterns in time-series analysis shows the need for a corresponding theory. We determine probabilities of order patterns in Gaussian and autoregressive moving-average (ARMA) processes. Two order functions are introduced which characterize a time series in a way similar to autocorrelation. For stationary ergodic processes, all finite-dimensional distributions are obtained from the one-dimensional distribution plus the order structure of a typical time series. Copyright 2007 The Authors Journal compilation 2007 Blackwell Publishing Ltd.
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paper_title: Measures of Statistical Complexity: Why?

paper_content:

We review several statistical complexity measures proposed over the last decade and a half as general indicators of structure or correlation. Recently, L\`opez-Ruiz, Mancini, and Calbet [Phys. Lett. A 209 (1995) 321] introduced another measure of statistical complexity $C_{\rm LMC}$ that, like others, satisfies the ``boundary conditions'' of vanishing in the extreme ordered and disordered limits. We examine some properties of $C_{\rm LMC}$ and find that it is neither an intensive nor an extensive thermodynamic variable. It depends nonlinearly on system size and vanishes exponentially in the thermodynamic limit for all one-dimensional finite-range spin systems. We propose a simple alteration of $C_{\rm LMC}$ that renders it extensive. however, this remedy results in a quantity that is a trivial function of the entropy density and hence of no use as a measure of structure or memory. We conclude by suggesting that a useful ``statistical complexity'' must not only obey the ordered-random bounary conditions of vanishing, it must also be defined in a setting that gives a clear interpretation to what structures are quantified.
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paper_title: True and false forbidden patterns in deterministic and random dynamics

paper_content:

In this letter we discuss some properties of order patterns both in deterministic and random orbit generation. As it turns out, the orbits of one-dimensional maps have always forbidden patterns, i.e., order patterns that cannot occur, in contrast with random time series, in which any order pattern appears with probability one. However, finite random sequences may exhibit "false" forbidden patterns with non-vanishing probability. In this case, forbidden patterns decay with the sequence length, thus unveiling the random nature of the sequence. Last but not least, true forbidden patterns are robust against noise and disintegrate with a rate that depends on the noise level. These properties can be embodied in a simple method to distinguish deterministic, finite time series with very high levels of observational noise, from random ones. We present numerical evidence for white noise.
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paper_title: Combinatorial detection of determinism in noisy time series

paper_content:

This paper deals with the distinction between white noise and deterministic chaos in multivariate noisy time series. Our method is combinatorial in the sense that it is based on the properties of topological permutation entropy, and it becomes especially interesting when the noise is so high that the standard denoising techniques fail, so a detection of determinism is the most one can hope for. It proceeds by i) counting the number of the so-called ordinal patterns in independent samples of length L from the data sequence and ii) performing a χ2 test based on the results of i), the null hypothesis being that the data are white noise. Holds the null hypothesis, so should all possible ordinal patterns of a given length be visible and evenly distributed over sufficiently many samples, contrarily to what happens in the case of noisy deterministic data. We present numerical evidence in two dimensions for the efficiency of this method. A brief comparison with two common tests for independence, namely, the calculation of the autocorrelation function and the BDS algorithm, is also performed.
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paper_title: Distinguishing noise from chaos.

paper_content:

Chaotic systems share with stochastic processes several properties that make them almost undistinguishable. In this communication we introduce a representation space, to be called the complexity-entropy causality plane. Its horizontal and vertical axis are suitable functionals of the pertinent probability distribution, namely, the entropy of the system and an appropriate statistical complexity measure, respectively. These two functionals are evaluated using the Bandt-Pompe recipe to assign a probability distribution function to the time series generated by the system. Several well-known model-generated time series, usually regarded as being of either stochastic or chaotic nature, are analyzed so as to illustrate the approach. The main achievement of this communication is the possibility of clearly distinguishing between them in our representation space, something that is rather difficult otherwise.
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paper_title: The organization of intrinsic computation: complexity-entropy diagrams and the diversity of natural information processing.

paper_content:

Intrinsic computation refers to how dynamical systems store, structure, and transform historical and spatial information. By graphing a measure of structural complexity against a measure of randomness, complexity-entropy diagrams display the different kinds of intrinsic computation across an entire class of systems. Here, we use complexity-entropy diagrams to analyze intrinsic computation in a broad array of deterministic nonlinear and linear stochastic processes, including maps of the interval, cellular automata, and Ising spin systems in one and two dimensions, Markov chains, and probabilistic minimal finite-state machines. Since complexity-entropy diagrams are a function only of observed configurations, they can be used to compare systems without reference to system coordinates or parameters. It has been known for some time that in special cases complexity-entropy diagrams reveal that high degrees of information processing are associated with phase transitions in the underlying process space, the so-called "edge of chaos." Generally, though, complexity-entropy diagrams differ substantially in character, demonstrating a genuine diversity of distinct kinds of intrinsic computation.
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paper_title: A Statistical Measure of Complexity

paper_content:

Abstract A measure of complexity based on a probabilistic description of physical systems is proposed. This measure incorporates the main features of the intuitive notion of such a magnitude. It can be applied to many physical situations and to different descriptions of a given system. Moreover, the calculation of its value does not require a considerable computational effort in many cases of physical interest.
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paper_title: Detecting and quantifying temporal correlations in stochastic resonance via information theory measures

paper_content:

We show that Information Theory quantifiers are suitable tools for detecting and for quantifying noise-induced temporal correlations in stochastic resonance phenomena. We use the Bandt & Pompe (BP) method [Phys. Rev. Lett. 88, 174102 (2002)] to define a probability distribution, P, that fully characterizes temporal correlations. The BP method is based on a comparison of neighboring values, and here is applied to the temporal sequence of residence-time intervals generated by the paradigmatic model of a Brownian particle in a sinusoidally modulated bistable potential. The probability distribution P generated via the BP method has associated a normalized Shannon entropy, H[P], and a statistical complexity measure, C[P], which is defined as proposed by Rosso et al. [Phys. Rev. Lett. 99, 154102 (2007)]. The statistical complexity quantifies not only randomness but also the presence of correlational structures, the two extreme circumstances of maximum knowledge (“perfect order") and maximum ignorance (“complete randomness") being regarded an “trivial", and in consequence, having complexity C = 0. We show that both, H and C, display resonant features as a function of the noise intensity, i.e., for an optimal level of noise the entropy displays a minimum and the complexity, a maximum. This resonant behavior indicates noise-enhanced temporal correlations in the sequence of residence-time intervals. The methodology proposed here has great potential for the precise detection of subtle signatures of noise-induced temporal correlations in real-world complex signals.
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paper_title: GENERALIZED STATISTICAL COMPLEXITY MEASURE

paper_content:

A generalized Statistical Complexity Measure (SCM) is a functional that characterizes the probability distribution P associated to the time series generated by a given dynamical system. It quantifies not only randomness but also the presence of correlational structures. We review here several fundamental issues in such a respect, namely, (a) the selection of the information measure ; (b) the choice of the probability metric space and associated distance ; (c) the question of defining the so-called generalized disequilibrium ; (d) the adequate way of picking up the probability distribution P associated to a dynamical system or time series under study, which is indeed a fundamental problem. In this communication we show (point d) that sensible improvements in the final results can be expected if the underlying probability distribution is "extracted" via appropriate consideration regarding causal effects in the system's dynamics.
---
paper_title: Detecting and quantifying stochastic and coherence resonances via information-theory complexity measurements

paper_content:

Statistical complexity measures are used to detect noise-induced order and to quantify stochastic and coherence resonances. We illustrate the method with two paradigmatic models, one of a Brownian particle in a sinusoidally modulated bistable potential, and the other, the FitzHugh-Nagumo model of excitable systems. The method can be employed for the precise detection of subtle signatures of noise-induced order in real-world complex signals.
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paper_title: Generalized statistical complexity measures: Geometrical and analytical properties

paper_content:

We discuss bounds on the values adopted by the generalized statistical complexity measures [M.T. Martin et al., Phys. Lett. A 311 (2003) 126; P.W. Lamberti et al., Physica A 334 (2004) 119] introduced by Lopez Ruiz et al. [Phys. Lett. A 209 (1995) 321] and Shiner et al. [Phys. Rev. E 59 (1999) 1459]. Several new theorems are proved and illustrated with reference to the celebrated logistic map.
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paper_title: Analysis of symbolic sequences using the Jensen-Shannon divergence

paper_content:

We study statistical properties of the Jensen-Shannon divergence D, which quantifies the difference between probability distributions, and which has been widely applied to analyses of symbolic sequences. We present three interpretations of D in the framework of statistical physics, information theory, and mathematical statistics, and obtain approximations of the mean, the variance, and the probability distribution of D in random, uncorrelated sequences. We present a segmentation method based on D that is able to segment a nonstationary symbolic sequence into stationary subsequences, and apply this method to DNA sequences, which are known to be nonstationary on a wide range of different length scales.
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paper_title: Quantifying the complexity of the delayed logistic map

paper_content:

Statistical complexity measures are used to quantify the degree of complexity of the delayed logistic map, with linear and nonlinear feedback. We employ two methods for calculating the complexity measures, one with the 'histogram-based' probability distribution function and the other one with ordinal patterns. We show that these methods provide complementary information about the complexity of the delay-induced dynamics: there are parameter regions where the histogram-based complexity is zero while the ordinal pattern complexity is not, and vice versa. We also show that the time series generated from the nonlinear delayed logistic map can present zero missing or forbidden patterns, i.e. all possible ordinal patterns are realized into orbits.
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paper_title: Article Distances in Probability Space and the Statistical Complexity Setup

paper_content:

Statistical complexity measures (SCM) are the composition of two ingredients: (i) entropies and (ii) distances in probability-space. In consequence, SCMs provide a simultaneous quantification of the randomness and the correlational structures present in the system under study. We address in this review important topics underlying the SCM structure, viz., (a) a good choice of probability metric space and (b) how to assess the best distance-choice, which in this context is called a “disequilibrium” and is denoted with the letter Q. Q, indeed the crucial SCM ingredient, is cast in terms of an associated distance D. Since out input data consists of time-series, we also discuss the best way of extracting from the time series a probability distribution P. As an illustration, we show just how these issues affect the description of the classical limit of quantum mechanics.
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paper_title: Commodity predictability analysis with a permutation information theory approach

paper_content:

It is widely known that commodity markets are not totally efficient. Long-range dependence is present, and thus the celebrated Brownian motion of prices can be considered only as a first approximation. In this work we analyzed the predictability in commodity markets by using a novel approach derived from Information Theory. The complexity–entropy causality plane has been recently shown to be a useful statistical tool to distinguish the stage of stock market development because differences between emergent and developed stock markets can be easily discriminated and visualized with this representation space [L. Zunino, M. Zanin, B.M. Tabak, D.G. Perez, O.A. Rosso, Complexity–entropy causality plane: a useful approach to quantify the stock market inefficiency, Physica A 389 (2010) 1891–1901]. By estimating the permutation entropy and permutation statistical complexity of twenty basic commodity future markets over a period of around 20 years (1991.01.02–2009.09.01), we can define an associated ranking of efficiency. This ranking is quantifying the presence of patterns and hidden structures in these prime markets. Moreover, the temporal evolution of the commodities in the complexity–entropy causality plane allows us to identify periods of time where the underlying dynamics is more or less predictable.
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paper_title: Measures of Statistical Complexity: Why?

paper_content:

We review several statistical complexity measures proposed over the last decade and a half as general indicators of structure or correlation. Recently, L\`opez-Ruiz, Mancini, and Calbet [Phys. Lett. A 209 (1995) 321] introduced another measure of statistical complexity $C_{\rm LMC}$ that, like others, satisfies the ``boundary conditions'' of vanishing in the extreme ordered and disordered limits. We examine some properties of $C_{\rm LMC}$ and find that it is neither an intensive nor an extensive thermodynamic variable. It depends nonlinearly on system size and vanishes exponentially in the thermodynamic limit for all one-dimensional finite-range spin systems. We propose a simple alteration of $C_{\rm LMC}$ that renders it extensive. however, this remedy results in a quantity that is a trivial function of the entropy density and hence of no use as a measure of structure or memory. We conclude by suggesting that a useful ``statistical complexity'' must not only obey the ordered-random bounary conditions of vanishing, it must also be defined in a setting that gives a clear interpretation to what structures are quantified.
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paper_title: Complexity-entropy causality plane: A useful approach to quantify the stock market inefficiency

paper_content:

The complexity-entropy causality plane has been recently introduced as a powerful tool for discriminating Gaussian from non-Gaussian process and different degrees of correlations [O.A. Rosso, H.A. Larrondo, M.T. Martin, A. Plastino, M.A. Fuentes, Distinguishing noise from chaos, Phys. Rev. Lett. 99 (2007) 154102]. We propose to use this representation space to distinguish the stage of stock market development. Our empirical results demonstrate that this statistical physics approach is useful, allowing a more refined classification of stock market dynamics.
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paper_title: Evolution of time delay signature of chaos generated in a mutually delay-coupled semiconductor lasers system

paper_content:

In this paper, evolution of time delay (TD) signature of chaos generated in a mutual delay-coupled semiconductor lasers (MDC-SLs) system is investigated experimentally and theoretically. Two statistical methods, including self-correlation function (SF) and permutation entropy (PE), are used to estimate the TD signature of chaotic time series. Through extracting the characteristic peak from the SF curve, a series of TD signature evolution maps are firstly obtained in the parameter space of coupled strength and frequency detuning. Meantime, the influences of injection current on the evolution maps of TD signature have been discussed, and the optimum scope of TD signature suppression is also specified. An overall qualitative agreement between our theoretical and experimental results is obtained.
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paper_title: Oscillation and chaos in physiological control systems

paper_content:

First-order nonlinear differential-delay equations describing physiological control systems are studied. The equations display a broad diversity of dynamical behavior including limit cycle oscillations, with a variety of wave forms, and apparently aperiodic or "chaotic" solutions. These results are discussed in relation to dynamical respiratory and hematopoietic diseases.
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paper_title: Permutation-information-theory approach to unveil delay dynamics from time-series analysis

paper_content:

In this paper an approach to identify delay phenomena from time series is developed. We show that it is possible to perform a reliable time delay identification by using quantifiers derived from information theory, more precisely, permutation entropy and permutation statistical complexity. These quantifiers show clear extrema when the embedding delay τ of the symbolic reconstruction matches the characteristic time delay τ(S) of the system. Numerical data originating from a time delay system based on the well-known Mackey-Glass equations operating in the chaotic regime were used as test beds. We show that our method is straightforward to apply and robust to additive observational and dynamical noise. Moreover, we find that the identification of the time delay is even more efficient in a noise environment. Our permutation approach is also able to recover the time delay in systems with low feedback rate or high nonlinearity.
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paper_title: Conceal time-delay signature of chaotic vertical-cavity surface-emitting lasers by variable-polarization optical feedback

paper_content:

Abstract Time-delay (TD) signatures of chaotic outputs generated by vertical-cavity surface-emitting lasers (VCSELs) are investigated quantitatively by using two proposed estimators, the peak signal to mean ratio (PSMR) calculated based on the auto-correlation function, and the valley signal to mean ratio (VSMR) calculated based on an information-theory-based estimator, the permutation entropy (PE) function. The VCSELs subject to variable-polarization optical feedback (VPOF) are considered, and the effects of polarizer angle, feedback strength, feedback delay and injection current are discussed. When the feedback strength is small, the TD signatures for the fully developed chaotic outputs of VCSELs can be well concealed for all polarizer angles. While for large feedback strength, the TD signatures cannot be concealed for any polarizer angle. However, for moderate feedback strength, the TD signatures of VCSELs with VPOF at intermediate polarizer angles can be better concealed than those of VCSELs with conventional polarization-selected optical feedback. Thus, the VCSELs with VPOF are extremely interesting for the security-enhanced chaotic optical communication systems.
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paper_title: Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator

paper_content:

In the dynamics of optical systems, one commonly needs to cope with the problem of coexisting deterministic and stochastic components. The separation of these components is an important, although difficult, task. Often the time scales at which determinism and noise dominate the system's dynamics differ. In this Letter we propose to use information-theory-derived quantifiers, more precisely, permutation entropy and statistical complexity, to distinguish between the two behaviors. Based on experiments of a paradigmatic opto-electronic oscillator, we demonstrate that the time scales at which deterministic or noisy behavior dominate can be identified. Supporting numerical simulations prove the accuracy of this identification.
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paper_title: Time Scales of a Chaotic Semiconductor Laser With Optical Feedback Under the Lens of a Permutation Information Analysis

paper_content:

We analyze the intrinsic time scales of the chaotic dynamics of a semiconductor laser subject to optical feedback by estimating quantifiers derived from a permutation information approach. Based on numerically and experimentally obtained times series, we find that permutation entropy and permutation statistical complexity allow the extraction of important characteristics of the dynamics of the system. We provide evidence that permutation statistical complexity is complementary to permutation entropy, giving valuable insights into the role of the different time scales involved in the chaotic regime of the semiconductor laser dynamics subject to delay optical feedback. The results obtained confirm that this novel approach is a conceptually simple and computationally efficient method to identify the characteristic time scales of this relevant physical system.
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paper_title: Direction of coupling from phases of interacting oscillators: A permutation information approach.

paper_content:

We introduce a directionality index for a time series based on a comparison of neighboring values. It can distinguish unidirectional from bidirectional coupling, as well as reveal and quantify asymmetry in bidirectional coupling. It is tested on a numerical model of coupled van der Pol oscillators, and applied to cardiorespiratory data from healthy subjects. There is no need for preprocessing and fine-tuning the parameters, which makes the method very simple, computationally fast and robust.
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paper_title: Using permutations to detect dependence between time series

paper_content:

Abstract In this paper, we propose an independence test between two time series which is based on permutations. The proposed test can be carried out by means of different common statistics such as Pearson’s chi-square or the likelihood ratio. We also point out why an exact test is necessary. Simulated and real data (return exchange rates between several currencies) reveal the capacity of this test to detect linear and nonlinear dependences.
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paper_title: A non-parametric independence test using permutation entropy

paper_content:

In the present paper we construct a new, simple, consistent and powerful test for independence by using symbolic dynamics and permutation entropy as a measure of serial dependence. We also give a standard asymptotic distribution of an affine transformation of the permutation entropy under the null hypothesis of independence. The test statistic and its standard limit distribution are invariant to any monotonic transformation. The test applies to time series with discrete or continuous distributions. Eventhough the test is based on entropy measures, it avoids smoothed non-parametric estimation. An application to several daily financial time series illustrates our approach.
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paper_title: A Non-Parametric Approach to Spatial Causality

paper_content:

The purpose of this paper is to show the capacity of a new non-parametric test based on symbolic entropy and symbolic dynamics to deal with the detection of linear and non-linear spatial causality. The good performance of the new test in detecting spatial causality and causal weighting matrix is notable and gives rise to an expectation that it may form a adequate tool for constructive specification searches.
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paper_title: Permutation Entropy: A Natural Complexity Measure for Time Series

paper_content:

We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.
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paper_title: Fine-grained permutation entropy as a measure of natural complexity for time series

paper_content:

In a recent paper [2002 Phys. Rev. Lett. 88 174102], Bandt and Pompe propose permutation entropy (PE) as a natural complexity measure for arbitrary time series which may be stationary or nonstationary, deterministic or stochastic. Their method is based on a comparison of neighbouring values. This paper further develops PE, and proposes the concept of fine-grained PE (FGPE) defined by the order pattern and magnitude of the difference between neighbouring values. This measure excludes the case where vectors with a distinct appearance are mistakenly mapped onto the same permutation type, and consequently FGPE becomes more sensitive to the dynamical change of time series than does PE, according to our simulation and experimental results.
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paper_title: Modified permutation-entropy analysis of heartbeat dynamics.

paper_content:

Heart rate variability (HRV) contains important information about the modulation of the cardiovascular system. Various methods of nonlinear dynamics (e.g., estimating Lyapunov exponents) and complexity measures (e.g., correlation dimension or entropies) have been applied to HRV analysis. Permutation entropy, which was proposed recently, has been widely used in many fields due to its conceptual and computational simplicity. It maps a time series onto a symbolic sequence of permutation ranks. The original permutation entropy assumes the time series under study has a continuous distribution, thus equal values are rare and can be ignored by ranking them according to their order of emergence, or broken by adding small random perturbations to ensure every symbol in a sequence is different. However, when the observed time series is digitized with lower resolution leading to a greater number of equal values, or the equalities represent certain characteristic sequential patterns of the system, it may not be rational to simply ignore or break them. In the present paper, a modified permutation entropy is proposed that, by mapping the equal value onto the same symbol (rank), allows for a more accurate characterization of system states. The application of the modified permutation entropy to the analysis of HRV is investigated using clinically collected data. Results show that modified permutation entropy can greatly improve the ability to distinguish the HRV signals under different physiological and pathological conditions. It can characterize the complexity of HRV more effectively than the original permutation entropy.
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paper_title: Direction of coupling from phases of interacting oscillators: A permutation information approach.

paper_content:

We introduce a directionality index for a time series based on a comparison of neighboring values. It can distinguish unidirectional from bidirectional coupling, as well as reveal and quantify asymmetry in bidirectional coupling. It is tested on a numerical model of coupled van der Pol oscillators, and applied to cardiorespiratory data from healthy subjects. There is no need for preprocessing and fine-tuning the parameters, which makes the method very simple, computationally fast and robust.
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paper_title: Nonlinear multivariate analysis of Neurophysiological Signals

paper_content:

Multivariate time series analysis is extensively used in neurophysiology with the aim of studying the relationship between simultaneously recorded signals. Recently, advances on information theory and nonlinear dynamical systems theory have allowed the study of various types of synchronization from time series. In this work, we first describe the multivariate linear methods most commonly used in neurophysiology and show that they can be extended to assess the existence of nonlinear interdependence between signals. We then review the concepts of entropy and mutual information followed by a detailed description of nonlinear methods based on the concepts of phase synchronization, generalized synchronization and event synchronization. In all cases, we show how to apply these methods to study different kinds of neurophysiological data. Finally, we illustrate the use of multivariate surrogate data test for the assessment of the strength (strong or weak) and the type (linear or nonlinear) of interdependence between neurophysiological signals.
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paper_title: Permutation entropy improves fetal behavioural state classification based on heart rate analysis from biomagnetic recordings in near term fetuses

paper_content:

The relevance of the complexity of fetal heart rate fluctuations with regard to the classification of fetal behavioural states has not been satisfyingly clarified so far. Because of the short behavioural states, the permutation entropy provides an advantageous complexity estimation leading to the Kullback–Leibler entropy (KLE). We test the hypothesis that parameters derived from KLE can improve the classification of fetal behaviour states based on classical heart rate fluctuation parameters (SDNN, RMSSD, ln(LF), ln(HF)). From measured heartbeat sequences (35 healthy fetuses at a gestational age between 35 and 40 completed weeks) representative intervals of 256 heartbeats were visually preclassified into fetal behavioural states. Employing discriminant analysis to separate the states 1F, 2F and 4F, the best classification result by classical parameters was 80.0% (SDNN). After additionally considering KLE parameters it was improved significantly (p<0.0005) to 94.3% (ln(LF), KLE_Mean). It could be confirmed that KLE can improve the state classification. This might reflect the consideration of different physiological aspects by classical and complexity measures.
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paper_title: Fast and Robust Detection of Epilepsy in Noisy EEG Signals Using Permutation Entropy

paper_content:

Permutation entropy (PE) is a new complexity measure which can extract important information from long, complex and high-dimensional time series. The advantages of this measure such as its fast calculation and robustness with respect to additive noise make it suitable for biomedical signal analysis. In this paper the ability of PE for characterizing the normal and epileptic EEG signals is investigated. Classification is performed using discriminant analysis. The effect of additive Gaussian noise on the discrimination performance is also studied and some parameters derived from PE are suggested to improve the classification accuracy when the signal is contaminated with noise. The results indicate that the proposed measures can distinguish normal and epileptic EEG signals with an accuracy of more than 97% for clean EEG and more than 85% for highly noised EEG signals.
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paper_title: Permutation Entropy: A Natural Complexity Measure for Time Series

paper_content:

We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.
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paper_title: Spontaneously emerging cortical representations of visual attributes

paper_content:

Spontaneous cortical activity—ongoing activity in the absence of intentional sensory input—has been studied extensively1, using methods ranging from EEG (electroencephalography)2,3,4, through voltage sensitive dye imaging5,6,7, down to recordings from single neurons8,9. Ongoing cortical activity has been shown to play a critical role in development10,11,12,13,14, and must also be essential for processing sensory perception, because it modulates stimulus-evoked activity5,15,16, and is correlated with behaviour17. Yet its role in the processing of external information and its relationship to internal representations of sensory attributes remains unknown. Using voltage sensitive dye imaging, we previously established a close link between ongoing activity in the visual cortex of anaesthetized cats and the spontaneous firing of a single neuron6. Here we report that such activity encompasses a set of dynamically switching cortical states, many of which correspond closely to orientation maps. When such an orientation state emerged spontaneously, it spanned several hypercolumns and was often followed by a state corresponding to a proximal orientation. We suggest that dynamically switching cortical states could represent the brain's internal context, and therefore reflect or influence memory, perception and behaviour.
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paper_title: Order patterns recurrence plots in the analysis of ERP data

paper_content:

Recurrence quantification analysis (RQA) is an established tool for data analysis in various behavioural sciences. In this article we present a refined notion of RQA based on order patterns. The use of order patterns is commonplace in time series analysis. Exploiting this concept in combination with recurrence plots (RP) and their quantification (RQA) allows for advances in contemporary EEG research, specifically in the analysis of event related potentials (ERP), as the method is known to be robust against non-stationary data. The use of order patterns recurrence plots (OPRPs) on EEG data recorded during a language processing experiment exemplifies the potentials of the method. We could show that the application of RQA to ERP data allows for a considerable reduction of the number of trials required in ERP research while still maintaining statistical validity.
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paper_title: Neuronal Avalanches in Neocortical Circuits

paper_content:

Networks of living neurons exhibit diverse patterns of activity, including oscillations, synchrony, and waves. Recent work in physics has shown yet another mode of activity in systems composed of many nonlinear units interacting locally. For example, avalanches, earthquakes, and forest fires all propagate in systems organized into a critical state in which event sizes show no characteristic scale and are described by power laws. We hypothesized that a similar mode of activity with complex emergent properties could exist in networks of cortical neurons. We investigated this issue in mature organotypic cultures and acute slices of rat cortex by recording spontaneous local field potentials continuously using a 60 channel multielectrode array. Here, we show that propagation of spontaneous activity in cortical networks is described by equations that govern avalanches. As predicted by theory for a critical branching process, the propagation obeys a power law with an exponent of -3/2 for event sizes, with a branching parameter close to the critical value of 1. Simulations show that a branching parameter at this value optimizes information transmission in feedforward networks, while preventing runaway network excitation. Our findings suggest that “neuronal avalanches” may be a generic property of cortical networks, and represent a mode of activity that differs profoundly from oscillatory, synchronized, or wave-like network states. In the critical state, the network may satisfy the competing demands of information transmission and network stability.
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paper_title: Detecting dynamical changes in time series using the permutation entropy

paper_content:

Timely detection of unusual and/or unexpected events in natural and man-made systems has deep scientific and practical relevance. We show that the recently proposed conceptually simple and easily calculated measure of permutation entropy can be effectively used to detect qualitative and quantitative dynamical changes. We illustrate our results on two model systems as well as on clinically characterized brain wave data from epileptic patients.
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paper_title: Comparison of anesthetic depth indexes based on thalamocortical local field potentials in rats.

paper_content:

BACKGROUND ::: Local field potentials may allow a more precise analysis of the brain electrical activity than the electroencephalogram. In this study, local field potentials were recorded in the thalamocortical axis of rats to (i) compare the performance of several indexes of anesthetic depth and (ii) investigate the existence of thalamocortical correlated or disrupted activity during isoflurane steady-state anesthesia. ::: ::: ::: METHODS ::: Five rats chronically implanted with microelectrodes were used to record local field potentials in the primary somatosensory cortex and ventroposterolateral thalamic nuclei at six periods: before induction of anesthesia; in the last 5 min of randomized 20-min steady-state end-tidal 0.8, 1.1, 1.4, and 1.7% isoflurane concentrations; and after recovery. The approximate entropy, the index of consciousness, the spectral edge frequency, and the permutation entropy were estimated using epochs of 8 s. A correction factor for burst suppression was applied to the spectral edge frequency and to the permutation entropy. The correlation between the derived indexes and the end-tidal isoflurane was calculated and compared for the two studied brain regions indexes. Coherence analysis was also performed. ::: ::: ::: RESULTS ::: The burst suppression-corrected permutation entropy showed the highest correlation with the end-tidal isoflurane concentration, and a high coherence was obtained between the two studied areas. ::: ::: ::: CONCLUSIONS ::: The permutation entropy corrected with the classic burst suppression ratio is a promising alternative to other indexes of anesthetic depth. Furthermore, high coherence level of activity exists between the somatosensory cortical and thalamic regions, even at deep isoflurane stages.
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paper_title: Forbidden ordinal patterns of periictal intracranial EEG indicate deterministic dynamics in human epileptic seizures

paper_content:

Epileptic seizures typically reveal a high degree of stereotypy, that is, for an individual patient they are characterized by an ordered and predictable sequence of symptoms and signs with typically little variability. Stereotypy implies that ictal neuronal dynamics might have deterministic characteristics, presumably most pronounced in the ictogenic parts of the brain, which may provide diagnostically and therapeutically important information. Therefore the goal of our study was to search for indications of determinism in periictal intracranial electroencephalography (EEG) studies recorded from patients with pharmacoresistent epilepsy.
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paper_title: Using permutation entropy to measure the electroencephalographic effects of sevoflurane.

paper_content:

BACKGROUND ::: Approximate entropy (AE) has been proposed as a measure of anesthetic drug effect in electroencephalographic data. Recently, a new method called permutation entropy (PE) based on symbolic dynamics was also proposed to measure the complexity in an electroencephalographic series. In this study, the AE and PE were applied to electroencephalographic recordings for revealing the effect of sevoflurane on brain activity. The dose-response relation of PE during sevoflurane anesthesia was compared with that of AE. ::: ::: ::: METHODS ::: Nineteen patients' electroencephalographic data were collected during the induction of general anesthesia with sevoflurane. PE and AE were applied to the electroencephalographic recordings, and the performance of both measures was assessed by pharmacokinetic-pharmacodynamic modeling and prediction probability. To ensure an accurate complexity measure of electroencephalographic recordings, a wavelet-based preprocessor was built in advance. ::: ::: ::: RESULTS ::: Both PE and AE could distinguish between the awake and anesthetized states and were highly correlated to each other (r = 0.8, P = 0.004). The pharmacokinetic-pharmacodynamic model adequately described the dose-response relation between PE and AE and sevoflurane effect site concentration. The coefficient R between PE and effect site concentration was 0.89 +/- 0.07 for all patients, compared with 0.60 +/- 0.14 for AE. Prediction probabilities of 0.86 +/- 0.04 and 0.79 +/- 0.09 for PE and AE showed that PE has a stronger ability to differentiate between the awake and anesthetic states. ::: ::: ::: CONCLUSION ::: The results show that PE can estimate the sevoflurane drug effect more effectively than AE. This method could be applied to design a new electroencephalographic monitoring system to estimate sevoflurane anesthetic drug effect.
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paper_title: Nonlinear dynamical analysis of EEG and MEG: Review of an emerging field

paper_content:

Abstract Many complex and interesting phenomena in nature are due to nonlinear phenomena. The theory of nonlinear dynamical systems, also called ‘chaos theory’, has now progressed to a stage, where it becomes possible to study self-organization and pattern formation in the complex neuronal networks of the brain. One approach to nonlinear time series analysis consists of reconstructing, from time series of EEG or MEG, an attractor of the underlying dynamical system, and characterizing it in terms of its dimension (an estimate of the degrees of freedom of the system), or its Lyapunov exponents and entropy (reflecting unpredictability of the dynamics due to the sensitive dependence on initial conditions). More recently developed nonlinear measures characterize other features of local brain dynamics (forecasting, time asymmetry, determinism) or the nonlinear synchronization between recordings from different brain regions. Nonlinear time series has been applied to EEG and MEG of healthy subjects during no-task resting states, perceptual processing, performance of cognitive tasks and different sleep stages. Many pathologic states have been examined as well, ranging from toxic states, seizures, and psychiatric disorders to Alzheimer's, Parkinson's and Cre1utzfeldt-Jakob's disease. Interpretation of these results in terms of ‘functional sources’ and ‘functional networks’ allows the identification of three basic patterns of brain dynamics: (i) normal, ongoing dynamics during a no-task, resting state in healthy subjects; this state is characterized by a high dimensional complexity and a relatively low and fluctuating level of synchronization of the neuronal networks; (ii) hypersynchronous, highly nonlinear dynamics of epileptic seizures; (iii) dynamics of degenerative encephalopathies with an abnormally low level of between area synchronization. Only intermediate levels of rapidly fluctuating synchronization, possibly due to critical dynamics near a phase transition, are associated with normal information processing, whereas both hyper—as well as hyposynchronous states result in impaired information processing and disturbed consciousness.
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paper_title: Deterministic dynamics of neural activity during absence seizures in rats.

paper_content:

The study of brain electrical activities in terms of deterministic nonlinear dynamics has recently received much attention. Forbidden ordinal patterns (FOP) is a recently proposed method to investigate the determinism of a dynamical system through the analysis of intrinsic ordinal properties of a nonstationary time series. The advantages of this method in comparison to others include simplicity and low complexity in computation without further model assumptions. In this paper, the FOP of the EEG series of genetic absence epilepsy rats from Strasbourg was examined to demonstrate evidence of deterministic dynamics during epileptic states. Experiments showed that the number of FOP of the EEG series grew significantly from an interictal to an ictal state via a preictal state. These findings indicated that the deterministic dynamics of neural networks increased significantly in the transition from the interictal to the ictal states and also suggested that the FOP measures of the EEG series could be considered as a predictor of absence seizures.
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paper_title: Distances of time series components by means of symbolic dynamics

paper_content:

In this note we describe a simple method for visualizing time-dependent similarities and dissimilarities between the components of a high-dimensional time series. On the base of symbolic dynamics, the time series is turned into a series of matrices whose rows quantify pattern types in the components of the original series. For different scales we introduce distances between the components via the obtained pattern type distributions and approximate them in a one-dimensional manner. The method is illustrated for 19-channel EEG data.
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paper_title: Ordinal pattern based similarity analysis for EEG recordings

paper_content:

Abstract Objective Ordinal patterns analysis such as permutation entropy of the EEG series has been found to usefully track brain dynamics and has been applied to detect changes in the dynamics of EEG data. In order to further investigate hidden nonlinear dynamical characteristics in EEG data for differentiating brain states, this paper proposes a novel dissimilarity measure based on the ordinal pattern distributions of EEG series. Methods Given a segment of EEG series, we first map this series into a phase space, then calculate the ordinal sequences and the distribution of these ordinal patterns. Finally, the dissimilarity between two EEG series can be qualified via a simple distance measure. A neural mass model was proposed to simulate EEG data and test the performance of the dissimilarity measure based on the ordinal patterns distribution. Furthermore, this measure was then applied to analyze EEG data from 24 Genetic Absence Epilepsy Rats from Strasbourg (GAERS), with the aim of distinguishing between interictal, preictal and ictal states. Results The dissimilarity measure of a pair of EEG signals within the same group and across different groups was calculated, respectively. As expected, the dissimilarity measures during different brain states were higher than internal dissimilarity measures. When applied to the preictal detection of absence seizures, the proposed dissimilarity measure successfully detected the preictal state prior to their onset in 109 out of 168 seizures (64.9%). Conclusions Our results showed that dissimilarity measures between EEG segments during the same brain state were significant smaller that those during different states. This suggested that the dissimilarity measure, based on the ordinal patterns in the time series, could be used to detect changes in the dynamics of EEG data. Moreover, our results suggested that ordinal patterns in the EEG might be a potential characteristic of brain dynamics. Significance This dissimilarity measure is a promising method to reveal dynamic changes in EEG, for example as occur in the transition of epileptic seizures. This method is simple and fast, so might be applied in designing an automated closed-loop seizure prevention system for absence epilepsy.
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paper_title: Permutation entropy to detect vigilance changes and preictal states from scalp EEG in epileptic patients. A preliminary study

paper_content:

Permutation entropy (PE) was recently introduced as a very fast and robust algorithm to detect dynamic complexity changes in time series. It was also suggested as a useful screening algorithm for epileptic events in EEG data. In the present work, we tested its efficacy on scalp EEG data recorded from three epileptic patients. With a receiver operating characteristics (ROC) analysis, we evaluated the separability of amplitude distributions of PE resulting from preictal and interictal phases. Moreover, the dependency of PE on vigilance state was tested by correlation coefficients. A good separability of interictal and preictal phase was found, nevertheless PE was shown to be sensitive to changes in vigilance state. The changes of PE during the preictal phase and at seizure onset coincided with changes in vigilance state, restricting its possible use for seizure prediction on scalp EEG; this finding however suggests its possible usefulness for an automated classification of vigilance states.
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paper_title: Modified permutation-entropy analysis of heartbeat dynamics.

paper_content:

Heart rate variability (HRV) contains important information about the modulation of the cardiovascular system. Various methods of nonlinear dynamics (e.g., estimating Lyapunov exponents) and complexity measures (e.g., correlation dimension or entropies) have been applied to HRV analysis. Permutation entropy, which was proposed recently, has been widely used in many fields due to its conceptual and computational simplicity. It maps a time series onto a symbolic sequence of permutation ranks. The original permutation entropy assumes the time series under study has a continuous distribution, thus equal values are rare and can be ignored by ranking them according to their order of emergence, or broken by adding small random perturbations to ensure every symbol in a sequence is different. However, when the observed time series is digitized with lower resolution leading to a greater number of equal values, or the equalities represent certain characteristic sequential patterns of the system, it may not be rational to simply ignore or break them. In the present paper, a modified permutation entropy is proposed that, by mapping the equal value onto the same symbol (rank), allows for a more accurate characterization of system states. The application of the modified permutation entropy to the analysis of HRV is investigated using clinically collected data. Results show that modified permutation entropy can greatly improve the ability to distinguish the HRV signals under different physiological and pathological conditions. It can characterize the complexity of HRV more effectively than the original permutation entropy.
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paper_title: Fractal dynamics in physiology: Alterations with disease and aging

paper_content:

According to classical concepts of physiologic control, healthy systems are self-regulated to reduce variability and maintain physiologic constancy. Contrary to the predictions of homeostasis, however, the output of a wide variety of systems, such as the normal human heartbeat, fluctuates in a complex manner, even under resting conditions. Scaling techniques adapted from statistical physics reveal the presence of long-range, power-law correlations, as part of multifractal cascades operating over a wide range of time scales. These scaling properties suggest that the nonlinear regulatory systems are operating far from equilibrium, and that maintaining constancy is not the goal of physiologic control. In contrast, for subjects at high risk of sudden death (including those with heart failure), fractal organization, along with certain nonlinear interactions, breaks down. Application of fractal analysis may provide new approaches to assessing cardiac risk and forecasting sudden cardiac death, as well as to monitoring the aging process. Similar approaches show promise in assessing other regulatory systems, such as human gait control in health and disease. Elucidating the fractal and nonlinear mechanisms involved in physiologic control and complex signaling networks is emerging as a major challenge in the postgenomic era.
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paper_title: Brain signal analysis based on recurrences

paper_content:

The EEG is one of the most commonly used tools in brain research. Though of high relevance in research, the data obtained is very noisy and nonstationary. In the present article we investigate the applicability of a nonlinear data analysis method, the recurrence quantification analysis (RQA), to such data. The method solely rests on the natural property of recurrence which is a phenomenon inherent to complex systems, such as the brain. We show that this method is indeed suitable for the analysis of EEG data and that it might improve contemporary EEG analysis.
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paper_title: Preplay of future place cell sequences by hippocampal cellular assemblies

paper_content:

Place cells in the hippocampus track an animal's position as it travels through space. Previous work contends that sequential place cell maps are produced upon the initial navigation of a new area and subsequently consolidated at rest or during sleep. Here, place-cell firing patterns during rest or sleep are observed before a novel spatial experience, a phenomenon termed 'preplay'. These sequences were separate from the replay of pervious experience and suggest that internal dynamics during rest may organize cell assemblies to be ready for any novel encoding that may occur in the immediate future.
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paper_title: Permutation entropy of the electroencephalogram: a measure of anaesthetic drug effect

paper_content:

Abstract Background It would be useful to have an open-source electroencephalographic (EEG) index of γ-amino-butyric acid (GABA)-ergic anaesthetic drug effect that is resistant to eye-blink artifact, responds rapidly to changes in EEG pattern, and can be linked to underlying neurophysiological and neuropharmacological mechanisms that control the conscious state. Methods The EEG waveform can be described as a sequence of ordinal patterns. The permutation entropy (PE) describes the relative occurrence of each of these patterns. It is high (≈1.0) when the signal has predominantly high frequencies and low (≈0.4) when the signal consists of only low frequencies. The response of the PE to various computer-generated EEG-like waveforms was assessed. A composite PE index (CPEI) was developed, which was the sum of two simple PEs and included a small measurement-noise threshold (0.5 μV). We also applied the CPEI to two small pilot EEG data sets from patients receiving sevoflurane ( n =21) or propofol ( n =9) anaesthesia. Results With minimal pre-processing or artifact rejection, the CPEI reliably tracked the anaesthetic-related EEG changes, namely loss of high frequencies, spindle-like waves, and delta waves. Using NONMEM, it was possible to construct adequate pharmacokinetic–pharmacodynamic models from the data. The CPEI was comparable with models derived using the bispectral index [BIS R 2 =0.88 (0.08) vs CPEI R 2 =0.91 (0.06) for the propofol data] and M-entropy indices [M-entropy R 2 =0.91 (0.06) vs CPEI R 2 =0.87 (0.09) for the sevoflurane data]. Conclusions PE of the EEG shows promise as a simple measure of GABAergic anaesthetic drug effect.
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paper_title: Classifying cardiac biosignals using ordinal pattern statistics and symbolic dynamics

paper_content:

The performance of (bio-)signal classification strongly depends on the choice of suitable features (also called parameters or biomarkers). In this article we evaluate the discriminative power of ordinal pattern statistics and symbolic dynamics in comparison with established heart rate variability parameters applied to beat-to-beat intervals. As an illustrative example we distinguish patients suffering from congestive heart failure from a (healthy) control group using beat-to-beat time series. We assess the discriminative power of individual features as well as pairs of features. These comparisons show that ordinal patterns sampled with an additional time lag are promising features for efficient classification.
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paper_title: Visualization of coupling in time series by order recurrence plots

paper_content:

We introduce a method to visualize dependencies between two time series by applying the concept of cross recurrence plots to the local ordinal structure. We derive a measure of the coupling strength which is robust against observational noise, nonlinear distortion of the amplitude, and low-frequency trends. Connections to the instantaneous phase and the determination of phase coupling of two coupled Rössler systems in the standard and funnel regimes are shown. An application to electroencephalogram data demonstrates that the method is robust with respect to artifacts.
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paper_title: Predictability analysis of absence seizures with permutation entropy

paper_content:

Summary In this study, we investigate permutation entropy as a tool to predict the absence seizures of genetic absence epilepsy rats from Strasbourg (GAERS) by using EEG recordings. The results show that permutation entropy can track the dynamical changes of EEG data, so as to describe transient dynamics prior to the absence seizures. Experiments demonstrate that permutation entropy can successfully detect pre-seizure state in 169 out of 314 seizures from 28 rats and the average anticipation time of permutation entropy is around 4.9 s. These findings could shed new light on the mechanism of absence seizure. In comparison with results of sample entropy, permutation entropy is better able to predict absence seizures.
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paper_title: Fast and Robust Detection of Epilepsy in Noisy EEG Signals Using Permutation Entropy

paper_content:

Permutation entropy (PE) is a new complexity measure which can extract important information from long, complex and high-dimensional time series. The advantages of this measure such as its fast calculation and robustness with respect to additive noise make it suitable for biomedical signal analysis. In this paper the ability of PE for characterizing the normal and epileptic EEG signals is investigated. Classification is performed using discriminant analysis. The effect of additive Gaussian noise on the discrimination performance is also studied and some parameters derived from PE are suggested to improve the classification accuracy when the signal is contaminated with noise. The results indicate that the proposed measures can distinguish normal and epileptic EEG signals with an accuracy of more than 97% for clean EEG and more than 85% for highly noised EEG signals.
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paper_title: Forbidden ordinal patterns of periictal intracranial EEG indicate deterministic dynamics in human epileptic seizures

paper_content:

Epileptic seizures typically reveal a high degree of stereotypy, that is, for an individual patient they are characterized by an ordered and predictable sequence of symptoms and signs with typically little variability. Stereotypy implies that ictal neuronal dynamics might have deterministic characteristics, presumably most pronounced in the ictogenic parts of the brain, which may provide diagnostically and therapeutically important information. Therefore the goal of our study was to search for indications of determinism in periictal intracranial electroencephalography (EEG) studies recorded from patients with pharmacoresistent epilepsy.
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paper_title: Detecting dynamical changes in time series using the permutation entropy

paper_content:

Timely detection of unusual and/or unexpected events in natural and man-made systems has deep scientific and practical relevance. We show that the recently proposed conceptually simple and easily calculated measure of permutation entropy can be effectively used to detect qualitative and quantitative dynamical changes. We illustrate our results on two model systems as well as on clinically characterized brain wave data from epileptic patients.
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paper_title: Detection of epileptic electroencephalogram based on Permutation Entropy and Support Vector Machines

paper_content:

The electroencephalogram (EEG) has proven a valuable tool in the study and detection of epilepsy. This paper investigates for the first time the use of Permutation Entropy (PE) as a feature for automated epileptic seizure detection. A Support Vector Machine (SVM) is used to classify segments of normal and epileptic EEG based on PE values. The proposed system utilizes the fact that the EEG during epileptic seizures is characterized by lower PE than normal EEG. It is shown that average sensitivity of 94.38% and average specificity of 93.23% is obtained by using PE as a feature to characterize epileptic and seizure-free EEG, while 100% sensitivity and specificity were also obtained in single-trial classifications.
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paper_title: Ordinal pattern based similarity analysis for EEG recordings

paper_content:

Abstract Objective Ordinal patterns analysis such as permutation entropy of the EEG series has been found to usefully track brain dynamics and has been applied to detect changes in the dynamics of EEG data. In order to further investigate hidden nonlinear dynamical characteristics in EEG data for differentiating brain states, this paper proposes a novel dissimilarity measure based on the ordinal pattern distributions of EEG series. Methods Given a segment of EEG series, we first map this series into a phase space, then calculate the ordinal sequences and the distribution of these ordinal patterns. Finally, the dissimilarity between two EEG series can be qualified via a simple distance measure. A neural mass model was proposed to simulate EEG data and test the performance of the dissimilarity measure based on the ordinal patterns distribution. Furthermore, this measure was then applied to analyze EEG data from 24 Genetic Absence Epilepsy Rats from Strasbourg (GAERS), with the aim of distinguishing between interictal, preictal and ictal states. Results The dissimilarity measure of a pair of EEG signals within the same group and across different groups was calculated, respectively. As expected, the dissimilarity measures during different brain states were higher than internal dissimilarity measures. When applied to the preictal detection of absence seizures, the proposed dissimilarity measure successfully detected the preictal state prior to their onset in 109 out of 168 seizures (64.9%). Conclusions Our results showed that dissimilarity measures between EEG segments during the same brain state were significant smaller that those during different states. This suggested that the dissimilarity measure, based on the ordinal patterns in the time series, could be used to detect changes in the dynamics of EEG data. Moreover, our results suggested that ordinal patterns in the EEG might be a potential characteristic of brain dynamics. Significance This dissimilarity measure is a promising method to reveal dynamic changes in EEG, for example as occur in the transition of epileptic seizures. This method is simple and fast, so might be applied in designing an automated closed-loop seizure prevention system for absence epilepsy.
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paper_title: Permutation entropy to detect vigilance changes and preictal states from scalp EEG in epileptic patients. A preliminary study

paper_content:

Permutation entropy (PE) was recently introduced as a very fast and robust algorithm to detect dynamic complexity changes in time series. It was also suggested as a useful screening algorithm for epileptic events in EEG data. In the present work, we tested its efficacy on scalp EEG data recorded from three epileptic patients. With a receiver operating characteristics (ROC) analysis, we evaluated the separability of amplitude distributions of PE resulting from preictal and interictal phases. Moreover, the dependency of PE on vigilance state was tested by correlation coefficients. A good separability of interictal and preictal phase was found, nevertheless PE was shown to be sensitive to changes in vigilance state. The changes of PE during the preictal phase and at seizure onset coincided with changes in vigilance state, restricting its possible use for seizure prediction on scalp EEG; this finding however suggests its possible usefulness for an automated classification of vigilance states.
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paper_title: Analysis and Synthesis of Pseudo-Periodic Job Arrivals in Grids: A Matching Pursuit Approach

paper_content:

Pseudo-periodicity is one of the basic job arrival patterns on data-intensive clusters and Grids. In this paper, a signal decomposition methodology called matching pursuit is applied for analysis and synthesis of pseudo-periodic job arrival processes. The matching pursuit decomposition is well localized both in time and frequency, and it is naturally suited for analyzing non-stationary as well as stationary signals. The stationarity of the processes can be quantitatively measured by permutation entropy, with which the relationship between stationarity and modeling complexity is excellently explained. Quantitative methods based on the power spectrum are also provided to measure the degree of periodicity present in the data. Matching pursuit is further shown to be able to extract patterns from signals, which is an attractive feature from a modeling perspective. Real world workload data from production clusters and Grids are used to empirically evaluate the proposed measures and methodologies.
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paper_title: Deterministic dynamics of neural activity during absence seizures in rats.

paper_content:

The study of brain electrical activities in terms of deterministic nonlinear dynamics has recently received much attention. Forbidden ordinal patterns (FOP) is a recently proposed method to investigate the determinism of a dynamical system through the analysis of intrinsic ordinal properties of a nonstationary time series. The advantages of this method in comparison to others include simplicity and low complexity in computation without further model assumptions. In this paper, the FOP of the EEG series of genetic absence epilepsy rats from Strasbourg was examined to demonstrate evidence of deterministic dynamics during epileptic states. Experiments showed that the number of FOP of the EEG series grew significantly from an interictal to an ictal state via a preictal state. These findings indicated that the deterministic dynamics of neural networks increased significantly in the transition from the interictal to the ictal states and also suggested that the FOP measures of the EEG series could be considered as a predictor of absence seizures.
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paper_title: From Intracerebral EEG Signals to Brain Connectivity: Identification of Epileptogenic Networks in Partial Epilepsy

paper_content:

Epilepsy is a complex neurological disorder characterized by recurring seizures. In 30% of patients, seizures are insufficiently reduced by anti-epileptic drugs. In the case where seizures originate from a relatively circumscribed region of the brain, epilepsy is said to be partial and surgery can be indicated. The success of epilepsy surgery depends on the accurate localization and delineation of the epileptogenic zone (which often involves several structures), responsible for seizures. It requires a comprehensive pre-surgical evaluation of patients that includes not only imaging data but also long-term monitoring of electrophysiological signals (scalp and intracerebral EEG). During the past decades, considerable effort has been devoted to the development of signal analysis techniques aimed at characterizing the functional connectivity among spatially distributed regions over interictal (outside seizures) or ictal (during seizures) periods from EEG data. Most of these methods rely on the measurement of statistical couplings among signals recorded from distinct brain sites. However, methods differ with respect to underlying theoretical principles (mostly coming from the field of statistics or the field of non-linear physics). The objectives of this paper are: (i) to provide an brief overview of methods aimed at characterizing functional brain connectivity from electrophysiological data, (ii) to provide concrete application examples in the context of drug-refractory partial epilepsies, and iii) to highlight some key points emerging from results obtained both on real intracerebral EEG signals and on signals simulated from physiologically plausible models in which the underlying connectivity patterns are known a priori (ground truth).
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paper_title: Estimating coupling direction between neuronal populations with permutation conditional mutual information

paper_content:

To further understand functional connectivity in the brain, we need to identify the coupling direction between neuronal signals recorded from different brain areas. In this paper, we present a novel methodology based on permutation analysis and conditional mutual information for estimation of a directionality index between two neuronal populations. First, the reliability of this method is numerically assessed with a coupled mass neural model; the simulations show that this method is superior to the conditional mutual information method and the Granger causality method for identifying the coupling direction between unidirectional or bidirectional neuronal populations that are generated by the mass neuronal model. The method is also applied to investigate the coupling direction between neuronal populations in CA1 and CA3 in the rat hippocampal tetanus toxin model of focal epilepsy; the propagation direction of the seizure events could be elucidated through this coupling direction estimation method. All together, these results suggest that the permutation conditional mutual information method is a promising technique for estimating directional coupling between mutually interconnected neuronal populations.
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paper_title: Symbolic transfer entropy.

paper_content:

We propose to estimate transfer entropy using a technique of symbolization. We demonstrate numerically that symbolic transfer entropy is a robust and computationally fast method to quantify the dominating direction of information flow between time series from structurally identical and nonidentical coupled systems. Analyzing multiday, multichannel electroencephalographic recordings from 15 epilepsy patients our approach allowed us to reliably identify the hemisphere containing the epileptic focus without observing actual seizure activity.
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paper_title: Forbidden ordinal patterns of periictal intracranial EEG indicate deterministic dynamics in human epileptic seizures

paper_content:

Epileptic seizures typically reveal a high degree of stereotypy, that is, for an individual patient they are characterized by an ordered and predictable sequence of symptoms and signs with typically little variability. Stereotypy implies that ictal neuronal dynamics might have deterministic characteristics, presumably most pronounced in the ictogenic parts of the brain, which may provide diagnostically and therapeutically important information. Therefore the goal of our study was to search for indications of determinism in periictal intracranial electroencephalography (EEG) studies recorded from patients with pharmacoresistent epilepsy.
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paper_title: Distances of time series components by means of symbolic dynamics

paper_content:

In this note we describe a simple method for visualizing time-dependent similarities and dissimilarities between the components of a high-dimensional time series. On the base of symbolic dynamics, the time series is turned into a series of matrices whose rows quantify pattern types in the components of the original series. For different scales we introduce distances between the components via the obtained pattern type distributions and approximate them in a one-dimensional manner. The method is illustrated for 19-channel EEG data.
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paper_title: Analysis and Synthesis of Pseudo-Periodic Job Arrivals in Grids: A Matching Pursuit Approach

paper_content:

Pseudo-periodicity is one of the basic job arrival patterns on data-intensive clusters and Grids. In this paper, a signal decomposition methodology called matching pursuit is applied for analysis and synthesis of pseudo-periodic job arrival processes. The matching pursuit decomposition is well localized both in time and frequency, and it is naturally suited for analyzing non-stationary as well as stationary signals. The stationarity of the processes can be quantitatively measured by permutation entropy, with which the relationship between stationarity and modeling complexity is excellently explained. Quantitative methods based on the power spectrum are also provided to measure the degree of periodicity present in the data. Matching pursuit is further shown to be able to extract patterns from signals, which is an attractive feature from a modeling perspective. Real world workload data from production clusters and Grids are used to empirically evaluate the proposed measures and methodologies.
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paper_title: Comparison of anesthetic depth indexes based on thalamocortical local field potentials in rats.

paper_content:

BACKGROUND ::: Local field potentials may allow a more precise analysis of the brain electrical activity than the electroencephalogram. In this study, local field potentials were recorded in the thalamocortical axis of rats to (i) compare the performance of several indexes of anesthetic depth and (ii) investigate the existence of thalamocortical correlated or disrupted activity during isoflurane steady-state anesthesia. ::: ::: ::: METHODS ::: Five rats chronically implanted with microelectrodes were used to record local field potentials in the primary somatosensory cortex and ventroposterolateral thalamic nuclei at six periods: before induction of anesthesia; in the last 5 min of randomized 20-min steady-state end-tidal 0.8, 1.1, 1.4, and 1.7% isoflurane concentrations; and after recovery. The approximate entropy, the index of consciousness, the spectral edge frequency, and the permutation entropy were estimated using epochs of 8 s. A correction factor for burst suppression was applied to the spectral edge frequency and to the permutation entropy. The correlation between the derived indexes and the end-tidal isoflurane was calculated and compared for the two studied brain regions indexes. Coherence analysis was also performed. ::: ::: ::: RESULTS ::: The burst suppression-corrected permutation entropy showed the highest correlation with the end-tidal isoflurane concentration, and a high coherence was obtained between the two studied areas. ::: ::: ::: CONCLUSIONS ::: The permutation entropy corrected with the classic burst suppression ratio is a promising alternative to other indexes of anesthetic depth. Furthermore, high coherence level of activity exists between the somatosensory cortical and thalamic regions, even at deep isoflurane stages.
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paper_title: Multiscale permutation entropy analysis of EEG recordings during sevoflurane anesthesia

paper_content:

Electroencephalogram (EEG) monitoring of the effect of anesthetic drugs on the central nervous system has long been used in anesthesia research. Several methods based on nonlinear dynamics, such as permutation entropy (PE), have been proposed to analyze EEG series during anesthesia. However, these measures are still single-scale based and may not completely describe the dynamical characteristics of complex EEG series. In this paper, a novel measure combining multiscale PE information, called CMSPE (composite multi-scale permutation entropy), was proposed for quantifying the anesthetic drug effect on EEG recordings during sevoflurane anesthesia. Three sets of simulated EEG series during awake, light and deep anesthesia were used to select the parameters for the multiscale PE analysis: embedding dimension m, lag τ and scales to be integrated into the CMSPE index. Then, the CMSPE index and raw single-scale PE index were applied to EEG recordings from 18 patients who received sevoflurane anesthesia. Pharmacokinetic/pharmacodynamic (PKPD) modeling was used to relate the measured EEG indices and the anesthetic drug concentration. Prediction probability (Pk) statistics and correlation analysis with the response entropy (RE) index, derived from the spectral entropy (M-entropy module; GE Healthcare, Helsinki, Finland), were investigated to evaluate the effectiveness of the new proposed measure. It was found that raw single-scale PE was blind to subtle transitions between light and deep anesthesia, while the CMSPE index tracked these changes accurately. Around the time of loss of consciousness, CMSPE responded significantly more rapidly than the raw PE, with the absolute slopes of linearly fitted response versus time plots of 0.12 (0.09–0.15) and 0.10 (0.06–0.13), respectively. The prediction probability Pk of 0.86 (0.85–0.88) and 0.85 (0.80–0.86) for CMSPE and raw PE indicated that the CMSPE index correlated well with the underlying anesthetic effect. The correlation coefficient for the comparison between the CMSPE index and RE index of 0.84 (0.80–0.88) was significantly higher than the raw PE index of 0.75 (0.66–0.84). The results show that the CMSPE outperforms the raw single-scale PE in reflecting the sevoflurane drug effect on the central nervous system.
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paper_title: Permutation entropy of the electroencephalogram: a measure of anaesthetic drug effect

paper_content:

Abstract Background It would be useful to have an open-source electroencephalographic (EEG) index of γ-amino-butyric acid (GABA)-ergic anaesthetic drug effect that is resistant to eye-blink artifact, responds rapidly to changes in EEG pattern, and can be linked to underlying neurophysiological and neuropharmacological mechanisms that control the conscious state. Methods The EEG waveform can be described as a sequence of ordinal patterns. The permutation entropy (PE) describes the relative occurrence of each of these patterns. It is high (≈1.0) when the signal has predominantly high frequencies and low (≈0.4) when the signal consists of only low frequencies. The response of the PE to various computer-generated EEG-like waveforms was assessed. A composite PE index (CPEI) was developed, which was the sum of two simple PEs and included a small measurement-noise threshold (0.5 μV). We also applied the CPEI to two small pilot EEG data sets from patients receiving sevoflurane ( n =21) or propofol ( n =9) anaesthesia. Results With minimal pre-processing or artifact rejection, the CPEI reliably tracked the anaesthetic-related EEG changes, namely loss of high frequencies, spindle-like waves, and delta waves. Using NONMEM, it was possible to construct adequate pharmacokinetic–pharmacodynamic models from the data. The CPEI was comparable with models derived using the bispectral index [BIS R 2 =0.88 (0.08) vs CPEI R 2 =0.91 (0.06) for the propofol data] and M-entropy indices [M-entropy R 2 =0.91 (0.06) vs CPEI R 2 =0.87 (0.09) for the sevoflurane data]. Conclusions PE of the EEG shows promise as a simple measure of GABAergic anaesthetic drug effect.
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paper_title: Permutation Entropy: A new feature for Brain-Computer Interfaces

paper_content:

This paper investigates the use of Permutation Entropy (PE) as a feature for mental task classification for a Brain-Computer Interface system. PE is a recently introduced measure which quantifies signal complexity by measuring the departure of a time series from a random one. More regular signals are characterized by lower PE values. Here, PE is utilized to characterize signals from electroencephalograms of 3 subjects performing 4 motor imagery tasks, which are then classified using a Support Vector Machine. Even though it is possible to obtain 100% single-trial classification accuracy, this is very much subject-dependent.
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paper_title: Order patterns recurrence plots in the analysis of ERP data

paper_content:

Recurrence quantification analysis (RQA) is an established tool for data analysis in various behavioural sciences. In this article we present a refined notion of RQA based on order patterns. The use of order patterns is commonplace in time series analysis. Exploiting this concept in combination with recurrence plots (RP) and their quantification (RQA) allows for advances in contemporary EEG research, specifically in the analysis of event related potentials (ERP), as the method is known to be robust against non-stationary data. The use of order patterns recurrence plots (OPRPs) on EEG data recorded during a language processing experiment exemplifies the potentials of the method. We could show that the application of RQA to ERP data allows for a considerable reduction of the number of trials required in ERP research while still maintaining statistical validity.
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paper_title: Brain signal analysis based on recurrences

paper_content:

The EEG is one of the most commonly used tools in brain research. Though of high relevance in research, the data obtained is very noisy and nonstationary. In the present article we investigate the applicability of a nonlinear data analysis method, the recurrence quantification analysis (RQA), to such data. The method solely rests on the natural property of recurrence which is a phenomenon inherent to complex systems, such as the brain. We show that this method is indeed suitable for the analysis of EEG data and that it might improve contemporary EEG analysis.
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paper_title: Direction of coupling from phases of interacting oscillators: A permutation information approach.

paper_content:

We introduce a directionality index for a time series based on a comparison of neighboring values. It can distinguish unidirectional from bidirectional coupling, as well as reveal and quantify asymmetry in bidirectional coupling. It is tested on a numerical model of coupled van der Pol oscillators, and applied to cardiorespiratory data from healthy subjects. There is no need for preprocessing and fine-tuning the parameters, which makes the method very simple, computationally fast and robust.
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paper_title: Permutation entropy improves fetal behavioural state classification based on heart rate analysis from biomagnetic recordings in near term fetuses

paper_content:

The relevance of the complexity of fetal heart rate fluctuations with regard to the classification of fetal behavioural states has not been satisfyingly clarified so far. Because of the short behavioural states, the permutation entropy provides an advantageous complexity estimation leading to the Kullback–Leibler entropy (KLE). We test the hypothesis that parameters derived from KLE can improve the classification of fetal behaviour states based on classical heart rate fluctuation parameters (SDNN, RMSSD, ln(LF), ln(HF)). From measured heartbeat sequences (35 healthy fetuses at a gestational age between 35 and 40 completed weeks) representative intervals of 256 heartbeats were visually preclassified into fetal behavioural states. Employing discriminant analysis to separate the states 1F, 2F and 4F, the best classification result by classical parameters was 80.0% (SDNN). After additionally considering KLE parameters it was improved significantly (p<0.0005) to 94.3% (ln(LF), KLE_Mean). It could be confirmed that KLE can improve the state classification. This might reflect the consideration of different physiological aspects by classical and complexity measures.
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paper_title: Modified permutation-entropy analysis of heartbeat dynamics.

paper_content:

Heart rate variability (HRV) contains important information about the modulation of the cardiovascular system. Various methods of nonlinear dynamics (e.g., estimating Lyapunov exponents) and complexity measures (e.g., correlation dimension or entropies) have been applied to HRV analysis. Permutation entropy, which was proposed recently, has been widely used in many fields due to its conceptual and computational simplicity. It maps a time series onto a symbolic sequence of permutation ranks. The original permutation entropy assumes the time series under study has a continuous distribution, thus equal values are rare and can be ignored by ranking them according to their order of emergence, or broken by adding small random perturbations to ensure every symbol in a sequence is different. However, when the observed time series is digitized with lower resolution leading to a greater number of equal values, or the equalities represent certain characteristic sequential patterns of the system, it may not be rational to simply ignore or break them. In the present paper, a modified permutation entropy is proposed that, by mapping the equal value onto the same symbol (rank), allows for a more accurate characterization of system states. The application of the modified permutation entropy to the analysis of HRV is investigated using clinically collected data. Results show that modified permutation entropy can greatly improve the ability to distinguish the HRV signals under different physiological and pathological conditions. It can characterize the complexity of HRV more effectively than the original permutation entropy.
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paper_title: Classifying cardiac biosignals using ordinal pattern statistics and symbolic dynamics

paper_content:

The performance of (bio-)signal classification strongly depends on the choice of suitable features (also called parameters or biomarkers). In this article we evaluate the discriminative power of ordinal pattern statistics and symbolic dynamics in comparison with established heart rate variability parameters applied to beat-to-beat intervals. As an illustrative example we distinguish patients suffering from congestive heart failure from a (healthy) control group using beat-to-beat time series. We assess the discriminative power of individual features as well as pairs of features. These comparisons show that ordinal patterns sampled with an additional time lag are promising features for efficient classification.
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paper_title: Forbidden patterns, permutation entropy and stock market inefficiency

paper_content:

In this paper we introduce two new quantifiers for the stock market inefficiency: the number of forbidden patterns and the normalized permutation entropy. They are model-independent measures, thus they have more general applicability. We find robust evidence that degree of market inefficiency is positively correlated with the number of forbidden patterns and negatively correlated with the permutation entropy. Our empirical results suggest that these two physical tools are useful to discriminate the stage of stock market development and can be easily implemented.
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paper_title: A New Approach to Measure Volatility in Energy Markets

paper_content:

Several measures of volatility have been developed in order to quantify the degree of uncertainty of an energy price series, which include historical volatility and price velocities, among others. This paper suggests using the permutation entropy, topological entropy and the modified permutation entropy as alternatives to measure volatility in energy markets. Simulated data show that these measures are more appropriate to quantify the uncertainty associated to a time series than those based on the standard deviation or other measures of dispersion. Finally, the proposed method is applied to some typical electricity markets: Nord Pool, Ontario, Omel and four Australian markets.
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paper_title: Forbidden patterns in financial time series

paper_content:

The existence of forbidden patterns, i.e., certain missing sequences in a given time series, is a recently proposed instrument of potential application in the study of time series. Forbidden patterns are related to the permutation entropy, which has the basic properties of classic chaos indicators, such as Lyapunov exponent or Kolmogorov entropy, thus allowing to separate deterministic (usually chaotic) from random series; however, it requires fewer values of the series to be calculated, and it is suitable for using with small datasets. In this paper, the appearance of forbidden patterns is studied in different economical indicators such as stock indices (Dow Jones Industrial Average and Nasdaq Composite), NYSE stocks (IBM and Boeing), and others (ten year Bond interest rate), to find evidence of deterministic behavior in their evolutions. Moreover, the rate of appearance of the forbidden patterns is calculated, and some considerations about the underlying dynamics are suggested.
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paper_title: Commodity predictability analysis with a permutation information theory approach

paper_content:

It is widely known that commodity markets are not totally efficient. Long-range dependence is present, and thus the celebrated Brownian motion of prices can be considered only as a first approximation. In this work we analyzed the predictability in commodity markets by using a novel approach derived from Information Theory. The complexity–entropy causality plane has been recently shown to be a useful statistical tool to distinguish the stage of stock market development because differences between emergent and developed stock markets can be easily discriminated and visualized with this representation space [L. Zunino, M. Zanin, B.M. Tabak, D.G. Perez, O.A. Rosso, Complexity–entropy causality plane: a useful approach to quantify the stock market inefficiency, Physica A 389 (2010) 1891–1901]. By estimating the permutation entropy and permutation statistical complexity of twenty basic commodity future markets over a period of around 20 years (1991.01.02–2009.09.01), we can define an associated ranking of efficiency. This ranking is quantifying the presence of patterns and hidden structures in these prime markets. Moreover, the temporal evolution of the commodities in the complexity–entropy causality plane allows us to identify periods of time where the underlying dynamics is more or less predictable.
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paper_title: A non-parametric independence test using permutation entropy

paper_content:

In the present paper we construct a new, simple, consistent and powerful test for independence by using symbolic dynamics and permutation entropy as a measure of serial dependence. We also give a standard asymptotic distribution of an affine transformation of the permutation entropy under the null hypothesis of independence. The test statistic and its standard limit distribution are invariant to any monotonic transformation. The test applies to time series with discrete or continuous distributions. Eventhough the test is based on entropy measures, it avoids smoothed non-parametric estimation. An application to several daily financial time series illustrates our approach.
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paper_title: Complexity-entropy causality plane: A useful approach to quantify the stock market inefficiency

paper_content:

The complexity-entropy causality plane has been recently introduced as a powerful tool for discriminating Gaussian from non-Gaussian process and different degrees of correlations [O.A. Rosso, H.A. Larrondo, M.T. Martin, A. Plastino, M.A. Fuentes, Distinguishing noise from chaos, Phys. Rev. Lett. 99 (2007) 154102]. We propose to use this representation space to distinguish the stage of stock market development. Our empirical results demonstrate that this statistical physics approach is useful, allowing a more refined classification of stock market dynamics.
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paper_title: Random number generators and causality

paper_content:

We advance a prescription to randomize physical or algorithmic Random Number Generators (RNG's) that do not pass Marsaglia's DIEHARD test suite and discuss a special physical quantifier, based on an intensive statistical complexity measure, that is able to adequately assess the improvements produced thereby. Eight RNG's are evaluated and the associated results are compared to those obtained by recourse to Marsaglia's DIEHARD test suite. Our quantifier, which is evaluated using causality arguments, can forecast whether a given RNG will pass the above mentioned test.
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paper_title: Analysis and Synthesis of Pseudo-Periodic Job Arrivals in Grids: A Matching Pursuit Approach

paper_content:

Pseudo-periodicity is one of the basic job arrival patterns on data-intensive clusters and Grids. In this paper, a signal decomposition methodology called matching pursuit is applied for analysis and synthesis of pseudo-periodic job arrival processes. The matching pursuit decomposition is well localized both in time and frequency, and it is naturally suited for analyzing non-stationary as well as stationary signals. The stationarity of the processes can be quantitatively measured by permutation entropy, with which the relationship between stationarity and modeling complexity is excellently explained. Quantitative methods based on the power spectrum are also provided to measure the degree of periodicity present in the data. Matching pursuit is further shown to be able to extract patterns from signals, which is an attractive feature from a modeling perspective. Real world workload data from production clusters and Grids are used to empirically evaluate the proposed measures and methodologies.
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paper_title: Complexity-entropy causality plane: a useful approach for distinguishing songs

paper_content:

Nowadays we are often faced with huge databases resulting from the rapid growth of data storage technologies. This is particularly true when dealing with music databases. In this context, it is essential to have techniques and tools able to discriminate properties from these massive sets. In this work, we report on a statistical analysis of more than ten thousand songs aiming to obtain a complexity hierarchy. Our approach is based on the estimation of the permutation entropy combined with an intensive complexity measure, building up the complexity-entropy causality plane. The results obtained indicate that this representation space is very promising to discriminate songs as well as to allow a relative quantitative comparison among songs. Additionally, we believe that the here-reported method may be applied in practical situations since it is simple, robust and has a fast numerical implementation.
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paper_title: Intensive statistical complexity measure of pseudorandom number generators

paper_content:

A Statistical Complexity measure has been recently proposed to quantify the performance of chaotic Pseudorandom number generators (PRNG) (Physica A 354 (2005) 281). Here we revisit this quantifier and introduce two important improvements: (i) consideration of an intensive statistical complexity (Physica A 334 (2004) 119), and (ii) following the prescription of Brand and Pompe (Phys. Rev. Lett. 88 (2002) 174102-1) in evaluating the probability distribution associated with the PRNG. The ensuing new measure is applied to a very well-tested PRNG advanced by Marsaglia.
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paper_title: Permutation Entropy: A Natural Complexity Measure for Time Series

paper_content:

We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.
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paper_title: Entropy analysis of the dynamics of El Niño/Southern Oscillation during the Holocene

paper_content:

This study explores temporal changes in the dynamics of the Holocene ENSO proxy record of the Laguna Pallcacocha sedimentary data using two entropy quantifiers. In particular, we analyze the possible connections between changes in entropy and epochs of rapid climate change (RCC). Our results indicate that the dynamics of the ENSO proxy record during the RCC interval 9000–8000 BP displays very low entropy (high predictability) that is remarkably different from that of the other RCCs of the Holocene. Both entropy quantifiers point out to the existence of cycles with a period close to 2000 years during the mid-to-late Holocene. Within these cycles, we find a tendency for entropy to increase (predictability to decrease) during the two longer RCC periods (6000–5000 and 3500–2500 BP) which might be associated with the reported increased aridity of the low tropics.
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paper_title: Characterizing the Hyperchaotic Dynamics of a Semiconductor Laser Subject to Optical Feedback Via Permutation Entropy

paper_content:

The time evolution of the output of a semiconductor laser subject to delayed optical feedback can exhibit high-dimensional chaotic fluctuations. In this contribution, our aim is to quantify the degree of unpredictability of this hyperchaotic time evolution. To that end, we estimate permutation entropy, a novel information-theory-derived quantifier particularly robust in a noisy environment. The permutation entropy is defined as a functional of a symbolic probability distribution, evaluated using the Bandt-Pompe recipe to assign a probability distribution function to the time series generated by the chaotic system. This measure quantifies the diversity of orderings present in the associated time series. In order to evaluate the performance of this novel quantifier, we compare with the results obtained by using a more standard chaos quantifier, namely the Kolmogorov-Sinai entropy. Here, we present numerical results showing that the permutation entropy, evaluated at specific time-scales involved in the chaotic regime of the semiconductor laser subject to optical feedback, give valuable information about the degree of unpredictability of the chaotic laser dynamics. The influence of additive observational noise on the proposed tool is also investigated.
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paper_title: Complexity measure of motor current signals for tool flute breakage detection in end milling

paper_content:

Automated tool condition monitoring is an important issue in the advanced machining process. Permutation entropy of a time series is a simple, robust and extremely fast complexity measure method for distinguishing the different conditions of a physical system. In this study, the permutation entropy of feed-motor current signals in end milling was applied to detect tool breakage. The detection method is composed of the estimation of permutation entropy and wavelet-based de-noising. To confirm the effectiveness and robustness of the method, typical experiments have been performed from the cutter runout and entry/exit cuts to cutting parameters variation. Results showed that the new method could successfully extract significant signature from the feed-motor current signals to effectively detect tool flute breakage during end milling. Whilst, this detection method was based on current sensors, so it possesses excellent potential for practical and real-time application at a low cost by comparison with the alternative sensors.
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paper_title: The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems

paper_content:

Abstract Permutation entropy quantifies the diversity of possible orderings of the values a random or deterministic system can take, as Shannon entropy quantifies the diversity of values. We show that the metric and permutation entropy rates—measures of new disorder per new observed value—are equal for ergodic finite-alphabet information sources (discrete-time stationary stochastic processes). With this result, we then prove that the same holds for deterministic dynamical systems defined by ergodic maps on n -dimensional intervals. This result generalizes a previous one for piecewise monotone interval maps on the real line [C. Bandt, G. Keller, B. Pompe, Entropy of interval maps via permutations, Nonlinearity 15 (2002) 1595–1602.] at the expense of requiring ergodicity and using a definition of permutation entropy rate differing modestly in the order of two limits. The case of non-ergodic finite-alphabet sources is also studied and an inequality developed. Finally, the equality of permutation and metric entropy rates is extended to ergodic non-discrete information sources when entropy is replaced by differential entropy in the usual way.
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paper_title: Permutation entropy of fractional Brownian motion and fractional Gaussian noise

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We have worked out theoretical curves for the permutation entropy of the fractional Brownian motion and fractional Gaussian noise by using the Bandt and Shiha [C. Bandt, F. Shiha, J. Time Ser. Anal. 28 (2007) 646] theoretical predictions for their corresponding relative frequencies. Comparisons with numerical simulations show an excellent agreement. Furthermore, the entropy-gap in the transition between these processes, observed previously via numerical results, has been here theoretically validated. Also, we have analyzed the behaviour of the permutation entropy of the fractional Gaussian noise for different time delays.
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paper_title: Chaos and complexity in the classical-quantum transition

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By recourse to the concept of Statistical Complexity we study here the classical-quantum transition in a special system that represents the matter-field interaction. Our work considers two distinct disequilibrium forms based on Euclidean norm and Jensen-Shannon divergence, on the one hand, and analyzes things, on the other one, by using two different numerical approaches for probability distribution, namely, relative wavelet energy and permutation patterns.
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paper_title: Information flow during the quantum-classical transition

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Abstract We have exhaustively investigated the classical limit of the semi-classical evolution with reference to a well-known model that represents the interaction between matter and a given field. In this Letter we approach this issue by recourse to a new statistical quantifier called the “symbolic transfer entropy” [T. Schreiber, Phys. Rev. Lett. 85 (2000) 461; M. Staniek, K. Lehnertz, Phys. Rev. Lett. 100 (2008) 158101]. We encounter that the quantum-classical transition gets thereby described as the sign reversal of the dominating direction of the information flow between classical and quantal variables. This can be considered as an evidence of the physical useful of this new statistical quantifier.
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paper_title: Extracting features of Gaussian self-similar stochastic processes via the Bandt-Pompe approach.

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By recourse to appropriate information theory quantifiers (normalized Shannon entropy and Martín-Plastino-Rosso intensive statistical complexity measure), we revisit the characterization of Gaussian self-similar stochastic processes from a Bandt-Pompe viewpoint. We show that the ensuing approach exhibits considerable advantages with respect to other treatments. In particular, clear quantifiers gaps are found in the transition between the continuous processes and their associated noises.
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paper_title: Mixing chaotic maps and electromagnetic interference reduction

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Spread-spectrum techniques for improving Electromagnetic Compatibilities are investigated. This requires generating Constant Envelope Wideband signals. We produce them via chaotic maps that yield pseudo random time series, used afterwards to modulate sinusoidal waves in frequency. How to assess a maps’s suitability for such task is also discussed.
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paper_title: Quantifying stochasticity in the dynamics of delay-coupled semiconductor lasers via forbidden patterns

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We quantify the level of stochasticity in the dynamics of two mutually coupled semiconductor lasers. Specifically, we concentrate on a regime in which the lasers synchronize their dynamics with a non-zero lag time, and the leader and laggard roles alternate irregularly between the lasers. We analyse this switching dynamics in terms of the number of forbidden patterns of the alternate time series. The results reveal that the system operates in a stochastic regime, with the level of stochasticity decreasing as the lasers are pumped further away from their lasing threshold. This behaviour is similar to that exhibited by a single semiconductor laser subject to external optical feedback, as its dynamics shifts from the regime of low-frequency fluctuations to coherence collapse.
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paper_title: Encryption test of pseudo-aleatory messages embedded on chaotic laser signals: An information theory approach

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In this Letter, we make use of two information-theory based indicators to measure the goodness of two encryption schemes commonly used within the context of chaotic communications. In particular, we have shown that the computation of the normalized Shannon entropy and the MPR-Statistical Complexity measure [M.T. Martin, A. Plastino, O.A. Rosso, Phys. Lett. A 311 (2003) 126, P.W. Lamberti, M.T. Martin, A. Plastino, O.A. Rosso, Physica A 334 (2004) 119] for different chaotic laser signals can lead to statistically significant criteria to assess the quality of several encryption techniques. The proposed measures allow, in some cases, to detect the presence of a message embedded within a chaotic carrier. They also reveal that the Chaos Modulation scheme is more reliable from the statistical point of view, when compared with the Chaos Shift Keying.
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paper_title: Fractional Brownian motion, fractional Gaussian noise, and Tsallis permutation entropy

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In this work, we analyze two important stochastic processes, the fractional Brownian motion and fractional Gaussian noise, within the framework of the Tsallis permutation entropy. This entropic measure, evaluated after using the Bandt & Pompe method to extract the associated probability distribution, is shown to be a powerful tool to characterize fractal stochastic processes. It allows for a better discrimination of the processes than the Shannon counterpart for appropriate ranges of values of the entropic index. Moreover, we find the optimum value of this entropic index for the stochastic processes under study.
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paper_title: Quantifiers for randomness of chaotic pseudo-random number generators

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We deal with randomness quantifiers and concentrate on their ability to discern the hallmark of chaos in time series used in connection with pseudo-random number generators (PRNGs). Workers in the field are motivated to use chaotic maps for generating PRNGs because of the simplicity of their implementation. Although there exist very efficient general-purpose benchmarks for testing PRNGs, we feel that the analysis provided here sheds additional didactic light on the importance of the main statistical characteristics of a chaotic map, namely (i) its invariant measure and (ii) the mixing constant. This is of help in answering two questions that arise in applications: (i) which is the best PRNG among the available ones? and (ii) if a given PRNG turns out not to be good enough and a randomization procedure must still be applied to it, which is the best applicable randomization procedure? Our answer provides a comparative analysis of several quantifiers advanced in the extant literature.
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paper_title: Sampling period, statistical complexity, and chaotic attractors

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We analyze the statistical complexity measure vs. entropy plane-representation of sampled chaotic attractors as a function of the sampling period τ and show that, if the Bandt and Pompe procedure is used to assign a probability distribution function (PDF) to the pertinent time series, the statistical complexity measure (SCM) attains a definite maximum for a specific sampling periodtM. On the contrary, the usual histogram approach for assigning PDFs to a time series leads to essentially constant SCM values for any sampling period τ. The significance of tM is further investigated by comparing it with typical times found in the literature for the two main reconstruction processes: the Takens’ one in a delay-time embedding, on one hand, and the exact Nyquist–Shannon reconstruction, on the other one. It is shown that tM is compatible with those times recommended as adequate delay ones in Takens’ reconstruction. The reported results correspond to three representative chaotic systems having correlation dimension 2
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paper_title: Influence of polarization mode competition on chaotic unpredictability of vertical-cavity surface-emitting lasers with polarization-rotated optical feedback

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The chaotic unpredictability of vertical-cavity surface-emitting lasers (VCSELs) with polarization-rotated optical feedback is evaluated quantitatively via the normalized permutation entropy H. The effect of polarization mode competition (PMC) is explored. For weak PMC, the H for X-polarization mode, Y-polarization mode, and total output are close to each other, while for strong PMC, the H for the total output is the highest one, which indicates that the permutation entropy is an effective tool for quantifying chaotic unpredictability of VCSELs and provides valuable information for choosing the proper chaotic carrier.
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paper_title: The complexity of gene expression dynamics revealed by permutation entropy

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BackgroundHigh complexity is considered a hallmark of living systems. Here we investigate the complexity of temporal gene expression patterns using the concept of Permutation Entropy (PE) first introduced in dynamical systems theory. The analysis of gene expression data has so far focused primarily on the identification of differentially expressed genes, or on the elucidation of pathway and regulatory relationships. We aim to study gene expression time series data from the viewpoint of complexity.ResultsApplying the PE complexity metric to abiotic stress response time series data in Arabidopsis thaliana, genes involved in stress response and signaling were found to be associated with the highest complexity not only under stress, but surprisingly, also under reference, non-stress conditions. Genes with house-keeping functions exhibited lower PE complexity. Compared to reference conditions, the PE of temporal gene expression patterns generally increased upon stress exposure. High-complexity genes were found to have longer upstream intergenic regions and more cis-regulatory motifs in their promoter regions indicative of a more complex regulatory apparatus needed to orchestrate their expression, and to be associated with higher correlation network connectivity degree. Arabidopsis genes also present in other plant species were observed to exhibit decreased PE complexity compared to Arabidopsis specific genes.ConclusionsWe show that Permutation Entropy is a simple yet robust and powerful approach to identify temporal gene expression profiles of varying complexity that is equally applicable to other types of molecular profile data.
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paper_title: The Quantum-Classical Transition as an Information Flow

paper_content:

We investigate the classical limit of the semiclassical evolution with reference to a well-known model that represents the interaction between matter and a given field. This is done by recourse to a special statistical quantifier called the "symbolic transfer entropy". We encounter that the quantum-classical transition gets thereby described as the sign-reversal of the dominating direction of the information flow between classical and quantal variables.
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paper_title: Randomizing nonlinear maps via symbolic dynamics

paper_content:

Pseudo Random Number Generators (PRNG) have attracted intense attention due to their obvious importance for many branches of science and technology. A randomizing technique is a procedure designed to improve the PRNG randomness degree according the specific requirements. It is obviously important to quantify its effectiveness. In order to classify randomizing techniques based on a symbolic dynamics’ approach, we advance a novel, physically motivated representation based on the statistical properties of chaotic systems. Recourse is made to a plane that has as coordinates (i) the Shannon entropy and (ii) a form of the statistical complexity measure. Each statistical quantifier incorporates a different probability distribution function, generating thus a representation that (i) sheds insight into just how each randomizing technique operates and also (ii) quantifies its effectiveness. Using the Logistic Map and the Three Way Bernoulli Map as typical examples of chaotic dynamics it is shown that our methodology allows for choosing the more convenient randomizing technique in each instance. Comparison with measures of complexity based on diagonal lines on the recurrence plots [N. Marwan, M.C. Romano, M. Thiel, J. Kurths, Phys. Rep. 438 (2007) 237] support the main conclusions of this paper.
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paper_title: Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback

paper_content:

Low-frequency fluctuations (LFFs) represent a dynamical instability that occurs in semiconductor lasers when they are operated near the lasing threshold and subject to moderate optical feedback. LFFs consist of sudden power dropouts followed by gradual, stepwise recoveries. We analyze experimental time series of intensity dropouts and quantify the complexity of the underlying dynamics employing two tools from information theory, namely, Shannon's entropy and the Martin, Plastino, and Rosso statistical complexity measure. These measures are computed using a method based on ordinal patterns, by which the relative length and ordering of consecutive interdropout intervals (i.e., the time intervals between consecutive intensity dropouts) are analyzed, disregarding the precise timing of the dropouts and the absolute durations of the interdropout intervals. We show that this methodology is suitable for quantifying subtle characteristics of the LFFs, and in particular the transition to fully developed chaos that takes place when the laser's pump current is increased. Our method shows that the statistical complexity of the laser does not increase continuously with the pump current, but levels off before reaching the coherence collapse regime. This behavior coincides with that of the first- and second-order correlations of the interdropout intervals, suggesting that these correlations, and not themore » chaotic behavior, are what determine the level of complexity of the laser's dynamics. These results hold for two different dynamical regimes, namely, sustained LFFs and coexistence between LFFs and steady-state emission.« less
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paper_title: Permutation entropy based real-time chatter detection using audio signal in turning process

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Machine tool chatter is an unfavorable phenomenon during metal cutting, which results in heavy vibration of cutting tool. With increase in depth of cut, the cutting regime changes from chatter-free cutting to one with chatter. In this paper, we propose the use of permutation entropy (PE), a conceptually simple and computationally fast measurement to detect the onset of chatter from the time series using sound signal recorded with a unidirectional microphone. PE can efficiently distinguish the regular and complex nature of any signal and extract information about the dynamics of the process by indicating sudden change in its value. Under situations where the data sets are huge and there is no time for preprocessing and fine-tuning, PE can effectively detect dynamical changes of the system. This makes PE an ideal choice for online detection of chatter, which is not possible with other conventional nonlinear methods. In the present study, the variation of PE under two cutting conditions is analyzed. Abrupt variation in the value of PE with increase in depth of cut indicates the onset of chatter vibrations. The results are verified using frequency spectra of the signals and the nonlinear measure, normalized coarse-grained information rate (NCIR).
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Title: Article Permutation Entropy and Its Main Biomedical and Econophysics Applications: A Review

Section 1: Introduction
Description 1: This section sets up the context of the paper, introducing the concept of information content in system observables, classical entropy measures, and the limitations of these methods that led to the development of permutation entropy.

Section 2: The Permutation Entropy
Description 2: This section explains the core concept of permutation entropy, detailing the method's application to time series, the construction of ordinal patterns, and the relevance of forbidden patterns to understand system dynamics.

Section 3: Distinguishing Noise from Chaos
Description 3: This section discusses how permutation entropy can differentiate between stochastic and chaotic systems, including methodologies for identifying determinism in the presence of noise.

Section 4: The Statistical Complexity and the Complexity-Entropy Plane
Description 4: This section delves into the measures of statistical complexity, their necessity for characterizing chaotic systems, and the use of the complexity-entropy plane to interpret system dynamics.

Section 5: Identification of Time Scales
Description 5: This section covers the usage of permutation entropy to identify characteristic time scales in complex systems, illustrated through various applications and case studies.

Section 6: Dependences between Time Series
Description 6: This section explores the methods for detecting dependencies between multiple time series using permutation entropy, including nonparametric tests and applications in various fields.

Section 7: Some Improvements on the PE Definition
Description 7: This section addresses enhancements to the original definition of permutation entropy, such as fine-grained permutation entropy and modified permutation entropy, to overcome specific limitations.

Section 8: Biomedical Applications
Description 8: This section reviews the major applications of permutation entropy in biomedical fields, particularly focusing on brain activity analysis, epilepsy research, anesthesiology, and heart rate variability studies.

Section 9: Econophysics Applications
Description 9: This section presents the use of permutation entropy in econophysics, including market efficiency assessment, stock market development, and analysis of market dependencies and volatilities.

Section 10: Conclusions
Description 10: This section summarizes the key findings of the review, highlighting the versatility, computational efficiency, and robustness of permutation entropy in various applications.