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

Deadline for manuscript submissions: closed (15 December 2019) | Viewed by 50802

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Prof. Dr. Lucas Lacasa

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School of Mathematical Sciences, Queen Mary University of London, Mile End, London E1 4NS, UK
Interests: complex systems; time series analysis; network science; nonlinear dynamics; statistical physics

Special Issue Information

Dear Colleagues,

Concepts such as ‘entropy’ or ‘complexity’ have been approached from many different angles in physics, mathematics, computer science and beyond. The interdisciplinary arena spanned by these concepts inherits ideas and tools from nonlinear dynamics (e.g. Kolmogorov–Sinai entropy, Renyi entropies), information theory (Shannon entropy, statistical complexity), statistical physics (Boltzmann entropy, Tsallis entropy), or network science (graph entropy), and make use of these to describe and understand the behaviour of complex systems in an amazingly wide range of contexts.

The aim of this Special Issue is to encourage researchers to present original and recent developments on topics closely related to entropy and complexity that emerge (typically) in nonlinear dynamical systems and related complex systems. The type of contributions can be theoretical or applied: they can address a particular fundamental open problem where the authors push forward the state of the art or can represent sensible examples that make efficient use of these tools in different contexts across physics, biology, economics or the computational social sciences, among others.

I look forward to reading your submissions.

Prof. Dr. Lucas Lacasa
Guest Editor

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Keywords

Entropy
Complexity measures
Complex systems
Networks
Time series analysis
Disordered systems
Nonlinear dynamical systems

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Published Papers (11 papers)

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Research

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15 pages, 589 KiB

Open AccessArticle

A Maximum Entropy Method for the Prediction of Size Distributions

by Cornelia Metzig and Caroline Colijn

Entropy 2020, 22(3), 312; https://doi.org/10.3390/e22030312 - 10 Mar 2020

Cited by 3 | Viewed by 3776

Abstract

We propose a method to derive the stationary size distributions of a system, and the degree distributions of networks, using maximisation of the Gibbs-Shannon entropy. We apply this to a preferential attachment-type algorithm for systems of constant size, which contains exit of balls and urns (or nodes and edges for the network case). Knowing mean size (degree) and turnover rate, the power law exponent and exponential cutoff can be derived. Our results are confirmed by simulations and by computation of exact probabilities. We also apply this entropy method to reproduce existing results like the Maxwell-Boltzmann distribution for the velocity of gas particles, the Barabasi-Albert model and multiplicative noise systems. Full article

(This article belongs to the Special Issue Entropy, Nonlinear Dynamics and Complexity)

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19 pages, 1175 KiB

Open AccessArticle

From Boltzmann to Zipf through Shannon and Jaynes

by Álvaro Corral and Montserrat García del Muro

Entropy 2020, 22(2), 179; https://doi.org/10.3390/e22020179 - 5 Feb 2020

Cited by 5 | Viewed by 3884

Abstract

The word-frequency distribution provides the fundamental building blocks that generate discourse in natural language. It is well known, from empirical evidence, that the word-frequency distribution of almost any text is described by Zipf’s law, at least approximately. Following Stephens and Bialek (2010), we interpret the frequency of any word as arising from the interaction potentials between its constituent letters. Indeed, Jaynes’ maximum-entropy principle, with the constrains given by every empirical two-letter marginal distribution, leads to a Boltzmann distribution for word probabilities, with an energy-like function given by the sum of the all-to-all pairwise (two-letter) potentials. The so-called improved iterative-scaling algorithm allows us finding the potentials from the empirical two-letter marginals. We considerably extend Stephens and Bialek’s results, applying this formalism to words with length of up to six letters from the English subset of the recently created Standardized Project Gutenberg Corpus. We find that the model is able to reproduce Zipf’s law, but with some limitations: the general Zipf’s power-law regime is obtained, but the probability of individual words shows considerable scattering. In this way, a pure statistical-physics framework is used to describe the probabilities of words. As a by-product, we find that both the empirical two-letter marginal distributions and the interaction-potential distributions follow well-defined statistical laws. Full article

(This article belongs to the Special Issue Entropy, Nonlinear Dynamics and Complexity)

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17 pages, 2243 KiB

Open AccessArticle

Spatial and Temporal Entropies in the Spanish Football League: A Network Science Perspective

by Johann H. Martínez, David Garrido, José L Herrera-Diestra, Javier Busquets, Ricardo Sevilla-Escoboza and Javier M. Buldú

Entropy 2020, 22(2), 172; https://doi.org/10.3390/e22020172 - 2 Feb 2020

Cited by 25 | Viewed by 6384

Abstract

We quantified the spatial and temporal entropy related to football teams and their players by means of a pass-based interaction. First, we calculated the spatial entropy associated to the positions of all passes made by a football team during a match, obtaining a spatial entropy ranking of Spanish teams during the 2017/2018 season. Second, we investigated how the player’s average location in the field is related to the amount of entropy of his passes. Next, we constructed the temporal passing networks of each team and computed the deviation of their network parameters along the match. For each network parameter, we obtained the permutation entropy and the statistical complexity of its temporal fluctuations. Finally, we investigated how the permutation entropy (and statistical complexity) of the network parameters was related to the total number of passes made by a football team. Our results show that (i) spatial entropy changes according to the position of players in the field, and (ii) the organization of passing networks change during a match and its evolution can be captured measuring the permutation entropy and statistical complexity of the network parameters, allowing to identify what parameters evolve more randomly. Full article

(This article belongs to the Special Issue Entropy, Nonlinear Dynamics and Complexity)

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13 pages, 3271 KiB

Open AccessArticle

Detection of Hypoglycemia Using Measures of EEG Complexity in Type 1 Diabetes Patients

by Maria Rubega, Fabio Scarpa, Debora Teodori, Anne-Sophie Sejling, Christian S. Frandsen and Giovanni Sparacino

Entropy 2020, 22(1), 81; https://doi.org/10.3390/e22010081 - 9 Jan 2020

Cited by 15 | Viewed by 4213

Abstract

Previous literature has demonstrated that hypoglycemic events in patients with type 1 diabetes (T1D) are associated with measurable scalp electroencephalography (EEG) changes in power spectral density. In the present study, we used a dataset of 19-channel scalp EEG recordings in 34 patients with T1D who underwent a hyperinsulinemic–hypoglycemic clamp study. We found that hypoglycemic events are also characterized by EEG complexity changes that are quantifiable at the single-channel level through empirical conditional and permutation entropy and fractal dimension indices, i.e., the Higuchi index, residuals, and tortuosity. Moreover, we demonstrated that the EEG complexity indices computed in parallel in more than one channel can be used as the input for a neural network aimed at identifying hypoglycemia and euglycemia. The accuracy was about 90%, suggesting that nonlinear indices applied to EEG signals might be useful in revealing hypoglycemic events from EEG recordings in patients with T1D. Full article

(This article belongs to the Special Issue Entropy, Nonlinear Dynamics and Complexity)

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13 pages, 1068 KiB

Open AccessArticle

Complexity Changes in the US and China’s Stock Markets: Differences, Causes, and Wider Social Implications

by Jianbo Gao, Yunfei Hou, Fangli Fan and Feiyan Liu

Entropy 2020, 22(1), 75; https://doi.org/10.3390/e22010075 - 6 Jan 2020

Cited by 10 | Viewed by 3907

Abstract

How different are the emerging and the well-developed stock markets in terms of efficiency? To gain insights into this question, we compared an important emerging market, the Chinese stock market, and the largest and the most developed market, the US stock market. Specifically, we computed the Lempel–Ziv complexity (LZ) and the permutation entropy (PE) from two composite stock indices, the Shanghai stock exchange composite index (SSE) and the Dow Jones industrial average (DJIA), for both low-frequency (daily) and high-frequency (minute-to-minute)stock index data. We found that the US market is basically fully random and consistent with efficient market hypothesis (EMH), irrespective of whether low- or high-frequency stock index data are used. The Chinese market is also largely consistent with the EMH when low-frequency data are used. However, a completely different picture emerges when the high-frequency stock index data are used, irrespective of whether the LZ or PE is computed. In particular, the PE decreases substantially in two significant time windows, each encompassing a rapid market rise and then a few gigantic stock crashes. To gain further insights into the causes of the difference in the complexity changes in the two markets, we computed the Hurst parameter H from the high-frequency stock index data of the two markets and examined their temporal variations. We found that in stark contrast with the US market, whose H is always close to 1/2, which indicates fully random behavior, for the Chinese market, H deviates from 1/2 significantly for time scales up to about 10 min within a day, and varies systemically similar to the PE for time scales from about 10 min to a day. This opens the door for large-scale collective behavior to occur in the Chinese market, including herding behavior and large-scale manipulation as a result of inside information. Full article

(This article belongs to the Special Issue Entropy, Nonlinear Dynamics and Complexity)

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15 pages, 3136 KiB

Open AccessArticle

Multiscale Entropy Analysis of Postural Stability for Estimating Fall Risk via Domain Knowledge of Timed-Up-And-Go Accelerometer Data for Elderly People Living in a Community

by Chi-Han Wu, Chia-Hsuan Lee, Bernard C. Jiang and Tien-Lung Sun

Entropy 2019, 21(11), 1076; https://doi.org/10.3390/e21111076 - 2 Nov 2019

Cited by 12 | Viewed by 2903

Abstract

As people in developed countries live longer, assessing the fall risk becomes more important. A major contributor to the risk of elderly people falling is postural instability. This study aimed to use the multiscale entropy (MSE) analysis to evaluate postural stability during a timed-up-and-go (TUG) test. This test was deemed a promising method for evaluating fall risk among the elderly in a community. The MSE analysis of postural instability can identify the elderly prone to falling, whereupon early medical rehabilitation can prevent falls. Herein, an objective approach is developed for assessing the postural stability of 85 community-dwelling elderly people (aged 76.12 ± 6.99 years) using the short-form Berg balance scale. Signals were collected from the TUG test using a triaxial accelerometer. A segment-based TUG (sTUG) test was designed, which can be obtained according to domain knowledge, including “Sit-to-Walk (STW),” “Walk,” “Turning,” and “Walk-to-Sit (WTS)” segments. Employing the complexity index (CI) of sTUG can reveal information about the physiological dynamics’ signal for postural stability assessment. Logistic regression was used to assess the fall risk based on significant features of CI related to sTUG. MSE curves for subjects at risk of falling (n = 19) exhibited different trends from those not at risk of falling (n = 66). Additionally, the CI values were lower for subjects at risk of falling than those not at risk of falling. Results show that the area under the curve for predicting fall risk among the elderly subjects with complexity index features from the overall TUG test is 0.797, which improves to 0.853 with the sTUG test. For the elderly living in a community, early assessment of the CI for sTUG using MSE can help predict the fall risk. Full article

(This article belongs to the Special Issue Entropy, Nonlinear Dynamics and Complexity)

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13 pages, 2018 KiB

Open AccessArticle

Multiscale Horizontal Visibility Graph Analysis of Higher-Order Moments for Estimating Statistical Dependency

by Keqiang Dong, Haowei Che and Zhi Zou

Entropy 2019, 21(10), 1008; https://doi.org/10.3390/e21101008 - 16 Oct 2019

Cited by 6 | Viewed by 4520

Abstract

The horizontal visibility graph is not only a powerful tool for the analysis of complex systems, but also a promising way to analyze time series. In this paper, we present an approach to measure the nonlinear interactions between a non-stationary time series based on the horizontal visibility graph. We describe how a horizontal visibility graph may be calculated based on second-order and third-order statistical moments. We compare the new methods with the first-order measure, and then give examples including stock markets and aero-engine performance parameters. These analyses suggest that measures derived from the horizontal visibility graph may be of particular relevance to the growing interest in quantifying the information exchange between time series. Full article

(This article belongs to the Special Issue Entropy, Nonlinear Dynamics and Complexity)

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22 pages, 12493 KiB

Open AccessArticle

A Nonvolatile Fractional Order Memristor Model and Its Complex Dynamics

by Jian Wu, Guangyi Wang, Herbert Ho-Ching Iu, Yiran Shen and Wei Zhou

Entropy 2019, 21(10), 955; https://doi.org/10.3390/e21100955 - 29 Sep 2019

Cited by 14 | Viewed by 3211

Abstract

It is found that the fractional order memristor model can better simulate the characteristics of memristors and that chaotic circuits based on fractional order memristors also exhibit abundant dynamic behavior. This paper proposes an active fractional order memristor model and analyzes the electrical characteristics of the memristor via Power-Off Plot and Dynamic Road Map. We find that the fractional order memristor has continually stable states and is therefore nonvolatile. We also show that the memristor can be switched from one stable state to another under the excitation of appropriate voltage pulse. The volt–ampere hysteretic curves, frequency characteristics, and active characteristics of integral order and fractional order memristors are compared and analyzed. Based on the fractional order memristor and fractional order capacitor and inductor, we construct a chaotic circuit, of which the dynamic characteristics with respect to memristor’s parameters, fractional order α, and initial values are analyzed. The chaotic circuit has an infinite number of equilibrium points with multi-stability and exhibits coexisting bifurcations and coexisting attractors. Finally, the fractional order memristor-based chaotic circuit is verified by circuit simulations and DSP experiments. Full article

(This article belongs to the Special Issue Entropy, Nonlinear Dynamics and Complexity)

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28 pages, 5978 KiB

Open AccessArticle

Remaining Useful Life Prediction with Similarity Fusion of Multi-Parameter and Multi-Sample Based on the Vibration Signals of Diesel Generator Gearbox

by Shenghan Zhou, Xingxing Xu, Yiyong Xiao, Wenbing Chang, Silin Qian and Xing Pan

Entropy 2019, 21(9), 861; https://doi.org/10.3390/e21090861 - 3 Sep 2019

Cited by 5 | Viewed by 3334

Abstract

The prediction of electrical machines’ Remaining Useful Life (RUL) can facilitate making electrical machine maintenance policies, which is important for improving their security and extending their life span. This paper proposes an RUL prediction model with similarity fusion of multi-parameter and multi-sample. Firstly, based on the time domain and frequency domain extraction of vibration signals, the performance damage indicator system of a gearbox is established to select the optimal damage indicators for RUL prediction. Low-pass filtering based on approximate entropy variance (Aev) is introduced in this process because of its stability. Secondly, this paper constructs Dynamic Time Warping Distance (DTWD) as a similarity measurement function, which belongs to the nonlinear dynamic programming algorithm. It performed better than the traditional Euclidean distance. Thirdly, based on DTWD, similarity fusion of multi-parameter and multi-sample methods is proposed here to achieve RUL prediction. Next, the performance evaluation indicator Q is adopted to evaluate the RUL prediction accuracy of different methods. Finally, the proposed method is verified by experiments, and the Multivariable Support Vector Machine (MSVM) and Principal Component Analysis (PCA) are introduced for comparative studies. The results show that the Mean Absolute Percentage Error (MAPE) of the similarity fusion of multi-parameter and multi-sample methods proposed here is below 14%, which is lower than MSVM’s and PCA’s. Additionally, the RUL prediction based on the DTWD function in multi-sample similarity fusion exhibits the best accuracy. Full article

(This article belongs to the Special Issue Entropy, Nonlinear Dynamics and Complexity)

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11 pages, 741 KiB

Open AccessArticle

Recurrence Networks in Natural Languages

by Edgar Baeza-Blancas, Bibiana Obregón-Quintana, Candelario Hernández-Gómez, Domingo Gómez-Meléndez, Daniel Aguilar-Velázquez, Larry S. Liebovitch and Lev Guzmán-Vargas

Entropy 2019, 21(5), 517; https://doi.org/10.3390/e21050517 - 23 May 2019

Cited by 4 | Viewed by 4016

Abstract

We present a study of natural language using the recurrence network method. In our approach, the repetition of patterns of characters is evaluated without considering the word structure in written texts from different natural languages. Our dataset comprises 85 ebookseBooks written in 17 different European languages. The similarity between patterns of length m is determined by the Hamming distance and a value r is considered to define a matching between two patterns, i.e., a repetition is defined if the Hamming distance is equal or less than the given threshold value r. In this way, we calculate the adjacency matrix, where a connection between two nodes exists when a matching occurs. Next, the recurrence network is constructed for the texts and some representative network metrics are calculated. Our results show that average values of network density, clustering, and assortativity are larger than their corresponding shuffled versions, while for metrics like such as closeness, both original and random sequences exhibit similar values. Moreover, our calculations show similar average values for density among languages which that belong to the same linguistic family. In addition, the application of a linear discriminant analysis leads to well-separated clusters of family languages based on based on the network-density properties. Finally, we discuss our results in the context of the general characteristics of written texts. Full article

(This article belongs to the Special Issue Entropy, Nonlinear Dynamics and Complexity)

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Review

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40 pages, 2261 KiB

Open AccessReview

Nonlinear Methods Most Applied to Heart-Rate Time Series: A Review

by Teresa Henriques, Maria Ribeiro, Andreia Teixeira, Luísa Castro, Luís Antunes and Cristina Costa-Santos

Entropy 2020, 22(3), 309; https://doi.org/10.3390/e22030309 - 9 Mar 2020

Cited by 93 | Viewed by 9723

Abstract

The heart-rate dynamics are one of the most analyzed physiological interactions. Many mathematical methods were proposed to evaluate heart-rate variability. These methods have been successfully applied in research to expand knowledge concerning the cardiovascular dynamics in healthy as well as in pathological conditions. Notwithstanding, they are still far from clinical practice. In this paper, we aim to review the nonlinear methods most used to assess heart-rate dynamics. We focused on methods based on concepts of chaos, fractality, and complexity: Poincaré plot, recurrence plot analysis, fractal dimension (and the correlation dimension), detrended fluctuation analysis, Hurst exponent, Lyapunov exponent entropies (Shannon, conditional, approximate, sample entropy, and multiscale entropy), and symbolic dynamics. We present the description of the methods along with their most notable applications. Full article

(This article belongs to the Special Issue Entropy, Nonlinear Dynamics and Complexity)

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