Entropy

2023

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27 pages, 536 KiB

Open AccessArticle

Convergence Rates for the Constrained Sampling via Langevin Monte Carlo

by Yuanzheng Zhu

Entropy 2023, 25(8), 1234; https://doi.org/10.3390/e25081234 - 18 Aug 2023

Viewed by 1685

Abstract

Sampling from constrained distributions has posed significant challenges in terms of algorithmic design and non-asymptotic analysis, which are frequently encountered in statistical and machine-learning models. In this study, we propose three sampling algorithms based on Langevin Monte Carlo with the Metropolis–Hastings steps to [...] Read more.

Sampling from constrained distributions has posed significant challenges in terms of algorithmic design and non-asymptotic analysis, which are frequently encountered in statistical and machine-learning models. In this study, we propose three sampling algorithms based on Langevin Monte Carlo with the Metropolis–Hastings steps to handle the distribution constrained within some convex body. We present a rigorous analysis of the corresponding Markov chains and derive non-asymptotic upper bounds on the convergence rates of these algorithms in total variation distance. Our results demonstrate that the sampling algorithm, enhanced with the Metropolis–Hastings steps, offers an effective solution for tackling some constrained sampling problems. The numerical experiments are conducted to compare our methods with several competing algorithms without the Metropolis–Hastings steps, and the results further support our theoretical findings. Full article

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22 pages, 2114 KiB

Open AccessArticle

Entropy Stable DGSEM Schemes of Gauss Points Based on Subcell Limiting

by Yang Liu, Huajun Zhu, Zhen-Guo Yan, Feiran Jia and Xinlong Feng

Entropy 2023, 25(6), 911; https://doi.org/10.3390/e25060911 - 8 Jun 2023

Viewed by 1363

Abstract

The discontinuous Galerkin spectral element method (DGSEM) is a compact and high-order method applicable to complex meshes. However, the aliasing errors in simulating under-resolved vortex flows and non-physical oscillations in simulating shock waves may lead to instability of the DGSEM. In this paper, [...] Read more.

The discontinuous Galerkin spectral element method (DGSEM) is a compact and high-order method applicable to complex meshes. However, the aliasing errors in simulating under-resolved vortex flows and non-physical oscillations in simulating shock waves may lead to instability of the DGSEM. In this paper, an entropy-stable DGSEM (ESDGSEM) based on subcell limiting is proposed to improve the non-linear stability of the method. First, we discuss the stability and resolution of the entropy-stable DGSEM based on different solution points. Second, a provably entropy-stable DGSEM based on subcell limiting is established on Legendre–Gauss (LG) solution points. Numerical experiments demonstrate that the ESDGSEM-LG scheme is superior in non-linear stability and resolution, and ESDGSEM-LG with subcell limiting is robust in shock-capturing. Full article

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19 pages, 2671 KiB

Open AccessArticle

Energy Stability Property of the CPR Method Based on Subcell Second-Order CNNW Limiting in Solving Conservation Laws

by Ran Liu, Zhen-Guo Yan, Huajun Zhu, Feiran Jia and Xinlong Feng

Entropy 2023, 25(5), 729; https://doi.org/10.3390/e25050729 - 28 Apr 2023

Viewed by 1419

Abstract

This paper studies the energy stability property of the correction procedure via reconstruction (CPR) method with staggered flux points based on second-order subcell limiting. The CPR method with staggered flux points uses the Gauss point as the solution point, dividing flux points based [...] Read more.

This paper studies the energy stability property of the correction procedure via reconstruction (CPR) method with staggered flux points based on second-order subcell limiting. The CPR method with staggered flux points uses the Gauss point as the solution point, dividing flux points based on Gauss weights, with the flux points being one more point than the solution points. For subcell limiting, a shock indicator is used to detect troubled cells where discontinuities may exist. Troubled cells are calculated by the second-order subcell compact nonuniform nonlinear weighted (CNNW2) scheme, which has the same solution points as the CPR method. The smooth cells are calculated by the CPR method. The linear energy stability of the linear CNNW2 scheme is proven theoretically. Through various numerical experiments, we demonstrate that the CNNW2 scheme and CPR method based on subcell linear CNNW2 limiting are energy-stable and that the CPR method based on subcell nonlinear CNNW2 limiting is nonlinearly stable. Full article

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33 pages, 1699 KiB

Open AccessArticle

On Magnetic Models in Wavefunction Ensembles

by Leonardo De Carlo and William D. Wick

Entropy 2023, 25(4), 564; https://doi.org/10.3390/e25040564 - 25 Mar 2023

Cited by 3 | Viewed by 3636

Abstract

In a wavefunction-only philosophy, thermodynamics must be recast in terms of an ensemble of wavefunctions. In this perspective we study how to construct Gibbs ensembles for magnetic quantum spin models. We show that with free boundary conditions and distinguishable “spins” there are no [...] Read more.

In a wavefunction-only philosophy, thermodynamics must be recast in terms of an ensemble of wavefunctions. In this perspective we study how to construct Gibbs ensembles for magnetic quantum spin models. We show that with free boundary conditions and distinguishable “spins” there are no finite-temperature phase transitions because of high dimensionality of the phase space. Then we focus on the simplest case, namely the mean-field (Curie–Weiss) model, in order to discover whether phase transitions are even possible in this model class. This strategy at least diminishes the dimensionality of the problem. We found that, even assuming exchange symmetry in the wavefunctions, no finite-temperature phase transitions appear when the Hamiltonian is given by the usual energy expression of quantum mechanics (in this case the analytical argument is not totally satisfactory and we relied partly on a computer analysis). However, a variant model with additional “wavefunction energy” does have a phase transition to a magnetized state. (With respect to dynamics, which we do not consider here, wavefunction energy induces a non-linearity which nevertheless preserves norm and energy. This non-linearity becomes significant only at the macroscopic level.) The three results together suggest that magnetization in large wavefunction spin chains appears if and only if we consider indistinguishable particles and block macroscopic dispersion (i.e., macroscopic superpositions) by energy conservation. Our principle technique involves transforming the problem to one in probability theory, then applying results from large deviations, particularly the Gärtner–Ellis Theorem. Finally, we discuss Gibbs vs. Boltzmann/Einstein entropy in the choice of the quantum thermodynamic ensemble, as well as open problems. Full article

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30 pages, 834 KiB

Open AccessArticle

High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States

by Constantino Sánchez Romero and Vicente Garzó

Entropy 2023, 25(2), 222; https://doi.org/10.3390/e25020222 - 24 Jan 2023

Cited by 1 | Viewed by 1787

Abstract

The Boltzmann equation for d-dimensional inelastic Maxwell models is considered to determine the collisional moments of the second, third and fourth degree in a granular binary mixture. These collisional moments are exactly evaluated in terms of the velocity moments of the distribution [...] Read more.

The Boltzmann equation for d-dimensional inelastic Maxwell models is considered to determine the collisional moments of the second, third and fourth degree in a granular binary mixture. These collisional moments are exactly evaluated in terms of the velocity moments of the distribution function of each species when diffusion is absent (mass flux of each species vanishes). The corresponding associated eigenvalues as well as cross coefficients are obtained as functions of the coefficients of normal restitution and the parameters of the mixture (masses, diameters and composition). The results are applied to the analysis of the time evolution of the moments (scaled with a thermal speed) in two different nonequilibrium situations: the homogeneous cooling state (HCS) and the uniform (or simple) shear flow (USF) state. In the case of the HCS, in contrast to what happens for simple granular gases, it is demonstrated that the third and fourth degree moments could diverge in time for given values of the parameters of the system. An exhaustive study on the influence of the parameter space of the mixture on the time behavior of these moments is carried out. Then, the time evolution of the second- and third-degree velocity moments in the USF is studied in the tracer limit (namely, when the concentration of one of the species is negligible). As expected, while the second-degree moments are always convergent, the third-degree moments of the tracer species can be also divergent in the long time limit. Full article

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2022

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21 pages, 1311 KiB

Open AccessArticle

Prior Distribution and Entropy in Computer Adaptive Testing Ability Estimation through MAP or EAP

by Joel Suárez-Cansino, Virgilio López-Morales, Luis Roberto Morales-Manilla, Adrián Alberto-Rodríguez and Julio César Ramos-Fernández

Entropy 2023, 25(1), 50; https://doi.org/10.3390/e25010050 - 27 Dec 2022

Viewed by 2090

Abstract

To derive a latent trait (for instance ability) in a computer adaptive testing (CAT) framework, the obtained results from a model must have a direct relationship to the examinees’ response to a set of items presented. The set of items is previously [...] Read more.

To derive a latent trait (for instance ability) in a computer adaptive testing (CAT) framework, the obtained results from a model must have a direct relationship to the examinees’ response to a set of items presented. The set of items is previously calibrated to decide which item to present to the examinee in the next evaluation question. Some useful models are more naturally based on conditional probability in order to involve previously obtained hits/misses. In this paper, we integrate an experimental part, obtaining the information related to the examinee’s academic performance, with a theoretical contribution of maximum entropy. Some academic performance index functions are built to support the experimental part and then explain under what conditions one can use constrained prior distributions. Additionally, we highlight that heuristic prior distributions might not properly work in all likely cases, and when to use personalized prior distributions instead. Finally, the inclusion of the performance index functions, arising from current experimental studies and historical records, are integrated into a theoretical part based on entropy maximization and its relationship with a CAT process. Full article

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28 pages, 5822 KiB

Open AccessArticle

Volatility Dynamics of Non-Linear Volatile Time Series and Analysis of Information Flow: Evidence from Cryptocurrency Data

by Muhammad Sheraz, Silvia Dedu and Vasile Preda

Entropy 2022, 24(10), 1410; https://doi.org/10.3390/e24101410 - 2 Oct 2022

Cited by 7 | Viewed by 2751

Abstract

This paper aims to empirically examine long memory and bi-directional information flow between estimated volatilities of highly volatile time series datasets of five cryptocurrencies. We propose the employment of Garman and Klass (GK), Parkinson’s, Rogers and Satchell (RS), and Garman and Klass-Yang and [...] Read more.

This paper aims to empirically examine long memory and bi-directional information flow between estimated volatilities of highly volatile time series datasets of five cryptocurrencies. We propose the employment of Garman and Klass (GK), Parkinson’s, Rogers and Satchell (RS), and Garman and Klass-Yang and Zhang (GK-YZ), and Open-High-Low-Close (OHLC) volatility estimators to estimate cryptocurrencies’ volatilities. The study applies methods such as mutual information, transfer entropy (TE), effective transfer entropy (ETE), and Rényi transfer entropy (RTE) to quantify the information flow between estimated volatilities. Additionally, Hurst exponent computations examine the existence of long memory in log returns and OHLC volatilities based on simple R/S, corrected R/S, empirical, corrected empirical, and theoretical methods. Our results confirm the long-run dependence and non-linear behavior of all cryptocurrency’s log returns and volatilities. In our analysis, TE and ETE estimates are statistically significant for all OHLC estimates. We report the highest information flow from BTC to LTC volatility (RS). Similarly, BNB and XRP share the most prominent information flow between volatilities estimated by GK, Parkinson’s, and GK-YZ. The study presents the practicable addition of OHLC volatility estimators for quantifying the information flow and provides an additional choice to compare with other volatility estimators, such as stochastic volatility models. Full article

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24 pages, 5488 KiB

Open AccessArticle

Influence of Transfer Entropy in the Short-Term Prediction of Financial Time Series Using an ∊-Machine

by José Crispín Zavala-Díaz, Joaquín Pérez-Ortega, Nelva Nely Almanza-Ortega, Rodolfo Pazos-Rangel and José María Rodríguez-Lelís

Entropy 2022, 24(8), 1049; https://doi.org/10.3390/e24081049 - 30 Jul 2022

Cited by 1 | Viewed by 2010

Abstract

Predicting the values of a financial time series is mainly a function of its price history, which depends on several factors, internal and external. With this history, it is possible to build an ∊-machine for predicting the financial time series. This work proposes [...] Read more.

Predicting the values of a financial time series is mainly a function of its price history, which depends on several factors, internal and external. With this history, it is possible to build an ∊-machine for predicting the financial time series. This work proposes considering the influence of a financial series through the transfer of entropy when the values of the other financial series are known. A method is proposed that considers the transfer of entropy for breaking the ties that occur when calculating the prediction with the ∊-machine. This analysis is carried out using data from six financial series: two American, the S&P 500 and the Nasdaq; two Asian, the Hang Seng and the Nikkei 225; and two European, the CAC 40 and the DAX. This work shows that it is possible to influence the prediction of the closing value of a series if the value of the influencing series is known. This work showed that the series that transfer the most information through entropy transfer are the American S&P 500 and Nasdaq, followed by the European DAX and CAC 40, and finally the Asian Nikkei 225 and Hang Seng. Full article

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22 pages, 357 KiB

Open AccessArticle

Exponential Families with External Parameters

by Marco Favretti

Entropy 2022, 24(5), 698; https://doi.org/10.3390/e24050698 - 14 May 2022

Cited by 1 | Viewed by 1980

Abstract

In this paper we introduce a class of statistical models consisting of exponential families depending on additional parameters, called external parameters. The main source for these statistical models resides in the Maximum Entropy framework where we have thermal parameters, corresponding to the natural [...] Read more.

In this paper we introduce a class of statistical models consisting of exponential families depending on additional parameters, called external parameters. The main source for these statistical models resides in the Maximum Entropy framework where we have thermal parameters, corresponding to the natural parameters of an exponential family, and mechanical parameters, here called external parameters. In the first part we we study the geometry of these models introducing a fibration of parameter space over external parameters. In the second part we investigate a class of evolution problems driven by a Fokker-Planck equation whose stationary distribution is an exponential family with external parameters. We discuss applications of these statistical models to thermodynamic length and isentropic evolution of thermodynamic systems and to a problem in the dynamic of quantitative traits in genetics. Full article

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13 pages, 1170 KiB

Open AccessArticle

Stability and Evolution of Synonyms and Homonyms in Signaling Game

by Dorota Lipowska and Adam Lipowski

Entropy 2022, 24(2), 194; https://doi.org/10.3390/e24020194 - 27 Jan 2022

Cited by 1 | Viewed by 2841

Abstract

Synonyms and homonyms appear in all natural languages. We analyze their evolution within the framework of the signaling game. Agents in our model use reinforcement learning, where probabilities of selection of a communicated word or of its interpretation depend on weights equal to [...] Read more.

Synonyms and homonyms appear in all natural languages. We analyze their evolution within the framework of the signaling game. Agents in our model use reinforcement learning, where probabilities of selection of a communicated word or of its interpretation depend on weights equal to the number of accumulated successful communications. When the probabilities increase linearly with weights, synonyms appear to be very stable and homonyms decline relatively fast. Such behavior seems to be at odds with linguistic observations. A better agreement is obtained when probabilities increase faster than linearly with weights. Our results may suggest that a certain positive feedback, the so-called Metcalfe’s Law, possibly drives some linguistic processes. Evolution of synonyms and homonyms in our model can be approximately described using a certain nonlinear urn model. Full article

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2021

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19 pages, 2773 KiB

Open AccessEditor’s ChoiceArticle

Gradient Profile Estimation Using Exponential Cubic Spline Smoothing in a Bayesian Framework

by Kushani De Silva, Carlo Cafaro and Adom Giffin

Entropy 2021, 23(6), 674; https://doi.org/10.3390/e23060674 - 27 May 2021

Viewed by 2790

Abstract

Attaining reliable gradient profiles is of utmost relevance for many physical systems. In many situations, the estimation of the gradient is inaccurate due to noise. It is common practice to first estimate the underlying system and then compute the gradient profile by taking [...] Read more.

Attaining reliable gradient profiles is of utmost relevance for many physical systems. In many situations, the estimation of the gradient is inaccurate due to noise. It is common practice to first estimate the underlying system and then compute the gradient profile by taking the subsequent analytic derivative of the estimated system. The underlying system is often estimated by fitting or smoothing the data using other techniques. Taking the subsequent analytic derivative of an estimated function can be ill-posed. This becomes worse as the noise in the system increases. As a result, the uncertainty generated in the gradient estimate increases. In this paper, a theoretical framework for a method to estimate the gradient profile of discrete noisy data is presented. The method was developed within a Bayesian framework. Comprehensive numerical experiments were conducted on synthetic data at different levels of noise. The accuracy of the proposed method was quantified. Our findings suggest that the proposed gradient profile estimation method outperforms the state-of-the-art methods. Full article

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12 pages, 3226 KiB

Open AccessArticle

Evolution towards Linguistic Coherence in Naming Game with Migrating Agents

by Dorota Lipowska and Adam Lipowski

Entropy 2021, 23(3), 299; https://doi.org/10.3390/e23030299 - 28 Feb 2021

Cited by 2 | Viewed by 2510

Abstract

As an integral part of our culture and way of life, language is intricately related to the migrations of people. To understand whether and how migration shapes language formation processes, we examine the dynamics of the naming game with migrating agents. (i) When [...] Read more.

As an integral part of our culture and way of life, language is intricately related to the migrations of people. To understand whether and how migration shapes language formation processes, we examine the dynamics of the naming game with migrating agents. (i) When all agents may migrate, the dynamics generates effective surface tension that drives the coarsening. Such behaviour is very robust and appears for a wide range of densities of agents and their migration rates. (ii) However, when only multilingual agents are allowed to migrate, monolingual islands are typically formed. In such a case, when the migration rate is sufficiently large, the majority of agents acquire a common language that spontaneously emerges with no indication of surface-tension-driven coarsening. Relatively slow coarsening that takes place in a dense static population is very fragile, and an arbitrarily small migration rate can most likely divert the system towards the quick formation of monolingual islands. Our work shows that migration influences language formation processes, but additional details such as density or mobility of agents are needed to more precisely specify this influence. Full article

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2020

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16 pages, 809 KiB

Open AccessReview

Statistical and Proactive Analysis of an Inter-Laboratory Comparison: The Radiocarbon Dating of the Shroud of Turin

by Paolo Di Lazzaro, Anthony C. Atkinson, Paola Iacomussi, Marco Riani, Marco Ricci and Peter Wadhams

Entropy 2020, 22(9), 926; https://doi.org/10.3390/e22090926 - 24 Aug 2020

Cited by 1 | Viewed by 9240

Abstract

We review the sampling and results of the radiocarbon dating of the archaeological cloth known as the Shroud of Turin, in the light of recent statistical analyses of both published and raw data. The statistical analyses highlight an inter-laboratory heterogeneity of the means [...] Read more.

We review the sampling and results of the radiocarbon dating of the archaeological cloth known as the Shroud of Turin, in the light of recent statistical analyses of both published and raw data. The statistical analyses highlight an inter-laboratory heterogeneity of the means and a monotone spatial variation of the ages of subsamples that suggest the presence of contaminants unevenly removed by the cleaning pretreatments. We consider the significance and overall impact of the statistical analyses on assessing the reliability of the dating results and the design of correct sampling. These analyses suggest that the 1988 radiocarbon dating does not match the current accuracy requirements. Should this be the case, it would be interesting to know the accurate age of the Shroud of Turin. Taking into account the whole body of scientific data, we discuss whether it makes sense to date the Shroud again. Full article

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2 pages, 210 KiB

Open AccessCorrection

Correction: Contreras-Reyes, J.E.; Cortés, D.D. Bounds on Rényi and Shannon Entropies for Finite Mixtures of Multivariate Skew-Normal Distributions: Application to Swordfish (Xiphias gladius Linnaeus). Entropy 2016, 18, 382

by Javier E. Contreras-Reyes and Daniel Devia Cortés

Entropy 2020, 22(8), 892; https://doi.org/10.3390/e22080892 - 14 Aug 2020

Viewed by 2576

Abstract

Section 3.3 of “Contreras-Reyes, J.E.; Cortés, D.D. Bounds on Rényi and Shannon Entropies for Finite Mixtures of Multivariate Skew-Normal Distributions: Application to Swordfish (Xiphias gladius Linnaeus). Entropy2016, 18, 382” contains errors. Therefore, this section is retracted. However, these [...] Read more.

Section 3.3 of “Contreras-Reyes, J.E.; Cortés, D.D. Bounds on Rényi and Shannon Entropies for Finite Mixtures of Multivariate Skew-Normal Distributions: Application to Swordfish (Xiphias gladius Linnaeus). Entropy2016, 18, 382” contains errors. Therefore, this section is retracted. However, these changes do not influence the conclusions and the other results of the paper. Full article

2019

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12 pages, 801 KiB

Open AccessArticle

European Option Based on Least-Squares Method under Non-Extensive Statistical Mechanics

by Limin Liu and Yingying Cui

Entropy 2019, 21(10), 933; https://doi.org/10.3390/e21100933 - 25 Sep 2019

Viewed by 2304

Abstract

This paper is devoted to the study of the pricing of European options under a non-Gaussian model. This model follows a non-extensive statistical mechanics which can better describe the fractal characteristics of price movement in the financial market. Moreover, we present a simple [...] Read more.

This paper is devoted to the study of the pricing of European options under a non-Gaussian model. This model follows a non-extensive statistical mechanics which can better describe the fractal characteristics of price movement in the financial market. Moreover, we present a simple but precise least-square method for approximation and obtain a closed-form solution of the price of European options. The advantages of this technique are illustrated by numerical simulation, which shows that the least-squares method is better compared with Borland’s two methods in 2002 and 2004. Full article

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19 pages, 649 KiB

Open AccessArticle

Asymptotic Behavior of Memristive Circuits

by Francesco Caravelli

Entropy 2019, 21(8), 789; https://doi.org/10.3390/e21080789 - 13 Aug 2019

Cited by 14 | Viewed by 6179

Abstract

The interest in memristors has risen due to their possible application both as memory units and as computational devices in combination with CMOS. This is in part due to their nonlinear dynamics, and a strong dependence on the circuit topology. We provide evidence [...] Read more.

The interest in memristors has risen due to their possible application both as memory units and as computational devices in combination with CMOS. This is in part due to their nonlinear dynamics, and a strong dependence on the circuit topology. We provide evidence that also purely memristive circuits can be employed for computational purposes. In the present paper we show that a polynomial Lyapunov function in the memory parameters exists for the case of DC controlled memristors. Such a Lyapunov function can be asymptotically approximated with binary variables, and mapped to quadratic combinatorial optimization problems. This also shows a direct parallel between memristive circuits and the Hopfield-Little model. In the case of Erdos-Renyi random circuits, we show numerically that the distribution of the matrix elements of the projectors can be roughly approximated with a Gaussian distribution, and that it scales with the inverse square root of the number of elements. This provides an approximated but direct connection with the physics of disordered system and, in particular, of mean field spin glasses. Using this and the fact that the interaction is controlled by a projector operator on the loop space of the circuit. We estimate the number of stationary points of the approximate Lyapunov function and provide a scaling formula as an upper bound in terms of the circuit topology only. Full article

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31 pages, 11783 KiB

Open AccessArticle

Effects of Advective-Diffusive Transport of Multiple Chemoattractants on Motility of Engineered Chemosensory Particles in Fluidic Environments

by Danielle King, Hakan Başağaoğlu, Hoa Nguyen, Frank Healy, Melissa Whitman and Sauro Succi

Entropy 2019, 21(5), 465; https://doi.org/10.3390/e21050465 - 4 May 2019

Cited by 2 | Viewed by 3814

Abstract

Motility behavior of an engineered chemosensory particle (ECP) in fluidic environments is driven by its responses to chemical stimuli. One of the challenges to understanding such behaviors lies in tracking changes in chemical signal gradients of chemoattractants and ECP-fluid dynamics as the fluid [...] Read more.

Motility behavior of an engineered chemosensory particle (ECP) in fluidic environments is driven by its responses to chemical stimuli. One of the challenges to understanding such behaviors lies in tracking changes in chemical signal gradients of chemoattractants and ECP-fluid dynamics as the fluid is continuously disturbed by ECP motion. To address this challenge, we introduce a new multiscale numerical model to simulate chemotactic swimming of an ECP in confined fluidic environments by accounting for motility-induced disturbances in spatiotemporal chemoattractant distributions. The model accommodates advective-diffusive transport of unmixed chemoattractants, ECP-fluid hydrodynamics at the ECP-fluid interface, and spatiotemporal disturbances in the chemoattractant concentrations due to particle motion. Demonstrative simulations are presented with an ECP, mimicking Escherichia coli (E. coli) chemotaxis, released into initially quiescent fluids with different source configurations of the chemoattractants N-methyl-L-aspartate and L-serine. Simulations demonstrate that initial distributions and temporal evolution of chemoattractants and their release modes (instantaneous vs. continuous, point source vs. distributed) dictate time histories of chemotactic motility of an ECP. Chemotactic motility is shown to be largely determined by spatiotemporal variation in chemoattractant concentration gradients due to transient disturbances imposed by ECP-fluid hydrodynamics, an observation not captured in previous numerical studies that relied on static chemoattractant concentration fields. Full article

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21 pages, 3427 KiB

Open AccessArticle

An Urban Scaling Estimation Method in a Heterogeneity Variance Perspective

by Wenjia Wu, Hongrui Zhao, Qifan Tan and Peichao Gao

Entropy 2019, 21(4), 337; https://doi.org/10.3390/e21040337 - 28 Mar 2019

Cited by 2 | Viewed by 3474

Abstract

Urban scaling laws describe powerful universalities of the scaling relationships between urban attributes and the city size across different countries and times. There are still challenges in precise statistical estimation of the scaling exponent; the properties of variance require further study. In this [...] Read more.

Urban scaling laws describe powerful universalities of the scaling relationships between urban attributes and the city size across different countries and times. There are still challenges in precise statistical estimation of the scaling exponent; the properties of variance require further study. In this paper, a statistical regression method based on the maximum likelihood estimation considering the lower bound constraints and the heterogeneous variance of error structure, termed as CHVR, is proposed for urban scaling estimation. In the CHVR method, the heterogeneous properties of variance are explored and modeled in the form of a power-of-the-mean variance model. The maximum likelihood fitting method is supplemented to satisfy the lower bound constraints in empirical data. The CHVR method has been applied to estimating the scaling exponents of six urban attributes covering three scaling regimes in China and compared with two traditional methods. Method evaluations based on three different criteria validate that compared with both classical methods, the CHVR method is more effective and robust. Moreover, a statistical test and long-term variations of the parameter in the variance function demonstrate that the proposed heterogeneous variance function can not only describe the heterogeneity in empirical data adequately but also provide more meaningful urban information. Therefore, the CHVR method shows great potential to provide a valuable tool for effective urban scaling studies across the world and be applied to scaling law estimation in other complex systems in the future. Full article

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14 pages, 546 KiB

Open AccessArticle

The Impact of Financial and Macroeconomic Shocks on the Entropy of Financial Markets

by Sorin Anagnoste and Petre Caraiani

Entropy 2019, 21(3), 316; https://doi.org/10.3390/e21030316 - 23 Mar 2019

Cited by 6 | Viewed by 3650

Abstract

We propose here a method to analyze whether financial and macroeconomic shocks influence the entropy of financial networks. We derive a measure of entropy using the correlation matrix of the stock market components of the DOW Jones Industrial Average (DJIA) index. Using VAR [...] Read more.

We propose here a method to analyze whether financial and macroeconomic shocks influence the entropy of financial networks. We derive a measure of entropy using the correlation matrix of the stock market components of the DOW Jones Industrial Average (DJIA) index. Using VAR models in different specifications, we show that shocks in production or the DJIA index lead to an increase in the entropy of the financial markets. Full article

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2017

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209 KiB

Open AccessArticle

On Entropy Dynamics for Active “Living” Particles

by Ahmed Elaiw, Mohammed Alghamdi and Nicola Bellomo

Entropy 2017, 19(10), 525; https://doi.org/10.3390/e19100525 - 2 Oct 2017

Cited by 1 | Viewed by 3805

Abstract

This paper presents a modeling approach, followed by entropy calculations of the dynamics of large systems of interacting active particles viewed as living—hence, complex—systems. Active particles are partitioned into functional subsystems, while their state is modeled by a discrete scalar variable, while the [...] Read more.

This paper presents a modeling approach, followed by entropy calculations of the dynamics of large systems of interacting active particles viewed as living—hence, complex—systems. Active particles are partitioned into functional subsystems, while their state is modeled by a discrete scalar variable, while the state of the overall system is defined by a probability distribution function over the state of the particles. The aim of this paper consists of contributing to a further development of the mathematical kinetic theory of active particles. Full article

1460 KiB

Open AccessArticle

Initial Results of Testing Some Statistical Properties of Hard Disks Workload in Personal Computers in Terms of Non-Extensive Entropy and Long-Range Dependencies

by Dominik Strzałka

Entropy 2017, 19(7), 335; https://doi.org/10.3390/e19070335 - 5 Jul 2017

Viewed by 3987

Abstract

The aim of this paper is to present some preliminary results and non-extensive statistical properties of selected operating system counters related to hard drive behaviour. A number of experiments have been carried out in order to generate the workload and analyse the behaviour [...] Read more.

The aim of this paper is to present some preliminary results and non-extensive statistical properties of selected operating system counters related to hard drive behaviour. A number of experiments have been carried out in order to generate the workload and analyse the behaviour of computers during man–machine interaction. All analysed computers were personal ones, worked under Windows operating systems. The research was conducted to demonstrate how the concept of non-extensive statistical mechanics can be helpful in the description of computer systems behaviour, especially in the context of statistical properties with scaling phenomena, long-term dependencies and statistical self-similarity. The studies have been made on the basis of perfmon tool that allows the user to trace operating systems counters during processing. Full article

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1195 KiB

Open AccessArticle

On the Simplification of Statistical Mechanics for Space Plasmas

by George Livadiotis

Entropy 2017, 19(6), 285; https://doi.org/10.3390/e19060285 - 18 Jun 2017

Cited by 13 | Viewed by 6118

Abstract

Space plasmas are frequently described by kappa distributions. Non-extensive statistical mechanics involves the maximization of the Tsallis entropic form under the constraints of canonical ensemble, considering also a dyadic formalism between the ordinary and escort probability distributions. This paper addresses the statistical origin [...] Read more.

Space plasmas are frequently described by kappa distributions. Non-extensive statistical mechanics involves the maximization of the Tsallis entropic form under the constraints of canonical ensemble, considering also a dyadic formalism between the ordinary and escort probability distributions. This paper addresses the statistical origin of kappa distributions, and shows that they can be connected with non-extensive statistical mechanics without considering the dyadic formalism of ordinary/escort distributions. While this concept does significantly simplify the usage of the theory, it costs the definition of a dyadic entropic formulation, in order to preserve the consistency between statistical mechanics and thermodynamics. Therefore, the simplification of the theory by means of avoiding dyadic formalism is impossible within the framework of non-extensive statistical mechanics. Full article

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243 KiB

Open AccessArticle

Peierls–Bogolyubov’s Inequality for Deformed Exponentials

by Frank Hansen, Jin Liang and Guanghua Shi

Entropy 2017, 19(6), 271; https://doi.org/10.3390/e19060271 - 12 Jun 2017

Cited by 2 | Viewed by 4394

Abstract

We study the convexity or concavity of certain trace functions for the deformed logarithmic and exponential functions, and in this way obtain new trace inequalities for deformed exponentials that may be considered as generalizations of Peierls–Bogolyubov’s inequality. We use these results to improve [...] Read more.

We study the convexity or concavity of certain trace functions for the deformed logarithmic and exponential functions, and in this way obtain new trace inequalities for deformed exponentials that may be considered as generalizations of Peierls–Bogolyubov’s inequality. We use these results to improve previously-known lower bounds for the Tsallis relative entropy. Full article

1567 KiB

Open AccessArticle

Quadratic Mutual Information Feature Selection

by Davor Sluga and Uroš Lotrič

Entropy 2017, 19(4), 157; https://doi.org/10.3390/e19040157 - 1 Apr 2017

Cited by 17 | Viewed by 6933

Abstract

We propose a novel feature selection method based on quadratic mutual information which has its roots in Cauchy–Schwarz divergence and Renyi entropy. The method uses the direct estimation of quadratic mutual information from data samples using Gaussian kernel functions, and can detect second [...] Read more.

We propose a novel feature selection method based on quadratic mutual information which has its roots in Cauchy–Schwarz divergence and Renyi entropy. The method uses the direct estimation of quadratic mutual information from data samples using Gaussian kernel functions, and can detect second order non-linear relations. Its main advantages are: (i) unified analysis of discrete and continuous data, excluding any discretization; and (ii) its parameter-free design. The effectiveness of the proposed method is demonstrated through an extensive comparison with mutual information feature selection (MIFS), minimum redundancy maximum relevance (MRMR), and joint mutual information (JMI) on classification and regression problem domains. The experiments show that proposed method performs comparably to the other methods when applied to classification problems, except it is considerably faster. In the case of regression, it compares favourably to the others, but is slower. Full article

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2016

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536 KiB

Open AccessArticle

Intra-Day Trading System Design Based on the Integrated Model of Wavelet De-Noise and Genetic Programming

by Hongguang Liu, Ping Ji and Jian Jin

Entropy 2016, 18(12), 435; https://doi.org/10.3390/e18120435 - 6 Dec 2016

Cited by 3 | Viewed by 8577

Abstract

Technical analysis has been proved to be capable of exploiting short-term fluctuations in financial markets. Recent results indicate that the market timing approach beats many traditional buy-and-hold approaches in most of the short-term trading periods. Genetic programming (GP) was used to generate short-term [...] Read more.

Technical analysis has been proved to be capable of exploiting short-term fluctuations in financial markets. Recent results indicate that the market timing approach beats many traditional buy-and-hold approaches in most of the short-term trading periods. Genetic programming (GP) was used to generate short-term trade rules on the stock markets during the last few decades. However, few of the related studies on the analysis of financial time series with genetic programming considered the non-stationary and noisy characteristics of the time series. In this paper, to de-noise the original financial time series and to search profitable trading rules, an integrated method is proposed based on the Wavelet Threshold (WT) method and GP. Since relevant information that affects the movement of the time series is assumed to be fully digested during the market closed periods, to avoid the jumping points of the daily or monthly data, in this paper, intra-day high-frequency time series are used to fully exploit the short-term forecasting advantage of technical analysis. To validate the proposed integrated approach, an empirical study is conducted based on the China Securities Index (CSI) 300 futures in the emerging China Financial Futures Exchange (CFFEX) market. The analysis outcomes show that the wavelet de-noise approach outperforms many comparative models. Full article

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2915 KiB

Open AccessArticle

Bounds on Rényi and Shannon Entropies for Finite Mixtures of Multivariate Skew-Normal Distributions: Application to Swordfish (Xiphias gladius Linnaeus)

by Javier E. Contreras-Reyes and Daniel Devia Cortés

Entropy 2016, 18(11), 382; https://doi.org/10.3390/e18110382 - 26 Oct 2016

Cited by 27 | Viewed by 6445 | Correction

Abstract

Mixture models are in high demand for machine-learning analysis due to their computational tractability, and because they serve as a good approximation for continuous densities. Predominantly, entropy applications have been developed in the context of a mixture of normal densities. In this paper, [...] Read more.

Mixture models are in high demand for machine-learning analysis due to their computational tractability, and because they serve as a good approximation for continuous densities. Predominantly, entropy applications have been developed in the context of a mixture of normal densities. In this paper, we consider a novel class of skew-normal mixture models, whose components capture skewness due to their flexibility. We find upper and lower bounds for Shannon and Rényi entropies for this model. Using such a pair of bounds, a confidence interval for the approximate entropy value can be calculated. In addition, an asymptotic expression for Rényi entropy by Stirling’s approximation is given, and upper and lower bounds are reported using multinomial coefficients and some properties and inequalities of

L^{p}

metric spaces. Simulation studies are then applied to a swordfish (Xiphias gladius Linnaeus) length dataset. Full article

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335 KiB

Open AccessArticle

Entropy for the Quantized Field in the Atom-Field Interaction: Initial Thermal Distribution

by Luis Amilca Andrade-Morales, Braulio M. Villegas-Martínez and Hector M. Moya-Cessa

Entropy 2016, 18(10), 346; https://doi.org/10.3390/e18100346 - 23 Sep 2016

Cited by 1 | Viewed by 4917

Abstract

We study the entropy of a quantized field in interaction with a two-level atom (in a pure state) when the field is initially in a mixture of two number states. We then generalise the result for a thermal state; i.e., an (infinite) statistical [...] Read more.

We study the entropy of a quantized field in interaction with a two-level atom (in a pure state) when the field is initially in a mixture of two number states. We then generalise the result for a thermal state; i.e., an (infinite) statistical mixture of number states. We show that for some specific interaction times, the atom passes its purity to the field and therefore the field entropy decreases from its initial value. Full article

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247 KiB

Open AccessLetter

Combinatorial Intricacies of Labeled Fano Planes

by Metod Saniga

Entropy 2016, 18(9), 312; https://doi.org/10.3390/e18090312 - 23 Aug 2016

Viewed by 6409

Abstract

Given a seven-element set

X = {1, 2, 3, 4, 5, 6, 7}

, there are 30 ways to define a Fano plane on it. Let us call a line of such a Fano plane—that [...] Read more.

Given a seven-element set

X = {1, 2, 3, 4, 5, 6, 7}

, there are 30 ways to define a Fano plane on it. Let us call a line of such a Fano plane—that is to say an unordered triple from X—ordinary or defective, according to whether the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of the order s,

0 \leq s \leq 3

, if there are s defective lines passing through it. With such structural refinement in mind, the 30 Fano planes are shown to fall into eight distinct types. Out of the total of 35 lines, nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types. It is shown that no labeled Fano plane can have all points of zero-th order, or feature just one point of order two. A connection with prominent configurations in Steiner triple systems is also pointed out. Full article

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290 KiB

Open AccessArticle

Common Probability Patterns Arise from Simple Invariances

by Steven A. Frank

Entropy 2016, 18(5), 192; https://doi.org/10.3390/e18050192 - 19 May 2016

Cited by 16 | Viewed by 5125

Abstract

Shift and stretch invariance lead to the exponential-Boltzmann probability distribution. Rotational invariance generates the Gaussian distribution. Particular scaling relations transform the canonical exponential and Gaussian patterns into the variety of commonly observed patterns. The scaling relations themselves arise from the fundamental invariances of [...] Read more.

Shift and stretch invariance lead to the exponential-Boltzmann probability distribution. Rotational invariance generates the Gaussian distribution. Particular scaling relations transform the canonical exponential and Gaussian patterns into the variety of commonly observed patterns. The scaling relations themselves arise from the fundamental invariances of shift, stretch and rotation, plus a few additional invariances. Prior work described the three fundamental invariances as a consequence of the equilibrium canonical ensemble of statistical mechanics or the Jaynesian maximization of information entropy. By contrast, I emphasize the primacy and sufficiency of invariance alone to explain the commonly observed patterns. Primary invariance naturally creates the array of commonly observed scaling relations and associated probability patterns, whereas the classical approaches derived from statistical mechanics or information theory require special assumptions to derive commonly observed scales. Full article

2015

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248 KiB

Open AccessArticle

Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems

by Constantino Tsallis

Entropy 2015, 17(5), 2853-2861; https://doi.org/10.3390/e17052853 - 5 May 2015

Cited by 18 | Viewed by 5099

Abstract

It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional (\(S_{BG}\)) is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's {\it Conceptual inadequacy of the Shannon information in quantum measurements}, among many [...] Read more.

It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional (\(S_{BG}\)) is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's {\it Conceptual inadequacy of the Shannon information in quantum measurements}, among many other systems exhibiting various forms of complexity. On the other hand, the Shannon and Khinchin axioms uniquely mandate the BG form \(S_{BG}=-k\sum_i p_i \ln p_i\); the Shore and Johnson axioms follow the same path. Many natural, artificial and social systems have been satisfactorily approached with nonadditive entropies such as the \(S_q=k \frac{1-\sum_i p_i^q}{q-1}\) one (\(q \in {\cal R}; \,S_1=S_{BG}\)), basis of nonextensive statistical mechanics. Consistently, the Shannon 1948 and Khinchine 1953 uniqueness theorems have already been generalized in the literature, by Santos 1997 and Abe 2000 respectively, in order to uniquely mandate \(S_q\). We argue here that the same remains to be done with the Shore and Johnson 1980 axioms. We arrive to this conclusion by analyzing specific classes of strongly correlated complex systems that await such generalization. Full article

762 KiB

Open AccessArticle

Optimum Accelerated Degradation Tests for the Gamma Degradation Process Case under the Constraint of Total Cost

by Heonsang Lim

Entropy 2015, 17(5), 2556-2572; https://doi.org/10.3390/e17052556 - 23 Apr 2015

Cited by 20 | Viewed by 5921

Abstract

An accelerated degradation test (ADT) is regarded as an effective alternative to an accelerated life test in the sense that an ADT can provide more accurate information on product reliability, even when few or no failures may be expected before the end of [...] Read more.

An accelerated degradation test (ADT) is regarded as an effective alternative to an accelerated life test in the sense that an ADT can provide more accurate information on product reliability, even when few or no failures may be expected before the end of a practical test period. In this paper, statistical methods for optimal designing ADT plans are developed assuming that the degradation characteristic follows a gamma process (GP). The GP-based approach has an advantage that it can deal with more frequently encountered situations in which the degradation should always be nonnegative and strictly increasing over time. The optimal ADT plan is developed under the total experimental cost constraint by determining the optimal settings of variables such as the number of measurements, the measurement times, the test stress levels and the number of units allocated to each stress level such that the asymptotic variance of the maximum likelihood estimator of the q-th quantile of the lifetime distribution at the use condition is minimized. In addition, compromise plans are developed to provide means to check the adequacy of the assumed acceleration model. Finally, sensitivity analysis procedures for assessing the effects of the uncertainties in the pre-estimates of unknown parameters are illustrated with an example. Full article

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2014

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217 KiB

Open AccessArticle

Extreme Value Laws for Superstatistics

by Pau Rabassa and Christian Beck

Entropy 2014, 16(10), 5523-5536; https://doi.org/10.3390/e16105523 - 20 Oct 2014

Cited by 7 | Viewed by 5752

Abstract

We study the extreme value distribution of stochastic processes modeled by superstatistics. Classical extreme value theory asserts that (under mild asymptotic independence assumptions) only three possible limit distributions are possible, namely: Gumbel, Fréchet and Weibull distribution. On the other hand, superstatistics contains three [...] Read more.

We study the extreme value distribution of stochastic processes modeled by superstatistics. Classical extreme value theory asserts that (under mild asymptotic independence assumptions) only three possible limit distributions are possible, namely: Gumbel, Fréchet and Weibull distribution. On the other hand, superstatistics contains three important universality classes, namely χ²-superstatistics, inverse χ²-superstatistics, and lognormal superstatistics, all maximizing different effective entropy measures. We investigate how the three classes of extreme value theory are related to the three classes of superstatistics. We show that for any superstatistical process whose local equilibrium distribution does not live on a finite support, the Weibull distribution cannot occur. Under the above mild asymptotic independence assumptions, we also show that χ²-superstatistics generally leads an extreme value statistics described by a Fréchet distribution, whereas inverse χ²-superstatistics, as well as lognormal superstatistics, lead to an extreme value statistics associated with the Gumbel distribution. Full article

233 KiB

Open AccessArticle

Nonlinearities in Elliptic Curve Authentication

by Ramzi Alsaedi, Nicolae Constantinescu and Vicenţiu Rādulescu

Entropy 2014, 16(9), 5144-5158; https://doi.org/10.3390/e16095144 - 25 Sep 2014

Cited by 9 | Viewed by 5224

Abstract

In order to construct the border solutions for nonsupersingular elliptic curve equations, some common used models need to be adapted from linear treated cases for use in particular nonlinear cases. There are some approaches that conclude with these solutions. Optimization in this area [...] Read more.

In order to construct the border solutions for nonsupersingular elliptic curve equations, some common used models need to be adapted from linear treated cases for use in particular nonlinear cases. There are some approaches that conclude with these solutions. Optimization in this area means finding the majority of points on the elliptic curve and minimizing the time to compute the solution in contrast with the necessary time to compute the inverse solution. We can compute the positive solution of PDE (partial differential equation) like oscillations of f(s)/s around the principal eigenvalue λ₁ of -Δ in

H_{0}^{1} (Ω)

.Translating mathematics into cryptographic applications will be relevant in everyday life, where in there are situations in which two parts that communicate need a third part to confirm this process. For example, if two persons want to agree on something they need an impartial person to confirm this agreement, like a notary. This third part does not influence in anyway the communication process. It is just a witness to the agreement. We present a system where the communicating parties do not authenticate one another. Each party authenticates itself to a third part who also sends the keys for the encryption/decryption process. Another advantage of such a system is that if someone (sender) wants to transmit messages to more than one person (receivers), he needs only one authentication, unlike the classic systems where he would need to authenticate himself to each receiver. We propose an authentication method based on zero-knowledge and elliptic curves. Full article

410 KiB

Open AccessArticle

Measures of Causality in Complex Datasets with Application to Financial Data

by Anna Zaremba and Tomaso Aste

Entropy 2014, 16(4), 2309-2349; https://doi.org/10.3390/e16042309 - 24 Apr 2014

Cited by 27 | Viewed by 8624

Abstract

This article investigates the causality structure of financial time series. We concentrate on three main approaches to measuring causality: linear Granger causality, kernel generalisations of Granger causality (based on ridge regression and the Hilbert–Schmidt norm of the cross-covariance operator) and transfer entropy, examining [...] Read more.

This article investigates the causality structure of financial time series. We concentrate on three main approaches to measuring causality: linear Granger causality, kernel generalisations of Granger causality (based on ridge regression and the Hilbert–Schmidt norm of the cross-covariance operator) and transfer entropy, examining each method and comparing their theoretical properties, with special attention given to the ability to capture nonlinear causality. We also present the theoretical benefits of applying non-symmetrical measures rather than symmetrical measures of dependence. We apply the measures to a range of simulated and real data. The simulated data sets were generated with linear and several types of nonlinear dependence, using bivariate, as well as multivariate settings. An application to real-world financial data highlights the practical difficulties, as well as the potential of the methods. We use two real data sets: (1) U.S. inflation and one-month Libor; (2) S&P data and exchange rates for the following currencies: AUDJPY, CADJPY, NZDJPY, AUDCHF, CADCHF, NZDCHF. Overall, we reach the conclusion that no single method can be recognised as the best in all circumstances, and each of the methods has its domain of best applicability. We also highlight areas for improvement and future research. Full article

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378 KiB

Open AccessArticle

From Random Motion of Hamiltonian Systems to Boltzmann’s H Theorem and Second Law of Thermodynamics: a Pathway by Path Probability

by Qiuping A. Wang and Aziz El Kaabouchiu

Entropy 2014, 16(2), 885-894; https://doi.org/10.3390/e16020885 - 13 Feb 2014

Cited by 5 | Viewed by 6986

Abstract

A numerical experiment of ideal stochastic motion of a particle subject to conservative forces and Gaussian noise reveals that the path probability depends exponentially on action. This distribution implies a fundamental principle generalizing the least action principle of the Hamiltonian/Lagrangian mechanics and yields [...] Read more.

A numerical experiment of ideal stochastic motion of a particle subject to conservative forces and Gaussian noise reveals that the path probability depends exponentially on action. This distribution implies a fundamental principle generalizing the least action principle of the Hamiltonian/Lagrangian mechanics and yields an extended formalism of mechanics for random dynamics. Within this theory, Liouville’s theorem of conservation of phase density distribution must be modified to allow time evolution of phase density and consequently the Boltzmann H theorem. We argue that the gap between the regular Newtonian dynamics and the random dynamics was not considered in the criticisms of the H theorem. Full article

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2013

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2213 KiB

Open AccessReview

Statistical Mechanics Ideas and Techniques Applied to Selected Problems in Ecology

by Hugo Fort

Entropy 2013, 15(12), 5237-5276; https://doi.org/10.3390/e15125237 - 27 Nov 2013

Cited by 8 | Viewed by 8132

Abstract

Ecosystem dynamics provides an interesting arena for the application of a plethora concepts and techniques from statistical mechanics. Here I review three examples corresponding each one to an important problem in ecology. First, I start with an analytical derivation of clumpy patterns for [...] Read more.

Ecosystem dynamics provides an interesting arena for the application of a plethora concepts and techniques from statistical mechanics. Here I review three examples corresponding each one to an important problem in ecology. First, I start with an analytical derivation of clumpy patterns for species relative abundances (SRA) empirically observed in several ecological communities involving a high number n of species, a phenomenon which have puzzled ecologists for decades. An interesting point is that this derivation uses results obtained from a statistical mechanics model for ferromagnets. Second, going beyond the mean field approximation, I study the spatial version of a popular ecological model involving just one species representing vegetation. The goal is to address the phenomena of catastrophic shifts—gradual cumulative variations in some control parameter that suddenly lead to an abrupt change in the system—illustrating it by means of the process of desertification of arid lands. The focus is on the aggregation processes and the effects of diffusion that combined lead to the formation of non trivial spatial vegetation patterns. It is shown that different quantities—like the variance, the two-point correlation function and the patchiness—may serve as early warnings for the desertification of arid lands. Remarkably, in the onset of a desertification transition the distribution of vegetation patches exhibits scale invariance typical of many physical systems in the vicinity a phase transition. I comment on similarities of and differences between these catastrophic shifts and paradigmatic thermodynamic phase transitions like the liquid-vapor change of state for a fluid. Third, I analyze the case of many species interacting in space. I choose tropical forests, which are mega-diverse ecosystems that exhibit remarkable dynamics. Therefore these ecosystems represent a research paradigm both for studies of complex systems dynamics as well as to unveil the mechanisms responsible for the assembly of species-rich communities. The more classical equilibrium approaches are compared versus non-equilibrium ones and in particular I discuss a recently introduced cellular automaton model in which species compete both locally in physical space and along a niche axis. Full article

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131 KiB

Open AccessArticle

The κ-Generalizations of Stirling Approximation and Multinominal Coefficients

by Tatsuaki Wada and Hiroki Suyari

Entropy 2013, 15(12), 5144-5153; https://doi.org/10.3390/e15125144 - 26 Nov 2013

Cited by 5 | Viewed by 5620

Abstract

Stirling approximation of the factorials and multinominal coefficients are generalized based on the κ-generalized functions introduced by Kaniadakis. We have related the κ-generalized multinominal coefficients to the κ-entropy by introducing a new κ-product operation, which exists only when κ ≠ 0. Full article

270 KiB

Open AccessArticle

Dynamics of Instantaneous Condensation in the ZRP Conditioned on an Atypical Current

by Rosemary J. Harris, Vladislav Popkov and Gunter M. Schütz

Entropy 2013, 15(11), 5065-5083; https://doi.org/10.3390/e15115065 - 19 Nov 2013

Cited by 29 | Viewed by 7422

Abstract

Using a generalized Doob’s h-transform we consider the zero-range process (ZRP) conditioned to carry an atypical current, with focus on the regime where the Gallavotti-Cohen symmetry loses its validity. For a single site we compute explicitly the boundary injection and absorption rates of [...] Read more.

Using a generalized Doob’s h-transform we consider the zero-range process (ZRP) conditioned to carry an atypical current, with focus on the regime where the Gallavotti-Cohen symmetry loses its validity. For a single site we compute explicitly the boundary injection and absorption rates of an effective process which maps to a biased random walk. Our approach provides a direct probabilistic confirmation of the theory of “instantaneous condensation” which was proposed some while ago to explain the dynamical origin of the the failure of the Gallavotti-Cohen symmetry for high currents in the ZRP. However, it turns out that for stochastic dynamics with infinite state space care needs to be taken in the application of the Doob’s transform—we discuss in detail the sense in which the effective dynamics can be interpreted as “typical” for different regimes of the current phase diagram. Full article

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1442 KiB

Open AccessReview

Statistical Mechanics and Information-Theoretic Perspectives on Complexity in the Earth System

by Georgios Balasis, Reik V. Donner, Stelios M. Potirakis, Jakob Runge, Constantinos Papadimitriou, Ioannis A. Daglis, Konstantinos Eftaxias and Jürgen Kurths

Entropy 2013, 15(11), 4844-4888; https://doi.org/10.3390/e15114844 - 7 Nov 2013

Cited by 76 | Viewed by 12190

Abstract

This review provides a summary of methods originated in (non-equilibrium) statistical mechanics and information theory, which have recently found successful applications to quantitatively studying complexity in various components of the complex system Earth. Specifically, we discuss two classes of methods: (i) entropies of [...] Read more.

This review provides a summary of methods originated in (non-equilibrium) statistical mechanics and information theory, which have recently found successful applications to quantitatively studying complexity in various components of the complex system Earth. Specifically, we discuss two classes of methods: (i) entropies of different kinds (e.g., on the one hand classical Shannon and R´enyi entropies, as well as non-extensive Tsallis entropy based on symbolic dynamics techniques and, on the other hand, approximate entropy, sample entropy and fuzzy entropy); and (ii) measures of statistical interdependence and causality (e.g., mutual information and generalizations thereof, transfer entropy, momentary information transfer). We review a number of applications and case studies utilizing the above-mentioned methodological approaches for studying contemporary problems in some exemplary fields of the Earth sciences, highlighting the potentials of different techniques. Full article

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255 KiB

Open AccessArticle

Group Invariance of Information Geometry on q-Gaussian Distributions Induced by Beta-Divergence

by Atsumi Ohara and Shinto Eguchi

Entropy 2013, 15(11), 4732-4747; https://doi.org/10.3390/e15114732 - 4 Nov 2013

Cited by 7 | Viewed by 5740

Abstract

We demonstrate that the q-exponential family particularly admits natural geometrical structures among deformed exponential families. The property is the invariance of structures with respect to a general linear group, which transitively acts on the space of positive definite matrices. We prove this [...] Read more.

We demonstrate that the q-exponential family particularly admits natural geometrical structures among deformed exponential families. The property is the invariance of structures with respect to a general linear group, which transitively acts on the space of positive definite matrices. We prove this property via the correspondence between information geometry induced by a deformed potential on the space and the one induced by what we call β-divergence defined on the q-exponential family with q = β + 1. The results are fundamental in robust multivariate analysis using the q-Gaussian family. Full article

591 KiB

Open AccessArticle

Law of Multiplicative Error and Its Generalization to the Correlated Observations Represented by the q-Product

by Hiroki Suyari

Entropy 2013, 15(11), 4634-4647; https://doi.org/10.3390/e15114634 - 28 Oct 2013

Cited by 3 | Viewed by 5169

Abstract

The law of multiplicative error is presented for independent observations and correlated observations represented by the q-product, respectively. We obtain the standard log-normal distribution in the former case and the log-q-normal distribution in the latter case. Queirós’ q-log normal distribution is also reconsidered [...] Read more.

The law of multiplicative error is presented for independent observations and correlated observations represented by the q-product, respectively. We obtain the standard log-normal distribution in the former case and the log-q-normal distribution in the latter case. Queirós’ q-log normal distribution is also reconsidered in the framework of the law of error. These results are presented with mathematical conditions to give rise to these distributions. Full article

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213 KiB

Open AccessArticle

The Phase Space Elementary Cell in Classical and Generalized Statistics

by Piero Quarati and Marcello Lissia

Entropy 2013, 15(10), 4319-4333; https://doi.org/10.3390/e15104319 - 15 Oct 2013

Cited by 11 | Viewed by 8711

Abstract

In the past, the phase-space elementary cell of a non-quantized system was set equal to the third power of the Planck constant; in fact, it is not a necessary assumption. We discuss how the phase space volume, the number of states and the [...] Read more.

In the past, the phase-space elementary cell of a non-quantized system was set equal to the third power of the Planck constant; in fact, it is not a necessary assumption. We discuss how the phase space volume, the number of states and the elementary-cell volume of a system of non-interacting N particles, changes when an interaction is switched on and the system becomes or evolves to a system of correlated non-Boltzmann particles and derives the appropriate expressions. Even if we assume that nowadays the volume of the elementary cell is equal to the cube of the Planck constant, h3, at least for quantum systems, we show that there is a correspondence between different values of h in the past, with important and, in principle, measurable cosmological and astrophysical consequences, and systems with an effective smaller (or even larger) phase-space volume described by non-extensive generalized statistics. Full article

278 KiB

Open AccessArticle

Examples of the Application of Nonparametric Information Geometry to Statistical Physics

by Giovanni Pistone

Entropy 2013, 15(10), 4042-4065; https://doi.org/10.3390/e15104042 - 25 Sep 2013

Cited by 22 | Viewed by 6393

Abstract

We review a nonparametric version of Amari’s information geometry in which the set of positive probability densities on a given sample space is endowed with an atlas of charts to form a differentiable manifold modeled on Orlicz Banach spaces. This nonparametric setting is [...] Read more.

We review a nonparametric version of Amari’s information geometry in which the set of positive probability densities on a given sample space is endowed with an atlas of charts to form a differentiable manifold modeled on Orlicz Banach spaces. This nonparametric setting is used to discuss the setting of typical problems in machine learning and statistical physics, such as black-box optimization, Kullback-Leibler divergence, Boltzmann-Gibbs entropy and the Boltzmann equation. Full article

303 KiB

Open AccessReview

Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions

by Giorgio Kaniadakis

Entropy 2013, 15(10), 3983-4010; https://doi.org/10.3390/e15103983 - 25 Sep 2013

Cited by 87 | Viewed by 9862

Abstract

We present the main features of the mathematical theory generated by the κ-deformed exponential function

\exp_{k} (x) = {(\sqrt{1 + k^{2} x^{2}} + k x)}^{\frac{1}{k}}

, with 0 ≤ κ < 1, developed [...] Read more.

We present the main features of the mathematical theory generated by the κ-deformed exponential function

\exp_{k} (x) = {(\sqrt{1 + k^{2} x^{2}} + k x)}^{\frac{1}{k}}

, with 0 ≤ κ < 1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The κ-mathematics has its roots in special relativity and furnishes the theoretical foundations of the κ-statistical mechanics predicting power law tailed statistical distributions, which have been observed experimentally in many physical, natural and artificial systems. After introducing the κ-algebra, we present the associated κ-differential and κ-integral calculus. Then, we obtain the corresponding κ-exponential and κ-logarithm functions and give the κ-version of the main functions of the ordinary mathematics. Full article

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163 KiB

Open AccessArticle

Solutions of Some Nonlinear Diffusion Equations and Generalized Entropy Framework

by Ervin K. Lenzi, Maike A. F. Dos Santos, Flavio S. Michels, Renio S. Mendes and Luiz R. Evangelista

Entropy 2013, 15(9), 3931-3940; https://doi.org/10.3390/e15093931 - 18 Sep 2013

Cited by 5 | Viewed by 6094

Abstract

We investigate solutions of a generalized diffusion equation that contains nonlinear terms in the presence of external forces and reaction terms. The solutions found here can have a compact or long tail behavior and can be expressed in terms of the q-exponential [...] Read more.

We investigate solutions of a generalized diffusion equation that contains nonlinear terms in the presence of external forces and reaction terms. The solutions found here can have a compact or long tail behavior and can be expressed in terms of the q-exponential functions present in the Tsallis framework. In the case of the long-tailed behavior, in the asymptotic limit, these solutions can also be connected with the L´evy distributions. In addition, from the results presented here, a rich class of diffusive processes, including normal and anomalous ones, can be obtained. Full article

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134 KiB

Open AccessArticle

Deformed Exponentials and Applications to Finance

by Barbara Trivellato

Entropy 2013, 15(9), 3471-3489; https://doi.org/10.3390/e15093471 - 2 Sep 2013

Cited by 48 | Viewed by 6706

Abstract

We illustrate some financial applications of the Tsallis and Kaniadakis deformed exponential. The minimization of the corresponding deformed divergence is discussed as a criterion to select a pricing measure in the valuation problems of incomplete markets. Moreover, heavy-tailed models for price processes are [...] Read more.

We illustrate some financial applications of the Tsallis and Kaniadakis deformed exponential. The minimization of the corresponding deformed divergence is discussed as a criterion to select a pricing measure in the valuation problems of incomplete markets. Moreover, heavy-tailed models for price processes are proposed, which generalized the well-known Black and Scholes model. Full article

4192 KiB

Open AccessArticle

Kinetic Theory Microstructure Modeling in Concentrated Suspensions

by Emmanuelle Abisset-Chavanne, Rabih Mezher, Steven Le Corre, Amine Ammar and Francisco Chinesta

Entropy 2013, 15(7), 2805-2832; https://doi.org/10.3390/e15072805 - 19 Jul 2013

Cited by 20 | Viewed by 6987

Abstract

When suspensions involving rigid rods become too concentrated, standard dilute theories fail to describe their behavior. Rich microstructures involving complex clusters are observed, and no model allows describing its kinematics and rheological effects. In previous works the authors propose a first attempt to [...] Read more.

When suspensions involving rigid rods become too concentrated, standard dilute theories fail to describe their behavior. Rich microstructures involving complex clusters are observed, and no model allows describing its kinematics and rheological effects. In previous works the authors propose a first attempt to describe such clusters from a micromechanical model, but neither its validity nor the rheological effects were addressed. Later, authors applied this model for fitting the rheological measurements in concentrated suspensions of carbon nanotubes (CNTs) by assuming a rheo-thinning behavior at the constitutive law level. However, three major issues were never addressed until now: (i) the validation of the micromechanical model by direct numerical simulation; (ii) the establishment of a general enough multi-scale kinetic theory description, taking into account interaction, diffusion and elastic effects; and (iii) proposing a numerical technique able to solve the kinetic theory description. This paper focuses on these three major issues, proving the validity of the micromechanical model, establishing a multi-scale kinetic theory description and, then, solving it by using an advanced and efficient separated representation of the cluster distribution function. These three aspects, never until now addressed in the past, constitute the main originality and the major contribution of the present paper. Full article

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211 KiB

Open AccessArticle

Statistical Properties of the Foreign Exchange Network at Different Time Scales: Evidence from Detrended Cross-Correlation Coefficient and Minimum Spanning Tree

by Gang-Jin Wang, Chi Xie, Yi-Jun Chen and Shou Chen

Entropy 2013, 15(5), 1643-1662; https://doi.org/10.3390/e15051643 - 6 May 2013

Cited by 79 | Viewed by 12227

Abstract

We investigate the statistical properties of the foreign exchange (FX) network at different time scales by two approaches, namely the methods of detrended cross-correlation coefficient (DCCA coefficient) and minimum spanning tree (MST). The daily FX rates of 44 major currencies in the period [...] Read more.

We investigate the statistical properties of the foreign exchange (FX) network at different time scales by two approaches, namely the methods of detrended cross-correlation coefficient (DCCA coefficient) and minimum spanning tree (MST). The daily FX rates of 44 major currencies in the period of 2007–2012 are chosen as the empirical data. Based on the analysis of statistical properties of cross-correlation coefficients, we find that the cross-correlation coefficients of the FX market are fat-tailed. By examining three MSTs at three special time scales (i.e., the minimum, medium, and maximum scales), we come to some conclusions: USD and EUR are confirmed as the predominant world currencies; the Middle East cluster is very stable while the Asian cluster and the Latin America cluster are not stable in the MSTs; the Commonwealth cluster is also found in the MSTs. By studying four evaluation criteria, we find that the MSTs of the FX market present diverse topological and statistical properties at different time scales. The scale-free behavior is observed in the FX network at most of time scales. We also find that most of links in the FX network survive from one time scale to the next. Full article

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