Entropy
http://mdpi.com/journal/entropy
Latest open access articles published in Entropy at http://mdpi.com/journal/entropy<![CDATA[Entropy, Vol. 17, Pages 1063-1089: Fully Bayesian Experimental Design for Pharmacokinetic Studies]]>
http://mdpi.com/1099-4300/17/3/1063
Utility functions in Bayesian experimental design are usually based on the posterior distribution. When the posterior is found by simulation, it must be sampled from for each future dataset drawn from the prior predictive distribution. Many thousands of posterior distributions are often required. A popular technique in the Bayesian experimental design literature, which rapidly obtains samples from the posterior, is importance sampling, using the prior as the importance distribution. However, importance sampling from the prior will tend to break down if there is a reasonable number of experimental observations. In this paper, we explore the use of Laplace approximations in the design setting to overcome this drawback. Furthermore, we consider using the Laplace approximation to form the importance distribution to obtain a more efficient importance distribution than the prior. The methodology is motivated by a pharmacokinetic study, which investigates the effect of extracorporeal membrane oxygenation on the pharmacokinetics of antibiotics in sheep. The design problem is to find 10 near optimal plasma sampling times that produce precise estimates of pharmacokinetic model parameters/measures of interest. We consider several different utility functions of interest in these studies, which involve the posterior distribution of parameter functions.Entropy2015-03-05173Article10.3390/e17031063106310891099-43002015-03-05doi: 10.3390/e17031063Elizabeth RyanChristopher DrovandiAnthony Pettitt<![CDATA[Entropy, Vol. 17, Pages 1054-1062: The Hosoya Entropy of a Graph]]>
http://mdpi.com/1099-4300/17/3/1054
This paper demonstrates properties of Hosoya entropy, a quantitative measure of graph complexity based on a decomposition of the vertices linked to partial Hosoya polynomials. Connections between the information content of a graph and Hosoya entropy are established, and the special case of Hosoya entropy of trees is investigated.Entropy2015-03-05173Article10.3390/e17031054105410621099-43002015-03-05doi: 10.3390/e17031054Abbe MowshowitzMatthias Dehmer<![CDATA[Entropy, Vol. 17, Pages 1042-1053: Tone Entropy Analysis of Foetal Heart Rate Variability]]>
http://mdpi.com/1099-4300/17/3/1042
Development of the foetal autonomic nervous system can be indirectly understood by looking at the changes in beat to beat variability in foetal heart rates. This study presents Tone-Entropy (T-E) analysis of foetal heart rate variability (HRV) at multiple lags (1–8) to understand the influence of gestational ages (early and late) on the development of the foetal autonomic nervous system (ANS). The analysis was based on foetal electrocardiograms (FECGs) of 46 healthy foetuses of 20–32 weeks (early group) and 22 foetuses of 35–41 weeks (late group). Tone represents sympatho-vagal balance and entropy the total autonomic activities. Results show that tone increases and entropy decreases at all lags for the late foetus group. On the other hand, tone decreases and entropy increases at lags 1–4 in the early foetus group. Increasing tone in late foetuses might represent significant maturation of sympathetic nervous systems because foetuses approaching to delivery period need increased sympathetic activity. T-E could be quantitative clinical index to determine the early foetuses from late ones on the basis of maturation of autonomic nervous system.Entropy2015-03-02173Article10.3390/e17031042104210531099-43002015-03-02doi: 10.3390/e17031042Ahsan KhandokerChandan KarmakarYoshitaka KimuraMiyuki EndoSayaka OshioMarimuthu Palaniswami<![CDATA[Entropy, Vol. 17, Pages 1023-1041: Mining Informative Hydrologic Data by Using Support Vector Machines and Elucidating Mined Data according to Information Entropy]]>
http://mdpi.com/1099-4300/17/3/1023
The support vector machine is used as a data mining technique to extract informative hydrologic data on the basis of a strong relationship between error tolerance and the number of support vectors. Hydrologic data of flash flood events in the Lan-Yang River basin in Taiwan were used for the case study. Various percentages (from 50% to 10%) of hydrologic data, including those for flood stage and rainfall data, were mined and used as informative data to characterize a flood hydrograph. Information on these mined hydrologic data sets was quantified using entropy indices, namely marginal entropy, joint entropy, transinformation, and conditional entropy. Analytical results obtained using the entropy indices proved that the mined informative data could be hydrologically interpreted and have a meaningful explanation based on information entropy. Estimates of marginal and joint entropies showed that, in view of flood forecasting, the flood stage was a more informative variable than rainfall. In addition, hydrologic models with variables containing more total information were preferable to variables containing less total information. Analysis results of transinformation explained that approximately 30% of information on the flood stage could be derived from the upstream flood stage and 10% to 20% from the rainfall. Elucidating the mined hydrologic data by applying information theory enabled using the entropy indices to interpret various hydrologic processes.Entropy2015-03-02173Article10.3390/e17031023102310411099-43002015-03-02doi: 10.3390/e17031023Shien-Tsung Chen<![CDATA[Entropy, Vol. 17, Pages 1007-1022: Entropy Measures in the Assessment of Heart Rate Variability in Patients with Cardiodepressive Vasovagal Syncope]]>
http://mdpi.com/1099-4300/17/3/1007
Sample entropy (SampEn) was reported to be useful in the assessment of the complexity of heart rate dynamics. Permutation entropy (PermEn) is a new measure based on the concept of order and was previously shown to be accurate for short, non-stationary datasets. The aim of the present study is to assess if SampEn and PermEn obtained from baseline recordings might differentiate patients with various outcomes of the head-up tilt test (HUTT). Time-domain heart rate variability (HRV) indices and several nonlinear parameters were calculated using 500 RR interval-long ECG recordings done before tilting in patients with a history suggesting vasovagal syncope. Groups of patients with so-called cardiodepressive vasovagal syncope (VVS_2) during HUTT and patients who did not faint during the test were compared. Two types of HUT tests were analyzed: with spontaneous (SB) or controlled breathing (CB). In our study, SampEn was higher in VVS_2 patients during SB, and PermEn was higher in VVS_2 patients during CB. Irrespective of the type of breathing during the test, SampEn and PermEn were similar in patients with the same type of reaction during HUTT. The use of several entropy-based parameters seems to be useful in HRV assessment in patients with vasovagal fainting.Entropy2015-03-02173Article10.3390/e17031007100710221099-43002015-03-02doi: 10.3390/e17031007Beata GraffGrzegorz GraffDanuta MakowiecAgnieszka KaczkowskaDorota WejerSzymon BudrejkoDariusz KozłowskiKrzysztof Narkiewicz<![CDATA[Entropy, Vol. 17, Pages 984-1006: Phytotoponyms, Geographical Features and Vegetation Coverage in Western Hubei, China]]>
http://mdpi.com/1099-4300/17/3/984
The purpose of this paper is to present and exploit fundamental information, such as semantic meanings and geographical features, of phytotoponyms (a type of toponym that includes plant names) in Western Hubei (China). Long-term vegetation degradation is also estimated. Toponym data for this study were obtained from the place names database of Hubei Province at the Civil Affairs Department of Hubei. In total, 1259 instances of phytotoponyms were recognised; 898 (71.3%) were woody plant toponyms, and 361 (28.7%) were herbaceous plant toponyms. Subsequently, we randomly selected a similar number (1250) of non-phytotoponyms to compare with the phytotoponyms. All toponyms were localised and geo-referenced. The results showed that the most common plant names recognisable in place names are common plants that have a close connection with daily life and positive morals in Chinese culture and literature. The occurrence of plant names can reflect the characteristic plants of a city. The vegetation coverage rate where phytotoponyms are located is higher than that in non-phytotoponym areas. Altitude has a stronger correlation with the number of phytotoponyms than slope and vegetation coverage degree. The identification of long-term vegetation degradation based on phytotoponyms is presented for reference only, and other methods and materials are needed to validate these results.Entropy2015-03-02173Article10.3390/e1703098498410061099-43002015-03-02doi: 10.3390/e17030984Guanghui ShiFu RenQingyun DuNan Gao<![CDATA[Entropy, Vol. 17, Pages 968-983: Do Transitive Preferences Always Result in Indifferent Divisions?]]>
http://mdpi.com/1099-4300/17/3/968
The transitivity of preferences is one of the basic assumptions used in the theory of games and decisions. It is often equated with the rationality of choice and is considered useful in building rankings. Intransitive preferences are considered paradoxical and undesirable. This problem is discussed by many social and natural scientists. A simple model of a sequential game in which two players choose one of the two elements in each iteration is discussed in this paper. The players make their decisions in different contexts defined by the rules of the game. It appears that the optimal strategy of one of the players can only be intransitive (the so-called relevant intransitive strategy)! On the other hand, the optimal strategy for the second player can be either transitive or intransitive. A quantum model of the game using pure one-qubit strategies is considered. In this model, an increase in the importance of intransitive strategies is observed: there is a certain course of the game where intransitive strategies are the only optimal strategies for both players. The study of decision-making models using quantum information theory tools may shed some new light on the understanding of mechanisms that drive the formation of types of preferences.Entropy2015-03-02173Article10.3390/e170309689689831099-43002015-03-02doi: 10.3390/e17030968Marcin MakowskiEdward PiotrowskiJan Sładkowski<![CDATA[Entropy, Vol. 17, Pages 950-967: Entropy Rate Maps of Complex Excitable Dynamics in Cardiac Monolayers]]>
http://mdpi.com/1099-4300/17/3/950
The characterization of spatiotemporal complexity remains a challenging task. This holds in particular for the analysis of data from fluorescence imaging (optical mapping), which allows for the measurement of membrane potential and intracellular calcium at high spatial and temporal resolutions and, therefore, allows for an investigation of cardiac dynamics. Dominant frequency maps and the analysis of phase singularities are frequently used for this type of excitable media. These methods address some important aspects of cardiac dynamics; however, they only consider very specific properties of excitable media. To extend the scope of the analysis, we present a measure based on entropy rates for determining spatiotemporal complexity patterns of excitable media. Simulated data generated by the Aliev–Panfilov model and the cubic Barkley model are used to validate this method. Then, we apply it to optical mapping data from monolayers of cardiac cells from chicken embryos and compare our findings with dominant frequency maps and the analysis of phase singularities. The studies indicate that entropy rate maps provide additional information about local complexity, the origins of wave breakup and the development of patterns governing unstable wave propagation.Entropy2015-02-26173Article10.3390/e170309509509671099-43002015-02-26doi: 10.3390/e17030950Alexander SchlemmerSebastian BergT. ShajahanStefan LutherUlrich Parlitz<![CDATA[Entropy, Vol. 17, Pages 928-949: Instantaneous 3D EEG Signal Analysis Based on Empirical Mode Decomposition and the Hilbert–Huang Transform Applied to Depth of Anaesthesia]]>
http://mdpi.com/1099-4300/17/3/928
Depth of anaesthesia (DoA) is an important measure for assessing the degree to which the central nervous system of a patient is depressed by a general anaesthetic agent, depending on the potency and concentration with which anaesthesia is administered during surgery. We can monitor the DoA by observing the patient’s electroencephalography (EEG) signals during the surgical procedure. Typically high frequency EEG signals indicates the patient is conscious, while low frequency signals mean the patient is in a general anaesthetic state. If the anaesthetist is able to observe the instantaneous frequency changes of the patient’s EEG signals during surgery this can help to better regulate and monitor DoA, reducing surgical and post-operative risks. This paper describes an approach towards the development of a 3D real-time visualization application which can show the instantaneous frequency and instantaneous amplitude of EEG simultaneously by using empirical mode decomposition (EMD) and the Hilbert–Huang transform (HHT). HHT uses the EMD method to decompose a signal into so-called intrinsic mode functions (IMFs). The Hilbert spectral analysis method is then used to obtain instantaneous frequency data. The HHT provides a new method of analyzing non-stationary and nonlinear time series data. We investigate this approach by analyzing EEG data collected from patients undergoing surgical procedures. The results show that the EEG differences between three distinct surgical stages computed by using sample entropy (SampEn) are consistent with the expected differences between these stages based on the bispectral index (BIS), which has been shown to be quantifiable measure of the effect of anaesthetics on the central nervous system. Also, the proposed filtering approach is more effective compared to the standard filtering method in filtering out signal noise resulting in more consistent results than those provided by the BIS. The proposed approach is therefore able to distinguish between key operational stages related to DoA, which is consistent with the clinical observations. SampEn can also be viewed as a useful index for evaluating and monitoring the DoA of a patient when used in combination with this approach.Entropy2015-02-20173Article10.3390/e170309289289491099-43002015-02-20doi: 10.3390/e17030928Mu-Tzu ShihFaiyaz DoctorShou-Zen FanKuo-Kuang JenJiann-Shing Shieh<![CDATA[Entropy, Vol. 17, Pages 914-927: Application of the Permutation Entropy over the Heart Rate Variability for the Improvement of Electrocardiogram-based Sleep Breathing Pause Detection]]>
http://mdpi.com/1099-4300/17/3/914
In this paper the permutation entropy (PE) obtained from heart rate variability (HRV) is analyzed in a statistical model. In this model we also integrate other feature extraction techniques, the cepstrum coefficients derived from the same HRV and a set of band powers obtained from the electrocardiogram derived respiratory (EDR) signal. The aim of the model is detecting obstructive sleep apnea (OSA) events. For this purpose, we apply two statistical classification methods: Logistic Regression (LR) and Quadratic Discriminant Analysis (QDA). For testing the models we use seventy ECG recordings from the Physionet database which are divided into equal-size learning and testing sets. Both sets consist of 35 recordings, each containing a single ECG signal. In our experiments we have found that the features extracted from the EDR signal present a sensitivity of 65.6% and specificity of 87.7% (auc = 85) in the LR classifier, and sensitivity of 59.4% and specificity of 90.3% (auc = 83.9) in the QDA classifier. The HRV-based cepstrum coefficients present a sensitivity of 63.8% and specificity of 89.2% (auc = 86) in the LR classifier, and sensitivity of 67.2% and specificity of 86.8% (auc = 86.9) in the QDA. Subsequent tests show that the contribution of the permutation entropy increases the performance of the classifiers, implying that the complexity of RR interval time series play an important role in the breathing pauses detection. Particularly, when all features are jointly used, the quantification task reaches a sensitivity of 71.9% and specificity of 92.1% (auc = 90.3) for LR. Similarly, for QDA the sensitivity is 75.1% and the specificity is 90.5% (auc = 91.7).Entropy2015-02-20173Article10.3390/e170309149149271099-43002015-02-20doi: 10.3390/e17030914Antonio Ravelo-GarcíaJuan Navarro-MesaUbay Casanova-BlancasSofia Martin-GonzalezPedro Quintana-MoralesIván Guerra-MorenoJosé Canino-RodríguezEduardo Hernández-Pérez<![CDATA[Entropy, Vol. 17, Pages 903-913: Thermophysical Characteristics of the Ferrofluid in a Vertical Rectangle]]>
http://mdpi.com/1099-4300/17/2/903
The article aimed to analytically investigate the thermophysical behaviors of a ferrofluid in a vertical rectangle with the variation of intensity of the magnetic field, viscosity of the ferrofluid and boundary conditions. The governing equations of the ferrofluid include the continuity, momentum and energy equations for describing the thermal-fluidic behaviors of the ferrofluid and the Maxwell equation and magnetization equation are also added to consider rotating effect of the nano-sized particles. The flow behavior and heat transfer characteristics of the ferrofluid with the intensity of the magnetic field, viscosities of the ferrofluid and boundary conditions were analyzed through isotherms, velocity profiles and both mean and local Nusselt numbers. As a result, the isotherms of the ferrofluid in the vertical rectangle increased with the increase of the magnetic volume fractions and magnetic field intensities. In addition, the mean Nusselt numbers increased with the increase of magnetite volume fractions at all magnetic field intensities because of the combined effects of both heat conduction by magnetite and the magnetic volume force.Entropy2015-02-16172Communication10.3390/e170209039039131099-43002015-02-16doi: 10.3390/e17020903Jae-Hyeong SeoByoung-Hee YouSang-Seuk KwenDong-Yeon LeeMoo-Yeon Lee<![CDATA[Entropy, Vol. 17, Pages 885-902: On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem]]>
http://mdpi.com/1099-4300/17/2/885
In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE) with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville) or a solution with increasing length of their support (Hukuhara difference). Then, in order to solve the FFDE analytically, we introduce the fuzzy Laplace transform of the Caputo H-derivative. To the best of our knowledge, there is limited research devoted to the analytical methods to solve the FFDE under the fuzzy Caputo fractional differentiability. An analytical solution is presented to confirm the capability of the proposed method.Entropy2015-02-16172Article10.3390/e170208858859021099-43002015-02-16doi: 10.3390/e17020885Soheil SalahshourAli AhmadianNorazak SenuDumitru BaleanuPraveen Agarwal<![CDATA[Entropy, Vol. 17, Pages 882-884: Entropy Best Paper Award 2015]]>
http://mdpi.com/1099-4300/17/2/882
We are pleased to announce the “Entropy Best Paper Award” for 2015. Nominations were selected by the Editor-in-Chief and designated Editorial Board Members from all the papers published in 2011. Reviews and research papers were evaluated separately. We gladly announce that the following three papers have won the Entropy Best Paper Award in 2015:[...]Entropy2015-02-16172Editorial10.3390/e170208828828841099-43002015-02-16doi: 10.3390/e17020882Kevin Knuth<![CDATA[Entropy, Vol. 17, Pages 866-881: Optimal Design of Magnetohydrodynamic Mixed Convection Flow in a Vertical Channel with Slip Boundary Conditions and Thermal Radiation Effects by Using an Entropy Generation Minimization Method]]>
http://mdpi.com/1099-4300/17/2/866
Investigation of the effect of thermal radiation on a fully developed magnetohydrodynamic (MHD) convective flow of a Newtonian, incompressible and electrically conducting fluid in a vertical microchannel bounded by two infinite vertical parallel plates with constant temperature walls through a lateral magnetic field of uniform strength is presented. The Rosseland model for the conduction radiation heat transfer in an absorbing medium and two plates with slip-flow and no-slip conditions are assumed. In addition, the induced magnetic field is neglected due to the assumption of a small magnetic Reynolds number. The non-dimensional governing equations are solved numerically using Runge–Kutta–Fehlberg method with a shooting technique. The channel is optimized based on the Second Law of Thermodynamics by changing various parameters such as the thermal radiation parameter, the temperature parameter, Hartmann number, Grashof to Reynolds ratio, velocity slip length, and temperature jump.Entropy2015-02-13172Article10.3390/e170208668668811099-43002015-02-13doi: 10.3390/e17020866Mohamad Abdollahzadeh JamalabadiJae ParkChang Lee<![CDATA[Entropy, Vol. 17, Pages 852-865: Relational Probabilistic Conditionals and Their Instantiations under Maximum Entropy Semantics for First-Order Knowledge Bases]]>
http://mdpi.com/1099-4300/17/2/852
For conditional probabilistic knowledge bases with conditionals based on propositional logic, the principle of maximum entropy (ME) is well-established, determining a unique model inductively completing the explicitly given knowledge. On the other hand, there is no general agreement on how to extend the ME principle to relational conditionals containing free variables. In this paper, we focus on two approaches to ME semantics that have been developed for first-order knowledge bases: aggregating semantics and a grounding semantics. Since they use different variants of conditionals, we define the logic PCI, which covers both approaches as special cases and provides a framework where the effects of both approaches can be studied in detail. While the ME models under PCI-grounding and PCI-aggregating semantics are different in general, we point out that parametric uniformity of a knowledge base ensures that both semantics coincide. Using some concrete knowledge bases, we illustrate the differences and common features of both approaches, looking in particular at the ground instances of the given conditionals.Entropy2015-02-13172Article10.3390/e170208528528651099-43002015-02-13doi: 10.3390/e17020852Christoph BeierleMarc FinthammerGabriele Kern-Isberner<![CDATA[Entropy, Vol. 17, Pages 841-851: Probabilistic Three-Party Sharing of Operation on a Remote Qubit]]>
http://mdpi.com/1099-4300/17/2/841
A probabilistic tripartite single-qubit operation sharing scheme is put forward by utilizing a two-qubit and a three-qubit non-maximally entangled state as quantum channels. Some specific comparisons between our scheme and another probabilistic scheme are made. It is found that, if the product of the two minimal coefficients characterizing channel entanglements is greater than 3/16, our scheme is more superior than the other one. Nonetheless, the price is that more classical and quantum resources are consumed, and the operation difficulty is rather increased. Moreover, some important features of the scheme, such as its security, probability and sharer symmetry, are revealed through concrete discussions. Additionally, the experimental feasibility of our scheme is analyzed and subsequently confirmed according to the current experimental techniques.Entropy2015-02-12172Article10.3390/e170208418418511099-43002015-02-12doi: 10.3390/e17020841Chuanmei XieYimin LiuHang XingZhanjun Zhang<![CDATA[Entropy, Vol. 17, Pages 818-840: Distributed Extreme Learning Machine for Nonlinear Learning over Network]]>
http://mdpi.com/1099-4300/17/2/818
Distributed data collection and analysis over a network are ubiquitous, especially over a wireless sensor network (WSN). To our knowledge, the data model used in most of the distributed algorithms is linear. However, in real applications, the linearity of systems is not always guaranteed. In nonlinear cases, the single hidden layer feedforward neural network (SLFN) with radial basis function (RBF) hidden neurons has the ability to approximate any continuous functions and, thus, may be used as the nonlinear learning system. However, confined by the communication cost, using the distributed version of the conventional algorithms to train the neural network directly is usually prohibited. Fortunately, based on the theorems provided in the extreme learning machine (ELM) literature, we only need to compute the output weights of the SLFN. Computing the output weights itself is a linear learning problem, although the input-output mapping of the overall SLFN is still nonlinear. Using the distributed algorithmto cooperatively compute the output weights of the SLFN, we obtain a distributed extreme learning machine (dELM) for nonlinear learning in this paper. This dELM is applied to the regression problem and classification problem to demonstrate its effectiveness and advantages.Entropy2015-02-12172Article10.3390/e170208188188401099-43002015-02-12doi: 10.3390/e17020818Songyan HuangChunguang Li<![CDATA[Entropy, Vol. 17, Pages 790-817: Factorization and Criticality in the Anisotropic XY Chain via Correlations]]>
http://mdpi.com/1099-4300/17/2/790
In this review, we discuss the zero and finite temperature behavior of various bipartite quantum and total correlation measures, the skew information-based quantum coherence and the local quantum uncertainty in the thermal ground state of the one-dimensional anisotropic XY model in a transverse magnetic field. We compare the ability of the considered measures to correctly detect or estimate the quantum critical point and the non-trivial factorization point possessed by the spin chain.Entropy2015-02-09172Review10.3390/e170207907908171099-43002015-02-09doi: 10.3390/e17020790Barış ÇakmakGöktuğ KarpatFelipe Fanchini<![CDATA[Entropy, Vol. 17, Pages 772-789: Quantropy]]>
http://mdpi.com/1099-4300/17/2/772
There is a well-known analogy between statistical and quantum mechanics. In statistical mechanics, Boltzmann realized that the probability for a system in thermal equilibrium to occupy a given state is proportional to \(\exp(-E/kT)\), where \(E\) is the energy of that state. In quantum mechanics, Feynman realized that the amplitude for a system to undergo a given history is proportional to \(\exp(-S/i\hbar)\), where \(S\) is the action of that history. In statistical mechanics, we can recover Boltzmann's formula by maximizing entropy subject to a constraint on the expected energy. This raises the question: what is the quantum mechanical analogue of entropy? We give a formula for this quantity, which we call ``quantropy''. We recover Feynman's formula from assuming that histories have complex amplitudes, that these amplitudes sum to one and that the amplitudes give a stationary point of quantropy subject to a constraint on the expected action. Alternatively, we can assume the amplitudes sum to one and that they give a stationary point of a quantity that we call ``free action'', which is analogous to free energy in statistical mechanics. We compute the quantropy, expected action and free action for a free particle and draw some conclusions from the results.Entropy2015-02-09172Article10.3390/e170207727727891099-43002015-02-09doi: 10.3390/e17020772John BaezBlake Pollard<![CDATA[Entropy, Vol. 17, Pages 763-771: Fokker-Planck Equation and Thermodynamic System Analysis]]>
http://mdpi.com/1099-4300/17/2/763
The non-linear Fokker-Planck equation or Kolmogorov forward equation is currently successfully applied for deep analysis of irreversibility and it gives an excellent approximation near the free energy minimum, just as Boltzmann’s definition of entropy follows from finding the maximum entropy state. A connection to Fokker-Planck dynamics and the free energy functional is presented and discussed—this approach has been particularly successful to deal with metastability. We focus our attention on investigating and discussing the fundamental role of dissipation analysis in metastable systems. The major novelty of our approach is that the obtained results enable us to reveal an appealing, and previously unexplored relationship between Fokker-Planck equation and the associated free energy functional. Namely, we point out that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy.Entropy2015-02-09172Article10.3390/e170207637637711099-43002015-02-09doi: 10.3390/e17020763Umberto LuciaGianpiero Gervino<![CDATA[Entropy, Vol. 17, Pages 755-762: Thermodynamic Analysis about Nucleation and Growth of Cubic Boron Nitride Crystals in the hBN-Li3N System under High Pressure and High Temperature]]>
http://mdpi.com/1099-4300/17/2/755
The nucleation of cubic boron nitride (cBN) single crystals synthesized with lithium nitride (Li3N) as a catalyst under high pressure and high temperature (HPHT) was analyzed. Many nanometer-sized cubic boron nitride nuclei formed in the near surface layer, as detected by high resolution transmission electron microscopy. Based on the experiment results, the transformation kinetics is described by a nucleation and growth process in the thermodynamic stability region of cBN. A theoretical description is developed based on the heterogeneous nucleation and layer growth mechanism, and the relevant parameters are estimated and discussed. The critical crystal radius, r*, increases with the temperature under constant pressure; the change with temperature more pronounced at lower pressure (such as 4.5 GPa). The crystal growth velocity increased with the temperature, and it is parabolic with temperature under certain pressure. These results are consistent with experimental data.Entropy2015-02-09172Article10.3390/e170207557557621099-43002015-02-09doi: 10.3390/e17020755Xiao-Fei GuoBin XuWen ZhangMei-Zhe LvHong-Mei YangXiao-Hong Fan<![CDATA[Entropy, Vol. 17, Pages 710-754: On Equivalence of Nonequilibrium Thermodynamic and Statistical Entropies]]>
http://mdpi.com/1099-4300/17/2/710
We review the concept of nonequilibrium thermodynamic entropy and observables and internal variables as state variables, introduced recently by us, and provide a simple first principle derivation of additive statistical entropy, applicable to all nonequilibrium states by treating thermodynamics as an experimental science. We establish their numerical equivalence in several cases, which includes the most important case when the thermodynamic entropy is a state function. We discuss various interesting aspects of the two entropies and show that the number of microstates in the Boltzmann entropy includes all possible microstates of non-zero probabilities even if the system is trapped in a disjoint component of the microstate space. We show that negative thermodynamic entropy can appear from nonnegative statistical entropy.Entropy2015-02-05172Review10.3390/e170207107107541099-43002015-02-05doi: 10.3390/e17020710Purushottam Gujrati<![CDATA[Entropy, Vol. 17, Pages 692-709: Parameters Estimation of Uncertain Fractional-Order Chaotic Systems via a Modified Artificial Bee Colony Algorithm]]>
http://mdpi.com/1099-4300/17/2/692
Parameter estimation for fractional-order chaotic systems has been an interesting and important issue in theory and various fields of application. In this paper, fractional orders, as well as systematic parameters of fractional-order chaotic systems are considered by treating fractional orders as additional parameters. The parameter estimation is transformed into a multidimensional optimization problem, and an effective modified artificial bee colony algorithm is proposed to solve this problem. Numerical simulations are conducted on two typical fractional-order chaotic systems to show the effectiveness of the proposed scheme.Entropy2015-02-04172Article10.3390/e170206926927091099-43002015-02-04doi: 10.3390/e17020692Wei HuYongguang YuSha Wang<![CDATA[Entropy, Vol. 17, Pages 669-691: Application of Entropy Measures on Intrinsic Mode Functions for the Automated Identification of Focal Electroencephalogram Signals]]>
http://mdpi.com/1099-4300/17/2/669
The brain is a complex structure made up of interconnected neurons, and its electrical activities can be evaluated using electroencephalogram (EEG) signals. The characteristics of the brain area affected by partial epilepsy can be studied using focal and non-focal EEG signals. In this work, a method for the classification of focal and non-focal EEG signals is presented using entropy measures. These entropy measures can be useful in assessing the nonlinear interrelation and complexity of focal and non-focal EEG signals. These EEG signals are first decomposed using the empirical mode decomposition (EMD) method to extract intrinsic mode functions (IMFs). The entropy features, namely, average Shannon entropy (ShEnAvg), average Renyi’s entropy (RenEnAvg ), average approximate entropy (ApEnAvg), average sample entropy (SpEnAvg) and average phase entropies (S1Avg and S2Avg), are computed from different IMFs of focal and non-focal EEG signals. These entropies are used as the input feature set for the least squares support vector machine (LS-SVM) classifier to classify into focal and non-focal EEG signals. Experimental results show that our proposed method is able to differentiate the focal and non-focal EEG signals with an average classification accuracy of 87% correct.Entropy2015-02-03172Article10.3390/e170206696696911099-43002015-02-03doi: 10.3390/e17020669Rajeev SharmaRam PachoriU. Acharya<![CDATA[Entropy, Vol. 17, Pages 658-668: Entropy Production of Main-Sequence Stars]]>
http://mdpi.com/1099-4300/17/2/658
The entropy production (inside the volume bounded by a photosphere) of main-sequence stars is calculated based on B–V photometry data. The entropy-production distribution function and the dependences of entropy production on temperature and luminosity are obtained for these stars for the first time. A very small range of variation of specific (per volume) entropy production discovered for main-sequence stars (only 0.5 to 1.8 solar magnitudes) is an interesting result that can be crucial for understanding thermodynamic processes of stars.Entropy2015-02-02172Article10.3390/e170206586586681099-43002015-02-02doi: 10.3390/e17020658Leonid MartyushevSergey Zubarev<![CDATA[Entropy, Vol. 17, Pages 646-657: Thermal and Structural Analysis of Mn49.3Ni43.7Sn7.0 Heusler Alloy Ribbons]]>
http://mdpi.com/1099-4300/17/2/646
The martensitic transformation and the solidification structures of Mn49.3Ni43.7Sn7.0 alloy ribbons prepared by melt-spinning were investigated by means of scanning electron microscopy, X-ray diffraction and differential scanning calorimetry. In those experiments special attention was given to melt spinning processing parameters such as the linear surface speed of the copper wheel rotating, the injection overpressure and the distance between wheel and injection quartz tube. Transformation entropy was found higher when increasing linear surface speed or the distance from injection point to wheel. The resulting samples showed chemical compositions close to the nominal ones and, at room temperature, crystallized in a monoclinic single-phase martensite with 14M modulation (without a significant variation in the cell parameters). Strong dependence of ribbon thickness on processing parameters was found. The average grain size varied between 1.6 and 6.6 μm, while the start temperature of the martensitic temperature varied from 394 to 430 K.Entropy2015-02-02172Article10.3390/e170206466466571099-43002015-02-02doi: 10.3390/e17020646Tarek BachagaRakia DalyMohamed KhitouniLluisa EscodaJoan SaurinaJoan Suñol<![CDATA[Entropy, Vol. 17, Pages 594-645: A Foundational Approach to Generalising the Maximum Entropy Inference Process to the Multi-Agent Context]]>
http://mdpi.com/1099-4300/17/2/594
The present paper seeks to establish a logical foundation for studying axiomatically multi-agent probabilistic reasoning over a discrete space of outcomes. We study the notion of a social inference process which generalises the concept of an inference process for a single agent which was used by Paris and Vencovská to characterise axiomatically the method of maximum entropy inference. Axioms for a social inference process are introduced and discussed, and a particular social inference process called the Social Entropy Process, or SEP, is defined which satisfies these axioms. SEP is justified heuristically by an information theoretic argument, and incorporates both the maximum entropy inference process for a single agent and the multi–agent normalised geometric mean pooling operator.Entropy2015-02-02172Article10.3390/e170205945946451099-43002015-02-02doi: 10.3390/e17020594George Wilmers<![CDATA[Entropy, Vol. 17, Pages 580-593: Study on Droplet Size and Velocity Distributions of a Pressure Swirl Atomizer Based on the Maximum Entropy Formalism]]>
http://mdpi.com/1099-4300/17/2/580
A predictive model for droplet size and velocity distributions of a pressure swirl atomizer has been proposed based on the maximum entropy formalism (MEF). The constraint conditions of the MEF model include the conservation laws of mass, momentum, and energy. The effects of liquid swirling strength, Weber number, gas-to-liquid axial velocity ratio and gas-to-liquid density ratio on the droplet size and velocity distributions of a pressure swirl atomizer are investigated. Results show that model based on maximum entropy formalism works well to predict droplet size and velocity distributions under different spray conditions. Liquid swirling strength, Weber number, gas-to-liquid axial velocity ratio and gas-to-liquid density ratio have different effects on droplet size and velocity distributions of a pressure swirl atomizer.Entropy2015-01-30172Article10.3390/e170205805805931099-43002015-01-30doi: 10.3390/e17020580Kai YanZhi NingMing LüChunhua Sun<![CDATA[Entropy, Vol. 17, Pages 560-579: Symbolic Entropy of the Amplitude rather than the Instantaneous Frequency of EEG Varies in Dementia]]>
http://mdpi.com/1099-4300/17/2/560
The dynamics of human electroencephalography (EEG) have been proved to be related to cognitive activities. This study separately assessed the two EEG components, amplitude and rhythm, aiming to capture their individual contributions to cognitive functions. We extracted the local peaks of EEGs under rest or photic stimulation and calculated the symbolic dynamics of their voltages (amplitude) and interpeak intervals (instantaneous frequency), individually. The sample consisted of 89 geriatric outpatients in three patient groups: 38 fresh cases of vascular dementia (VD), 22 fresh cases of Alzheimer’s disease (AD) and 29 controls. Both sample entropy and number of forbidden words revealed significantly less regular symbolic dynamics in the whole EEG tracings of the VD than the AD and control groups. We found consistent results between groups with the symbolic dynamics in the local-peak voltage sequence rather than the interpeak interval sequence. Photic stimulation amplified the differences between groups. These results suggest that the EEG dynamics which relates to either cognitive functions or the underlying pathologies of dementia are embedded within the dynamics of the amount of but not the interval between each synchronized firing of adjacent cerebral neurons.Entropy2015-01-29172Article10.3390/e170205605605791099-43002015-01-29doi: 10.3390/e17020560Pei-Feng LinJenho TsaoMen-Tzung LoChen LinYi-Chung Chang<![CDATA[Entropy, Vol. 17, Pages 539-559: Effects of Direct Fuel Injection Strategies on Cycle-by-Cycle Variability in a Gasoline Homogeneous Charge Compression Ignition Engine: Sample Entropy Analysis]]>
http://mdpi.com/1099-4300/17/2/539
In this study we summarize and analyze experimental observations of cyclic variability in homogeneous charge compression ignition (HCCI) combustion in a single-cylinder gasoline engine. The engine was configured with negative valve overlap (NVO) to trap residual gases from prior cycles and thus enable auto-ignition in successive cycles. Correlations were developed between different fuel injection strategies and cycle average combustion and work output profiles. Hypothesized physical mechanisms based on these correlations were then compared with trends in cycle-by-cycle predictability as revealed by sample entropy. The results of these comparisons help to clarify how fuel injection strategy can interact with prior cycle effects to affect combustion stability and so contribute to design control methods for HCCI engines.Entropy2015-01-29172Article10.3390/e170205395395591099-43002015-01-29doi: 10.3390/e17020539Jacek HuniczAlejandro MedinaGrzegorz LitakPedro Curto-RissoLev Guzmán-Vargas<![CDATA[Entropy, Vol. 17, Pages 528-538: A New Telegrapher’s-Poisson System in Semiconductor Theory: A Singular Perturbation Approach]]>
http://mdpi.com/1099-4300/17/2/528
In the theory of energy and momentum relaxation in semiconductor devices, the introduction of two temperatures and two mean velocities for electron and phonons is required. A new model, based on an asymptotic procedure for solving the kinetic equations of electrons and phonons is proposed, which naturally gives the displaced Maxwellian at the leading order. After that, balance equations for the electron number, energy densities and momentum densities are constructed, which constitute now a system of five equations for the chemical potential of electrons, the temperatures and the drift velocities. Moreover, Poisson’s equation is coupled, in order to calculate the self-consistent electric field. In Bloch’s approximation, we derive a telegrapher’s-Poisson system for the electron number density and the electric potential, which could allow simple semiconductor calculations, but still including wave propagation effects.Entropy2015-01-29172Article10.3390/e170205285285381099-43002015-01-29doi: 10.3390/e17020528Alberto Rossani<![CDATA[Entropy, Vol. 17, Pages 509-527: EWnFM: An Environment States Oriented Web Service Non-Functional Property Model]]>
http://mdpi.com/1099-4300/17/2/509
A proper model of Web service non-functional properties is the key foundation to the evaluation of non-functional properties of Adaptive Service Based Software (ASBS) systems. As the environment in which a Web service is deployed may keep changing, environmental factors would affect the non-functional properties of a Web service a lot. However, available non-functional property models usually ignore the impact of environmental factors, leading to insufficient modeling power of non-functional properties, limited effect of system wide non-functional property evaluation based on these models, and the inability to support environment states oriented specifications of ASBS. This paper propose an environment states oriented Web service non-functional property model. By considering the differences of a non-functional property under different environment states, environment states of a Web service is analyzed using a Dirichlet process based method. With such a foundation, an environment states oriented Web service non-functional property model is introduced, together with the parameter estimation methods based on historical monitor data. Experiment results have shown that compared to the evaluated methods, our model could generate data that are much close to real monitored data.Entropy2015-01-28172Article10.3390/e170205095095271099-43002015-01-28doi: 10.3390/e17020509Yin ZhangLiang GeKening GaoBin ZhangZhuyin Xue<![CDATA[Entropy, Vol. 17, Pages 502-508: Reshaping the Science of Reliability with the Entropy Function]]>
http://mdpi.com/1099-4300/17/2/502
The present paper revolves around two argument points. As first, we have observed a certain parallel between the reliability of systems and the progressive disorder of thermodynamical systems; and we import the notion of reversibility/irreversibility into the reliability domain. As second, we note that the reliability theory is a very active area of research which although has not yet become a mature discipline. This is due to the majority of researchers who adopt the inductive logic instead of the deductive logic typical of mature scientific sectors. The deductive approach was inaugurated by Gnedenko in the reliability domain. We mean to continue Gnedenko’s work and we use the Boltzmann-like entropy to pursue this objective. This paper condenses the papers published in the past decade which illustrate the calculus of the Boltzmann-like entropy. It is demonstrated how the every result complies with the deductive logic and are consistent with Gnedenko’s achievements.Entropy2015-01-26172Article10.3390/e170205025025081099-43002015-01-26doi: 10.3390/e17020502Paolo RocchiGiulia Capacci<![CDATA[Entropy, Vol. 17, Pages 483-501: Analyses of Heart Rate, Respiration and Cardiorespiratory Coupling in Patients with Schizophrenia]]>
http://mdpi.com/1099-4300/17/2/483
Schizophrenia is a severe mental disorder associated with a significantly increased cardiovascular mortality rate. However, the underlying mechanisms leading to this cardiovascular disease (CVD) are not fully known. Therefore, the objective of this study was to characterize the cardiorespiratory influence by investigating heart rate, respiration and the causal strength and direction of cardiorespiratory coupling (CRC), based mainly on entropy measures. We investigated 23 non-medicated patients with schizophrenia (SZ), comparing them to 23 age- and gender-matched healthy controls (CO). A significantly reduced complexity was found for the heart rate and a significantly increased complexity in respiration and CRC in SZ patients when compared to corresponding measurements from CO (p &lt; 0.001). CRC analyses revealed a clear coupling, with a driver-responder relationship from respiration to heart rate in SZ patients. Moreover, a slight driver-responder relationship from heart rate to respiration could be recognized. These findings lead to the assumption that SZ should be considered to be a high-risk group for CVD. We hypothesize that the varying cardiorespiratory regulation contributes to the increased risk for cardiac mortality. Therefore, regular monitoring of the cardiorespiratory status of SZ is suggested to identify autonomic regulation impairment at an early stage—to develop timely and effective treatment and intervention strategies.Entropy2015-01-23172Article10.3390/e170204834835011099-43002015-01-23doi: 10.3390/e17020483Steffen SchulzKarl-Jürgen BärAndreas Voss<![CDATA[Entropy, Vol. 17, Pages 471-482: Modeling and Analyzing the Interaction between Network Rumors and Authoritative Information]]>
http://mdpi.com/1099-4300/17/1/471
In this paper, we propose a novel two-stage rumor spreading Susceptible-Infected-Authoritative-Removed (SIAR) model for complex homogeneous and heterogeneous networks. The interaction Markov chains (IMC) mean-field equations based on the SIAR model are derived to describe the dynamic interaction between the rumors and authoritative information. We use a Monte Carlo simulation method to characterize the dynamics of the Susceptible-Infected-Removed (SIR) and SIAR models, showing that the SIAR model with consideration of authoritative information gives a more realistic description of propagation features of rumors than the SIR model. The simulation results demonstrate that the critical threshold λc of the SIAR model has the tiniest increase than the threshold of SIR model. The sooner the authoritative information is introduced, the less negative impact the rumors will bring. We also get the result that heterogeneous networks are more prone to the spreading of rumors. Additionally, the inhibition of rumor spreading, as one of the characteristics of the new SIAR model itself, is instructive for later studies on the rumor spreading models and the controlling strategies.Entropy2015-01-19171Article10.3390/e170104714714821099-43002015-01-19doi: 10.3390/e17010471Lingling XiaGuoping JiangYurong SongBo Song<![CDATA[Entropy, Vol. 17, Pages 438-470: A Recipe for the Estimation of Information Flow in a Dynamical System]]>
http://mdpi.com/1099-4300/17/1/438
Information-theoretic quantities, such as entropy and mutual information (MI), can be used to quantify the amount of information needed to describe a dataset or the information shared between two datasets. In the case of a dynamical system, the behavior of the relevant variables can be tightly coupled, such that information about one variable at a given instance in time may provide information about other variables at later instances in time. This is often viewed as a flow of information, and tracking such a flow can reveal relationships among the system variables. Since the MI is a symmetric quantity; an asymmetric quantity, called Transfer Entropy (TE), has been proposed to estimate the directionality of the coupling. However, accurate estimation of entropy-based measures is notoriously difficult. Every method has its own free tuning parameter(s) and there is no consensus on an optimal way of estimating the TE from a dataset. We propose a new methodology to estimate TE and apply a set of methods together as an accuracy cross-check to provide a reliable mathematical tool for any given data set. We demonstrate both the variability in TE estimation across techniques as well as the benefits of the proposed methodology to reliably estimate the directionality of coupling among variables. Entropy2015-01-19171Article10.3390/e170104384384701099-43002015-01-19doi: 10.3390/e17010438Deniz GencagaKevin KnuthWilliam Rossow<![CDATA[Entropy, Vol. 17, Pages 425-437: Self-Similarity in Population Dynamics: Surname Distributions and Genealogical Trees]]>
http://mdpi.com/1099-4300/17/1/425
The frequency distribution of surnames turns out to be a relevant issue not only in historical demography but also in population biology, and especially in genetics, since surnames tend to behave like neutral genes and propagate like Y chromosomes. The stochastic dynamics leading to the observed scale-invariant distributions has been studied as a Yule process, as a branching phenomenon and also by field-theoretical renormalization group techniques. In the absence of mutations the theoretical models are in good agreement with empirical evidence, but when mutations are present a discrepancy between the theoretical and the experimental exponents is observed. Hints for the possible origin of the mismatch are discussed, with some emphasis on the difference between the asymptotic frequency distribution of a full population and the frequency distributions observed in its samples. A precise connection is established between surname distributions and the statistical properties of genealogical trees. Ancestors tables, being obviously self-similar, may be investigated theoretically by renormalization group techniques, but they can also be studied empirically by exploiting the large online genealogical databases concerning European nobility.Entropy2015-01-19171Article10.3390/e170104254254371099-43002015-01-19doi: 10.3390/e17010425Paolo Rossi<![CDATA[Entropy, Vol. 17, Pages 407-424: Entropy, Age and Time Operator]]>
http://mdpi.com/1099-4300/17/1/407
The time operator and internal age are intrinsic features of entropy producing innovation processes. The innovation spaces at each stage are the eigenspaces of the time operator. The internal age is the average innovation time, analogous to lifetime computation. Time operators were originally introduced for quantum systems and highly unstable dynamical systems. Extending the time operator theory to regular Markov chains allows one to relate internal age with norm distances from equilibrium. The goal of this work is to express the evolution of internal age in terms of Lyapunov functionals constructed from entropies. We selected the Boltzmann–Gibbs–Shannon entropy and more general entropy functions, namely the Tsallis entropies and the Kaniadakis entropies. Moreover, we compare the evolution of the distance of initial distributions from equilibrium to the evolution of the Lyapunov functionals constructed from norms with the evolution of Lyapunov functionals constructed from entropies. It is remarkable that the entropy functionals evolve, violating the second law of thermodynamics, while the norm functionals evolve thermodynamically.Entropy2015-01-19171Article10.3390/e170104074074241099-43002015-01-19doi: 10.3390/e17010407Ilias GialampoukidisIoannis Antoniou<![CDATA[Entropy, Vol. 17, Pages 401-406: On an Objective Basis for the Maximum Entropy Principle]]>
http://mdpi.com/1099-4300/17/1/401
In this letter, we elaborate on some of the issues raised by a recent paper by Neapolitan and Jiang concerning the maximum entropy (ME) principle and alternative principles for estimating probabilities consistent with known, measured constraint information. We argue that the ME solution for the “problematic” example introduced by Neapolitan and Jiang has stronger objective basis, rooted in results from information theory, than their alternative proposed solution. We also raise some technical concerns about the Bayesian analysis in their work, which was used to independently support their alternative to the ME solution. The letter concludes by noting some open problems involving maximum entropy statistical inference.Entropy2015-01-19171Letter10.3390/e170104014014061099-43002015-01-19doi: 10.3390/e17010401David MillerHossein Soleimani<![CDATA[Entropy, Vol. 17, Pages 384-400: Tsallis Distribution Decorated with Log-Periodic Oscillation]]>
http://mdpi.com/1099-4300/17/1/384
In many situations, in all branches of physics, one encounters the power-like behavior of some variables, which is best described by a Tsallis distribution characterized by a nonextensivity parameter q and scale parameter T. However, there exist experimental results that can be described only by a Tsallis distributions, which are additionally decorated by some log-periodic oscillating factor. We argue that such a factor can originate from allowing for a complex nonextensivity parameter q. The possible information conveyed by such an approach (like the occurrence of complex heat capacity, the notion of complex probability or complex multiplicative noise) will also be discussed.Entropy2015-01-14171Article10.3390/e170103843844001099-43002015-01-14doi: 10.3390/e17010384Grzegorz WilkZbigniew Włodarczyk<![CDATA[Entropy, Vol. 17, Pages 368-383: Message Authentication over Noisy Channels]]>
http://mdpi.com/1099-4300/17/1/368
The essence of authentication is the transmission of unique and irreproducible information. In this paper, the authentication becomes a problem of the secure transmission of the secret key over noisy channels. A general analysis and design framework for message authentication is presented based on the results of Wyner’s wiretap channel. Impersonation and substitution attacks are primarily investigated. Information-theoretic lower and upper bounds on the opponent’s success probability are derived, and the lower bound and the upper bound are shown to match. In general, the fundamental limits on message authentication over noisy channels are fully characterized. Analysis results demonstrate that introducing noisy channels is a reliable way to enhance the security of authentication.Entropy2015-01-14171Article10.3390/e170103683683831099-43002015-01-14doi: 10.3390/e17010368Fanfan ZhengZhiqing XiaoShidong ZhouJing WangLianfen Huang<![CDATA[Entropy, Vol. 17, Pages 346-367: Robust H∞ Finite-Time Control for Discrete Markovian Jump Systems with Disturbances of Probabilistic Distributions]]>
http://mdpi.com/1099-4300/17/1/346
This paper is concerned with the robust H∞ finite-time control for discrete delayed nonlinear systems with Markovian jumps and external disturbances. It is usually assumed that the disturbance affects the system states and outputs with the same influence degree of 100%, which is not evident enough to reflect the situation where the disturbance affects these two parts by different influence degrees. To tackle this problem, a probabilistic distribution denoted by binomial sequences is introduced to describe the external disturbance. Throughout the paper, the definitions of the finite-time boundedness (FTB) and the H∞ FTB are firstly given respectively. To extend the results further, a model which combines a linear dynamic system and a static nonlinear operator is referred to describe the system under discussion. Then by virtue of state feedback control method, some new sufficient criteria are derived which guarantee the FTB and H∞ FTB performances for the considered system. Finally, an example is provided to demonstrate the effectiveness of the developed control laws.Entropy2015-01-14171Article10.3390/e170103463463671099-43002015-01-14doi: 10.3390/e17010346Haiyang ChenMeiqin LiuSenlin Zhang<![CDATA[Entropy, Vol. 17, Pages 304-345: Black-Box Optimization Using Geodesics in Statistical Manifolds]]>
http://mdpi.com/1099-4300/17/1/304
Information geometric optimization (IGO) is a general framework for stochastic optimization problems aiming at limiting the influence of arbitrary parametrization choices: the initial problem is transformed into the optimization of a smooth function on a Riemannian manifold, defining a parametrization-invariant first order differential equation and, thus, yielding an approximately parametrization-invariant algorithm (up to second order in the step size). We define the geodesic IGO update, a fully parametrization-invariant algorithm using the Riemannian structure, and we compute it for the manifold of Gaussians, thanks to Noether’s theorem. However, in similar algorithms, such as CMA-ES (Covariance Matrix Adaptation - Evolution Strategy) and xNES (exponential Natural Evolution Strategy), the time steps for the mean and the covariance are decoupled. We suggest two ways of doing so: twisted geodesic IGO (GIGO) and blockwise GIGO. Finally, we show that while the xNES algorithm is not GIGO, it is an instance of blockwise GIGO applied to the mean and covariance matrix separately. Therefore, xNES has an almost parametrization-invariant description.Entropy2015-01-13171Article10.3390/e170103043043451099-43002015-01-13doi: 10.3390/e17010304Jérémy Bensadon<![CDATA[Entropy, Vol. 17, Pages 277-303: Information Decomposition in Bivariate Systems: Theory and Application to Cardiorespiratory Dynamics]]>
http://mdpi.com/1099-4300/17/1/277
In the framework of information dynamics, the temporal evolution of coupled systems can be studied by decomposing the predictive information about an assigned target system into amounts quantifying the information stored inside the system and the information transferred to it. While information storage and transfer are computed through the known self-entropy (SE) and transfer entropy (TE), an alternative decomposition evidences the so-called cross entropy (CE) and conditional SE (cSE), quantifying the cross information and internal information of the target system, respectively. This study presents a thorough evaluation of SE, TE, CE and cSE as quantities related to the causal statistical structure of coupled dynamic processes. First, we investigate the theoretical properties of these measures, providing the conditions for their existence and assessing the meaning of the information theoretic quantity that each of them reflects. Then, we present an approach for the exact computation of information dynamics based on the linear Gaussian approximation, and exploit this approach to characterize the behavior of SE, TE, CE and cSE in benchmark systems with known dynamics. Finally, we exploit these measures to study cardiorespiratory dynamics measured from healthy subjects during head-up tilt and paced breathing protocols. Our main result is that the combined evaluation of the measures of information dynamics allows to infer the causal effects associated with the observed dynamics and to interpret the alteration of these effects with changing experimental conditions.Entropy2015-01-12171Article10.3390/e170102772773031099-43002015-01-12doi: 10.3390/e17010277Luca FaesAlberto PortaGiandomenico Nollo<![CDATA[Entropy, Vol. 17, Pages 264-276: The Entropy of an Armco Iron under Irreversible Deformation]]>
http://mdpi.com/1099-4300/17/1/264
This work is devoted to the development of a theoretical approach for the calculation of entropy in metals under plastic deformation. The thermodynamic analysis of the plastic deformation process allowed us to obtain the expression for determination of the entropy production. The value of the entropy production in an Armco iron specimen under plastic deformation as calculated the basis of the proposed technique and infrared thermography data. This method also lets us define the inelastic strain caused by the initiation and growth of the defects which was used as the internal variable in the considered thermomechanical model from the experimental data. In order to verify the obtained results a theoretical analysis of the modeled situation was carried out.Entropy2015-01-12171Article10.3390/e170102642642761099-43002015-01-12doi: 10.3390/e17010264Anastasiia KostinaOleg Plekhov<![CDATA[Entropy, Vol. 17, Pages 244-263: A Comparative Study on Energy and Exergy Analyses of a CI Engine Performed with Different Multiple Injection Strategies at Part Load: Effect of Injection Pressure]]>
http://mdpi.com/1099-4300/17/1/244
In this study, a four stroke four cylinder direct injection CI engine was run using three different injection pressures. In all measurements, the fuel quantity per cycle, the pre injection and main injection timing, the boost pressure and the engine speed were kept constant. The motor tests were performed under 130, 140 and 150 MPa rail pressure. During the theoretical part of the study, combustion, emission, energy and exergy analysis were made using the test results. An increase in the injection pressure increases combustion efficiency. The results show that combustion efficiency is not enough by itself, because the increase in the power need of the injection pump, decreases the thermal efficiency. The increase in the combustion temperature, increases the cooling loss and decreases the exergetic efficiency. In addition, the NOx emissions increased by 12% and soot emissions decreased 44% via increasing injection pressure by 17%. The thermal and exergetic efficiencies are found inversely proportional with injection pressure. Exergy destruction is found independent of the injection pressure and its value is obtained as ~6%.Entropy2015-01-12171Article10.3390/e170102442442631099-43002015-01-12doi: 10.3390/e17010244Muammer Özkan<![CDATA[Entropy, Vol. 17, Pages 231-243: Multiscale Entropy Analysis of Heart Rate Variability for Assessing the Severity of Sleep Disordered Breathing]]>
http://mdpi.com/1099-4300/17/1/231
Obstructive sleep apnea (OSA) is an independent cardiovascular risk factor to which autonomic nervous dysfunction has been reported to be an important contributor. Ninety subjects recruited from the sleep center of a single medical center were divided into four groups: normal snoring subjects without OSA (apnea hypopnea index, AHI &lt; 5, n = 11), mild OSA (5 ≤ AHI &lt; 15, n = 10), moderate OSA (15 ≤ AHI &lt; 30, n = 24), and severe OSA (AHI ≥ 30, n = 45). Demographic (i.e., age, gender), anthropometric (i.e., body mass index, neck circumference), and polysomnographic (PSG) data were recorded and compared among the different groups. For each subject, R-R intervals (RRI) from 10 segments of 10-minute electrocardiogram recordings during non-rapid eye movement sleep at stage N2 were acquired and analyzed for heart rate variability (HRV) and sample entropy using multiscale entropy index (MEI) that was divided into small scale (MEISS, scale 1–5) and large scale (MEILS, scale 6–10). Our results not only demonstrated that MEISS could successfully distinguish normal snoring subjects and those with mild OSA from those with moderate and severe disease, but also revealed good correlation between MEISS and AHI with Spearman correlation analysis (r = −0.684, p &lt; 0.001). Therefore, using the two parameters of EEG and ECG, MEISS may serve as a simple preliminary screening tool for assessing the severity of OSA before proceeding to PSG analysis.Entropy2015-01-12171Article10.3390/e170102312312431099-43002015-01-12doi: 10.3390/e17010231Wen-Yao PanMao-Chang SuHsien-Tsai WuMeng-Chih LinI-Ting TsaiCheuk-Kwan Sun<![CDATA[Entropy, Vol. 17, Pages 214-230: An 18 Moments Model for Dense Gases: Entropy and Galilean Relativity Principles without Expansions]]>
http://mdpi.com/1099-4300/17/1/214
The 14 moments model for dense gases, introduced in the last few years by Arima, Taniguchi, Ruggeri and Sugiyama, is here extended up to 18 moments. They have found the closure of the balance equations up to a finite order with respect to equilibrium; it is also possible to impose for that model the entropy and Galilean relativity principles up to whatever order with respect to equilibrium, but by using Taylor’s expansion. Here, the exact solution is found, without expansions, but a bigger number of moments has to be considered and reasons will be shown suggesting that this number is at least 18.Entropy2015-01-09171Article10.3390/e170102142142301099-43002015-01-09doi: 10.3390/e17010214M. CarrisiSebastiano Pennisi<![CDATA[Entropy, Vol. 17, Pages 197-213: Deduction of Lorentz Transformations from Classical Thermodynamics]]>
http://mdpi.com/1099-4300/17/1/197
The Lorentz transformations are obtained by assuming that the laws of classical thermodynamics are invariant under changes of inertial reference frames. As Maxwell equations are used in order to deduce a wave equation that shows the constancy of the speed of light, by means of the laws of classical thermodynamics, the invariance of the Carnot cycle is deduced under reference frame changes. Starting with this result and the blackbody particle number density in a rest frame, the Lorentz transformations are obtained. A discussion about the universality of classical thermodynamics is given.Entropy2015-01-08171Article10.3390/e170101971972131099-43002015-01-08doi: 10.3390/e17010197Angela de PargaFernado Angulo-BrownGonzalo de Parga<![CDATA[Entropy, Vol. 17, Pages 181-196: An Image Encryption Scheme Based on Hyperchaotic Rabinovich and Exponential Chaos Maps]]>
http://mdpi.com/1099-4300/17/1/181
This paper proposes a new four-dimensional hyperchaotic map based on the Rabinovich system to realize chaotic encryption in higher dimension and improve the security. The chaotic sequences generated by Runge-Kutta method are combined with the chaotic sequences generated by an exponential chaos map to generate key sequences. The key sequences are used for image encryption. The security test results indicate that the new hyperchaotic system has high security and complexity. The comparison between the new hyperchaotic system and the several low-dimensional chaotic systems shows that the proposed system performs more efficiently.Entropy2015-01-08171Article10.3390/e170101811811961099-43002015-01-08doi: 10.3390/e17010181Xiaojun TongYang LiuMiao ZhangHui XuZhu Wang<![CDATA[Entropy, Vol. 17, Pages 151-180: A Clustering Method Based on the Maximum Entropy Principle]]>
http://mdpi.com/1099-4300/17/1/151
Clustering is an unsupervised process to determine which unlabeled objects in a set share interesting properties. The objects are grouped into k subsets (clusters) whose elements optimize a proximity measure. Methods based on information theory have proven to be feasible alternatives. They are based on the assumption that a cluster is one subset with the minimal possible degree of “disorder”. They attempt to minimize the entropy of each cluster. We propose a clustering method based on the maximum entropy principle. Such a method explores the space of all possible probability distributions of the data to find one that maximizes the entropy subject to extra conditions based on prior information about the clusters. The prior information is based on the assumption that the elements of a cluster are “similar” to each other in accordance with some statistical measure. As a consequence of such a principle, those distributions of high entropy that satisfy the conditions are favored over others. Searching the space to find the optimal distribution of object in the clusters represents a hard combinatorial problem, which disallows the use of traditional optimization techniques. Genetic algorithms are a good alternative to solve this problem. We benchmark our method relative to the best theoretical performance, which is given by the Bayes classifier when data are normally distributed, and a multilayer perceptron network, which offers the best practical performance when data are not normal. In general, a supervised classification method will outperform a non-supervised one, since, in the first case, the elements of the classes are known a priori. In what follows, we show that our method’s effectiveness is comparable to a supervised one. This clearly exhibits the superiority of our method.Entropy2015-01-07171Article10.3390/e170101511511801099-43002015-01-07doi: 10.3390/e17010151Edwin Aldana-BobadillaAngel Kuri-Morales<![CDATA[Entropy, Vol. 17, Pages 142-150: Acknowledgement to Reviewers of Entropy in 2014]]>
http://mdpi.com/1099-4300/17/1/142
The editors of Entropy would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2014:[...]Entropy2015-01-07171Editorial10.3390/e170101421421501099-43002015-01-07doi: 10.3390/e17010142 Entropy Editorial Office<![CDATA[Entropy, Vol. 17, Pages 123-141: Assessment of Time and Frequency Domain Entropies to Detect Sleep Apnoea in Heart Rate Variability Recordings from Men and Women]]>
http://mdpi.com/1099-4300/17/1/123
Heart rate variability (HRV) provides useful information about heart dynamics both under healthy and pathological conditions. Entropy measures have shown their utility to characterize these dynamics. In this paper, we assess the ability of spectral entropy (SE) and multiscale entropy (MsE) to characterize the sleep apnoea-hypopnea syndrome (SAHS) in HRV recordings from 188 subjects. Additionally, we evaluate eventual differences in these analyses depending on the gender. We found that the SE computed from the very low frequency band and the low frequency band showed ability to characterize SAHS regardless the gender; and that MsE features may be able to distinguish gender specificities. SE and MsE showed complementarity to detect SAHS, since several features from both analyses were automatically selected by the forward-selection backward-elimination algorithm. Finally, SAHS was modelled through logistic regression (LR) by using optimum sets of selected features. Modelling SAHS by genders reached significant higher performance than doing it in a jointly way. The highest diagnostic ability was reached by modelling SAHS in women. The LR classifier achieved 85.2% accuracy (Acc) and 0.951 area under the ROC curve (AROC). LR for men reached 77.6% Acc and 0.895 AROC, whereas LR for the whole set reached 72.3% Acc and 0.885 AROC. Our results show the usefulness of the SE and MsE analyses of HRV to detect SAHS, as well as suggest that, when using HRV, SAHS may be more accurately modelled if data are separated by gender.Entropy2015-01-06171Article10.3390/e170101231231411099-43002015-01-06doi: 10.3390/e17010123Gonzalo Gutiérrez-TobalDaniel ÁlvarezJavier Gomez-PilarFélix del CampoRoberto Hornero<![CDATA[Entropy, Vol. 17, Pages 102-122: Estimating the Entropy of a Weibull Distribution under Generalized Progressive Hybrid Censoring]]>
http://mdpi.com/1099-4300/17/1/102
Recently, progressive hybrid censoring schemes have become quite popular in a life-testing problem and reliability analysis. However, the limitation of the progressive hybrid censoring scheme is that it cannot be applied when few failures occur before time T. Therefore, a generalized progressive hybrid censoring scheme was introduced. In this paper, the estimation of the entropy of a two-parameter Weibull distribution based on the generalized progressively censored sample has been considered. The Bayes estimators for the entropy of the Weibull distribution based on the symmetric and asymmetric loss functions, such as the squared error, linex and general entropy loss functions, are provided. The Bayes estimators cannot be obtained explicitly, and Lindley’s approximation is used to obtain the Bayes estimators. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, a real dataset has been analyzed for illustrative purposes.Entropy2015-01-05171Article10.3390/e170101021021221099-43002015-01-05doi: 10.3390/e17010102Youngseuk ChoHokeun SunKyeongjun Lee<![CDATA[Entropy, Vol. 17, Pages 74-101: Entropy-Based Characterization of Internet Background Radiation]]>
http://mdpi.com/1099-4300/17/1/74
Network security requires real-time monitoring of network traffic in order to detect new and unexpected attacks. Attack detection methods based on deep packet inspection are time consuming and costly, due to their high computational demands. This paper proposes a fast, lightweight method to distinguish different attack types observed in an IP darkspace monitor. The method is based on entropy measures of traffic-flow features and machine learning techniques. The explored data belongs to a portion of the Internet background radiation from a large IP darkspace, i.e., real traffic captures that exclusively contain unsolicited traffic, ongoing attacks, attack preparation activities and attack aftermaths. Results from an in-depth traffic analysis based on packet headers and content are used as a reference to label data and to evaluate the quality of the entropy-based classification. Full IP darkspace traffic captures from a three-week observation period in April, 2012, are used to compare the entropy-based classification with the in-depth traffic analysis. Results show that several traffic types present a high correlation to the respective traffic-flow entropy signals and can even fit polynomial regression models. Therefore, sudden changes in traffic types caused by new attacks or attack preparation activities can be identified based on entropy variations.Entropy2014-12-31171Article10.3390/e17010074741011099-43002014-12-31doi: 10.3390/e17010074Félix IglesiasTanja Zseby<![CDATA[Entropy, Vol. 17, Pages 52-73: The Big World of Nanothermodynamics]]>
http://mdpi.com/1099-4300/17/1/52
Nanothermodynamics extends standard thermodynamics to facilitate finite-size effects on the scale of nanometers. A key ingredient is Hill’s subdivision potential that accommodates the non-extensive energy of independent small systems, similar to how Gibbs’ chemical potential accommodates distinct particles. Nanothermodynamics is essential for characterizing the thermal equilibrium distribution of independently relaxing regions inside bulk samples, as is found for the primary response of most materials using various experimental techniques. The subdivision potential ensures strict adherence to the laws of thermodynamics: total energy is conserved by including an instantaneous contribution from the entropy of local configurations, and total entropy remains maximized by coupling to a thermal bath. A unique feature of nanothermodynamics is the completely-open nanocanonical ensemble. Another feature is that particles within each region become statistically indistinguishable, which avoids non-extensive entropy, and mimics quantum-mechanical behavior. Applied to mean-field theory, nanothermodynamics gives a heterogeneous distribution of regions that yields stretched-exponential relaxation and super-Arrhenius activation. Applied to Monte Carlo simulations, there is a nonlinear correction to Boltzmann’s factor that improves agreement between the Ising model and measured non-classical critical scaling in magnetic materials. Nanothermodynamics also provides a fundamental mechanism for the 1/f noise found in many materials.Entropy2014-12-31171Review10.3390/e1701005252731099-43002014-12-31doi: 10.3390/e17010052Ralph Chamberlin<![CDATA[Entropy, Vol. 17, Pages 39-51: Finite-Time Synchronization of Chaotic Complex Networks with Stochastic Disturbance]]>
http://mdpi.com/1099-4300/17/1/39
This paper is concerned with the problem of finite-time synchronization in complex networks with stochastic noise perturbations. By using a novel finite-time ℒ -operator differential inequality and other inequality techniques, some novel sufficient conditions are obtained to ensure finite-time stochastic synchronization for the complex networks concerned, where the coupling matrix need not be symmetric. The effects of control parameters on synchronization speed and time are also analyzed, and the synchronization time in this paper is shorter than that in the existing literature. The results here are also applicable to both directed and undirected weighted networks without any information of the coupling matrix. Finally, an example with numerical simulations is given to demonstrate the effectiveness of the proposed method.Entropy2014-12-30171Article10.3390/e1701003939511099-43002014-12-30doi: 10.3390/e17010039Liangliang LiJigui Jian<![CDATA[Entropy, Vol. 17, Pages 28-38: A Color Image Encryption Algorithm Based on a Fractional-Order Hyperchaotic System]]>
http://mdpi.com/1099-4300/17/1/28
In this paper, a new color image encryption algorithm based on a fractional-order hyperchaotic system is proposed. Firstly, four chaotic sequences are generated by a fractional-order hyperchaotic system. The parameters of such a system, together with the initial value, are regarded as the secret keys and the plain image is encrypted by performing the XOR and shufﬂing operations simultaneously. The proposed encryption scheme is described in detail with security analyses, including correlation analysis, histogram analysis, differential attacks, and key sensitivity analysis. Experimental results show that the proposed encryption scheme has big key space, and high sensitivity to keys properties, and resists statistical analysis and differential attacks, so it has high security and is suitable for color image encryption.Entropy2014-12-23171Article10.3390/e1701002828381099-43002014-12-23doi: 10.3390/e17010028Xia HuangTiantian SunYuxia LiJinling Liang<![CDATA[Entropy, Vol. 17, Pages 1-27: Complexity-Regularized Regression for Serially-Correlated Residuals with Applications to Stock Market Data]]>
http://mdpi.com/1099-4300/17/1/1
A popular approach in the investigation of the short-term behavior of a non-stationary time series is to assume that the time series decomposes additively into a long-term trend and short-term fluctuations. A first step towards investigating the short-term behavior requires estimation of the trend, typically via smoothing in the time domain. We propose a method for time-domain smoothing, called complexity-regularized regression (CRR). This method extends recent work, which infers a regression function that makes residuals from a model “look random”. Our approach operationalizes non-randomness in the residuals by applying ideas from computational mechanics, in particular the statistical complexity of the residual process. The method is compared to generalized cross-validation (GCV), a standard approach for inferring regression functions, and shown to outperform GCV when the error terms are serially correlated. Regression under serially-correlated residuals has applications to time series analysis, where the residuals may represent short timescale activity. We apply CRR to a time series drawn from the Dow Jones Industrial Average and examine how both the long-term and short-term behavior of the market have changed over time.Entropy2014-12-23171Article10.3390/e170100011271099-43002014-12-23doi: 10.3390/e17010001David DarmonMichelle Girvan<![CDATA[Entropy, Vol. 16, Pages 6722-6738: A Large Deviation Principle and an Expression of the Rate Function for a Discrete Stationary Gaussian Process]]>
http://mdpi.com/1099-4300/16/12/6722
We prove a large deviation principle for a stationary Gaussian process over Rb,indexed by Ζd (for some positive integers d and b), with positive definite spectral density, andprovide an expression of the corresponding rate function in terms of the mean of the processand its spectral density. This result is useful in applications where such an expression isneeded.Entropy2014-12-221612Article10.3390/e16126722672267381099-43002014-12-22doi: 10.3390/e16126722Olivier FaugerasJames MacLaurin<![CDATA[Entropy, Vol. 16, Pages 6705-6721: A Representation of the Relative Entropy with Respect to a Diffusion Process in Terms of Its Infinitesimal Generator]]>
http://mdpi.com/1099-4300/16/12/6705
In this paper we derive an integral (with respect to time) representation of the relative entropy (or Kullback–Leibler Divergence) R(μ||P), where μ and P are measures on C([0,T];Rd). The underlying measure P is a weak solution to a martingale problem with continuous coefficients. Our representation is in the form of an integral with respect to its infinitesimal generator. This representation is of use in statistical inference (particularly involving medical imaging). Since R(μ||P) governs the exponential rate of convergence of the empirical measure (according to Sanov’s theorem), this representation is also of use in the numerical and analytical investigation of finite-size effects in systems of interacting diffusions.Entropy2014-12-221612Article10.3390/e16126705670567211099-43002014-12-22doi: 10.3390/e16126705Oliver FaugerasJames MacLaurin<![CDATA[Entropy, Vol. 16, Pages 6686-6704: Effect of the Postural Challenge on the Dependence of the Cardiovascular Control Complexity on Age]]>
http://mdpi.com/1099-4300/16/12/6686
Short-term complexity of heart period (HP) and systolic arterial pressure (SAP) was computed to detect age and gender influences over cardiovascular control in resting supine condition (REST) and during standing (STAND). Healthy subjects (n = 110, men = 55) were equally divided into five groups (21–30; 31–40; 41–50; 51–60; and 61–70 years of age). HP and SAP series were recorded for 15 min at REST and during STAND. A normalized complexity index (NCI) based on conditional entropy was assessed. At REST we found that both NCIHP and NCISAP decreased with age in the overall population, but only women were responsible for this trend. During STAND we observed that both NCIHP and NCISAP were unrelated to age in the overall population, even when divided by gender. When the variation of NCI in response to STAND (ΔNCI = NCI at REST-NCI during STAND) was computed individually, we found that ΔNCIHP progressively decreased with age in the overall population, and women were again responsible for this trend. Conversely, ΔNCISAP was unrelated to age and gender. This study stresses that the complexity of cardiovascular control and its ability to respond to stressors are more importantly lost with age in women than in men.Entropy2014-12-221612Article10.3390/e16126686668667041099-43002014-12-22doi: 10.3390/e16126686Aparecida CataiAnielle TakahashiNatália PerseguiniJuliana MilanVinicius MinatelPatrícia Rehder-SantosAndrea MarchiVlasta BariAlberto Porta<![CDATA[Entropy, Vol. 16, Pages 6667-6685: Fast Rate Estimation for RDO Mode Decision in HEVC]]>
http://mdpi.com/1099-4300/16/12/6667
The latter-day H.265/HEVC video compression standard is able to provide two-times higher compression efficiency compared to the current industrial standard, H.264/AVC. However, coding complexity also increased. The main bottleneck of the compression process is the rate-distortion optimization (RDO) stage, as it involves numerous sequential syntax-based binary arithmetic coding (SBAC) loops. In this paper, we present an entropy-based RDO estimation technique for H.265/HEVC compression, instead of the common approach based on the SBAC. Our RDO implementation reduces RDO complexity, providing an average bit rate overhead of 1.54%. At the same time, elimination of the SBAC from the RDO estimation reduces block interdependencies, thus providing an opportunity for the development of the compression system with parallel processing of multiple blocks of a video frame.Entropy2014-12-191612Article10.3390/e16126667666766851099-43002014-12-19doi: 10.3390/e16126667Maxim SharabaykoOleg Ponomarev<![CDATA[Entropy, Vol. 16, Pages 6654-6666: The Effects of Spontaneous Random Activity on Information Transmission in an Auditory Brain Stem Neuron Model]]>
http://mdpi.com/1099-4300/16/12/6654
This paper presents the effects of spontaneous random activity on information transmission in an auditory brain stem neuron model. In computer simulations, the supra-threshold synaptic current stimuli ascending from auditory nerve fibers (ANFs) were modeled by a filtered inhomogeneous Poisson process modulated by sinusoidal functions at a frequency of 220–3520 Hz with regard to the human speech spectrum. The stochastic sodium and stochastic high- and low-threshold potassium channels were incorporated into a single compartment model of the soma in spherical bushy neurons, so as to realize threshold fluctuations or a variation of spike firing times. The results show that the information rates estimated from the entropy of inter-spike intervals of spike trains tend toward a convex function of the spontaneous rates when the intensity of sinusoidal functions decreases. Furthermore, the results show that a convex function of the spontaneous rates tends to disappear as the frequency of the sinusoidal function increases, such that the phase-locked response can be unobserved. It is concluded that this sort of stochastic resonance (SR) phenomenon, which depends on the spontaneous rates with supra-threshold stimuli, can better enhance information transmission in a smaller intensity of sinusoidal functions within the human speech spectrum, like the situation in which the regular SR can enhance weak signals.Entropy2014-12-191612Article10.3390/e16126654665466661099-43002014-12-19doi: 10.3390/e16126654Hiroyuki Mino<![CDATA[Entropy, Vol. 16, Pages 6624-6653: The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals]]>
http://mdpi.com/1099-4300/16/12/6624
For the simplest colored branching process, we prove an analog to the McMillan theorem and calculate the Hausdorff dimensions of random fractals defined in terms of the limit behavior of empirical measures generated by finite genetic lines. In this setting, the role of Shannon’s entropy is played by the Kullback–Leibler divergence, and the Hausdorff dimensions are computed by means of the so-called Billingsley–Kullback entropy, defined in the paper.Entropy2014-12-191612Article10.3390/e16126624662466531099-43002014-12-19doi: 10.3390/e16126624Victor Bakhtin<![CDATA[Entropy, Vol. 16, Pages 6602-6623: Detection and Modeling of Cyber Attacks with Petri Nets]]>
http://mdpi.com/1099-4300/16/12/6602
The aim of this article is to present an approach to develop and verify a method of formal modeling of cyber threats directed at computer systems. Moreover, the goal is to prove that the method enables one to create models resembling the behavior of malware that support the detection process of selected cyber attacks and facilitate the application of countermeasures. The most common cyber threats targeting end users and terminals are caused by malicious software, called malware. The malware detection process can be performed either by matching their digital signatures or analyzing their behavioral models. As the obfuscation techniques make the malware almost undetectable, the classic signature-based anti-virus tools must be supported with behavioral analysis. The proposed approach to modeling of malware behavior is based on colored Petri nets. This article is addressed to cyber defense researchers, security architects and developers solving up-to-date problems regarding the detection and prevention of advanced persistent threats.Entropy2014-12-191612Article10.3390/e16126602660266231099-43002014-12-19doi: 10.3390/e16126602Bartosz JasiulMarcin SzpyrkaJoanna Śliwa<![CDATA[Entropy, Vol. 16, Pages 6590-6601: Depth Image Coding Using Entropy-Based Adaptive Measurement Allocation]]>
http://mdpi.com/1099-4300/16/12/6590
Differently from traditional two-dimensional texture images, the depth images of three-dimensional (3D) video systems have significant sparse characteristics under the certain transform basis, which make it possible for compressive sensing to represent depth information efficiently. Therefore, in this paper, a novel depth image coding scheme is proposed based on a block compressive sensing method. At the encoder, in view of the characteristics of depth images, the entropy of pixels in each block is employed to represent the sparsity of depth signals. Then according to the different sparsity in the pixel domain, the measurements can be adaptively allocated to each block for higher compression efficiency. At the decoder, the sparse transform can be combined to achieve the compressive sensing reconstruction. Experimental results have shown that at the same sampling rate, the proposed scheme can obtain higher PSNR values and better subjective quality of the rendered virtual views, compared with the method using a uniform sampling rate.Entropy2014-12-171612Article10.3390/e16126590659066011099-43002014-12-17doi: 10.3390/e16126590Huihui BaiMengmeng ZhangMeiqin LiuAnhong WangYao Zhao<![CDATA[Entropy, Vol. 16, Pages 6573-6589: Automatic Sleep Stages Classification Using EEG Entropy Features and Unsupervised Pattern Analysis Techniques]]>
http://mdpi.com/1099-4300/16/12/6573
Sleep is a growing area of research interest in medicine and neuroscience. Actually, one major concern is to find a correlation between several physiologic variables and sleep stages. There is a scientific agreement on the characteristics of the five stages of human sleep, based on EEG analysis. Nevertheless, manual stage classification is still the most widely used approach. This work proposes a new automatic sleep classification method based on unsupervised feature classification algorithms recently developed, and on EEG entropy measures. This scheme extracts entropy metrics from EEG records to obtain a feature vector. Then, these features are optimized in terms of relevance using the Q-α algorithm. Finally, the resulting set of features is entered into a clustering procedure to obtain a final segmentation of the sleep stages. The proposed method reached up to an average of 80% correctly classified stages for each patient separately while keeping the computational cost low.Entropy2014-12-171612Article10.3390/e16126573657365891099-43002014-12-17doi: 10.3390/e16126573Jose Rodríguez-SoteloAlejandro Osorio-ForeroAlejandro Jiménez-RodríguezDavid Cuesta-FrauEva Cirugeda-RoldánDiego Peluffo<![CDATA[Entropy, Vol. 16, Pages 6553-6572: Enhanced Automatic Wavelet Independent Component Analysis for Electroencephalographic Artifact Removal]]>
http://mdpi.com/1099-4300/16/12/6553
Electroencephalography (EEG) is a fundamental diagnostic instrument for many neurological disorders, and it is the main tool for the investigation of the cognitive or pathological activity of the brain through the bioelectromagnetic fields that it generates. The correct interpretation of the EEG is misleading, both for clinicians’ visual evaluation and for automated procedures, because of artifacts. As a consequence, artifact rejection in EEG is a key preprocessing step, and the quest for reliable automatic processors has been quickly growing in the last few years. Recently, a promising automatic methodology, known as automatic wavelet-independent component analysis (AWICA), has been proposed. In this paper, a more efficient and sensitive version, called enhanced-AWICA (EAWICA), is proposed, and an extensive performance comparison is carried out by a step of tuning the different parameters that are involved in artifact detection. EAWICA is shown to minimize information loss and to outperform AWICA in artifact removal, both on simulated and real experimental EEG recordings.Entropy2014-12-171612Article10.3390/e16126553655365721099-43002014-12-17doi: 10.3390/e16126553Nadia MammoneFrancesco Morabito<![CDATA[Entropy, Vol. 16, Pages 6539-6552: Chaos Control and Synchronization of a Hyperchaotic Zhou System by Integral Sliding Mode control]]>
http://mdpi.com/1099-4300/16/12/6539
In this paper, an adaptive integral sliding mode control scheme is proposed for synchronization of hyperchaotic Zhou systems. In the proposed scheme, an integral sliding mode control is designed to stabilize a hyperchaotic Zhou system with known parameters to its unstable equilibrium at the origin. The control is then applied to the synchronization of two identical systems, i.e., a slave and a master hyperchaotic Zhou system with unknown parameters. The adaptive control mechanism introduced synchronizes the systems by estimating the unknown parameters. Simulation results have shown that the proposed method has an excellent convergence from both speed and accuracy points of view, and it outperforms Vaidyanathan’s scheme, which is a well-recognized scheme in this area.Entropy2014-12-121612Article10.3390/e16126539653965521099-43002014-12-12doi: 10.3390/e16126539Yashar ToopchiJidong Wang<![CDATA[Entropy, Vol. 16, Pages 6524-6538: Consensus of Discrete Multiagent System with Various Time Delays and Environmental Disturbances]]>
http://mdpi.com/1099-4300/16/12/6524
In this paper, the consensus problem of discrete multiagent systems with time varying sampling periods is studied. Firstly, with thorough analysis of various delays among agents, the control input of each agent is designed with consideration of sending delay and receiving delay. With construction of discrete dynamics of state error vector, it is proved by applying Halanay inequality that consensus of the system can be reached. Further, the definition of bounded consensus is proposed in the situation where environmental disturbances exist. In order to handle this problem, the Halanay inequality is extended into a more general one with boundedness property. Based on the new Halanay inequality obtained, the boundedness of consensus error is guaranteed. At last, simulation examples are presented to demonstrate the theoretical conclusions.Entropy2014-12-111612Article10.3390/e16126524652465381099-43002014-12-11doi: 10.3390/e16126524Zheping YanDi WuYibo Liu<![CDATA[Entropy, Vol. 16, Pages 6515-6523: Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature]]>
http://mdpi.com/1099-4300/16/12/6515
In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying microscopic model of a thermodynamic system, which poses a limitation on the usefulness of Ruppeiner’s metric and conjecture. Finally, we address the problem of the choice of coordinates in black hole thermodynamics. We propose an alternative energy representation for Kerr-Newman black holes that mimics fully Weinhold’s approach. The corresponding Ruppeiner’s metrics become degenerate only at absolute zero and have non-vanishing scalar curvatures.Entropy2014-12-111612Article10.3390/e16126515651565231099-43002014-12-11doi: 10.3390/e16126515Miguel García-ArizaMerced MontesinosGerardo Torres del Castillo<![CDATA[Entropy, Vol. 16, Pages 6497-6514: Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle]]>
http://mdpi.com/1099-4300/16/12/6497
Certain fluctuations in particle number, \(n\), at fixed total energy, \(E\), lead exactly to a cut-power law distribution in the one-particle energy, \(\omega\), via the induced fluctuations in the phase-space volume ratio, \(\Omega_n(E-\omega)/\Omega_n(E)=(1-\omega/E)^n\). The only parameters are \(1/T=\langle \beta \rangle=\langle n \rangle/E\) and \(q=1-1/\langle n \rangle + \Delta n^2/\langle n \rangle^2\). For the binomial distribution of \(n\) one obtains \(q=1-1/k\), for the negative binomial \(q=1+1/(k+1)\). These results also represent an approximation for general particle number distributions in the reservoir up to second order in the canonical expansion \(\omega \ll E\). For general systems the average phase-space volume ratio \(\langle e^{S(E-\omega)}/e^{S(E)}\rangle\) to second order delivers \(q=1-1/C+\Delta \beta^2/\langle \beta \rangle^2\) with \(\beta=S^{\prime}(E)\) and \(C=dE/dT\) heat capacity. However, \(q \ne 1\) leads to non-additivity of the Boltzmann–Gibbs entropy, \(S\). We demonstrate that a deformed entropy, \(K(S)\), can be constructed and used for demanding additivity, i.e., \(q_K=1\). This requirement leads to a second order differential equation for \(K(S)\). Finally, the generalized \(q\)-entropy formula, \(K(S)=\sum p_i K(-\ln p_i)\), contains the Tsallis, Rényi and Boltzmann–Gibbs–Shannon expressions as particular cases. For diverging variance, \(\Delta\beta^2\) we obtain a novel entropy formula.Entropy2014-12-091612Article10.3390/e16126497649765141099-43002014-12-09doi: 10.3390/e16126497Tamás BiróPéter VánGergely BarnaföldiKároly Ürmössy<![CDATA[Entropy, Vol. 16, Pages 6477-6496: An Evolutionary Algorithm for the Texture Analysis of Cubic System Materials Derived by the Maximum Entropy Principle]]>
http://mdpi.com/1099-4300/16/12/6477
Based on the principle of maximum entropy method (MEM) for quantitative texture analysis, the differential evolution (DE) algorithm was effectively introduced. Using a DE-optimized algorithm with a faster but more stable convergence rate of iteration reliable complete orientation distributions (C-ODF) have been obtained for deep-drawn IF steel sheets and the recrystallized aluminum foils after cold-rolling, which are designated as showing a macroscopic cubic-orthogonal symmetry. With special reference to the data processing, no more other assumptions are required for DE-optimized MEM except that the system entropy approach the maximum.Entropy2014-12-091612Article10.3390/e16126477647764961099-43002014-12-09doi: 10.3390/e16126477Dapeng WangDazhi WangBaolin WuFu WangZhide Liang<![CDATA[Entropy, Vol. 16, Pages 6464-6476: A Memristive Diode Bridge-Based Canonical Chua’s Circuit]]>
http://mdpi.com/1099-4300/16/12/6464
A novel memristor circuit is presented, which is generated from the canonical Chua’s circuit by replacing the Chua’s diode with a first order memristive diode bridge. The circuit dynamical characteristics with the variations of circuit parameters are investigated both theoretically and numerically. It can be found that the circuit has three determined equilibrium points, including a zero saddle point and two nonzero saddle-foci with index 2. Specially, the circuit is non-dissipative in the neighborhood of the zero saddle point, and there exists complex nonlinear phenomena of coexisting bifurcation modes and coexisting chaotic attractors. Experimental observations are performed to verify the numerical simulation results.Entropy2014-12-081612Article10.3390/e16126464646464761099-43002014-12-08doi: 10.3390/e16126464Mo ChenJingjing YuQing YuChangdi LiBocheng Bao<![CDATA[Entropy, Vol. 16, Pages 6434-6463: An Entropy-Based Damage Characterization]]>
http://mdpi.com/1099-4300/16/12/6434
This paper presents a scientific basis for the description of the causes of damage within an irreversible thermodynamic framework and the effects of damage as observable variables that signify degradation of structural integrity. The approach relies on the fundamentals of irreversible thermodynamics and specifically the notion of entropy generation as a measure of degradation and damage. We first review the state-of-the-art advances in entropic treatment of damage followed by a discussion on generalization of the entropic concept to damage characterization that may offers a better definition of damage metric commonly used for structural integrity assessment. In general, this approach provides the opportunity to described reliability and risk of structures in terms of fundamental science concepts. Over the years, many studies have focused on materials damage assessment by determining physics-based cause and affect relationships, the goal of this paper is to put this work in perspective and encourage future work of materials damage based on the entropy concept.Entropy2014-12-051612Article10.3390/e16126434643464631099-43002014-12-05doi: 10.3390/e16126434Mehdi AmiriMohammad Modarres<![CDATA[Entropy, Vol. 16, Pages 6415-6433: Detecting Chaos from Agricultural Product Price Time Series]]>
http://mdpi.com/1099-4300/16/12/6415
Analysis of the characteristics of agricultural product price volatility and trend forecasting are necessary to formulate and implement agricultural price control policies. Taking wholesale cabbage prices as an example, a multiple test methodology has been adopted to identify the nonlinearity, fractality, and chaos of the data. The approaches used include the R/S analysis, the BDS test, the power spectra, the recurrence plot, the largest Lyapunov exponent, the Kolmogorov entropy, and the correlation dimension. The results show that there is chaos in agricultural wholesale price data, which provides a good theoretical basis for selecting reasonable forecasting models as prediction techniques based on chaos theory can be applied to forecasting agricultural prices.Entropy2014-12-051612Article10.3390/e16126415641564331099-43002014-12-05doi: 10.3390/e16126415Xin SuYi WangShengsen DuanJunhai Ma<![CDATA[Entropy, Vol. 16, Pages 6394-6414: Determining Common Weights in Data Envelopment Analysis with Shannon’s Entropy]]>
http://mdpi.com/1099-4300/16/12/6394
Data Envelopment Analysis (DEA) is a non-parametric method for evaluating the efficiency of Decision Making Units (DMUs) with multiple inputs and outputs. In the traditional DEA models, the DMU is allowed to use its most favorable multiplier weights to maximize its efficiency. There is usually more than one efficient DMU which cannot be further discriminated. Evaluating DMUs with different multiplier weights would also be somewhat irrational in practice. The common weights DEA model is an effective method for solving these problems. In this paper, we propose a methodology combining the common weights DEA with Shannon’s entropy. In our methodology, we propose a modified weight restricted DEA model for calculating non-zero optimal weights. Then these non-zero optimal weights would be aggregated to be the common weights using Shannon’s entropy. Compared with the traditional models, our proposed method is more powerful in discriminating DMUs, especially when the inputs and outputs are numerous. Our proposed method also keeps in accordance with the basic DEA method considering the evaluation of the most efficient and inefficient DMUs. Numerical examples are provided to examine the validity and effectiveness of our proposed methodology.Entropy2014-12-041612Article10.3390/e16126394639464141099-43002014-12-04doi: 10.3390/e16126394Xiao-Guang QiBo Guo<![CDATA[Entropy, Vol. 16, Pages 6384-6393: Entropy Analysis of RR and QT Interval Variability during Orthostatic and Mental Stress in Healthy Subjects]]>
http://mdpi.com/1099-4300/16/12/6384
Autonomic activity affects beat-to-beat variability of heart rate and QT interval. The aim of this study was to explore whether entropy measures are suitable to detect changes in neural outflow to the heart elicited by two different stress paradigms. We recorded short-term ECG in 11 normal subjects during an experimental protocol that involved head-up tilt and mental arithmetic stress and computed sample entropy, cross-sample entropy and causal interactions based on conditional entropy from RR and QT interval time series. Head-up tilt resulted in a significant reduction in sample entropy of RR intervals and cross-sample entropy, while mental arithmetic stress resulted in a significant reduction in coupling directed from RR to QT. In conclusion, measures of entropy are suitable to detect changes in neural outflow to the heart and decoupling of repolarisation variability from heart rate variability elicited by orthostatic or mental arithmetic stress.Entropy2014-12-031612Article10.3390/e16126384638463931099-43002014-12-03doi: 10.3390/e16126384Mathias BaumertBarbora CzippelovaAnand GanesanMartin SchmidtSebastian ZaunsederMichal Javorka<![CDATA[Entropy, Vol. 16, Pages 6382-6383: Correction on Iliyasu, A.M. et al. Hybrid Quantum-Classical Protocol for Storage and Retrieval of Discrete-Valued Information. Entropy, 2014, 16, 3537-3551]]>
http://mdpi.com/1099-4300/16/12/6382
The authors wish to make the following corrections to this paper [1]: The correct name of the fourth author is: Ahmed S. Salama. In the Acknowledgment Section, we added the research Project No. 2014/01/2079. Below is the corrected version of the section.[...]Entropy2014-12-021612Correction10.3390/e16126382638263831099-43002014-12-02doi: 10.3390/e16126382Abdullah IliyasuSalvador Venegas-AndracaFei YanAhmed Salama<![CDATA[Entropy, Vol. 16, Pages 6338-6381: The Information Geometry of Bregman Divergences and Some Applications in Multi-Expert Reasoning]]>
http://mdpi.com/1099-4300/16/12/6338
The aim of this paper is to develop a comprehensive study of the geometry involved in combining Bregman divergences with pooling operators over closed convex sets in a discrete probabilistic space. A particular connection we develop leads to an iterative procedure, which is similar to the alternating projection procedure by Csiszár and Tusnády. Although such iterative procedures are well studied over much more general spaces than the one we consider, only a few authors have investigated combining projections with pooling operators. We aspire to achieve here a comprehensive study of such a combination. Besides, pooling operators combining the opinions of several rational experts allows us to discuss possible applications in multi-expert reasoning.Entropy2014-12-011612Article10.3390/e16126338633863811099-43002014-12-01doi: 10.3390/e16126338Martin Adamčík<![CDATA[Entropy, Vol. 16, Pages 6313-6337: Landscape Analysis of Geographical Names in Hubei Province, China]]>
http://mdpi.com/1099-4300/16/12/6313
Hubei Province is the hub of communications in central China, which directly determines its strategic position in the country’s development. Additionally, Hubei Province is well-known for its diverse landforms, including mountains, hills, mounds and plains. This area is called “The Province of Thousand Lakes” due to the abundance of water resources. Geographical names are exclusive names given to physical or anthropogenic geographic entities at specific spatial locations and are important signs by which humans understand natural and human activities. In this study, geographic information systems (GIS) technology is adopted to establish a geodatabase of geographical names with particular characteristics in Hubei Province and extract certain geomorphologic and environmental factors. We carry out landscape analysis of mountain-related geographical names and water-related geographical names respectively. In the end, we calculate the information entropy of geographical names of each county to describe the diversity and inhomogeneity of place names in Hubei province. Our study demonstrates that geographical names represent responses to the cultural landscape and physical environment. The geographical names are more interesting in specific landscapes, such as mountains and rivers.Entropy2014-12-011612Article10.3390/e16126313631363371099-43002014-12-01doi: 10.3390/e16126313Xixi ChenTao HuFu RenDeng ChenLan LiNan Gao<![CDATA[Entropy, Vol. 16, Pages 6300-6312: Ab intio Investigation of the Thermochemistry and Kinetics of the SO2 + O3− → SO3− + O2 Reaction in Aircraft Engines and the Environment]]>
http://mdpi.com/1099-4300/16/12/6300
In the present work, the mechanisms, thermochemistry and kinetics of the reaction of SO2 with O3− have been studied using the CCSD(T)/6-31G(d) + CF method. It has been shown that there exist two possible pathways A and B of the SO2 + O3− → SO3− + O2 reaction. The two pathways’ A and B barrier heights are 0.61 kcal mol−1 and 3.40 kcal mol−1, respectively, while the energy of the SO2 + O3− → SO3− + O2 reaction is −25.25 kcal mol−1. The canonical variational transition state theory with small-curvature tunneling (CVT/SCT) has been applied to study the reaction kinetics. The CVT/SCT study shows that the rate constants K for pathways A and B, KA = 1.11 × 10−12exp(−2526.13/T) and KB = 2.7 × 10−14exp(−1029.25/T), respectively, grow as the temperature increases and are much larger than those of the SO2 + O3 → SO3 + O2 reaction over the entire temperature range of 200–1500 K. This indicates that ionization of O3 and high temperatures are favorable for the SO2 oxidation via the reaction with ozone. The new data obtained in the present study can be utilized directly for the evaluation of experiments and model predictions concerning SO2 oxidation and kinetic modeling of gas-phase chemistry of pollutants/nucleation precursors formed in aircraft engines and the Earth’s atmosphere.Entropy2014-12-011612Article10.3390/e16126300630063121099-43002014-12-01doi: 10.3390/e16126300Xuechao GuoAlexey NadyktoYisheng XuQingzhu ZhangJingtian Hu<![CDATA[Entropy, Vol. 16, Pages 6286-6299: Adaptive Synchronization of Fractional Neural Networks with Unknown Parameters and Time Delays]]>
http://mdpi.com/1099-4300/16/12/6286
In this paper, the parameters identification and synchronization problem of fractional-order neural networks with time delays are investigated. Based on some analytical techniques and an adaptive control method, a simple adaptive synchronization controller and parameter update laws are designed to synchronize two uncertain complex networks with time delays. Besides, the system parameters in the uncertain network can be identified in the process of synchronization. To demonstrate the validity of the proposed method, several illustrative examples are presented.Entropy2014-12-011612Article10.3390/e16126286628662991099-43002014-12-01doi: 10.3390/e16126286Weiyuan MaChangpin LiYujiang WuYongqing Wu<![CDATA[Entropy, Vol. 16, Pages 6263-6285: A Thermodynamical Selection-Based Discrete Differential Evolution for the 0-1 Knapsack Problem]]>
http://mdpi.com/1099-4300/16/12/6263
Many problems in business and engineering can be modeled as 0-1 knapsack problems. However, the 0-1 knapsack problem is one of the classical NP-hard problems. Therefore, it is valuable to develop effective and efficient algorithms for solving 0-1 knapsack problems. Aiming at the drawbacks of the selection operator in the traditional differential evolution (DE), we present a novel discrete differential evolution (TDDE) for solving 0-1 knapsack problem. In TDDE, an enhanced selection operator inspired by the principle of the minimal free energy in thermodynamics is employed, trying to balance the conflict between the selective pressure and the diversity of population to some degree. An experimental study is conducted on twenty 0-1 knapsack test instances. The comparison results show that TDDE can gain competitive performance on the majority of the test instances.Entropy2014-11-281612Article10.3390/e16126263626362851099-43002014-11-28doi: 10.3390/e16126263Zhaolu GuoXuezhi YueKejun ZhangShenwen WangZhijian Wu<![CDATA[Entropy, Vol. 16, Pages 6254-6262: On a Local Fractional Wave Equation under Fixed Entropy Arising in Fractal Hydrodynamics]]>
http://mdpi.com/1099-4300/16/12/6254
In this paper, based on fixed entropy, the adiabatic equation of state in fractal flow is discussed. The local fractional wave equation for the velocity potential is also obtained by using the non-differential perturbations for the pressure and density of fractal hydrodynamics.Entropy2014-11-281612Article10.3390/e16126254625462621099-43002014-11-28doi: 10.3390/e16126254Yu ZhangDumitru BaleanuXiaojun Yang<![CDATA[Entropy, Vol. 16, Pages 6240-6253: Generation and Nonlinear Dynamical Analyses of Fractional-Order Memristor-Based Lorenz Systems]]>
http://mdpi.com/1099-4300/16/12/6240
In this paper, four fractional-order memristor-based Lorenz systems with the flux-controlled memristor characterized by a monotone-increasing piecewise linear function, a quadratic nonlinearity, a smooth continuous cubic nonlinearity and a quartic nonlinearity are presented, respectively. The nonlinear dynamics are analyzed by using numerical simulation methods, including phase portraits, bifurcation diagrams, the largest Lyapunov exponent and power spectrum diagrams. Some interesting phenomena, such as inverse period-doubling bifurcation and intermittent chaos, are found to exist in the proposed systems.Entropy2014-11-281612Article10.3390/e16126240624062531099-43002014-11-28doi: 10.3390/e16126240Huiling XiYuxia LiXia Huang<![CDATA[Entropy, Vol. 16, Pages 6212-6239: Ordinal Patterns, Entropy, and EEG]]>
http://mdpi.com/1099-4300/16/12/6212
In this paper we illustrate the potential of ordinal-patterns-based methods for analysis of real-world data and, especially, of electroencephalogram (EEG) data. We apply already known (empirical permutation entropy, ordinal pattern distributions) and new (empirical conditional entropy of ordinal patterns, robust to noise empirical permutation entropy) methods for measuring complexity, segmentation and classification of time series.Entropy2014-11-271612Article10.3390/e16126212621262391099-43002014-11-27doi: 10.3390/e16126212Karsten KellerAnton UnakafovValentina Unakafova<![CDATA[Entropy, Vol. 16, Pages 6195-6211: Complex Modified Hybrid Projective Synchronization of Different Dimensional Fractional-Order Complex Chaos and Real Hyper-Chaos]]>
http://mdpi.com/1099-4300/16/12/6195
This paper introduces a type of modified hybrid projective synchronization with complex transformationmatrix (CMHPS) for different dimensional fractional-order complex chaos and fractional-order real hyper-chaos. The transformationmatrix in this type of chaotic synchronization is a non-square matrix, and its elements are complex numbers. Based on the stability theory of fractional-order systems, by employing the feedback control technique, necessary and sufficient criteria on CMHPS are derived. Furthermore, CMHPS between fractional-order real hyper-chaotic Rössler system and other two different dimensional fractional-order complex Lorenz-like chaotic systems is provided as two examples to discuss reduced order and increased order synchronization, respectively.Entropy2014-11-271612Article10.3390/e16126195619562111099-43002014-11-27doi: 10.3390/e16126195Jian Liu<![CDATA[Entropy, Vol. 16, Pages 6186-6194: What You See Is What You Get]]>
http://mdpi.com/1099-4300/16/11/6186
This paper corrects three widely held misunderstandings about Maxent when used in common sense reasoning: That it is language dependent; That it produces objective facts; That it subsumes, and so is at least as untenable as, the paradox-ridden Principle of Insufficient Reason.Entropy2014-11-211611Article10.3390/e16116186618661941099-43002014-11-21doi: 10.3390/e16116186Jeff Paris<![CDATA[Entropy, Vol. 16, Pages 6166-6185: Self-oscillating Water Chemiluminescence Modes and Reactive Oxygen Species Generation Induced by Laser Irradiation; Effect of the Exclusion Zone Created by Nafion]]>
http://mdpi.com/1099-4300/16/11/6166
Samples of water inside and outside an exclusion zone (EZ), created by Nafion swollen in water, were irradiated at the wavelength l = 1264 nm, which stimulates the electronic transition of dissolved oxygen from the triplet state to the excited singlet state. This irradiation induces, after a long latent period, chemiluminescence self-oscillations in the visible and near UV spectral range, which last many hours. It occurs that this effect is EZ-specific: the chemiluminescence intensity is twice lower than that from the bulk water, while the latent period is longer for the EZ. Laser irradiation causes accumulation of H2O2, which is also EZ-specific: its concentration inside the EZ is less than that in the bulk water. These phenomena can be interpreted in terms of a model of decreasing O2 content in the EZ due to increased chemical activity of bisulfite anions (HSO3−), arisen as the result of dissociation of terminal sulfonate groups of the Nafion. The wavelet transform analysis of the chemiluminescence intensity from the EZ and the bulk water gives, that self-oscillations regimes occurring in the liquid after the latent period are the determinate processes. It occurred that the chemiluminescence dynamics in case of EZ is characterized by a single-frequency self-oscillating regime, whereas in case of the bulk water, the self-oscillation spectrum consists of three spectral bands.Entropy2014-11-211611Article10.3390/e16116166616661851099-43002014-11-21doi: 10.3390/e16116166Sergey GudkovMaxim AstashevVadim BruskovValeriy KozlovStanislav ZakharovNikolai Bunkin<![CDATA[Entropy, Vol. 16, Pages 6152-6165: Improving the Authentication Scheme and Access Control Protocol for VANETs]]>
http://mdpi.com/1099-4300/16/11/6152
Privacy and security are very important in vehicular ad hoc networks (VANETs). VANETs are negatively affected by any malicious user’s behaviors, such as bogus information and replay attacks on the disseminated messages. Among various security threats, privacy preservation is one of the new challenges of protecting users’ private information. Existing authentication protocols to secure VANETs raise challenges, such as certificate distribution and reduction of the strong reliance on tamper-proof devices. In 2011, Yeh et al. proposed a PAACP: a portable privacy-preserving authentication and access control protocol in vehicular ad hoc networks. However, PAACP in the authorization phase is breakable and cannot maintain privacy in VANETs. In this paper, we present a cryptanalysis of an attachable blind signature and demonstrate that the PAACP’s authorized credential (AC) is not secure and private, even if the AC is secretly stored in a tamper-proof device. An eavesdropper can construct an AC from an intercepted blind document. Any eavesdropper can determine who has which access privileges to access which service. For this reason, this paper copes with these challenges and proposes an efficient scheme. We conclude that an improving authentication scheme and access control protocol for VANETs not only resolves the problems that have appeared, but also is more secure and efficient.Entropy2014-11-191611Article10.3390/e16116152615261651099-43002014-11-19doi: 10.3390/e16116152Wei-Chen WuYi-Ming Chen<![CDATA[Entropy, Vol. 16, Pages 6133-6151: Application of Entropy and Fractal Dimension Analyses to the Pattern Recognition of Contaminated Fish Responses in Aquaculture]]>
http://mdpi.com/1099-4300/16/11/6133
The objective of the work was to develop a non-invasive methodology for image acquisition, processing and nonlinear trajectory analysis of the collective fish response to a stochastic event. Object detection and motion estimation were performed by an optical flow algorithm in order to detect moving fish and simultaneously eliminate background, noise and artifacts. The Entropy and the Fractal Dimension (FD) of the trajectory followed by the centroids of the groups of fish were calculated using Shannon and permutation Entropy and the Katz, Higuchi and Katz-Castiglioni’s FD algorithms respectively. The methodology was tested on three case groups of European sea bass (Dicentrarchus labrax), two of which were similar (C1 control and C2 tagged fish) and very different from the third (C3, tagged fish submerged in methylmercury contaminated water). The results indicate that Shannon entropy and Katz-Castiglioni were the most sensitive algorithms and proved to be promising tools for the non-invasive identification and quantification of differences in fish responses. In conclusion, we believe that this methodology has the potential to be embedded in online/real time architecture for contaminant monitoring programs in the aquaculture industry.Entropy2014-11-191611Article10.3390/e16116133613361511099-43002014-11-19doi: 10.3390/e16116133Harkaitz EguiraunKarmele López-de-IpiñaIciar Martinez<![CDATA[Entropy, Vol. 16, Pages 6116-6132: Entropy Generation during Turbulent Flow of Zirconia-water and Other Nanofluids in a Square Cross Section Tube with a Constant Heat Flux]]>
http://mdpi.com/1099-4300/16/11/6116
The entropy generation based on the second law of thermodynamics is investigated for turbulent forced convection flow of ZrO2-water nanofluid through a square pipe with constant wall heat flux. Effects of different particle concentrations, inlet conditions and particle sizes on entropy generation of ZrO2-water nanofluid are studied. Contributions from frictional and thermal entropy generations are investigated, and the optimal working condition is analyzed. The results show that the optimal volume concentration of nanoparticles to minimize the entropy generation increases when the Reynolds number decreases. It was also found that the thermal entropy generation increases with the increase of nanoparticle size whereas the frictional entropy generation decreases. Finally, the entropy generation of ZrO2-water was compared with that from other nanofluids (including Al2O3, SiO2 and CuO nanoparticles in water). The results showed that the SiO2 provided the highest entropy generation.Entropy2014-11-191611Article10.3390/e16116116611661321099-43002014-11-19doi: 10.3390/e16116116Hooman YarmandGoodarz AhmadiSamira GharehkhaniSalim KaziMohammad SafaeiMaryam AlehashemAbu Mahat<![CDATA[Entropy, Vol. 16, Pages 6099-6115: Image Fusion Based on the \({\Delta ^{ - 1}} - T{V_0}\) Energy Function]]>
http://mdpi.com/1099-4300/16/11/6099
This article proposes a \({\Delta^{-1}}-T{V_0}\) energy function to fuse a multi-spectral image with a panchromatic image. The proposed energy function consists of two components, a \(TV_0\) component and a \(\Delta^{-1}\) component. The \(TV_0\) term uses the sparse priority to increase the detailed spatial information; while the \({\Delta ^{ - 1}}\) term removes the block effect of the multi-spectral image. Furthermore, as the proposed energy function is non-convex, we also adopt an alternative minimization algorithm and the \(L_0\) gradient minimization to solve it. Experimental results demonstrate the improved performance of the proposed method over existing methods.Entropy2014-11-181611Article10.3390/e16116099609961151099-43002014-11-18doi: 10.3390/e16116099Qiwei XieChao MaChunzhao GuoVijay JohnSeiichi MitaQian Long<![CDATA[Entropy, Vol. 16, Pages 6059-6098: How to Read Probability Distributions as Statements about Process]]>
http://mdpi.com/1099-4300/16/11/6059
Probability distributions can be read as simple expressions of information. Each continuous probability distribution describes how information changes with magnitude. Once one learns to read a probability distribution as a measurement scale of information, opportunities arise to understand the processes that generate the commonly observed patterns. Probability expressions may be parsed into four components: the dissipation of all information, except the preservation of average values, taken over the measurement scale that relates changes in observed values to changes in information, and the transformation from the underlying scale on which information dissipates to alternative scales on which probability pattern may be expressed. Information invariances set the commonly observed measurement scales and the relations between them. In particular, a measurement scale for information is defined by its invariance to specific transformations of underlying values into measurable outputs. Essentially all common distributions can be understood within this simple framework of information invariance and measurement scale.Entropy2014-11-181611Review10.3390/e16116059605960981099-43002014-11-18doi: 10.3390/e16116059Steven Frank<![CDATA[Entropy, Vol. 16, Pages 6042-6058: Informational Non-Differentiable Entropy and Uncertainty Relations in Complex Systems]]>
http://mdpi.com/1099-4300/16/11/6042
Considering that the movements of complex system entities take place on continuous, but non-differentiable, curves, concepts, like non-differentiable entropy, informational non-differentiable entropy and informational non-differentiable energy, are introduced. First of all, the dynamics equations of the complex system entities (Schrödinger-type or fractal hydrodynamic-type) are obtained. The last one gives a specific fractal potential, which generates uncertainty relations through non-differentiable entropy. Next, the correlation between informational non-differentiable entropy and informational non-differentiable energy implies specific uncertainty relations through a maximization principle of the informational non-differentiable entropy and for a constant value of the informational non-differentiable energy. Finally, for a harmonic oscillator, the constant value of the informational non-differentiable energy is equivalent to a quantification condition.Entropy2014-11-181611Article10.3390/e16116042604260581099-43002014-11-18doi: 10.3390/e16116042Maricel AgopAlina GavriluțGabriel CrumpeiBogdan Doroftei<![CDATA[Entropy, Vol. 16, Pages 6033-6041: Effect of Atmospheric Ions on Interfacial Water]]>
http://mdpi.com/1099-4300/16/11/6033
The effect of atmospheric positivity on the electrical properties of interfacial water was explored. Interfacial, or exclusion zone (EZ) water was created in the standard way, next to a sheet of Nafion placed horizontally at the bottom of a water-filled chamber. Positive atmospheric ions were created from a high voltage source placed above the chamber. Electrical potential distribution in the interfacial water was measured using microelectrodes. We found that beyond a threshold, the positive ions diminished the magnitude of the negative electrical potential in the interfacial water, sometimes even turning it to positive. Additionally, positive ions produced by an air conditioner were observed to generate similar effects; i.e., the electrical potential shifted in the positive direction but returned to negative when the air conditioner stopped blowing. Sometimes, the effect of the positive ions from the air conditioner was strong enough to destroy the structure of interfacial water by turning the potential decidedly positive. Thus, positive air ions can compromise interfacial water negativity and may explain the known negative impact of positive ions on health.Entropy2014-11-181611Communication10.3390/e16116033603360411099-43002014-11-18doi: 10.3390/e16116033Chien-Chang Kurt KungGerald H. Pollack<![CDATA[Entropy, Vol. 16, Pages 6006-6032: Entropy Generation through a Deterministic Boundary-Layer Structure in Warm Dense Plasma]]>
http://mdpi.com/1099-4300/16/11/6006
The computational prediction of nonlinear interactive instabilities in three-dimensional boundary layers is obtained for a warm dense plasma boundary layer environment. The method is applied to the Richtmyer–Meshkov flow over the rippled surface of a laser-driven warm dense plasma experiment. Coupled, nonlinear spectral velocity equations of Lorenz form are solved with the mean boundary-layer velocity gradients as input control parameters. The nonlinear time series solutions indicate that after an induction period, a sharp instability occurs in the solutions. The power spectral density yields the available kinetic energy dissipation rates within the instability. The application of the singular value decomposition technique to the nonlinear time series solution yields empirical entropies. Empirical entropic indices are then obtained from these entropies. The intermittency exponents obtained from the entropic indices thus allow the computation of the entropy generation through the deterministic structure to the final dissipation of the initial fluctuating kinetic energy into background thermal energy, representing the resulting entropy increase.Entropy2014-11-171611Article10.3390/e16116006600660321099-43002014-11-17doi: 10.3390/e16116006LaVar Isaacson<![CDATA[Entropy, Vol. 16, Pages 5976-6005: A Quantitative Analysis of an EEG Epileptic Record Based on MultiresolutionWavelet Coefficients]]>
http://mdpi.com/1099-4300/16/11/5976
The characterization of the dynamics associated with electroencephalogram (EEG) signal combining an orthogonal discrete wavelet transform analysis with quantifiers originated from information theory is reviewed. In addition, an extension of this methodology based on multiresolution quantities, called wavelet leaders, is presented. In particular, the temporal evolution of Shannon entropy and the statistical complexity evaluated with different sets of multiresolution wavelet coefficients are considered. Both methodologies are applied to the quantitative EEG time series analysis of a tonic-clonic epileptic seizure, and comparative results are presented. In particular, even when both methods describe the dynamical changes of the EEG time series, the one based on wavelet leaders presents a better time resolution.Entropy2014-11-171611Article10.3390/e16115976597660051099-43002014-11-17doi: 10.3390/e16115976Mariel RosenblattAlejandra FigliolaGustavo PaccosiEduardo SerranoOsvaldo Rosso