Entropy
http://mdpi.com/journal/entropy
Latest open access articles published in Entropy at http://mdpi.com/journal/entropy<![CDATA[Entropy, Vol. 17, Pages 1734-1754: Research on the Stability of Open Financial System]]>
http://mdpi.com/1099-4300/17/4/1734
We propose a new herd mechanism and embed it into an open financial market system, which allows traders to get in and out of the system based on some transition rates. Moreover, the novel mechanism can avoid the volatility disappearance when the population scale increases. There are three kinds of heterogeneous agents in the system: optimistic, pessimistic and fundamental. Interactions especially occur among three different groups of agents instead of two, which makes the artificial financial market more close to the real one. By the simulation results of this complex system, we can explain stylized facts like volatility clustering and find the key parameters of market bubbles and market collapses.Entropy2015-03-27174Article10.3390/e17041734173417541099-43002015-03-27doi: 10.3390/e17041734Haijun YangLin LiDeshen Wang<![CDATA[Entropy, Vol. 17, Pages 1701-1733: Synchronicity from Synchronized Chaos]]>
http://mdpi.com/1099-4300/17/4/1701
The synchronization of loosely-coupled chaotic oscillators, a phenomenon investigated intensively for the last two decades, may realize the philosophical concept of “synchronicity”—the commonplace notion that related eventsmysteriously occur at the same time. When extended to continuous media and/or large discrete arrays, and when general (non-identical) correspondences are considered between states, intermittent synchronous relationships indeed become ubiquitous. Meaningful synchronicity follows naturally if meaningful events are identified with coherent structures, defined by internal synchronization between remote degrees of freedom; a condition that has been posited as necessary for synchronizability with an external system. The important case of synchronization between mind and matter is realized if mind is analogized to a computer model, synchronizing with a sporadically observed system, as in meteorological data assimilation. Evidence for the ubiquity of synchronization is reviewed along with recent proposals that: (1) synchronization of different models of the same objective process may be an expeditious route to improved computational modeling and may also describe the functioning of conscious brains; and (2) the nonlocality in quantum phenomena implied by Bell’s theorem may be explained in a variety of deterministic (hidden variable) interpretations if the quantum world resides on a generalized synchronization “manifold”.Entropy2015-03-27174Article10.3390/e17041701170117331099-43002015-03-27doi: 10.3390/e17041701Gregory Duane<![CDATA[Entropy, Vol. 17, Pages 1690-1700: Maximum Entropy and Probability Kinematics Constrained by Conditionals]]>
http://mdpi.com/1099-4300/17/4/1690
Two open questions of inductive reasoning are solved: (1) does the principle of maximum entropy (PME) give a solution to the obverse Majerník problem; and (2) isWagner correct when he claims that Jeffrey’s updating principle (JUP) contradicts PME? Majerník shows that PME provides unique and plausible marginal probabilities, given conditional probabilities. The obverse problem posed here is whether PME also provides such conditional probabilities, given certain marginal probabilities. The theorem developed to solve the obverse Majerník problem demonstrates that in the special case introduced by Wagner PME does not contradict JUP, but elegantly generalizes it and offers a more integrated approach to probability updating.Entropy2015-03-27174Article10.3390/e17041690169017001099-43002015-03-27doi: 10.3390/e17041690Stefan Lukits<![CDATA[Entropy, Vol. 17, Pages 1673-1689: Analysis of Data Complexity in Human DNA for Gene-Containing Zone Prediction]]>
http://mdpi.com/1099-4300/17/4/1673
This study delves further into the analysis of genomic data by computing a variety of complexity measures. We analyze the effect of window size and evaluate the precision and recall of the prediction of gene zones, aided with a much larger dataset (full chromosomes). A technique based on the separation of two cases (gene-containing and non-gene-containing) has been developed as a basic gene predictor for automated DNA analysis. This predictor was tested on various sequences of human DNA obtained from public databases, in a set of three experiments. The first one covers window size and other parameters; the second one corresponds to an analysis of a full human chromosome (198 million nucleic acids); and the last one tests subject variability (with five different individual subjects). All three experiments have high-quality results, in terms of recall and precision, thus indicating the effectiveness of the predictor.Entropy2015-03-27174Article10.3390/e17041673167316891099-43002015-03-27doi: 10.3390/e17041673Ricardo MongeJuan Crespo<![CDATA[Entropy, Vol. 17, Pages 1660-1672: Evolutionary Voluntary Prisoner’s Dilemma Game under Deterministic and Stochastic Dynamics]]>
http://mdpi.com/1099-4300/17/4/1660
The voluntary prisoner’s dilemma (VPD) game has sparked interest from various fields since it was proposed as an effective mechanism to incentivize cooperative behavior. Current studies show that the inherent cyclic dominance of the strategies of the VPD game results in periodic oscillations in population. This paper investigated the influence of the level of individual rationality and the size of a population on the evolutionary dynamics of the VPD game. Different deterministic dynamics, such as the replicator dynamic, the Smith dynamic, the Brown-von Neumann-Nash (BNN) dynamic and the best response (BR) dynamic, for the evolutionary VPD game were modeled and simulated. The stochastic evolutionary dynamics based on quasi birth and death (QBD) process was proposed for the evolutionary VPD game and compared with deterministic dynamics. The results indicated that with the increase of the loners’ fixed payoff, the loner is more likely to remain in the stable state of a VPD game under any of the dynamics mentioned above. However, the different speeds of motion under the dynamics in the cycle dominance proved to be diverse under different evolutionary dynamics and also highly sensitive to the rationality of individuals in a population. Furthermore, in QBD stochastic dynamics, the size of the population has a remarkable effect on the possibility distribution. When the population size increases, the limited distribution of the QBD process will be in accordance with the results in the deterministic dynamics.Entropy2015-03-26174Article10.3390/e17041660166016721099-43002015-03-26doi: 10.3390/e17041660Qian YuRan ChenXiaoyan Wen<![CDATA[Entropy, Vol. 17, Pages 1634-1659: Quantum Discord and Information Deficit in Spin Chains]]>
http://mdpi.com/1099-4300/17/4/1634
We examine the behavior of quantum correlations of spin pairs in a finite anisotropic XY spin chain immersed in a transverse magnetic field, through the analysis of the quantum discord and the conventional and quadratic one-way information deficits. We first provide a brief review of these measures, showing that the last ones can be obtained as particular cases of a generalized information deficit based on general entropic forms. All of these measures coincide with an entanglement entropy in the case of pure states, but can be non-zero in separable mixed states, vanishing just for classically correlated states. It is then shown that their behavior in the exact ground state of the chain exhibits similar features, deviating significantly from that of the pair entanglement below the critical field. In contrast with entanglement, they reach full range in this region, becoming independent of the pair separation and coupling range in the immediate vicinity of the factorizing field. It is also shown, however, that significant differences between the quantum discord and the information deficits arise in the local minimizing measurement that defines them. Both analytical and numerical results are provided.Entropy2015-03-26174Article10.3390/e17041634163416591099-43002015-03-26doi: 10.3390/e17041634Norma CanosaLeonardo CilibertiRaúl Rossignoli<![CDATA[Entropy, Vol. 17, Pages 1606-1633: A Fundamental Scale of Descriptions for Analyzing Information Content of Communication Systems]]>
http://mdpi.com/1099-4300/17/4/1606
The complexity of the description of a system is a function of the entropy of its symbolic description. Prior to computing the entropy of the system’s description, an observation scale has to be assumed. In texts written in artificial and natural languages, typical scales are binary, characters, and words. However, considering languages as structures built around certain preconceived set of symbols, like words or characters, limits the level of complexity that can be revealed analytically. This study introduces the notion of the fundamental description scale to analyze the essence of the structure of a language. The concept of Fundamental Scale is tested for English and musical instrument digital interface (MIDI) music texts using an algorithm developed to split a text in a collection of sets of symbols that minimizes the observed entropy of the system. This Fundamental Scale reflects more details of the complexity of the language than using bits, characters or words. Results show that this Fundamental Scale allows to compare completely different languages, such as English and MIDI coded music regarding its structural entropy. This comparative power facilitates the study of the complexity of the structure of different communication systems.Entropy2015-03-25174Article10.3390/e17041606160616331099-43002015-03-25doi: 10.3390/e17041606Gerardo FebresKlaus Jaffe<![CDATA[Entropy, Vol. 17, Pages 1581-1605: Kählerian Information Geometry for Signal Processing]]>
http://mdpi.com/1099-4300/17/4/1581
We prove the correspondence between the information geometry of a signal filter and a Kähler manifold. The information geometry of a minimum-phase linear system with a finite complex cepstrum norm is a Kähler manifold. The square of the complex cepstrum norm of the signal filter corresponds to the Kähler potential. The Hermitian structure of the Kähler manifold is explicitly emergent if and only if the impulse response function of the highest degree in z is constant in model parameters. The Kählerian information geometry takes advantage of more efficient calculation steps for the metric tensor and the Ricci tensor. Moreover, α-generalization on the geometric tensors is linear in α . It is also robust to find Bayesian predictive priors, such as superharmonic priors, because Laplace–Beltrami operators on Kähler manifolds are in much simpler forms than those of the non-Kähler manifolds. Several time series models are studied in the Kählerian information geometry.Entropy2015-03-25174Article10.3390/e17041581158116051099-43002015-03-25doi: 10.3390/e17041581Jaehyung ChoiAndrew Mullhaupt<![CDATA[Entropy, Vol. 17, Pages 1558-1580: High Recharge Areas in the Choushui River Alluvial Fan (Taiwan) Assessed from Recharge Potential Analysis and Average Storage Variation Indexes]]>
http://mdpi.com/1099-4300/17/4/1558
High recharge areas significantly influence the groundwater quality and quantity in regional groundwater systems. Many studies have applied recharge potential analysis (RPA) to estimate groundwater recharge potential (GRP) and have delineated high recharge areas based on the estimated GRP. However, most of these studies define the RPA parameters with supposition, and this represents a major source of uncertainty for applying RPA. To objectively define the RPA parameter values without supposition, this study proposes a systematic method based on the theory of parameter identification. A surrogate variable, namely the average storage variation (ASV) index, is developed to calibrate the RPA parameters, because of the lack of direct GRP observations. The study results show that the correlations between the ASV indexes and computed GRP values improved from 0.67 before calibration to 0.85 after calibration, thus indicating that the calibrated RPA parameters represent the recharge characteristics of the study area well; these data also highlight how defining the RPA parameters with ASV indexes can help to improve the accuracy. The calibrated RPA parameters were used to estimate the GRP distribution of the study area, and the GRP values were graded into five levels. High and excellent level areas are defined as high recharge areas, which composed 7.92% of the study area. Overall, this study demonstrates that the developed approach can objectively define the RPA parameters and high recharge areas of the Choushui River alluvial fan, and the results should serve as valuable references for the Taiwanese government in their efforts to conserve the groundwater quality and quantity of the study area.Entropy2015-03-24174Article10.3390/e17041558155815801099-43002015-03-24doi: 10.3390/e17041558Jui-Pin TsaiYu-Wen ChenLiang-Cheng ChangYi-Ming KuoYu-Hsuan TuChen-Che Pan<![CDATA[Entropy, Vol. 17, Pages 1549-1557: Thermodynamics in Curved Space-Time and Its Application to Holography]]>
http://mdpi.com/1099-4300/17/4/1549
The thermodynamic behaviors of a system living in a curved space-time are different from those of a system in a flat space-time. We have investigated the thermodynamics for a system consisting of relativistic massless bosons. We show that a strongly curved metric will produce a large enhancement of the degrees of freedom in the formulae of energy and entropy of the system, as a comparison to the case in a flat space-time. We are mainly concerned with its implications to holography, including the derivations of holographic entropy and holographic screen.Entropy2015-03-24174Article10.3390/e17041549154915571099-43002015-03-24doi: 10.3390/e17041549Yong XiaoLi-Hua FengLi Guan<![CDATA[Entropy, Vol. 17, Pages 1535-1548: Clustering Heterogeneous Data with k-Means by Mutual Information-Based Unsupervised Feature Transformation]]>
http://mdpi.com/1099-4300/17/3/1535
Traditional centroid-based clustering algorithms for heterogeneous data with numerical and non-numerical features result in different levels of inaccurate clustering. This is because the Hamming distance used for dissimilarity measurement of non-numerical values does not provide optimal distances between different values, and problems arise from attempts to combine the Euclidean distance and Hamming distance. In this study, the mutual information (MI)-based unsupervised feature transformation (UFT), which can transform non-numerical features into numerical features without information loss, was utilized with the conventional k-means algorithm for heterogeneous data clustering. For the original non-numerical features, UFT can provide numerical values which preserve the structure of the original non-numerical features and have the property of continuous values at the same time. Experiments and analysis of real-world datasets showed that, the integrated UFT-k-means clustering algorithm outperformed others for heterogeneous data with both numerical and non-numerical features.Entropy2015-03-23173Article10.3390/e17031535153515481099-43002015-03-23doi: 10.3390/e17031535Min WeiTommy ChowRosa Chan<![CDATA[Entropy, Vol. 17, Pages 1508-1534: Space-Time Quantum Imaging]]>
http://mdpi.com/1099-4300/17/3/1508
We report on an experimental and theoretical investigation of quantum imaging where the images are stored in both space and time. Ghost images of remote objects are produced with either one or two beams of chaotic laser light generated by a rotating ground glass and two sensors measuring the reference field and bucket field at different space-time points. We further observe that the ghost images translate depending on the time delay between the sensor measurements. The ghost imaging experiments are performed both with and without turbulence. A discussion of the physics of the space-time imaging is presented in terms of quantum nonlocal two-photon analysis to support the experimental results. The theoretical model includes certain phase factors of the rotating ground glass. These experiments demonstrated a means to investigate the time and space aspects of ghost imaging and showed that ghost imaging contains more information per measured photon than was previously recognized where multiple ghost images are stored within the same ghost imaging data sets. This suggests new pathways to explore quantum information stored not only in multi-photon coincidence information but also in time delayed multi-photon interference. The research is applicable to making enhanced space-time quantum images and videos of moving objects where the images are stored in both space and time.Entropy2015-03-23173Article10.3390/e17031508150815341099-43002015-03-23doi: 10.3390/e17031508Ronald MeyersKeith Deacon<![CDATA[Entropy, Vol. 17, Pages 1477-1507: Application of Divergence Entropy to Characterize the Structure of the Hydrophobic Core in DNA Interacting Proteins]]>
http://mdpi.com/1099-4300/17/3/1477
The fuzzy oil drop model, a tool which can be used to study the structure of the hydrophobic core in proteins, has been applied in the analysis of proteins belonging to the jumonji group—JARID2, JARID1A, JARID1B and JARID1D—proteins that share the property of being able to interact with DNA. Their ARID and PHD domains, when analyzed in the context of the fuzzy oil drop model, are found to exhibit structural variability regarding the status of their secondary folds, including the β-hairpin which determines their biological function. Additionally, the structure of disordered fragments which are present in jumonji proteins (as confirmed by the DisProt database) is explained on the grounds of the hydrophobic core model, suggesting that such fragments contribute to tertiary structural stabilization. This conclusion is supported by divergence entropy measurements, expressing the degree of ordering in each protein’s hydrophobic core.Entropy2015-03-23173Article10.3390/e17031477147715071099-43002015-03-23doi: 10.3390/e17031477Barbara KalinowskaMateusz BanachLeszek KoniecznyIrena Roterman<![CDATA[Entropy, Vol. 17, Pages 1466-1476: The Solute-Exclusion Zone: A Promising Application for Mirofluidics]]>
http://mdpi.com/1099-4300/17/3/1466
While unique phenomena exist at fluid-solid phase intersections, many interfacial phenomena manifest solely on limited scales—i.e., the nm-mm ranges—which stifles their application potential. Here, we constructed microfluidic chips that utilize the unique long-distance interface effects of the Solute-Exclusion Zone (EZ) phenomenon to mix, separate, and guide samples in desired directions within microfluidic channels. On our “EZ Chip”, we utilized the interfacial force generated by EZs to transport specimens across streamlines without the need of an off-chip power source. The advantages of easy-integration, low fabrication cost, and no off-chip energy input make the EZ suitable for independent, portable lab-on-chip system applications.Entropy2015-03-23173Article10.3390/e17031466146614761099-43002015-03-23doi: 10.3390/e17031466Chi-Shuo ChenErik FarrJesse AnayaEric ChenWei-Chun Chin<![CDATA[Entropy, Vol. 17, Pages 1452-1465: Thermodynamic Analysis of a Waste Heat Driven Vuilleumier Cycle Heat Pump]]>
http://mdpi.com/1099-4300/17/3/1452
A Vuilleumier (VM) cycle heat pump is a closed gas cycle driven by heat energy. It has the highest performance among all known heat driven technologies. In this paper, two thermodynamic analyses, including energy and exergy analysis, are carried out to evaluate the application of a VM cycle heat pump for waste heat utilization. For a prototype VM cycle heat pump, equations for theoretical and actual cycles are established. Under the given conditions, the exergy efficiency for the theoretical cycle is 0.23 compared to 0.15 for the actual cycle. This is due to losses taking place in the actual cycle. Reheat losses and flow friction losses account for almost 83% of the total losses. Investigation of the effect of heat source temperature, cycle pressure and speed on the exergy efficiency indicate that the low temperature waste heat is a suitable heat source for a VM cycle heat pump. The selected cycle pressure should be higher than 100 MPa, and 200–300 rpm is the optimum speed.Entropy2015-03-20173Article10.3390/e17031452145214651099-43002015-03-20doi: 10.3390/e17031452Yingbai XieXuejie Sun<![CDATA[Entropy, Vol. 17, Pages 1441-1451: Approximated Information Analysis in Bayesian Inference]]>
http://mdpi.com/1099-4300/17/3/1441
In models with nuisance parameters, Bayesian procedures based on Markov Chain Monte Carlo (MCMC) methods have been developed to approximate the posterior distribution of the parameter of interest. Because these procedures require burdensome computations related to the use of MCMC, approximation and convergence in these procedures are important issues. In this paper, we explore Gibbs sensitivity by using an alternative to the full conditional distribution of the nuisance parameter. The approximate sensitivity of the posterior distribution of interest is studied in terms of an information measure, including Kullback–Leibler divergence. As an illustration, we then apply these results to simple spatial model settings.Entropy2015-03-20173Article10.3390/e17031441144114511099-43002015-03-20doi: 10.3390/e17031441Jung SeoYongku Kim<![CDATA[Entropy, Vol. 17, Pages 1425-1440: A Comparison of Nonlinear Measures for the Detection of Cardiac Autonomic Neuropathy from Heart Rate Variability]]>
http://mdpi.com/1099-4300/17/3/1425
In this work we compare three multiscale measures for their ability to discriminate between participants having cardiac autonomic neuropathy (CAN) and aged controls. CAN is a disease that involves nerve damage leading to an abnormal control of heart rate, so one would expect disease progression to manifest in changes to heart rate variability (HRV). We applied multiscale entropy (MSE), multi fractal detrended fluctuation analysis (MFDFA), and Renyi entropy (RE) to recorded datasets of RR intervals. The latter measure provided the best separation (lowest p-value in Mann–Whitney tests) between classes of participants having CAN, early CAN or no CAN (controls). This comparison suggests the efficacy of RE as a measure for diagnosis of CAN and its progression, when compared to the other multiscale measures.Entropy2015-03-19173Article10.3390/e17031425142514401099-43002015-03-19doi: 10.3390/e17031425David CornforthHerbert JelinekMika Tarvainen<![CDATA[Entropy, Vol. 17, Pages 1411-1424: Entropy Generation Analysis for a CNT Suspension Nanofluid in Plumb Ducts with Peristalsis]]>
http://mdpi.com/1099-4300/17/3/1411
The purpose of the current investigation was to discuss the entropy generation analysis for a carbon nanotube (CNT) suspension nanofluid in a plumb duct with peristalsis. The entropy generation number due to heat transfer and fluid friction is formulated. The velocity and temperature distributions across the tube are presented along with pressure attributes. Exact analytical solution for velocity and temperature profile are obtained. It is found that the entropy generation number attains high values in the region close to the walls of the tube, while it attains low values near the center of the tube.Entropy2015-03-19173Article10.3390/e17031411141114241099-43002015-03-19doi: 10.3390/e17031411Noreen Akbar<![CDATA[Entropy, Vol. 17, Pages 1387-1410: Applied Cryptography Using Chaos Function for Fast Digital Logic-Based Systems in Ubiquitous Computing]]>
http://mdpi.com/1099-4300/17/3/1387
Recently, chaotic dynamics-based data encryption techniques for wired and wireless networks have become a topic of active research in computer science and network security such as robotic systems, encryption, and communication. The main aim of deploying a chaos-based cryptosystem is to provide encryption with several advantages over traditional encryption algorithms such as high security, speed, and reasonable computational overheads and computational power requirements. These challenges have motivated researchers to explore novel chaos-based data encryption techniques with digital logics dealing with hiding information for fast secure communication networks. This work provides an overview of how traditional data encryption techniques are revised and improved to achieve good performance in a secure communication network environment. A comprehensive survey of existing chaos-based data encryption techniques and their application areas are presented. The comparative tables can be used as a guideline to select an encryption technique suitable for the application at hand. Based on the limitations of the existing techniques, an adaptive chaos based data encryption framework of secure communication for future research is proposedEntropy2015-03-19173Review10.3390/e17031387138714101099-43002015-03-19doi: 10.3390/e17031387Piyush ShuklaAnkur KhareMurtaza RizviShalini StalinSanjay Kumar<![CDATA[Entropy, Vol. 17, Pages 1379-1386: The Optimal Fix-Free Code for Anti-Uniform Sources]]>
http://mdpi.com/1099-4300/17/3/1379
An \(n\) symbol source which has a Huffman code with codelength vector \(L_{n}=(1,2,3,\cdots,n-2,n-1,n-1)\) is called an anti-uniform source. In this paper, it is shown that for this class of sources, the optimal fix-free code and symmetric fix-free code is \(C_{n}^{*}=(0,11,101,1001,\cdots,1\overbrace{0\cdots0}^{n-2}1) \)Entropy2015-03-19173Article10.3390/e17031379137913861099-43002015-03-19doi: 10.3390/e17031379Ali ZaghianAdel AghajanT. Gulliver<![CDATA[Entropy, Vol. 17, Pages 1358-1378: Hidden State Conditional Random Field for Abnormal Activity Recognition in Smart Homes]]>
http://mdpi.com/1099-4300/17/3/1358
As the number of elderly people has increased worldwide, there has been a surge of research into assistive technologies to provide them with better care by recognizing their normal and abnormal activities. However, existing abnormal activity recognition (AAR) algorithms rarely consider sub-activity relations when recognizing abnormal activities. This paper presents an application of the Hidden State Conditional Random Field (HCRF) method to detect and assess abnormal activities that often occur in elderly persons’ homes. Based on HCRF, this paper designs two AAR algorithms, and validates them by comparing them with a feature vector distance based algorithm in two experiments. The results demonstrate that the proposed algorithms favorably outperform the competitor, especially when abnormal activities have same sensor type and sensor number as normal activities.Entropy2015-03-18173Article10.3390/e17031358135813781099-43002015-03-18doi: 10.3390/e17031358Yu TongRong ChenJian Gao<![CDATA[Entropy, Vol. 17, Pages 1347-1357: Geometric Shrinkage Priors for Kählerian Signal Filters]]>
http://mdpi.com/1099-4300/17/3/1347
We construct geometric shrinkage priors for Kählerian signal filters. Based on the characteristics of Kähler manifolds, an efficient and robust algorithm for finding superharmonic priors which outperform the Jeffreys prior is introduced. Several ansätze for the Bayesian predictive priors are also suggested. In particular, the ansätze related to Kähler potential are geometrically intrinsic priors to the information manifold of which the geometry is derived from the potential. The implication of the algorithm to time series models is also provided.Entropy2015-03-17173Article10.3390/e17031347134713571099-43002015-03-17doi: 10.3390/e17031347Jaehyung ChoiAndrew Mullhaupt<![CDATA[Entropy, Vol. 17, Pages 1329-1346: Metriplectic Algebra for Dissipative Fluids in Lagrangian Formulation]]>
http://mdpi.com/1099-4300/17/3/1329
The dynamics of dissipative fluids in Eulerian variables may be derived from an algebra of Leibniz brackets of observables, the metriplectic algebra, that extends the Poisson algebra of the frictionless limit of the system via a symmetric semidefinite component, encoding dissipative forces. The metriplectic algebra includes the conserved total Hamiltonian H, generating the non-dissipative part of dynamics, and the entropy S of those microscopic degrees of freedom draining energy irreversibly, which generates dissipation. This S is a Casimir invariant of the Poisson algebra to which the metriplectic algebra reduces in the frictionless limit. The role of S is as paramount as that of H, but this fact may be underestimated in the Eulerian formulation because S is not the only Casimir of the symplectic non-canonical part of the algebra. Instead, when the dynamics of the non-ideal fluid is written through the parcel variables of the Lagrangian formulation, the fact that entropy is symplectically invariant clearly appears to be related to its dependence on the microscopic degrees of freedom of the fluid, that are themselves in involution with the position and momentum of the parcel.Entropy2015-03-16173Article10.3390/e17031329132913461099-43002015-03-16doi: 10.3390/e17031329Massimo Materassi<![CDATA[Entropy, Vol. 17, Pages 1309-1328: A Link between Nano- and Classical Thermodynamics: Dissipation Analysis (The Entropy Generation Approach in Nano-Thermodynamics)]]>
http://mdpi.com/1099-4300/17/3/1309
The interest in designing nanosystems is continuously growing. Engineers apply a great number of optimization methods to design macroscopic systems. If these methods could be introduced into the design of small systems, a great improvement in nanotechnologies could be achieved. To do so, however, it is necessary to extend classical thermodynamic analysis to small systems, but irreversibility is also present in small systems, as the Loschmidt paradox highlighted. Here, the use of the recent improvement of the Gouy-Stodola theorem to complex systems (GSGL approach), based on the use of entropy generation, is suggested to obtain the extension of classical thermodynamics to nanothermodynamics. The result is a new approach to nanosystems which avoids the difficulties highlighted in the usual analysis of the small systems, such as the definition of temperature for nanosystems.Entropy2015-03-16173Article10.3390/e17031309130913281099-43002015-03-16doi: 10.3390/e17031309Umberto Lucia<![CDATA[Entropy, Vol. 17, Pages 1278-1308: Ricci Curvature, Isoperimetry and a Non-additive Entropy]]>
http://mdpi.com/1099-4300/17/3/1278
Searching for the dynamical foundations of Havrda-Charvát/Daróczy/ Cressie-Read/Tsallis non-additive entropy, we come across a covariant quantity called, alternatively, a generalized Ricci curvature, an N-Ricci curvature or a Bakry-Émery-Ricci curvature in the configuration/phase space of a system. We explore some of the implications of this tensor and its associated curvature and present a connection with the non-additive entropy under investigation. We present an isoperimetric interpretation of the non-extensive parameter and comment on further features of the system that can be probed through this tensor.Entropy2015-03-16173Article10.3390/e17031278127813081099-43002015-03-16doi: 10.3390/e17031278Nikos Kalogeropoulos<![CDATA[Entropy, Vol. 17, Pages 1273-1277: Symmetry, Probabiliy, Entropy: Synopsis of the Lecture at MAXENT 2014]]>
http://mdpi.com/1099-4300/17/3/1273
In this discussion, we indicate possibilities for (homological and non-homological) linearization of basic notions of the probability theory and also for replacing the real numbers as values of probabilities by objects of suitable combinatorial categories.Entropy2015-03-13173Meeting Report10.3390/e17031273127312771099-43002015-03-13doi: 10.3390/e17031273Misha Gromov<![CDATA[Entropy, Vol. 17, Pages 1253-1272: Entropic Measures of Complexity of Short-Term Dynamics of Nocturnal Heartbeats in an Aging Population]]>
http://mdpi.com/1099-4300/17/3/1253
Two entropy-based approaches are investigated to study patterns describing differences in time intervals between consecutive heartbeats. The first method explores matrices arising from networks of transitions constructed following events represented by a time series. The second method considers distributions of ordinal patterns of length three, whereby patterns with repeated values are counted as different patterns. Both methods provide estimators of dynamical aspects of short-term heartbeat signals obtained from nocturnal Holter electrocardiogram (ECG) recordings of healthy people of different ages and genders. The deceleration capacity, arising from the adjacency matrix of the network, and the entropy rate, resulting from the transition matrix of the network, are also calculated, and both significantly decay with aging. As people age, the permutation entropy grows, due to the increase in patterns with repeated values. All of these estimators describe in a consistent way changes in the beat-to-beat heart period dynamics caused by aging. An overall slowing down of heart period changes is observed, and an increase of permutation entropy results from the progressive increase of patterns with repeated values. This result points to the sympathetic drive becoming dominant in cardiac regulation of nocturnal heart rate with age.Entropy2015-03-13173Article10.3390/e17031253125312721099-43002015-03-13doi: 10.3390/e17031253Danuta MakowiecAgnieszka KaczkowskaDorota WejerMarta Żarczyńska-BuchowieckaZbigniew Struzik<![CDATA[Entropy, Vol. 17, Pages 1236-1252: Analysis of the Magnetocaloric Effect in Heusler Alloys: Study of Ni50CoMn36Sn13 by Calorimetric Techniques]]>
http://mdpi.com/1099-4300/17/3/1236
Direct determinations of the isothermal entropy increment, \(-\Delta S_T\), in the Heusler alloy Ni\(_{50}\)CoMn\(_{36}\)Sn\(_{13}\) on demagnetization gave positive values, corresponding to a normal magnetocaloric effect. These values contradict the results derived from heat-capacity measurements and also previous results obtained from magnetization measurements, which indicated an inverse magnetocaloric effect, but showing different values depending on the technique employed. The puzzle is solved, and the apparent incompatibilities are quantitatively explained considering the hysteresis, the width of the martensitic transition and the detailed protocol followed to obtain each datum. The results show that these factors should be analyzed in detail when dealing with Heusler alloys.Entropy2015-03-12173Article10.3390/e17031236123612521099-43002015-03-12doi: 10.3390/e17031236Elias PalaciosJuan BartoloméGaofeng WangRamon BurrielKonstantin SkokovSergey TaskaevVladimir Khovaylo<![CDATA[Entropy, Vol. 17, Pages 1218-1235: Information Hiding Method Using Best DCT and Wavelet Coefficients and ItsWatermark Competition]]>
http://mdpi.com/1099-4300/17/3/1218
In recent years, information hiding and its evaluation criteria have been developed by the IHC (Information Hiding and its Criteria) Committee of Japan. This committee was established in 2011 with the aim of establishing standard evaluation criteria for robust watermarks. In this study, we developed an information hiding method that satisfies the IHC evaluation criteria. The proposed method uses the difference of the frequency coefficients derived from a discrete cosine transform or a discrete wavelet transform. The algorithm employs a statistical analysis to find the best positions in the frequency domains for watermark insertion. In particular, we use the BCH (Bose-Chaudhuri-Hocquenghem) (511,31,109) code to error correct the watermark bits and the BCH (63,16,11) code as the sync signal to withstand JPEG (Joint Photographic Experts Group) compression and cropping attacks. Our experimental results showed that there were no errors in 10 HDTV-size areas after the second decompression. It should be noted that after the second compression, the file size should be less than 1 25 of the original size to satisfy the IHC evaluation criteria.Entropy2015-03-12173Article10.3390/e17031218121812351099-43002015-03-12doi: 10.3390/e17031218Hyunho KangKeiichi Iwamura<![CDATA[Entropy, Vol. 17, Pages 1204-1217: Information Geometry on the \(\kappa\)-Thermostatistics]]>
http://mdpi.com/1099-4300/17/3/1204
We explore the information geometric structure of the statistical manifold generated by the \(\kappa\)-deformed exponential family. The dually-flat manifold is obtained as a dualistic Hessian structure by introducing suitable generalization of the Fisher metric and affine connections. As a byproduct, we obtain the fluctuation-response relations in the \(\kappa\)-formalism based on the \(\kappa\)-generalized exponential family.Entropy2015-03-12173Article10.3390/e17031204120412171099-43002015-03-12doi: 10.3390/e17031204Tatsuaki WadaAntonio Scarfone<![CDATA[Entropy, Vol. 17, Pages 1197-1203: Generalized Multiscale Entropy Analysis: Application to Quantifying the Complex Volatility of Human Heartbeat Time Series]]>
http://mdpi.com/1099-4300/17/3/1197
We introduce a generalization of multiscale entropy (MSE) analysis. The method is termed MSEn, where the subscript denotes the moment used to coarse-grain a time series. MSEμ, described previously, uses the mean value (first moment). Here, we focus on MSEσ2 , which uses the second moment, i.e., the variance. MSEσ2 quantifies the dynamics of the volatility (variance) of a signal over multiple time scales. We use the method to analyze the structure of heartbeat time series. We find that the dynamics of the volatility of heartbeat time series obtained from healthy young subjects is highly complex. Furthermore, we find that the multiscale complexity of the volatility, not only the multiscale complexity of the mean heart rate, degrades with aging and pathology. The “bursty” behavior of the dynamics may be related to intermittency in energy and information flows, as part of multiscale cycles of activation and recovery. Generalized MSE may also be useful in quantifying the dynamical properties of other physiologic and of non-physiologic time series.Entropy2015-03-12173Communication10.3390/e17031197119712031099-43002015-03-12doi: 10.3390/e17031197Madalena CostaAry Goldberger<![CDATA[Entropy, Vol. 17, Pages 1181-1196: Entropy of Quantum Measurement]]>
http://mdpi.com/1099-4300/17/3/1181
A notion of entropy of a normal state on a finite von Neumann algebra in Segal’s sense is considered, and its superadditivity is proven together with a necessary and sufficient condition for its additivity. Bounds on the entropy of the state after measurement are obtained, and it is shown that a weakly repeatable measurement gives minimal entropy and that a minimal state entropy measurement satisfying some natural additional conditions is repeatable.Entropy2015-03-12173Article10.3390/e17031181118111961099-43002015-03-12doi: 10.3390/e17031181Hanna Podsȩdkowska<![CDATA[Entropy, Vol. 17, Pages 1165-1180: Distributed Consensus for Metamorphic Systems Using a GossipAlgorithm for CAT(0) Metric Spaces]]>
http://mdpi.com/1099-4300/17/3/1165
We present an application of distributed consensus algorithms to metamorphic systems. A metamorphic system is a set of identical units that can self-assemble to form a rigid structure. For instance, one can think of a robotic arm composed of multiple links connected by joints. The system can change its shape in order to adapt to different environments via reconfiguration of its constituting units. We assume in this work that several metamorphic systems form a network: two systems are connected whenever they are able to communicate with each other. The aim of this paper is to propose a distributed algorithm that synchronizes all of the systems in the network. Synchronizing means that all of the systems should end up having the same configuration. This aim is achieved in two steps: (i) we cast the problem as a consensus problem on a metric space; and (ii) we use a recent distributed consensus algorithm that only makes use of metrical notions.Entropy2015-03-12173Article10.3390/e17031165116511801099-43002015-03-12doi: 10.3390/e17031165Anass Bellachehab<![CDATA[Entropy, Vol. 17, Pages 1146-1164: Maximum Relative Entropy Updating and the Value of Learning]]>
http://mdpi.com/1099-4300/17/3/1146
We examine the possibility of justifying the principle of maximum relative entropy (MRE) considered as an updating rule by looking at the value of learning theorem established in classical decision theory. This theorem captures an intuitive requirement for learning: learning should lead to new degrees of belief that are expected to be helpful and never harmful in making decisions. We call this requirement the value of learning. We consider the extent to which learning rules by MRE could satisfy this requirement and so could be a rational means for pursuing practical goals. First, by representing MRE updating as a conditioning model, we show that MRE satisfies the value of learning in cases where learning prompts a complete redistribution of one’s degrees of belief over a partition of propositions. Second, we show that the value of learning may not be generally satisfied by MRE updates in cases of updating on a change in one’s conditional degrees of belief. We explain that this is so because, contrary to what the value of learning requires, one’s prior degrees of belief might not be equal to the expectation of one’s posterior degrees of belief. This, in turn, points towards a more general moral: that the justification of MRE updating in terms of the value of learning may be sensitive to the context of a given learning experience. Moreover, this lends support to the idea that MRE is not a universal nor mechanical updating rule, but rather a rule whose application and justification may be context-sensitive.Entropy2015-03-11173Article10.3390/e17031146114611641099-43002015-03-11doi: 10.3390/e17031146Patryk Dziurosz-Serafinowicz<![CDATA[Entropy, Vol. 17, Pages 1135-1145: Comparing Security Notions of Secret Sharing Schemes]]>
http://mdpi.com/1099-4300/17/3/1135
Different security notions of secret sharing schemes have been proposed by different information measures. Entropies, such as Shannon entropy and min entropy, are frequently used in the setting security notions for secret sharing schemes. Different to the entropies, Kolmogorov complexity was also defined and used in study the security of individual instances for secret sharing schemes. This paper is concerned with these security notions for secret sharing schemes defined by the variational measures, including Shannon entropy, guessing probability, min entropy and Kolmogorov complexity.Entropy2015-03-10173Article10.3390/e17031135113511451099-43002015-03-10doi: 10.3390/e17031135Songsong DaiDonghui Guo<![CDATA[Entropy, Vol. 17, Pages 1123-1134: Projective Synchronization for a Class of Fractional-Order Chaotic Systems with Fractional-Order in the (1, 2) Interval]]>
http://mdpi.com/1099-4300/17/3/1123
In this paper, a projective synchronization approach for a class of fractional-order chaotic systems with fractional-order 1 &lt; q &lt; 2 is demonstrated. The projective synchronization approach is established through precise theorization. To illustrate the effectiveness of the proposed scheme, we discuss two examples: (1) the fractional-order Lorenz chaotic system with fractional-order q = 1.1; (2) the fractional-order modified Chua’s chaotic system with fractional-order q = 1.02. The numerical simulations show the validity and feasibility of the proposed scheme.Entropy2015-03-10173Article10.3390/e17031123112311341099-43002015-03-10doi: 10.3390/e17031123Ping ZhouRongji BaiJiming Zheng<![CDATA[Entropy, Vol. 17, Pages 1103-1122: Weakest-Link Scaling and Extreme Events in Finite-Sized Systems]]>
http://mdpi.com/1099-4300/17/3/1103
Weakest-link scaling is used in the reliability analysis of complex systems. It is characterized by the extensivity of the hazard function instead of the entropy. The Weibull distribution is the archetypical example of weakest-link scaling, and it describes variables such as the fracture strength of brittle materials, maximal annual rainfall, wind speed and earthquake return times. We investigate two new distributions that exhibit weakest-link scaling, i.e., a Weibull generalization known as the κ-Weibull and a modified gamma probability function that we propose herein. We show that in contrast with the Weibull and the modified gamma, the hazard function of the κ -Weibull is non-extensive, which is a signature of inter-dependence between the links. We also investigate the impact of heterogeneous links, modeled by means of a stochastic Weibull scale parameter, on the observed probability distribution.Entropy2015-03-09173Article10.3390/e17031103110311221099-43002015-03-09doi: 10.3390/e17031103Dionissios HristopulosManolis PetrakisGiorgio Kaniadakis<![CDATA[Entropy, Vol. 17, Pages 1090-1102: Speed Gradient and MaxEnt Principles for Shannon and Tsallis Entropies]]>
http://mdpi.com/1099-4300/17/3/1090
In this paper we consider dynamics of non-stationary processes that follow the MaxEnt principle. We derive a set of equations describing dynamics of a system for Shannon and Tsallis entropies. Systems with discrete probability distribution are considered under mass conservation and energy conservation constraints. The existence and uniqueness of solution are established and asymptotic stability of the equilibrium is proved. Equations are derived based on the speed-gradient principle originated in control theory.Entropy2015-03-06173Article10.3390/e17031090109011021099-43002015-03-06doi: 10.3390/e17031090Alexander FradkovDmitry Shalymov<![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