Entropy, Volume 20, Issue 9 (September 2018) – 97 articles

In this paper, we obtain an integral formula for the rate distortion function (RDF) of any Gaussian asymptotically wide sense stationary (AWSS) vector process. Applying this result, we also obtain an integral formula for the RDF of Gaussian moving average (MA) vector processes and of Gaussian autoregressive MA (ARMA) AWSS vector processes. Full article

The Belt and Road initiative (BRI) was announced in 2013 by the Chinese government. Its goal is to promote the cooperation between European and Asian countries, as well as enhancing the trust between members and unifying the market. Since its creation, more and more developing countries are joining the initiative. Based on the geographical location characteristics of the countries in this initiative, we propose an improvement of a popular recommendation algorithm that includes geographic location information. This recommendation algorithm is able to make suitable recommendations of products for countries in the BRI. Then, Fitness and Complexity metrics are used to evaluate the impact of the recommendation results and measure the country’s competitiveness. The aim of this work is to provide countries’ insights on the ideal development direction. By following the recommendations, the countries can quickly increase their international competitiveness. Full article

We proposed in this work the introduction of a new vision of stochastic processes through geometry induced by dilation. The dilation matrices of a given process are obtained by a composition of rotation matrices built in with respect to partial correlation coefficients. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, only one dilation matrix is obtained and it corresponds therefore to Naimark dilation. When the process is nonstationary, a set of dilation matrices is obtained. They correspond to Kolmogorov decomposition. In this work, the nonstationary class of periodically correlated processes was of interest. The underlying periodicity of correlation coefficients is then transmitted to the set of dilation matrices. Because this set lives on the Lie group of rotation matrices, we can see them as points of a closed curve on the Lie group. Geometrical aspects can then be investigated through the shape of the obtained curves, and to give a complete insight into the space of curves, a metric and the derived geodesic equations are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices, and because the metric in the space of curve naturally extends to the space of shapes; this enables a comparison between curves’ shapes and allows then the classification of random processes’ measures. Full article

In order to overcome the difficulty of key management in “one time pad” encryption schemes and also resist the attack of chosen plaintext, a new image encryption algorithm based on chaos and SHA-256 is proposed in this paper. The architecture of confusion and diffusion is adopted. Firstly, the surrounding of a plaintext image is surrounded by a sequence generated from the SHA-256 hash value of the plaintext to ensure that each encrypted result is different. Secondly, the image is scrambled according to the random sequence obtained by adding the disturbance term associated with the plaintext to the chaotic sequence. Third, the cyphertext (plaintext) feedback mechanism of the dynamic index in the diffusion stage is adopted, that is, the location index of the cyphertext (plaintext) used for feedback is dynamic. The above measures can ensure that the algorithm can resist chosen plaintext attacks and can overcome the difficulty of key management in “one time pad” encryption scheme. Also, experimental results such as key space analysis, key sensitivity analysis, differential analysis, histograms, information entropy, and correlation coefficients show that the image encryption algorithm is safe and reliable, and has high application potential. Full article

The sustainability of regional water resources has important supporting data needed for establishing policies on the sustainable development of the social economy. The purpose of this paper is to propose an assessment method to accurately reflect the sustainability of regional water resources in various areas. The method is based on the relative entropy of the information entropy theory. The steps are as follows. Firstly, the pretreatment of the evaluation sample data is required, before the relative entropy of each standard evaluation sample and evaluation grade (SEG) is calculated to obtain the entropy weight of each evaluation index. After this, the entropy weighted comprehensive index (WCI) of the standard evaluation grade sample is obtained. The function relation between WCI and SEG can be fitted by the cubic polynomial to construct the evaluation function. Using the above steps, a generalized entropy method (GEM) for the sustainable assessment of regional water resources is established and it is used to evaluate the sustainability of water resources in the Pingba and Huai River areas in China. The results show that the proposed GEM model can accurately reflect the sustainable water resources in the two regions. Compared with the other evaluation models, such as the Shepherd method, Artificial Neural Network and Fuzzy comprehensive evaluation, the GEM model has larger differences in its evaluation results, which are more reasonable. Thus, the proposed GEM model can provide scientific data support for coordinating the relationship between the sustainable development and utilization of regional water resources in order to improve the development of regional population, society and economy. Full article

The aim of this paper is to investigate the generalization of the Sierpinski gasket through the harmonic metric. In particular, this work presents an antenna based on such a generalization. In fact, the harmonic Sierpinski gasket is used as a geometric configuration of small antennas. As with fractal antennas and Rényi entropy, their performance is characterized by the associated entropy that is studied and discussed here. Full article

We present a new characterization of propositional formulas called entropy, which approximates the freedom we have in assigning the variables. Like several other such measures (e.g., back-door and back-door-key variables), it is computationally expensive to compute. Nevertheless, for small and medium-size satisfiable formulas, it enables us to study the effect of this freedom on the impact of various SAT heuristics, following up on a recent study by C. Oh (Oh, SAT’15, LNCS 9340, 307–323). Oh’s findings were that the expected success of various heuristics depends on whether the input formula is satisfiable or not. With entropy, and also with the measure of solution density, we are able to refine these findings for the case of satisfiable formulas. Specifically, we found empirically that satisfiable formulas with small entropy “behave” similarly to unsatisfiable formulas. Full article

The notion of three-phase (line) tension remains one of the most disputable notions in surface science. A very broad range of its values has been reported. Experts even do not agree on the sign of line tension. The polymer-chain-like model of three-phase (triple) line enables rough estimation of entropic input into the value of line tension, estimated as ${\Gamma }_{en}\cong \frac{k_{B}T}{d_m}\cong {10}^{-11}N$ , where $d_m$ is the diameter of the liquid molecule. The introduction of the polymer-chain-like model of the triple line is justified by the “water string” model of the liquid state, predicting strong orientation effects for liquid molecules located near hydrophobic moieties. The estimated value of the entropic input into the line tension is close to experimental findings, reported by various groups, and seems to be relevant for the understanding of elastic properties of biological membranes. Full article

Network virtualization can offer more flexibility and better manageability for next generation Internet. With the increasing deployments of virtual networks in military and commercial networks, a major challenge is to ensure virtual network survivability against hybrid multiple failures. In this paper, we study the problem of recovering virtual networks affected by hybrid multiple failures in substrate networks and provide an integer linear programming formulation to solve it. We propose a heuristic algorithm to tackle the complexity of the integer linear programming formulation, which includes a faulty virtual network reconfiguration ranking method based on weighted relative entropy, a hybrid multiple failures ranking algorithm, and a virtual node migration method based on weighted relative entropy. In the faulty virtual network reconfiguration ranking method based on weighted relative entropy and virtual node migration method based on weighted relative entropy, multiple ranking indicators are combined in a suitable way based on weighted relative entropy. In the hybrid multiple failures ranking algorithm, the virtual node and its connective virtual links are re-embedded, firstly. Evaluation results show that our heuristic method not only has the best acceptance ratio and normal operation ratio, but also achieves the highest long-term average revenue to cost ratio compared with other virtual network reconfiguration methods. Full article

This paper is concerned with the co-existence of different synchronization types for fractional-order discrete-time chaotic systems with different dimensions. In particular, we show that through appropriate nonlinear control, projective synchronization (PS), full state hybrid projective synchronization (FSHPS), and generalized synchronization (GS) can be achieved simultaneously. A second nonlinear control scheme is developed whereby inverse full state hybrid projective synchronization (IFSHPS) and inverse generalized synchronization (IGS) are shown to co-exist. Numerical examples are presented to confirm the findings. Full article

This paper is devoted to change-point detection using only the ordinal structure of a time series. A statistic based on the conditional entropy of ordinal patterns characterizing the local up and down in a time series is introduced and investigated. The statistic requires only minimal a priori information on given data and shows good performance in numerical experiments. By the nature of ordinal patterns, the proposed method does not detect pure level changes but changes in the intrinsic pattern structure of a time series and so it could be interesting in combination with other methods. Full article

Cities and their infrastructure networks are always in motion and permanently changing in structure and function. This paper presents a methodology for automatically creating future water distribution networks (WDNs) that are stressed step-by-step by disconnection and connection of WDN parts. The associated effects of demand shifting and flow rearrangements are simulated and assessed with hydraulic performances. With the methodology, it is possible to test various planning and adaptation options of the future WDN, where the unknown (future) network is approximated via the co-located and known (future) road network, and hence different topological characteristics (branched vs. strongly looped layout) can be investigated. The reliability of the planning options is evaluated with the flow entropy, a measure based on Shannon’s informational entropy. Uncertainties regarding future water consumption and water loss management are included in a scenario analysis. To avoid insufficient water supply to customers during the transition process from an initial to a final WDN state, an adaptation concept is proposed where critical WDN components are replaced over time. Finally, the method is applied to the drastic urban transition of Kiruna, Sweden. Results show that without adaptation measures severe performance drops will occur after the WDN state 2023, mainly caused by the disconnection of WDN parts. However, with low adaptation efforts that consider 2–3% pipe replacement, sufficient pressure performances are achieved. Furthermore, by using an entropy-cost comparison, the best planning options are determined. Full article

A stochastic system under the influence of a stochastic environment is correlated with both present and future states of the environment. Such a system can be seen as implicitly implementing a predictive model of future environmental states. The non-predictive model complexity has been shown to lower-bound the thermodynamic dissipation. Here we explore these statistical and physical quantities at steady state in simple models. We show that under quasi-static driving this model complexity saturates the dissipation. Beyond the quasi-static limit, we demonstrate a lower bound on the ratio of this model complexity to total dissipation, that is realized in the limit of weak driving. Full article

We examine how information theory has been used to study cognition over the last seven decades. After an initial burst of activity in the 1950s, the backlash that followed stopped most work in this area. The last couple of decades has seen both a revival of interest, and a more firmly grounded, experimentally justified use of information theory. We can view cognition as the process of transforming perceptions into information—where we use information in the colloquial sense of the word. This last clarification is one of the problems we run into when trying to use information theoretic principles to understand or analyze cognition. Information theory is mathematical, while cognition is a subjective phenomenon. It is relatively easy to discern a subjective connection between cognition and information; it is a different matter altogether to apply the rigor of information theory to the process of cognition. In this paper, we will look at the many ways in which people have tried to alleviate this problem. These approaches range from narrowing the focus to only quantifiable aspects of cognition or borrowing conceptual machinery from information theory to address issues of cognition. We describe applications of information theory across a range of cognition research, from neural coding to cognitive control and predictive coding. Full article

A basic pattern in the body plan architecture of many animals, plants and some molecular and cellular systems is five-part units. This pattern has been understood as a result of genetic blueprints in development and as a widely conserved evolutionary character. Despite some efforts, a definitive explanation of the abundance of pentagonal symmetry at so many levels of complexity is still missing. Based on both, a computational platform and a statistical spatial organization argument, we show that five-fold morphology is substantially different from other abundant symmetries like three-fold, four-fold and six-fold symmetries in terms of spatial interacting elements. We develop a measuring system to determine levels of spatial organization in 2D polygons (homogeneous or heterogeneous partition of defined areas) based on principles of regularity in a morphospace. We found that spatial organization of five-fold symmetry is statistically higher than all other symmetries studied here (3 to 10-fold symmetries) in terms of spatial homogeneity. The significance of our findings is based on the statistical constancy of geometrical constraints derived from spatial organization of shapes, beyond the material or complexity level of the many different systems where pentagonal symmetry occurs. Full article

A response is given to a comment of Zanotto and Mauro on our paper published in Entropy 20, 103 (2018). Our arguments presented in this paper are widely ignored by them, and no new considerations are outlined in the comment, which would require a revision of our conclusions. For this reason, we restrict ourselves here to a brief response, supplementing it by some additional arguments in favor of our point of view not included in our above-cited paper. Full article

In a recent article, Schmelzer and Tropin [Entropy 2018, 20, 103] presented a critique of several aspects of modern glass science, including various features of glass transition and relaxation, crystallization, and the definition of glass itself. We argue that these criticisms are at odds with well-accepted knowledge in the field from both theory and experiments. The objective of this short comment is to clarify several of these issues. Full article

The energy use analysis of coal-fired power plant units is of significance for energy conservation and consumption reduction. One of the most serious problems attributed to Chinese coal-fired power plants is coal waste. Several units in one plant may experience a practical rated output situation at the same time, which may increase the coal consumption of the power plant. Here, we propose a new hybrid methodology for plant-level load optimization to minimize coal consumption for coal-fired power plants. The proposed methodology includes two parts. One part determines the reference value of the controllable operating parameters of net coal consumption under typical load conditions, based on an improved K-means algorithm and the Hadoop platform. The other part utilizes a support vector machine to determine the sensitivity coefficients of various operating parameters for the net coal consumption under different load conditions. Additionally, the fuzzy rough set attribute reduction method was employed to obtain the minimalist properties reduction method parameters to reduce the complexity of the dataset. This work is based on continuously-measured information system data from a 600 MW coal-fired power plant in China. The results show that the proposed strategy achieves high energy conservation performance. Taking the 600 MW load optimization value as an example, the optimized power supply coal consumption is 307.95 g/(kW·h) compared to the actual operating value of 313.45 g/(kW·h). It is important for coal-fired power plants to reduce their coal consumption. Full article

In view of the nonlinear characteristics of electroencephalography (EEG) signals collected in the driving fatigue state recognition research and the issue that the recognition accuracy of the driving fatigue state recognition method based on EEG is still unsatisfactory, this paper proposes a driving fatigue recognition method based on sample entropy (SE) and kernel principal component analysis (KPCA), which combines the advantage of the high recognition accuracy of sample entropy and the advantages of KPCA in dimensionality reduction for nonlinear principal components and the strong non-linear processing capability. By using support vector machine (SVM) classifier, the proposed method (called SE_KPCA) is tested on the EEG data, and compared with those based on fuzzy entropy (FE), combination entropy (CE), three kinds of entropies including SE, FE and CE that merged with KPCA. Experiment results show that the method is effective. Full article

The time evolution of stochastic reaction networks can be modeled with the chemical master equation of the probability distribution. Alternatively, the numerical problem can be reformulated in terms of probability moment equations. Herein we present a new alternative method for numerically solving the time evolution of stochastic reaction networks. Based on the assumption that the entropy of the reaction network is maximum, Lagrange multipliers are introduced. The proposed method derives equations that model the time derivatives of these Lagrange multipliers. We present detailed steps to transform moment equations to Lagrange multiplier equations. In order to demonstrate the method, we present examples of non-linear stochastic reaction networks of varying degrees of complexity, including multistable and oscillatory systems. We find that the new approach is as accurate and significantly more efficient than Gillespie’s original exact algorithm for systems with small number of interacting species. This work is a step towards solving stochastic reaction networks accurately and efficiently. Full article

The Koch curve exciting coil eddy current sensor is a kind of novel flexible planar eddy current probe. In this study, an intersection angle spectrum entropy index and a radial direction energy spectrum entropy were proposed to evaluate the eddy current distribution. Eddy current distributions induced by one turn of a circular coil and one turn of a second order Koch curve coil feed with different exciting frequency alternative currents and at different lift-off distances, were simulated and the eddy current distributions varying with lift-off distance in different exciting frequencies were compared by the two proposed indices. With the increase of the lift-off distance or the decrease of exciting frequency, the similarity between the shape of the Koch curve and the eddy current distribution became weakened and the degree of the concentration of the eddy current distribution in the specimen under the exciting coil was loosened. Full article

The combination of different sources of information is a problem that arises in several situations, for instance, when data are analysed using different similarity measures. Often, each source of information is given as a similarity, distance, or a kernel matrix. In this paper, we propose a new class of methods which consists of producing, for anomaly detection purposes, a single Mercer kernel (that acts as a similarity measure) from a set of local entropy kernels and, at the same time, avoids the task of model selection. This kernel is used to build an embedding of data in a variety that will allow the use of a (modified) one-class Support Vector Machine to detect outliers. We study several information combination schemes and their limiting behaviour when the data sample size increases within an Information Geometry context. In particular, we study the variety of the given positive definite kernel matrices to obtain the desired kernel combination as belonging to that variety. The proposed methodology has been evaluated on several real and artificial problems. Full article

This paper brings analysis of the multiple-valued memory system (MVMS) composed by a pair of the resonant tunneling diodes (RTD). Ampere-voltage characteristic (AVC) of both diodes is approximated in operational voltage range as common in practice: by polynomial scalar function. Mathematical model of MVMS represents autonomous deterministic dynamical system with three degrees of freedom and smooth vector field. Based on the very recent results achieved for piecewise-linear MVMS numerical values of the parameters are calculated such that funnel and double spiral chaotic attractor is observed. Existence of such types of strange attractors is proved both numerically by using concept of the largest Lyapunov exponents (LLE) and experimentally by computer-aided simulation of designed lumped circuit using only commercially available active elements. Full article

A general framework for inference in dynamical systems is described, based on the language of Bayesian probability theory and making use of the maximum entropy principle. Taking the concept of a path as fundamental, the continuity equation and Cauchy’s equation for fluid dynamics arise naturally, while the specific information about the system can be included using the maximum caliber (or maximum path entropy) principle. Full article

The 37th edition of MaxEnt was held in Brazil, hosting several distinguished researchers and students. The workshop offered four tutorials, nine invited talks, twenty four oral presentations and twenty seven poster presentations. All submissions received their first choice between oral and poster presentations. The event held a celebration to Julio Stern’s 60th anniversary and awarded two prizes to young researchers. As customary, the workshop had one free afternoon, in which participants visited the city’s surroundings and experienced Brazilian food and traditions. Full article

According to the keyword abstract extraction function in the Natural Language Processing and Information Retrieval Sharing Platform (NLPIR), the design method of a dynamic rounds chaotic block cipher is presented in this paper, which takes into account both the security and efficiency. The cipher combines chaotic theory with the Feistel structure block cipher, and uses the randomness of chaotic sequence and the nonlinearity of chaotic S-box to dynamically generate encrypted rounds, realizing more numbers of dynamic rounds encryption for the important information marked by NLPIR, while less numbers of dynamic rounds encryption for the non-important information that is not marked. Through linear and differential cryptographic analysis, ciphertext information entropy, “0–1” balance and National Institute of Science and Technology (NIST) tests and the comparison with other traditional and lightweight block ciphers, the results indicate that the dynamic variety of encrypted rounds can achieve different levels of encryption for different information, which can achieve the purpose of enhancing the anti-attack ability and reducing the number of encrypted rounds. Therefore, the dynamic rounds chaotic block cipher can guarantee the security of information transmission and realize the lightweight of the cryptographic algorithm. Full article

We study an analog of Bayes’ formula and the nonnegativity property of mutual information for systems with one random variable. For single-qudit states, we present new entropic inequalities in the form of the subadditivity and condition corresponding to hidden correlations in quantum systems. We present qubit states in the quantum suprematism picture, where these states are identified with three probability distributions, describing the states of three classical coins, and illustrate the states by Triada of Malevich’s squares with areas satisfying the quantum constraints. We consider arbitrary quantum states belonging to N-dimensional Hilbert space as $(N^2-1)$ fair probability distributions describing the states of $(N^2-1)$ classical coins. We illustrate the geometrical properties of the qudit states by a set of Triadas of Malevich’s squares. We obtain new entropic inequalities for matrix elements of an arbitrary density N×N-matrix of qudit systems using the constructed maps of the density matrix on a set of the probability distributions. In addition, to construct the bijective map of the qudit state onto the set of probabilities describing the positions of classical coins, we show that there exists a bijective map of any quantum observable onto the set of dihotomic classical random variables with statistics determined by the above classical probabilities. Finally, we discuss the physical meaning and possibility to check derived inequalities in the experiments with superconducting circuits based on Josephson junction devices. Full article

The neural network approach is proposed for studying very-low- and low-frequency (VLF and LF) subionospheric radio wave variations in the time vicinities of magnetic storms and earthquakes, with the purpose of recognizing anomalies of different types. We also examined the days with quiet geomagnetic conditions in the absence of seismic activity, in order to distinguish between the disturbed signals and the quiet ones. To this end, we trained the neural network (NN) on the examples of the representative database. The database included both the VLF/LF data that was measured during four-year monitoring at the station in Petropavlovsk-Kamchatsky, and the parameters of seismicity in the Kuril-Kamchatka and Japan regions. It was shown that the neural network can distinguish between the disturbed and undisturbed signals. Furthermore, the prognostic behavior of the VLF/LF variations indicative of magnetic and seismic activity has a different appearance in the time vicinity of the earthquakes and magnetic storms. Full article

In this paper, we obtain the upper bounds for the normalized $\delta$ -Casorati curvatures and generalized normalized $\delta$ -Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds with constant curvature. Further, we discuss the equality case of the inequalities. Moreover, we give the necessary and sufficient condition for a Sasaki-like statistical manifold to be $\eta$ -Einstein. Finally, we provide the condition under which the metric of Sasaki-like statistical manifolds with constant curvature is a solution of vacuum Einstein field equations. Full article