Entropy Principle Research Articles

On the statistical behavior of homogenized properties and ultrasonic phase velocities in random polycrystals

Most theoretical studies on homogenized properties of polycrystals consider infinite textureless media with orientations characterized by independent Euler angles. However, microstructural analyses of polycrystals reveal spatially correlated orientations of grains whose sizes often follow lognormal distributions. Moreover, experimental investigations show that the single-crystal elastic constants (SEC) in the crystallite’s local coordinates could exhibit variabilities. To the best of our knowledge, in the context of our study, these have never been considered in the literature. In this paper, the crystal orientations are simulated using random fields (RFs) with different correlation parameters. A maximum entropy principle is used to simulate realizations of the local stiffness matrices. Numerical results indicate that generating Euler angles using independent random variables is legitimate when correlation lengths of orientations are close enough to the average grain size. Analytical formulas are derived to estimate the statistical behavior of effective elastic moduli and the phase velocities considering either unimodal or bimodal grain size distributions and fluctuations in local tensors for both two- and three-dimensional polycrystals. The former highlight the important roles of the coefficient of variations of the grain sizes and that of the elastic constants. This work contributes to microstructural characterization research associated with ultrasonic phase velocity measurements.

Fracture network organization with continuation of injections in the Habanero field (Cooper Basin, Australia) disclosed by coda wave fluctuations

Hydraulic stimulation of a geothermal reservoir aims to increase the permeability of the rock formations by creating/enhancing a fracture network, to take advantage of the geothermal energy hosted by deep hot formations. A common side effect of the fracturing process is the small-in-magnitude but abundant induced seismicity. In this work, we study the seismicity induced during the hydraulic stimulation of the Habanero 4 well, in the Cooper Basin (Australia), where an extensive network of sensors has been deployed to monitor the stimulation process. The detailed monitoring of the induced seismic events enabled the study of the dynamic characteristics of the enhanced fracture network by making use of the seismic coda waves scattering properties. Since the Earth’s crust has inhomogeneities with fractal properties, manifested in the seismic coda waves, we address the fracture network as a complex, non-equilibrium system. In this line, the framework of Non-Extensive Statistical Physics (NESP) and the maximum entropy principle were used to study the fluctuations of the seismic coda waves and their possible variations during the hydraulic stimulation. The analysis shows that the probability density functions of the normalized increments of S-coda waves present heavy tails with asymptotic power-law behavior, enhancing the probabilities of large fluctuations. The results further indicate that as the stimulation progresses, this enhancement becomes even more pronounced. The observed scaling behavior can exactly be approximated with the q-Gaussian distribution, derived in the framework of NESP, showing a modest uprising trend of the entropic index (q-value) with continuation of injections. These results suggest that the system of fractures progressively becomes more organized, with stronger memory effects and long-range interactions, as the hydraulic stimulation progresses.

Parameter Estimation of the Dirichlet Distribution Based on Entropy

The Dirichlet distribution as a multivariate generalization of the beta distribution is especially important for modeling categorical distributions. Hence, its applications vary within a wide range from modeling cell probabilities of contingency tables to modeling income inequalities. Thus, it is commonly used as the conjugate prior of the multinomial distribution in Bayesian statistics. In this study, the parameters of a bivariate Dirichlet distribution are estimated by entropy formalism. As an alternative to maximum likelihood and the method of moments, two methods based on the principle of maximum entropy are used, namely the ordinary entropy method and the parameter space expansion method. It is shown that in estimating the parameters of the bivariate Dirichlet distribution, the ordinary entropy method and the parameter space expansion method give the same results as the method of maximum likelihood. Thus, we emphasize that these two methods can be used alternatively in modeling bivariate and multinomial Dirichlet distributions.

Bayesian analysis of systematic errors in the determination of the constant of gravitation

Measurements of the gravitational constant G are notoriously difficult. Individual state-of-the-art experiments have managed to determine the value of G with high precision: although, when considered collectively, the range in the measured values of G far exceeds individual uncertainties, suggesting the presence of unaccounted for systematic effects. Here, we propose a Bayesian framework to account for the presence of systematic errors in the various measurement of G while proposing a consensus value, following two paths: a parametric approach, based on the maximum entropy principle, and a non-parametric one, the latter being a very flexible approach not committed to any specific functional form. With both our methods, we find that the uncertainty on this fundamental constant, once systematics are included, is significantly larger than what quoted in CODATA 2018. Moreover, the morphology of the non-parametric distribution hints towards the presence of several sources of unaccounted for systematics. In light of this, we recommend a consensus value for the gravitational constant G = 6.6740^{+0.0015}_{-0.0015} times 10^{-11} textrm{m}^3 textrm{kg}^{-1} textrm{s}^{-2}.

Maximum-entropy principle in knowledge-enhanced machine learning in acoustics

Recent acoustics investigations increasingly apply model-based methods, they purposely empower learning machines to analyze intrinsic parameters encapsulated in well-established models from experimental data. These data-driven analyses incorporate the well-understood /established models to the machine learning process as an important part of prior information, in addition to another part of prior information on the model parameters of interest. A data-driven machine learning method represents a learning process from data to update our knowledge, namely, learning from the data. This consistently represents the core of probabilistic inference within Bayesian framework on how one's prior knowledge is improved upon the experimental data. This paper discusses the data-driven machine learning methods in a way of thinking like a Bayesian. It emphasizes that knowledge-enhanced incorporation of the well-understood models in many learning processes can rigorously resort to the principle of maximum entropy, including Gaussian process due to their Bayesian nature. Benefits of the data-driven machine learning methods are highlighted when incorporating well understood/established models. This paper discusses physics-informed, knowledge-enhanced methods based on recent acoustic investigations, such as estimation of acoustic boundary conditions in wave-based numerical simulations and diffusion-equation-informed reverberation analysis.

Propagation of material uncertainty in modal parameters and its influence in damage quantification of shear buildings

Structural health monitoring (SHM) plays a crucial role in post-disaster mitigation in investigating the serviceability of an existing structure. The physical properties, namely mass density, cross-sectional area, Poisson’s ratio, modulous of elasticity, etc., vary spatially throughout a real system. These uncertainties propagate in the modal parameters involved in SHM techniques. In this study, the elastic modulous has been considered the primary source of uncertainty. A detailed literature survey has been performed to identify the suitable probability distribution recommended for defining material uncertainty. Although the Gaussian distribution cannot replicate realistic material behaviour, its explicit implementation makes it prevalent. However, based on the principle of maximum entropy, the gamma distribution is the most appropriate recommendation among all non-Gaussian continuous distributions ranging [0, ∞). The Karhunen–Loève (KL) expansion has been adopted to discretize the stochastic field to determine the spectral mass and stiffness of a multi-damaged axially vibrating prismatic cantilever bar-type structure. Closed-form orthogonal pairs of eigensolutions of the KL expansion considering (Adhikari and Friswell, 2010) exponential covariance function have been derived. The expressions for spectral mass and stiffness for a multi-damaged cantilever bar considering material uncertainty have been developed. The uncertainty propagation in the modal properties has been estimated. The complete procedure has been followed to identify uncertainty propagation in damage severity of a common class of civil structures like shear buildings. The influence of material uncertainty in determining the damage severity of a structure with numerable degrees of freedom using a non-destructive approach has been explored. Extensive simulations have been performed to evaluate the effectiveness of the present study in a multi-damaged bar as well as shear building. Statistical parameters, namely mean and standard deviation of natural frequencies, modal displacement and damage severity have been determined. The uncertainty propagation has been reflected in the mean value for varying input stochasticity. However, the standard deviation remained consistent for undamaged as well as damaged structure irrespective of material uncertainty type. Considering gamma distribution in material stochasticity, the spread of uncertainty propagation has been more than the Gaussian distribution. However, experimental studies need to be performed further to investigate the stochastic behaviour in modal parameters of a real structure.

Multi-level Uncertain Fatigue Analysis of a Truss under Incomplete Available Information

We predict the fatigue life of a planar tubular truss when geometrical parameters, material properties, and live loads are non-deterministic. A multi-level calculation uncertainty quantification framework code was designed to aggregate the finite element method and fatigue-induced sequential failures. Due to the incompleteness of the aleatory-type inputs, the maximum entropy principle was applied. Two sensitivity analyses were performed to report the most influencing factors. In terms of variance, the results suggest that the slope of the curve crack growth rate × stress intensity factor range is the most influencing factor related to fatigue life. Furthermore, due to the application of the entropy concept, the fatigue crack growth boundaries and fatigue crack size boundaries obtained provide the most unbiased fatigue crack design mapping. These boundaries allow the designer to select the worst-case fatigue scenario, besides being able to predict the crack behavior at a required confidence level.

Predicting the current and future distribution of Monochamus carolinensis (Coleoptera: Cerambycidae) based on the maximum entropy model.

Monochamus carolinensis is an important vector of pinewood nematodes in North America that is under quarantine in several countries worldwide. The distribution of M. carolinensis was previously thought to be limited to North America; however, we discovered it during trapping in China in 2022. Using this discovery and information regarding the area of origin, we applied a machine-learning algorithm based on the maximum entropy principle to predict the current and future (2050s, 2070s) potential distribution areas of M. carolinensis using bioclimatic variables. The biological suitability of M. carolinensis was mainly driven by precipitation factors (BIO18, BIO15, BIO19), with 87.18% of the potential distribution areas located in South America, Asia, North America and Africa. Future potential distribution areas of M. carolinensis are predicted to expand to high latitudes, with an average increase of 10 245 874.88 km2 , and only 6.89% of the current suitable areas will become unsuitable. The potential distribution areas in 2070 are largest under the SSP585 scenario, with a 41.40% predicted increase (52 309 803.61 km2 ) above the current distribution, mainly reflecting an increase of the marginally and highly suitable areas. The determination of dominant climatic factors and potential distribution areas will help provide an early warning for an M. carolinensis invasion, as well as provide a scientific basis for the spread and outbreak, facilitating development of effective governmental prevention and control measures. © 2023 Society of Chemical Industry.

Synergy Prediction Model of Information Entropy Based on Zone Safety Degree and Stope Roof Weighting Step Analysis

During the underground mining of coal resources, the rock pressure emerges acutely and the mine geological disasters occur frequently. It is of great significance to grasp the manifestation law of rock pressure in time to guide the safety production and operation in the pit. In this research, the calculation equations and concept of information entropy based on zone safety degree are primarily defined, and the synergetic theory of the maximum information entropy principle is combined to put forward the synergy prediction model of information entropy based on zone safety degree. In the meantime, the synergy prediction model of information entropy based on zone safety degree is employed to calculate and predict the first weighting step and the periodic weighting step of the main roof of the 9203 working face of Hengsheng Coal Mine in China’s Shanxi Province, as well as verifying the validity and reliability of the synergy prediction model of information entropy based on zone safety degree by the comparison of similar simulation test results, which has presented a scientific basis for the effective control of rock pressure and roof management.

Coal Classification Based on Reflection Spectroscopy and the IAT-TELM Algorithm.

As a principal energy globally, coal's quality and variety critically influence the effectiveness of industrial processes. Different coal types cater to specific industrial requirements due to their unique attributes. Traditional methods for coal classification, typically relying on manual examination and chemical assays, lack efficiency and fail to offer consistent accuracy. Addressing these challenges, this work introduces an algorithm based on the reflectance spectrum of coal and machine learning. This method approach facilitates the rapid and accurate classification of coal types through the analysis of coal spectral data. First, the reflection spectra of three types of coal, namely, bituminous coal, anthracite, and lignite, were collected and preprocessed. Second, a model utilizing two hidden layer extreme learning machine (TELM) and affine transformation function is introduced, which is called affine transformation function TELM (AT-TELM). AT-TELM introduces an affine transformation function on the basis of TELM, so that the hidden layer output satisfies the maximum entropy principle and improves the recognition performance of the model. Third, we improve AT-TELM by optimizing the weight matrix and bias of AT-TELM to address the issue of highly skewed distribution caused by randomly assigned weights and biases. The method is named the improved affine transformation function (IAT-TELM). The experimental findings demonstrate that IAT-TELM achieves a remarkable coal classification accuracy of 97.8%, offering a cost-effective, rapid, and precise method for coal classification.

What constrains food webs? A maximum entropy framework for predicting their structure with minimal biases.

Food webs are complex ecological networks whose structure is both ecologically and statistically constrained, with many network properties being correlated with each other. Despite the recognition of these invariable relationships in food webs, the use of the principle of maximum entropy (MaxEnt) in network ecology is still rare. This is surprising considering that MaxEnt is a statistical tool precisely designed for understanding and predicting many types of constrained systems. This principle asserts that the least-biased probability distribution of a system's property, constrained by prior knowledge about that system, is the one with maximum information entropy. MaxEnt has been proven useful in many ecological modeling problems, but its application in food webs and other ecological networks is limited. Here we show how MaxEnt can be used to derive many food-web properties both analytically and heuristically. First, we show how the joint degree distribution (the joint probability distribution of the numbers of prey and predators for each species in the network) can be derived analytically using the number of species and the number of interactions in food webs. Second, we present a heuristic and flexible approach of finding a network's adjacency matrix (the network's representation in matrix format) based on simulated annealing and SVD entropy. We built two heuristic models using the connectance and the joint degree sequence as statistical constraints, respectively. We compared both models' predictions against corresponding null and neutral models commonly used in network ecology using open access data of terrestrial and aquatic food webs sampled globally (N = 257). We found that the heuristic model constrained by the joint degree sequence was a good predictor of many measures of food-web structure, especially the nestedness and motifs distribution. Specifically, our results suggest that the structure of terrestrial and aquatic food webs is mainly driven by their joint degree distribution.

Involutory Negator of Basic Belief Assignments

Abstract This paper analyzes the different definitions of a negator of a probability mass function (pmf) and a Basic Belief Assignment (BBA) available in the literature. To overcome their limitations we propose an involutory negator of BBA, and we present a new indirect information fusion method based on this negator which can simplify the conflict management problem. The direct and indirect information fusion strategies are analyzed for three interesting examples of fusion of two BBAs. We also propose two methods for using the whole available information (the original BBAs and their negators) for decision-making support. The first method is based on the combination of the direct and indirect fusion strategies, and the second method selects the most reasonable fusion strategy to apply (direct, or indirect) based on the maximum entropy principle.

The motion reliability analysis of a series system with multiple coordinated manipulators

Coordinated manipulators can construct a series system, such that all the position error of the end-effector is required to remain within the permissible limit. This article proposed a new method to handle the problem of reliability analysis involving in multiple failure modes. The extreme value of the position error is introduced to construct a surrogate model to replace the original failure event, the moment with fraction order instead of the traditional integer is applied to derive the best-fit distribution of the extreme value based on the maximum entropy principle. The innovations mainly include: (1) the high efficient dimension reduction via the surrogate model constructed with the extreme value and (2) a better probability density function is derived through the fractional moment. A series system consists of three manipulators is used as an example to demonstrate the efficiency and accuracy of the proposed method.

Synergetic Theory of Information Entropy Based on Failure Approach Index for Stability Analysis of Surrounding Rock System.

It is generally acknowledged that the stability evaluation of surrounding rock denotes nonlinear complex system engineering. In order to accurately and quantitatively assess the safety states of surrounding rock and provide a scientific basis for the prevention and control of surrounding rock stability, the analysis method of the synergetic theory of information entropy using the failure approach index has been proposed. By means of deriving the general relationship between the total two-dimensional plastic shear strain and the total three-dimensional plastic shear strain and obtaining the numerical limit analysis step of the plastic shear strain, the threshold value of the ultimate plastic shear strain can be determined, which has provided the key criterion for the calculation of the information entropy based on the failure approach index. In addition, combining with the synergetic theory of the principle of maximum information entropy, the evolution equation of the excavation step and information entropy based on the failure approach index of the surrounding rock system in underground mining space are established, and the equations of the general solution and particular solution as well as the expression of the destabilizing excavation step are given. To account for this, the method is applied to analyze the failure states of the floor surrounding rock after the mining of the 71 coal seam in Xutuan Coal Mine and involve the disturbance effect and stability control method of the underlying 72 coal seam roof from the macroscopic and microscopic aspects. Consequently, the validity of the analysis method of synergetic theory of information entropy based on the failure approach index has been verified, which presents an updated approach for the stability evaluation of surrounding rock systems that is of satisfactory capability and value in engineering applications.

Coupled coordination analysis of green finance on economic growth based on big data

Abstract Exploring the coupled and coordinated relationship between green finance and economic growth is to achieve the high-quality dual development of green and economy. This paper studies the rough set and information entropy attribute approximate calculation based on MapReduce parallel computing framework under big data technology. The basic principles of rough set and information entropy are described, and the MapReduce algorithm is used to calculate information entropy, mutual information, and attribute kernel, which shows that MapReduce calculation is always on the key-value pairs. For the correlation analysis of green finance and economic growth, the coupled coordination model of green finance and economic growth is constructed using the MapReduce algorithm, and the coupled coordination analysis of green finance and economic growth is carried out for this model. From the comprehensive evaluation index, the comprehensive evaluation index of green finance and economic growth increased by 6308.44% and 2242.21%, respectively. The coupling degree of green finance and economic growth increased by 164.66% and 835.84% from the coupling degree and coordination degree, respectively. Thus, the coupling relationship between green finance and economic growth based on big data is relatively low, but the coordination degree is high, indicating that the two influence each other and interact.

Optimizing molecular potential models by imposing kinetic constraints with path reweighting.

Empirical force fields employed in molecular dynamics simulations of complex systems are often optimized to reproduce experimentally determined structural and thermodynamic properties. In contrast, experimental knowledge about the interconversion rates between metastable states in such systems is hardly ever incorporated in a force field due to a lack of an efficient approach. Here, we introduce such a framework based on the relationship between dynamical observables, such as rate constants, and the underlying molecular model parameters using the statistical mechanics of trajectories. Given a prior ensemble of molecular dynamics trajectories produced with imperfect force field parameters, the approach allows for the optimal adaption of these parameters such that the imposed constraint of equally predicted and experimental rate constant is obeyed. To do so, the method combines the continuum path ensemble maximum caliber approach with path reweighting methods for stochastic dynamics. When multiple solutions are found, the method selects automatically the combination that corresponds to the smallest perturbation of the entire path ensemble, as required by the maximum entropy principle. To show the validity of the approach, we illustrate the method on simple test systems undergoing rare event dynamics. Next to simple 2D potentials, we explore particle models representing molecular isomerization reactions and protein-ligand unbinding. Besides optimal interaction parameters, the methodology gives physical insights into what parts of the model are most sensitive to the kinetics. We discuss the generality and broad implications of the methodology.

On the impact of dimensionally-consistent and physics-based inner products for POD-Galerkin and least-squares model reduction of compressible flows

Model reduction of the compressible Euler equations based on proper orthogonal decomposition (POD) and Galerkin orthogonality or least-squares residual minimization requires the selection of inner product spaces in which to perform projections and measure norms. The most popular choice is the vector-valued L2(Ω) inner product space. This choice, however, yields dimensionally-inconsistent reduced-order model (ROM) formulations which often lack robustness. In this work, we try to address this weakness by studying a set of dimensionally-consistent inner products with application to the compressible Euler equations. First, we demonstrate that non-dimensional inner products have a positive impact on both POD and Galerkin/least-squares ROMs. Second, we further demonstrate that physics-based inner products based on entropy principles result in drastically more accurate and robust ROM formulations than those based on non-dimensional L2(Ω) inner products. As test cases, we consider the following problems: the one-dimensional Sod shock tube, a two-dimensional Riemann problem, a two-dimensional Kelvin-Helmholtz instability, and two-dimensional homogeneous isotropic turbulence.

Harmonic State Estimation for Distribution Networks Based on Multi-Measurement Data

Accurate harmonic state estimation (HSE) is necessary to mitigate harmonic issues in power distribution networks. However, existing HSE methods face difficulties in solution of state equation that the measured data is insufficient with limited monitors. Therefore, this study proposes a HSE algorithm for distribution networks based on multi-measurement data. Firstly, a harmonic source location method based on mutual information and distribution network partition is proposed to reduce the number of state variables and to solve the state equation. Thus, HSE with low resolution (15 min) is realized. Secondly, a HSE method with enhanced resolution is developed. This method establishes maximum entropy principle models for harmonic currents injected into buses, calculates the probability density function (PDF) of the harmonic current, and realizes optimal interpolation of the harmonic data to enhance the resolution of the HSE (3 min) according to the PDF. Finally, the applicability and effectiveness of the proposed method are verified using an IEEE-33 test system simulation and a real power system in South China.

Information geometry of Markov Kernels: a survey

Information geometry and Markov chains are two powerful tools used in modern fields such as finance, physics, computer science, and epidemiology. In this survey, we explore their intersection, focusing on the theoretical framework. We attempt to provide a self-contained treatment of the foundations without requiring a solid background in differential geometry. We present the core concepts of information geometry of Markov chains, including information projections and the pivotal information geometric construction of Nagaoka. We then delve into recent advances in the field, such as geometric structures arising from time reversibility, lumpability of Markov chains, or tree models. Finally, we highlight practical applications of this framework, such as parameter estimation, hypothesis testing, large deviation theory, and the maximum entropy principle.

Bertrand’s Paradox Resolution and Its Implications for the Bing–Fisher Problem

Bertrand’s paradox is a problem in geometric probability that has resisted resolution for more than one hundred years. Bertrand provided three seemingly reasonable solutions to his problem — hence the paradox. Bertrand’s paradox has also been influential in philosophical debates about frequentist versus Bayesian approaches to statistical inference. In this paper, the paradox is resolved (1) by the clarification of the primary variate upon which the principle of maximum entropy is employed and (2) by imposing constraints, based on a mathematical analysis, on the random process for any subsequent nonlinear transformation to a secondary variable. These steps result in a unique solution to Bertrand’s problem, and this solution differs from the classic answers that Bertrand proposed. It is shown that the solutions proposed by Bertrand and others reflected sampling processes that are not purely random. It is also shown that the same two steps result in the resolution of the Bing–Fisher problem, which has to do with the selection of a consistent prior for Bayesian inference. The resolution of Bertrand’s paradox and the Bing–Fisher problem rebuts philosophical arguments against the Bayesian approach to statistical inference, which were based on those two ostensible problems.