最大熵原理及改进方法的研究现状

熵作为信息论中的一个基本概念受到了广泛的关注与研究.以熵为基础的最大熵原理也在众多学者的研究与应用中不断发展,逐渐形成了自有的理论体系并得到了广泛应用,例如在使用贝叶斯方法进行测量不确定度评定时,最大熵原理可以估计先验分布.本文以Shannon熵为主要对象,对目前研究中涉及的不同约束下最大熵原理进行了归纳,并整理为形式...     

Entropy propagation analysis in stochastic structural dynamics: application to a beam with uncertain cross sectional area

This paper investigates the impact of different probabilistic models of uncertain parameters on the response of a dynamical structure. The probabilistic models of the uncertain parameters are constructed using the maximum entropy principle, where different information is considered, such as bounds, mean value, etc. Nested probabilistic models are constructed with increasing information; as the information given increases, the level of entropy of the input model decreases. The response of the linear dynamical model is given in the frequency domain, and the propagation of the input uncertainty throughout the computational model is analyzed in terms of Shannon’s entropy. Low and high frequencies are analyzed because uncertainties propagate differently depending on the frequency band. A beam discretized by means of the finite element method with random cross sectional area (random field) is the application analyzed.     

Spectral color constancy using a maximum entropy approach

This paper proposes a solution to the spectral color constancy problem. The method is based on a statistical model for the surface reflectance spectrum and applies a maximum entropy constraint. Unlike prior methods based on linear models, the solution process does not require a set of basis functions to be defined, nor does it require a database of spectra to be specified in advance. Experiments on simulated and real data show that spectral estimation using the maximum entropy approach is feasible and performs similarly to existing spectral methods in spite of the lower level of a priori information required.     

Probabilistic modeling of apparent tensors in elastostatics: A MaxEnt approach under material symmetry and stochastic boundedness constraints

In this work, we address the stochastic modeling of apparent elasticity tensors, for which both material symmetry and stochastic boundedness constraints have to be taken into account, in addition to the classical constraint of invertibility. We first introduce a stochastic measure of anisotropy, which is defined using metrics in the set of elasticity tensors and used for quantitatively characterizing the fulfillment of material symmetry constraints. After having defined a numerical approximation for the stochastic boundedness constraint, we then propose a methodology allowing one to unify maximum entropy based models that have been previously derived by considering some of these constraints and which consists in constructing a probabilistic model for an auxiliary random variable. The latter can be interpreted as a stochastic compliance tensor, for which the available information to be used in the maximum entropy formulation can be readily deduced from the one considered for the elasticity tensor. A numerical illustration of the approach to an elastic microstructure is finally provided.     

Constrained noninformative priors in risk assessment

A constrained noninformative prior distribution, a generalization of the Jeffreys noninformative prior, is defined for a single unknown parameter as the distribution corresponding to the maximum entropy distribution, subject to the assumed constraint(s), in the transformed model where the unknown parameter is approximately a location parameter. This note illustrates this idea with binomial and Poisson data models, and gives an example from risk assessment showing the practical usefulness of the constrained noninformative prior.     

Modelling enthalpy and entropy of pure and mixed refrigerants with an innovative corresponding states method

In this work an original improvement of the Corresponding States technique is developed and a new model, based on a three parameters CS format, is proposed to predict the enthalpy and the entropy of the new generation halogenated alkanes fluids together with some alkanes. Limiting the analysis of the selected fluids to a specific thermodynamic property behaviour, an appropriate conformality approach can be deduced, which allows to set up a predictive model of high accuracy level on a wide range of the enthalpy and entropy surfaces. The fundamentals of the model are innovative scaling parameters deduced from the enthalpy of vaporization and from two dedicated equations, belonging to the selected family of fluids. This allows to set up innovative models following a CS format. Through the introduction of advanced mixing rules, the models can be simply extended to calculate the corresponding properties for mixtures. The proposed models allow also the calculation of VLE for systems of rather regular behaviour. The required inputs for a pure target fluid are an ideal gas isobaric heat capacity correlation, a single value of saturated liquid density and of vaporization enthalpy; if the last one is lacking, a single value of vapor pressure can be alternatively supplied. For non azeotropic mixtures the enthalpy and entropy models are predictive, whereas in case of azeotropy VLE calculations are possibly only applying regressed interaction coefficients. Due to the lack of accurate experimental enthalpy data and to the particular nature of the entropy function, the validation of the models is proposed against fundamental dedicated EoS available, both for pure and mixtures, for a significant number of the studied family of fluids. The predictive character of the proposed approach as well as the high performances reached, make these models particularly suitable for the new families of fluids regarding advanced technological applications.     

Second Law Analysis and Optimization of Elliptical Pin Fin Heat Sinks Using Firefly Algorithm

One of the most significant considerations in the design of a heat sink is thermal management due to increasing thermal flux and miniature in size. These heat sinks utilize plate or pin fins depending upon the required heat dissipation rate. They are designed to optimize overall performance. Elliptical pin fin heat sinks enhance heat transfer rates and reduce the pumping power. In this study, the Firefly Algorithm is implemented to optimize heat sinks with elliptical pin-fins. The pin-fins are arranged in an inline fashion. The natureinspired metaheuristic algorithm performs powerfully and efficiently in solving numerical global optimization problems. Based on mass, energy, and entropy balance, three models are developed for thermal resistance, hydraulic resistance, and entropy generation rate in the heat sink. The major axis is used as the characteristic length, and the maximum velocity is used as the reference velocity. The entropy generation rate comprises the combined effect of thermal resistance and pressure drop. The total EGR is minimized by utilizing the firefly algorithm. The optimization model utilizes analytical/empirical correlations for the heat transfer coefficients and friction factors. It is shown that both thermal resistance and pressure drop can be simultaneously optimized using this algorithm. It is demonstrated that the performance of FFA is much better than PPA.     

Zubair Lomax Distribution: Properties and Estimation Based on Ranked Set Sampling

In this article, we offer a new adapted model with three parameters, called Zubair Lomax distribution. The new model can be very useful in analyzing and modeling real data and provides better fits than some others new models. Primary properties of the Zubair Lomax model are determined by moments, probability weighted moments, Renyi entropy, quantile function and stochastic ordering, among others. Maximum likelihood method is used to estimate the population parameters, owing to simple random sample and ranked set sampling schemes. The behavior of the maximum likelihood estimates for the model parameters is studied using Monte Carlo simulation. Criteria measures including biases, mean square errors and relative efficiencies are used to compare estimates. Regarding the simulation study, we observe that the estimates based on ranked set sampling are more efficient compared to the estimates based on simple random sample. The importance and flexibility of Zubair Lomax are validated empirically in modeling two types of lifetime data.     

基于云模型的信任评估研究

讨论了信任关系的随机性和模糊性共存以及相互融合问题。分析云模型描述不确定性概念的方法和实现定性语意与定量数值相互转换的算法,提出了基于云理论的信任评估模型—信任云。该模型提出云特征参数表达的信任传递和合并算法,在精确描述信任期望值的同时,通过熵和超熵刻画了信任的不确定性。相对于传统的信任评估策略,该模型获取的信任值包含更多的语意信息,更适合作为信任决策的依据。     

Calibration and validation of coarse-grained models of atomic systems: application to semiconductor manufacturing

Coarse-grained models of atomic systems, created by aggregating groups of atoms into molecules to reduce the number of degrees of freedom, have been used for decades in important scientific and technological applications. In recent years, interest in developing a more rigorous theory for coarse graining and in assessing the predictivity of coarse-grained models has arisen. In this work, Bayesian methods for the calibration and validation of coarse-grained models of atomistic systems in thermodynamic equilibrium are developed. For specificity, only configurational models of systems in canonical ensembles are considered. Among major challenges in validating coarse-grained models are (1) the development of validation processes that lead to information essential in establishing confidence in the model’s ability predict key quantities of interest and (2), above all, the determination of the coarse-grained model itself; that is, the characterization of the molecular architecture, the choice of interaction potentials and thus parameters, which best fit available data. The all-atom model is treated as the “ground truth,” and it provides the basis with respect to which properties of the coarse-grained model are compared. This base all-atom model is characterized by an appropriate statistical mechanics framework in this work by canonical ensembles involving only configurational energies. The all-atom model thus supplies data for Bayesian calibration and validation methods for the molecular model. To address the first challenge, we develop priors based on the maximum entropy principle and likelihood functions based on Gaussian approximations of the uncertainties in the parameter-to-observation error. To address challenge (2), we introduce the notion of model plausibilities as a means for model selection. This methodology provides a powerful approach toward constructing coarse-grained models which are most plausible for given all-atom data. We demonstrate the theory and methods through applications to representative atomic structures and we discuss extensions to the validation process for molecular models of polymer structures encountered in certain semiconductor nanomanufacturing processes. The powerful method of model plausibility as a means for selecting interaction potentials for coarse-grained models is discussed in connection with a coarse-grained hexane molecule. Discussions of how all-atom information is used to construct priors are contained in an appendix.     

Peculiarities of calculating entropy in self-consistent field models

The calculation of entropy in the self-consistent field models faces some difficulties related to the entropy behavior peculiarities at low temperatures in the models with different degrees of specification of the electron energy spectrum. This paper considers the behavior of the electron entropy in two self-consistent field models that have been widely used lately, namely the Liberman model and the quasi-zone model. Special attention is focused on low temperatures. In particular, it is shown that the entropy of electrons in the Liberman model at low densities and the temperature T = 0 takes on nonzero values which differ in the relativistic and nonrelativistic versions of the model. The electron entropy in the quasi-zone model at low densities of the matter and T = 0 is zero. In addition, it is shown that at low temperatures and high densities the entropy in the Liberman model smoothly turns to the asymptotic form of the Thomas-Fermi (TF) model, while such a smooth transition to the TF asymptotic form is not always observed in the quasi-zone model.     

The description of morphologically stable regimes for steady state solidification based on the maximum entropy production rate postulate

The maximum entropy production rate (MEPR) in the solid–liquid zone is developed and tested as a possible postulate for predicting the stable morphology for the special case of steady state directional solidification (DS). The principle of MEPR states that, if there are sufficient degrees of freedom within a system, it will adopt a stable state at which the entropy generation (production) rate is maximized. Where feasible, the system will also try and adopt a steady state. The MEPR postulate determines the most probable state and therefore allows pathway selections to occur in an open thermodynamic system. In the context of steady state solidification, pathway selections are reflected in the corresponding morphological selections made by the system in the solid–liquid (mushy) zone in order to cope with the required entropy production. Steady state solidification is feasible at both close to, and far from equilibrium conditions. Based on MEPR, a model is proposed for examining the stability of various morphologies that have been experimentally observed during steady state directional solidification. This model employs a control volume approach for entropy balance, including the entropy generation term (S _gen), which depends on the diffuse zone and average temperature of the solid–liquid region within the control volume. In this manner, the model takes a different approach from the successful kinetic models that have been able to predict key features of stable morphological patterns. Unstable planar interfaces, faceted cellular arrays, cell–dendrite transitions, half cells both faceted and smooth, and other transitions such as the absolute stability transition at high solid/liquid velocities are examined with the model. Uncommon solidification morphological features such as non-crystallographic dendrites and discontinuous cell-tip splitting are also examined with the model. The preferred morphological change-direction for the emergence of the stable morphological feature is inferred with the MEPR postulate in a manner analogous to the free energy minimization principle(s) when used for predicting phase stability and metastable phase formation. Aspects of mixed-mode order transformation characteristics are also discussed for non-equilibrium solidification containing a diffuse interface, in contrast to classifying solidification as purely a first order transformation. The MEPR model predictions are shown to follow the experimental transitions observed to date in several historical studies.     

基于最大幅值变分模态分解和均方根熵的滚动轴承故障诊断

针对变分模态分解中模态个数的设定会对分解结果产生重要影响的问题,提出一种求取最优分解层数的方法,该方法以瞬时频率的幅值特性为依据,通过分析变分模态分解过程中,各分量最大幅值之间的关系来确定最佳分解参数;均方根熵可以反映不同振动信号的能量值,以信号均方根熵为故障特征参量,通过优化支持向量机建立故障分类模型,实现故障模式分类。将基于最大幅值变分模态分解和均方根熵的故障诊断方法应用于滚动轴承实测信号中,实验结果表明基于最大幅值变分模态分解和均方根熵的方法能够有效识别滚动轴承运行状态,识别准确率高达98.75%。     

The Principle of Maximum Entropy Applied in the Evaluation of the Measurement Uncertainty

The maximum entropy approach is a flexible and powerful tool for assigning a probability distribution to a measurable quantity treated as a random variable subjected to known moment constraints. The aim of this paper is to describe how the principle of maximum entropy may be used to transform information about the value of a quantity into a probability density function (pdf) that reflects exactly that information and nothing else. This principle will be applied to common cases of metrological interest, where different kinds of information are available. The derivation of the pdf is given in each case, and two practical examples with numerical results are reported to demonstrate the efficiency of the maximum entropy method     

Structural uncertainty modeling for nonlinear geometric response using nonintrusive reduced order models

The focus of the present investigation is on the introduction of uncertainty directly in reduced order models of the nonlinear geometric response of structures following maximum entropy concepts. While the approach was formulated and preliminary validated in an earlier paper, its broad application to a variety of structures based on their finite element models from commercial software was impeded by two key challenges. The first of these involves an indeterminacy in the mapping of the nonlinear stiffness coefficients identified from the finite element model to those of the reduced order model form that is suitable for the uncertainty analysis. The second challenge is that a key matrix in the uncertainty modeling was expected to be positive definite but was numerically observed not to be. This latter issue is shown here to be rooted in differences in nonlinear finite element modeling between the commercial software and the theoretical developments. Both of these challenges are successfully resolved and applications examples are presented that confirm the broad applicability of the methodology.     

Second law analysis of an automotive air conditioning system using HFO-1234yf,an environmentally friendly refrigerant

In this paper, a comparative study of the second law of thermodynamics is presented to determine the possibility of using HFO-1234yf, an environmentally friendly refrigerant, as a drop-in replacement of HFC-134a in automotive air conditioning system. For the thermodynamic analysis, a computer program is written to simulate the operating conditions of automobile air conditioning system. The thermodynamic properties of the refrigerants are extracted from the REFPROP 8.0 software. For calculating the coefficient of performance (COP), exergy destruction, exergy efficiency and entropy generation, computational models are used to evaluate the effects of different parameters on their changes. It is found that using HFO-1234yf as the air conditioning refrigerant leads to higher exergy efficiency compared to HFC-134a. Also, maximum entropy generation and exergy destruction occur in the compressor. The exergy destruction and entropy generation of the cycle components are less in the case of using HFO-1234yf refrigerant instead of HFC-134a.     

Maximum entropy principle and non-stationary distributions of stochastic systems

The principle of maximum entropy (in its classical form), successfully applied in many fields (e.g. statistics, reliability, estimation), has recently been extended to analyze the systems governed by stochastic differential equations and especially to determining the stationary probability distribution of the solution process. In this paper we develop the maximum entropy approach to characterize non-stationary probability distributions of the solutions of stochastic systems. The variational problem for the entropy functional includes time-dependent constraints in the form of differential equations for moments. The general scheme of the method is given along with the effective treatment of a number of first and second order stochastic systems. The maximum entropy probability distributions are compared with the exact solutions and with the simulation results.     

A Constraint‐based Maximum Entropy Sampling Method for Kriging Models in Fuel Cell Applications

A metamodel replaces the simulation model with an approximation model to make design optimization computationally achievable. The accuracy of a metamodel depends highly on the choice of sampling points. This article proposes a constraint‐based maximum entropy sampling method that locates most sampling points within a feasible constraint domain represented as a complex nonlinear function. As a robust measure of information, a maximum entropy criterion is used to select sampling points for constructing the Kriging model. The violation ratio from the feasible domain is incorporated into the covariance function in the Kriging model. The constraint‐based maximum entropy sampling method is applied to reduce the weight of a bipolar plate in a vanadium redox battery by optimizing its channel design. The proposed sampling method rapidly approximates the boundary of the feasible domain with a relatively small number of sampling points. Final optimal design results for the plate channel using the proposed method indicate a significant reduction in the plate weight compared with the existing bipolar plate design. Copyright © 2012 John Wiley & Sons, Ltd.     

生产资源组织模式结构与运行评价的熵测度法

针对不同生产资源组织模式的结构与运行状态的差异,目前尚缺乏系统的定量评价方法。基于信息熵理论及有序度定义,建立了结构熵模型与运行熵模型来分别评价生产资源组织模式在结构上和运行中的有序度。其中,结构熵模型包括时效熵和质量熵两部分;运行熵模型包括静态运行熵和动态运行熵两部分。对某压缩机制造企业的加工车间做了实例研究,以实施动态单元组织模式前后的结构与运行状态作对比分析,分别求解出2种不同模式下的结构熵与运行熵。结果表明,动态单元组织模式通过合理规划人力、设备资源及生产流程,能够极大减少企业生产中零件的转运与设备空闲时间,具有更优的结构有序度和运行有序度,能获得更好的零件加工效率。