A modified method to calculate reliability index using maximum entropy principle

Routine reliability index method, first order second moment (FOSM), may not ensure convergence of iteration when the performance function is strongly nonlinear. A modified method was proposed to calculate reliability index based on maximum entropy (MaxEnt) principle. To achieve this goal, the complicated iteration of first order second moment (FOSM) method was replaced by the calculation of entropy density function. Local convergence of Newton iteration method utilized to calculate entropy density function was proved, which ensured the convergence of iteration when calculating reliability index. To promote calculation efficiency, Newton down-hill algorithm was incorporated into calculating entropy density function and Monte Carlo simulations (MCS) were performed to assess the efficiency of the presented method. Two numerical examples were presented to verify the validation of the presented method. Moreover, the execution and advantages of the presented method were explained. From Example 1, after seven times iteration, the proposed method is capable of calculating the reliability index when the performance function is strongly nonlinear and at the same time the proposed method can preserve the calculation accuracy; From Example 2, the reliability indices calculated using the proposed method, FOSM and MCS are 3.823 9, 3.813 0 and 3.827 6, respectively, and the according iteration times are 5, 36 and 10 6 , which shows that the presented method can improve calculation accuracy without increasing computational cost for the performance function of which the reliability index can be calculated using first order second moment (FOSM) method.

Surface Elevation Distribution of Sea Waves Based on the Maximum Entropy Principle

A probability density function of surface elevation is obtained through improvement of the method introduced by Cieslikiewicz who employed the maximum entropy principle to investigate the surface elevation distribution. The density function can be easily extended to higher order according to demand and is non-negative everywhere, satisfying the basic behavior of the probability, Moreover because the distribution is derived without any assumption about sea waves, it is found from comparison with several accepted distributions that the new form of distribution can be applied in a wider range of wave conditions, In addition, the density function can be used to fit some observed distributions of surface vertical acceleration although something remains unsolved.

Tree Network Formation in Poisson Equation Models and the Implications for the Maximum Entropy Production Principle

This paper presents not only practical but also instructive mathematical models to simulate tree network formation using the Poisson equation and the Finite Difference Method (FDM). Then, the implications for entropic theories are discussed from the viewpoint of Maximum Entropy Production (MEP). According to the MEP principle, open systems existing in the state far from equilibrium are stabilized when entropy production is maximized, creating dissipative structures with low entropy such as the tree-shaped network. We prepare two simulation models: one is the Poisson equation model that simulates the state far from equilibrium, and the other is the Laplace equation model that simulates the isolated state or the state near thermodynamic equilibrium. The output of these equations is considered to be positively correlated to entropy production of the system. Setting the Poisson equation model so that entropy production is maximized, tree network formation is advanced. We suppose that this is due to the invocation of the MEP principle, that is, entropy of the system is lowered by emitting maximal entropy out of the system. On the other hand, tree network formation is not observed in the Laplace equation model. Our simulation results will offer the persuasive evidence that certifies the effect of the MEP principle.

A microscopic ancient river channel identification method based on maximum entropy principle and Wigner-Ville Distribution and its application

In view of the problem of fine characterization of narrow and thin channels,the maximum entropy criterion is used to enhance the focusing characteristics of Wigner-Ville Distribution.On the basis of effectively improving the time-frequency resolution of seismic signal,a new method of microscopic ancient river channel identification is established.Based on the principle of the equivalence between the maximum entropy power spectrum and the AR model power spectrum,the prediction error and the autoregression coefficient of AR model are obtained using the Burg algorithm and Levinson-Durbin recurrence rule.Under the condition of the first derivative of autocorrelation function being 0,the Wigner-Ville Distribution of seismic signal is calculated,and the Wigner-Ville Distribution time-frequency power spectrum(MEWVD)is obtained under the maxi-mum entropy criterion of the microscopic ancient river channel.Through analysis of emulational seismic signal and forward numerical simulation signal of narrow thin model,it is found that MEWVD can effectively avoid the interference of cross term of Wigner-Ville Distribution,and obtain more accurate spectral characteristics than STFT and CWT signal analysis methods.It is also proved that the narrow and thin river channels of different scales can be identified effectively by MEWVD of different frequencies.The method is applied to the third member of Jurassic Shaximiao Formation(J2s33-2)gas reservoir of the Zhongji-ang gas field in Sichuan Basin.The spatial information of width and direction of narrow and thin river channels with width less than 500 m and sandstone thickness less than 35 m is accurately identified,providing bases for well deployment and horizontal well fracturing section selection.

A Maximum-Entropy Compound Distribution Model for Extreme Wave Heights of Typhoon-Affected Sea Areas

A new compound distribution model for extreme wave heights of typhoon-affected sea areas is proposed on the basis of the maximum-entropy principle. The new model is formed by nesting a discrete distribution in a continuous one, having eight parameters which can be determined in terms of observed data of typhoon occurrence-frequency and extreme wave heights by numerically solving two sets of equations derived in this paper. The model is examined by using it to predict the N-year return-period wave height at two hydrology stations in the Yellow Sea, and the predicted results are compared with those predicted by use of some other compound distribution models. Examinations and comparisons show that the model has some advantages for predicting the N-year return-period wave height in typhoon-affected sea areas.

Extraction of Information from Crowdsourcing: Experimental Test Employing Bayesian, Maximum Likelihood, and Maximum Entropy Methods

A crowdsourcing experiment in which viewers (the “crowd”) of a British Broadcasting Corporation (BBC) television show submitted estimates of the number of coins in a tumbler was shown in an antecedent paper (Part 1) to follow a log-normal distribution ∧(m,s2). The coin-estimation experiment is an archetype of a broad class of image analysis and object counting problems suitable for solution by crowdsourcing. The objective of the current paper (Part 2) is to determine the location and scale parameters (m,s) of ∧(m,s2) by both Bayesian and maximum likelihood (ML) methods and to compare the results. One outcome of the analysis is the resolution, by means of Jeffreys’ rule, of questions regarding the appropriate Bayesian prior. It is shown that Bayesian and ML analyses lead to the same expression for the location parameter, but different expressions for the scale parameter, which become identical in the limit of an infinite sample size. A second outcome of the analysis concerns use of the sample mean as the measure of information of the crowd in applications where the distribution of responses is not sought or known. In the coin-estimation experiment, the sample mean was found to differ widely from the mean number of coins calculated from ∧(m,s2). This discordance raises critical questions concerning whether, and under what conditions, the sample mean provides a reliable measure of the information of the crowd. This paper resolves that problem by use of the principle of maximum entropy (PME). The PME yields a set of equations for finding the most probable distribution consistent with given prior information and only that information. If there is no solution to the PME equations for a specified sample mean and sample variance, then the sample mean is an unreliable statistic, since no measure can be assigned to its uncertainty. Parts 1 and 2 together demonstrate that the information content of crowdsourcing resides in the distribution of responses (very often log-normal in form), which can be obtained empirically or by appropriate modeling.

Maximum Entropy and Bayesian Inference for the Monty Hall Problem

We devise an approach to Bayesian statistics and their applications in the analysis of the Monty Hall problem. We combine knowledge gained through applications of the Maximum Entropy Principle and Nash equilibrium strategies to provide results concerning the use of Bayesian approaches unique to the Monty Hall problem. We use a model to describe Monty’s decision process and clarify that Bayesian inference results in an “irrelevant, therefore invariant” hypothesis. We discuss the advantages of Bayesian inference over the frequentist inference in tackling the uneven prior probability Monty Hall variant. We demonstrate that the use of Bayesian statistics conforms to the Maximum Entropy Principle in information theory and Bayesian approach successfully resolves dilemmas in the uneven probability Monty Hall variant. Our findings have applications in the decision making, information theory, bioinformatics, quantum game theory and beyond.

基于能量系数的电力变压器热故障严重性评估方法

运行电力变压器内部的热点温度直接影响其使用寿命和运行可靠性。为准确评估电力变压器内部过热故障的严重程度,文章借助焓变理论提出一种计算故障能量系数的方法,为故障赋予一定的权值系数;并基于能量系数这一概念,以电力变压器各类不同程度的热故障为研究对象,建立其严重程度分级评估基准,此基准是以563组电力变压器热故障油色谱分析数据所计算出的能量系数值作为样本数据,利用最大熵原理得出最符合实际分布的电力变压器热故障能量系数概率分布函数,并求取其分位值Q_(20)、Q_(70)和Q_(95)作为热故障严重程度的分级评估值。经实际算例分析,采用文中所提方法,不仅可以实现对热故障严重程度的量化评估,还可以合理地划分其严重等级,可为检修人员制定和执行维护计划提供参考。

熵权模糊改进法在宁夏黄河水质评价中的应用

为了解宁夏境内黄河流域的水质状况,选取银古公路桥、平罗黄河大桥、中卫香山湖、吴忠金沙湾、固原沟圈5个监测断面在2021年4月、6月、8月、12月4个时期的溶解氧、高锰酸盐指数、氨氮、总磷、总氮5项评价指标,共获得80组水样。针对熵权模糊评价法在权重趋于1时其微小变化会引起熵权成倍变化的病态问题和由于最大隶属度原则不能普遍适用而导致评价失效的问题,提出了一种熵权模糊改进的评价方法,通过改进熵权法确定权重和加权等级,建立改进的模糊综合评价模型,对水质样品进行水质分析和综合评价。结果表明,在权重分配方面,相比于隶属度法和传统的熵值法,改进熵权法分配的权重更为合理;在水质评价方面,加权等级法评定出的结果更符合研究区域的真实水质状况;宁夏境内黄河流域Ⅰ类和Ⅱ类水质占比超过90%,水质状况良好,5个断面4个时期的水质综合评价结果显示,12月份的水质最优,Ⅰ类水质占比85%;对水质的时空分析显示氨氮为全局性污染因子,溶解氧为区域性污染因子。改进的模糊综合评价模型对宁夏黄河水质的评价结果更为客观合理。

基于最大熵原理的谐波评估指标研究

随着新型电力系统的加速构建,电网的谐波特性日益复杂,研究谐波数据的有效统计管理,对评估电网电能质量好坏具有重要意义。提出一种基于最大熵原理的谐波评估指标统计方法,通过记录并保存谐波数据的平均值、中心距等数字特征,使用最大熵原理对谐波进行概率分布拟合,达到保存并识别谐波特征的目的,方便数据储存;利用拟合的概率分布获取谐波95%概率值和99%概率值两种谐波评估标准,保证了指标的准确性和一致性。最后通过谐波实测数据进行算法实例分析,验证了所提方法的有效性。

基于数据归一化原点矩约束最大熵原理的小样本数据测量不确定度评定

测量检验是生产制造的重要环节,测量不确定度是测量质量的重要评价指标,表征测量数据的分散程度。最大熵原理常用于测量不确定度评定,基于最大熵原理获得测量样本数据的概率密度函数,能够避免主观因素,反映数据客观的分布情况。小批量生产中获得的测量数据往往较少,在小样本数据下,基于原点矩约束最大熵原理求得的测量不确定度不够准确可靠。针对这一问题,提出基于数据归一化原点矩约束最大熵原理的小样本数据测量不确定度评定方法,并通过数值仿真和应用案例验证所提测量不确定度评定方法的有效性。

基于双重自表达与最大熵原理的深度子空间聚类算法

深度子空间聚类算法使用深度神经网络将原始输入数据映射至潜在空间,并利用数据的自表达性作为数据相似程度的度量,从而实现对高维数据的有效聚类。然而,这类算法仅关注潜在空间中的自表达关系,导致其性能严重依赖于深度神经网络所提取特征的质量。此外,正则化过程忽略各空间内的连通性,影响谱聚类算法的性能。针对这些问题,提出了基于双重自表达与最大熵原理的深度子空间聚类算法。该算法同时学习潜在空间与输入空间的自表达关系,以引导深度神经网络获得适合于子空间聚类的数据表示。通过最大化相似度矩阵的熵,确保同一子空间的元素分布均匀且密集,从而提升数据聚类性能。在5个数据集上进行大量实验,验证了所提算法的有效性。

基于最大熵Copula方法的古黄河宿迁段流域月降雨量模拟

为了解决Copula函数方法在拟合过程中要先选择函数类型和多变量水文模拟中保留相关性的问题,基于最大熵原理(POME),直接构建Copula函数,而无需进行Copula函数类型的优选,简化了联合概率分布函数的拟合过程。以古黄河宿迁段流域的皂河站、金锁站和泗阳站3个雨量站点39年(1980~2018年)的月降雨量实测资料为研究对象,进行实例分析。通过建立不同站点间的降雨量联合分布,检验最大熵Copula方法的有效性,并利用Gibbs采样法对月降雨量进行随机模拟。结果表明,基于最大熵Copula方法能保留各站点降雨序列的统计特性,以及不同站点间的降雨相关性,验证了该方法的有效性,可为流域多站点间的降雨量相关性分析和水资源规划与利用提供参考。

可靠性的科学探索:从经典到量子

作为设备稳定服役能力的一种衡量,可靠性对工程技术发展和创新十分重要。基于物理、信息论和统计方法的耦合,可靠性研究正逐步发展并有望成为一门新的科学。同时随着量子技术的发展,量子相干器件及设备将会被大量地植入到传统装备系统中。这不仅给可靠性自身的研究带来新的机遇和挑战,也会使量子物理和可靠性研究交叉起来,导致交叉领域思想上的原始创新。文章首先介绍可靠性研究从工程向科学迈进的历程,然后从可靠性工程需求的视角介绍量子可靠性的基本概念和逻辑,强调量子自洽历史对寿命统计的内在描述,最后结合目前量子技术所面临的退相干挑战,展望可靠性工程对量子技术的潜在应用。

基于Copula函数的区域雨潮遭遇风险概率分析

随着气候变化、人类活动影响加剧以及城市化进程等一系列事件的加快,我国的城市内涝问题日渐突出。以扬州市仪邗区域为例,由于扬州南濒长江,受长江洪水、潮水顶托,当面临暴雨等灾害时,仪邗区域无法快速外排,进而增大了内涝风险,针对内涝灾害的研究也因此成为了重要内容。本文利用Copula函数构建了研究区域最大三日面雨量和长江最大三日平均潮位的联合概率分布函数,并对不同雨潮组合情况下的遭遇概率进行分析,最终得到了雨潮的联合风险概率和同现风险概率。最终结果显示,Copula函数在雨潮联合概率分布方面拟合良好。这一拟合结果不仅可以为解决仪邗山洪在长江高水位时出路不足的问题提供一定的理论支持和科学依据,也可以为其他感潮河段的防洪决策提供参考。

基于Copula-POME的负荷与气象因素相关性度量研究

负荷与气象因素的关系是非线性且模糊的,针对传统的线性相关系数不能准确刻画负荷与其气象成因的相依结构。在分析负荷对气象因素响应的基础上,提出了结合Copula函数与最大熵原理(POME)的负荷与气象因素相关性度量方法,该方法基于POME建立了负荷与气象因素的边缘分布,利用Copula函数拟合了负荷与气象多变量系统中的非线性相依结构,并推导了度量相关性的Kendall秩相关系数、Spearman秩相关系数和Copula熵。在实际的负荷和气象系统中的应用表明,Copula-POME方法在分析负荷与其气象成因关系时无先验分布假定,具有灵活的函数形式,能准确表达多变量系统的相依结构;秩相关系数和Copula熵弥补了线性相关系数在度量尾部相关中的不足,能准确度量负荷与气象因素的相关性。

Principle of maximum entropy for reliability analysis in the design of machine components

We studied the reliability of machine components with parameters that follow an arbitrary statistical distribution using the principle of maximum entropy(PME).We used PME to select the statistical distribution that best fits the available information.We also established a probability density function(PDF)and a failure probability model for the parameters of mechanical components using the concept of entropy and the PME.We obtained the first four moments of the state function for reliability analysis and design.Furthermore,we attained an estimate of the PDF with the fewest human bias factors using the PME.This function was used to calculate the reliability of the machine components,including a connecting rod,a vehicle half-shaft,a front axle,a rear axle housing,and a leaf spring,which have parameters that typically follow a non-normal distribution.Simulations were conducted for comparison.This study provides a design methodology for the reliability of mechanical components for practical engineering projects.

APPLICATION OF MAXIMUM ENTROPY PRINCIPLE METHOD TO THE STUDY OF WAVE CLIMATE STATISTICAL CHARACTERISTICS

The Maximum Entropy Principle (MEP) method is elaborated, and thecorresponding probability density evaluation method for the random fluctuation system is introduced,the goal of the article is to find the best fitting method for the wave climate statisticaldistribution. For the first time, a kind of new maximum entropy probability distribution (MEPdistribution) expression is deduced in accordance with the second order moment of a random process.Different from all the fitting methods in the past, the MEP distribution can describe theprobability distribution of any random fluctuation system conveniently and reasonably. If themoments of the random signal is limited to the second order, that is, the ratio of theroot-mean-square value to the mean value of the random variable is obtained from the random sample,the corresponding MEP distribution can be computed according to the deduced expression in thisessay. The concept of the wave climate is introduced here, and the MEP distribution is applied tofit the probability density distributions of the significant wave height and spectral peak period.Take the Mexico Gulf as an example, three stations at different locations, depths and wind wavestrengths are chosen in the half-closed gulf, the significant wave height and spectral peak perioddistributions at each station are fitted with the MEP distribution, the Weibull distribution and theLog-normal distribution respectively, the fitted results are compared with the field observations,the results show that the MEP distribution is the best fitting method, and the Weibull distributionis the worst one when applied to the significant wave height and spectral peak period distributionsat different locations, water depths and wind wave strengths in the Gulf. The conclusion shows thefeasibility and reasonability of fitting wave climate statistical distributions with the deduced MEPdistributions in this essay, and furthermore proves the great potential of MEP method to the studyof wave statistical properties.

小样本数据下圆柱薄壳初始缺陷不确定性量化的极大熵方法

具有不确定性特征的初始缺陷被认为是导致薄壳结构实际临界载荷值与理论解不相符并呈现分散特征的主要原因.对实际薄壳结构初始缺陷的建模至少需要考虑两个方面的不确定性量化,一是对缺陷分布形式和幅值等固有随机性的量化,二是对小样本量和不准确测量所导致缺陷统计量的不确定性的量化.本文在利用随机场的Karhunen-Loeve展开法对薄壳初始几何缺陷建模的基础上,提出一种基于极大熵原理的缺陷建模方法.首先,采用极大熵分布来估计Karhunen-Loeve随机变量的概率密度函数,以适应不能使用高斯随机场进行缺陷随机场建模的情况.随后,通过将经典的等式约束极大熵模型扩展为区间约束极大熵模型,实现对实际工程中仅能获得少量薄壳结构几何缺陷样本数据所导致的认知不确定性的量化.最后,将所提方法用于对国际缺陷数据库的A-Shell进行缺陷建模和临界载荷预测.研究表明,所提基于区间约束极大熵原理的随机场建模方法在能够有效表征实测数据高阶矩信息的同时,还具备量化小样本数据导致的认知不确定性的能力,并且高斯随机场模型和基于等式约束极大熵原理的随机场模型是本文所提建模方法的两种特殊情况.

信息熵原理探讨及其在二元系统的应用

信息熵通常用以衡量随机变量的不确定性,在物理学、统计学、信息论等领域发挥着重要作用,帮助我们理解和解释各种现象。与热力学中的熵不同,信息熵呈现一定的秩序性,这与熵所代表的无序和复杂性形成鲜明对比。以二元统计模型为例,探讨信息熵与随机事件之间的联系,定量描述信息熵的演化过程,揭示信息熵秩序性的本质规律。