Mathematical representation of uncertainty

Oberkampf, William L.; Helton, Jon C.; Sentz, Kari

doi:10.2514/6.2001-1645

Cited by 225 publications

(125 citation statements)

References 48 publications

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“…Other alternative theories exist 7 ; however, linkages between them are a matter of continued research. Complex decision problems such as verification and validation of computational and simulation models are replete with uncertainties (aleatory and epistemic) and errors in:…”

Section: Discussionmentioning

confidence: 99%

Linking Probability Theory and Fuzzy Sets: A Study in Uncertainty Assessment

Booker

Bement

Sellers

et al. 2002

43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

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Uncertainties enter into a complex problem from many sources: variability, errors, and lack of knowledge. A fundamental question arises in how to characterize the various kinds of uncertainty and then combine within a problem such as the verification and validation of a computer model, reliability of a dynamic system, or a complex decision problem. Our aim is to explore how probability theory and fuzzy set theory can be made to work in concert, so that uncertainty of outcomes and imprecision can be treated in a unified and coherent manner. Both the theoretical and application of a linkage between the two theories will be presented. An example from a reliability application will illustrate how the linkage between the theories is accomplished through the use of Bayes Theorem and the probability of fuzzy sets.

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Section: Discussionmentioning

confidence: 99%

Linking Probability Theory and Fuzzy Sets: A Study in Uncertainty Assessment

Booker

Bement

Sellers

et al. 2002

43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

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“…[2-4] ，模糊集 [5] ，凸模型 [6][7][8][9][10][11] ，证据理论 [12][13][14][15][16] 等。在上述理论中，证据理论是一种较为一般的建模方法 [17] ，它可以灵活处理各类认知不确定性。在不同的条件下，证据理论可以分别等效为经典概率理论，可能性理论，模糊集理论和凸模型理论等。近几年证据理论在结构可靠性分析领域逐渐得到重视，并出现了一系列研究成果。OBERKAMPF 等 [17] 通过简单的代数函数研究了证据理论在结构可靠性方面的应用，并总结了其优缺点。SOUNDAPPAN 等 [18] 对决策问题中量化认知不确定性的证据理论和贝叶斯方法进行了比较。MOURELATOS 等 [19] 发展了一种基于证据理论的高效可靠性优化方法。 DU [20] 结合证据理论和概率方法，提出了一种能够同时处理随机不确定性和认知不确定性的混合模型。李玲玲等 [21] 基于证据理论提出了一种能同时处理随机信息和模糊信息的可靠性度量方法。GUO 等 [22] 构建了一种基于证据理论和区间分析的可靠性优化设计方法，并将其用于压力容器设计。BAI 等 [23] 将证据理论运用于结构的动静态线弹性响应分析。JIANG 等 [24] 对证据变量进行均匀化处理，提出了一种基于证据理论的高效可靠性分析方法。从上述工作可以看出，以证据理论为基础的结构可靠性分析已经取得了一些进展。然而，现有方法大都假定各证据变量相互独立，其联合基本可信度分配函数(Basic probability assignment, BPA)直接由各边缘 BPA 相乘得到。而在许多实际问题中，设计变量间具有相关性 [25][26] ，变量间的相关性很多时候会对结构可靠性分析结果产生显著影响。从本文算例分析中也将看到，在某些情况下独立性假定将给计算结果带来较大误差。所以，建立一种考虑相关性的证据理论可靠性分析模型对于实际产品或结构的设计是很有必要的。本文基于 Copula 函数，提出了一种考虑相关性的证据理论模型及结构可靠性分析方法。下文主要包含四个部分：介绍证据理论的基本原理；引入 Copula 函数，建立一种考虑相关性的证据理论模型；构建基于证据理论相关性模型的可靠性分析方法；最后，本文方法应用于悬臂梁、汽车转向机构和汽车侧碰工况的可靠性分析。 1 证据理论基本原理证据理论又称为 Dempster-Shafer 理论 [12] ，可考虑由于信息不完备造成的不确定性，并用可信度 …”

Section: 要考虑另一类不确定性，即认知不确定性。处理认知不确定性的常用方法有可能性理论unclassified

A Copula Function Based Evidence Theory Model For Correlation Analysis and Corresponding Structural Reliability Method

Jiang¹

2017

JME

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Abstract：A Copula function based evidence theory model for correlation analysis and corresponding structural reliability method is proposed to deal with reliability design problems with dependent evidence variables. The Copula function is used to describe the correlation between evidence variables, and to identify the best copula, a Bayesian method which is usually used in probabilistic problem is expanded to evidence theory, the empirical distribution is used to convert the evidence variables to standard uniform variables, and the weight is calculated from the samples to identify the best copula function. The joint basic probability assignment (BPA) is gained by making the difference between the marginal BPAs using copula function. Then the reliability interval is gained by calculating the cumulative BPA of the focal elements in the reliable domain. Three examples are investigated to verify the effectiveness of the proposed method, the results show that the correlation between evidence variables may influence the reliability results significantly, and the commonly used independence assumption may lead to big error of the reliability result.

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“…There are 15 many sources of uncertainty in the modeling of a problem, and different classifications of the uncertainties have been proposed (Kennedy and O'Hagan, 2001;Oberkampf et al, 2001;Beyer and Sendhoff, 2007). A common classification is to divide the uncertainty into aleatory and epistemic uncertainties (Oberkampf et al, 2001).…”

Section: Uncertainty Quantificationmentioning

confidence: 99%

Polynomial chaos to efficiently compute the annual energy production in wind farm layout optimization

Padron¹,

Thomas²,

Stanley³

et al. 2018

Preprint

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Abstract.In this paper, we develop computationally-efficient techniques to calculate statistics used in wind farm optimization with the goal of enabling the use of higher-fidelity models and larger wind farm optimization problems. We apply these techniques to maximize the Annual Energy Production (AEP) of a wind farm by optimizing the position of the individual wind turbines. The AEP (a statistic itself) is the expected power produced by the wind farm over a period of one year subject to uncertainties in 5 the wind conditions (wind direction and wind speed) that are described with empirically-determined probability distributions.To compute the AEP of the wind farm, we use a wake model to simulate the power at different input conditions composed of wind direction and wind speed pairs. We use polynomial chaos (PC), an uncertainty quantification method, to construct a polynomial approximation of the power over the entire stochastic space and to efficiently (using as few simulations as possible) compute the expected power (AEP). We explore both regression and quadrature approaches to compute the PC coefficients. 10PC based on regression is significantly more efficient than the rectangle rule (the method most commonly used to compute the expected power). With PC based on regression, we have reduced by as much as an order of magnitude the number of simulations required to accurately compute the AEP, thus enabling the use of more expensive, higher-fidelity models or larger wind farm optimizations. We perform a large suite of gradient-based optimizations with different initial turbine locations and with different numbers of samples to compute the AEP. The optimizations with PC based on regression result in optimized 15 layouts that produce the same AEP as the optimized layouts found with the rectangle rule but using only one-third of the samples. Furthermore, for the same number of samples, the AEP of the optimal layouts found with PC is 1 % higher than the AEP of the layouts found with the rectangle rule.

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Mathematical representation of uncertainty

Cited by 225 publications

References 48 publications

Linking Probability Theory and Fuzzy Sets: A Study in Uncertainty Assessment

Linking Probability Theory and Fuzzy Sets: A Study in Uncertainty Assessment

A Copula Function Based Evidence Theory Model For Correlation Analysis and Corresponding Structural Reliability Method

Polynomial chaos to efficiently compute the annual energy production in wind farm layout optimization

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