An efficient epistemic uncertainty analysis method using evidence theory

https://doi.org/10.1016/j.cma.2018.04.033Get rights and content

Highlights

  • An efficient epistemic uncertainty analysis method using evidence theory is proposed.

  • Johnson p-boxes are constructed to equivalently represent evidence variables.

  • Uncertainty analysis using evidence theory is converted to two times of probabilistic uncertainty analysis.

  • The computational issue resulted from the discrete property of evidence variables is solved to a large extent.

Abstract

Evidence theory is widely regarded as a promising mathematical tool for epistemic uncertainty analysis. However, the heavy computational burden has severely hindered its application in practical engineering problems, which is essentially caused by the discrete uncertainty quantification mechanism of evidence variables. In this paper, an efficient epistemic uncertainty analysis method using evidence theory is proposed, based on a probabilistic and continuous representation of the epistemic uncertainty presented in evidence variables. Firstly, each evidence variable is equivalently transformed to a Johnson p-box which is a family of Johnson distributions enveloped by the CDF bounds. Subsequently, the probability bound analysis is conducted for the input Johnson p-box and the response CDF based on monotonicity analysis. Finally, the CDF bounds of the response are directly calculated using the CDF bounds of the input Johnson p-boxes, by which a high computational efficiency is achieved for the proposed method. Two mathematical problems and two engineering applications are presented to demonstrate the effectiveness of the proposed method.

Introduction

Uncertainties widely exist in practical engineering problems, which should be appropriately quantified and controlled for the reliability and safety of a product. Generally, uncertainties are distinguished into two categories: aleatory uncertainty and epistemic uncertainty [[1], [2]]. Aleatory uncertainty describes the inherent randomness in the behavior of a system or its environment, which has been well treated using probability theory [[3], [4], [5], [6]]. Epistemic uncertainty is resulted from the lack of information in some phases of the design process, which, therefore, can be reduced with the collection of more data. In recent years, the research on epistemic uncertainty has become a hot topic. A series of non-probabilistic models have been proposed to characterize epistemic uncertainty such as evidence theory [[7], [8]], fuzzy sets [[9], [10], [11]], interval analysis theory [[12], [13], [14], [15]], etc.

Evidence theory is regarded as a promising complement to probability theory in uncertainty analysis when only several possible continuous or discontinuous intervals can be obtained to roughly describe the distribution of the uncertain variable [16]. At different situations, evidence theory can be equivalent to probability theory, fuzzy sets, interval analysis theory, which therefore provides a more general framework for uncertainty quantification [17]. The basic axioms of evidence theory enable it to deal with aleatory and epistemic uncertainty together in a straightforward way without any baseless assumptions [18]. Interval information from multiple sources or experts can be flexibly fused using combination rules of evidence theory [19]. The above properties of evidence theory are very appealing for engineering designers, thus it has been extensively applied in the structural uncertainty analysis in recent years.

Evidence theory was first explored for parametric uncertainty analysis of a simple system, the strengths and weaknesses of which were discussed thoroughly compared with probability theory [[20], [21]]. By establishing a multi-point approximation of the limit-state surface, a reliability analysis method with high efficiency was proposed for structures with epistemic uncertainty [[22], [23]]. A sampling-based approach [24] and a semi-analytic approach [25] were developed for evidence-theory-based sensitivity analysis, which are useful to analyze the significant contributing factors in engineering design. A numerical method based on evidence theory was developed for the structural uncertainty analysis in the presence of mixed aleatory and epistemic uncertainties [18]. However, it requires a large-scale computational cost for complex systems, thus enhanced uncertainty analysis methods using stochastic expansions and Kriging model were developed [[26], [27]]. Evidence theory was extended to quantify the model-form uncertainty in the case that two or more models can be provided to predict responses of a physical system [[28], [29]]. An uncertainty propagation algorithm using evidence theory [30] was developed for functions with multidimensional output, by formulating the problem as multiple one-dimensional optimization problems. An uncertainty quantification approach based on evidence theory [[31], [32]] was developed for determining epistemic uncertainty involved in concrete fatigue life prediction problems. Evidence theory was extended to predict the uncertain response of acoustic fields with the help of orthogonal polynomial approximation theory [33].

Though some important achievements have been introduced above, it is still a challenge to apply evidence theory in the uncertainty analysis of practical engineering problems. One of the most critical issues is the high computational cost [22], which in essence is caused by the discrete uncertainty quantification mechanism of evidence variables. In evidence-theory-based uncertainty analysis, only a series of intervals with probability assignments is obtained to describe the epistemic uncertainty, rather than an explicit and continuous function like the probability density function in probability theory. It results in repeated interval analysis of the response function in the uncertainty analysis, which will inevitably bring about expensive computational cost. Furthermore, the computational cost of evidence-theory-based uncertainty analysis increases exponentially with the number of evidence variables and that of the intervals of each variable, e.g. the combination explosion issue, which seems unacceptable for practical engineering problems that generally have a large number of variables.

A series of numerical methods have been proposed to alleviate the computational burden of evidence-theory-based uncertainty analysis. In Refs. [[22], [34], [35]], the response surface method was adopted to approximate the response function, by which the uncertainty analysis problems can be solved efficiently with an explicit approximated function. However, the precision of these methods is not stable since the response surface is influenced by many factors such as the selection of design of experiment (DOE) techniques and approximation model types. Furthermore, some of the response surface methods tend to smooth the accurate response function, which may result in large computational errors. In Refs. [[36], [37]], some numerical methods were proposed to improve the computational efficiency by reducing the number of focal elements that require function evaluations. However, the efficiency of these methods is not satisfied enough, especially for problems with high dimensionality. In this paper, we tend to solve the evidence-theory-based uncertainty analysis problems from a novel perspective. As introduced above, it is the discrete property of evidence variables that results in repeated interval analysis and hence expensive computational cost in the evidence-theory-based uncertainty analysis. It would be reasonable to believe that if we can deal with the evidence-theory-based uncertainty analysis problem using a continuous approach, high computational efficiency will be obtained.

In this paper, an efficient epistemic uncertainty analysis method using evidence theory is proposed, based on a probabilistic representation of the epistemic uncertainty present in evidence variables. The remainder of this paper is organized as follows. The conventional uncertainty analysis method based on evidence theory is introduced in Section 2. The proposed uncertainty analysis method is formulated in Section 3. Four numerical examples are investigated in Section 4. Finally, conclusions are summarized in Section 5.

Section snippets

Conventional uncertainty analysis based on evidence theory

Consider the following response function: Y=g(X)where X denotes a vector of n independent evidence variables, Y denotes the structural response. Generally, the conventional evidence-theory-based uncertainty analysis for g(X) contains three main contents: (1) Definition of the Frame of Discernment; (2) Construction of the Basic Probability Assignment; (3) Computation of the Belief and Plausibility Functions.

The proposed uncertainty analysis method

As introduced previously, evidence theory employs a collection of discrete intervals with probability assignments to describe epistemic uncertainty, which essentially results in the repeated extreme analysis and hence high computational cost of evidence-theory-based uncertainty analysis. For example, there exist only four uncertain evidence parameters in a structure and each parameter contains only six focal elements, then a total of 64=1296 focal elements will be involved in the uncertainty

Mathematical problem

Consider the following response function: g(X1,X2)=X12X22where X1[3,4] and X2[1,2] are two independent evidence variables. Two different BPA structures are considered for X1 and X2, as shown in Table 1.

To conduct uncertainty analysis using the proposed method, the bounds [MlL,MlU] of the first-order raw moment, and the second-order, third-order and fourth-order central moments of X1 and X2 are first calculated according to Eqs. (13) and (14), the results of which are shown in Table 2. Here,

Conclusions

In this paper, an efficient epistemic uncertainty analysis method based on evidence theory is proposed. It converts the conventional evidence-theory-based uncertainty analysis problem to two probabilistic uncertainty analysis problems, by which the computational issue resulted from the discrete property of evidence variables is considered to be solved to a large extent. The computational efficiency and accuracy of the proposed method are compared with that of the conventional method and MCS

Acknowledgments

This work is supported by the Science Challenge Project (Grant No. TZ2018007), National Natural Science Foundation of China for Distinguished Young Scholars (Grant No. 51725502), National Natural Science Foundation of China (Grant No. 51490662), National Key Research and Development Plan (Grant No. 2016YFD0701105).

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