A new method of level2 uncertainty analysis in risk assessment based on uncertainty theory
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Abstract
The objective of this study is to present a novel method of level2 uncertainty analysis in risk assessment by means of uncertainty theory. In the proposed method, aleatory uncertainty is characterized by probability distributions, whose parameters are affected by epistemic uncertainty. These parameters are described as uncertain variables. For monotone risk models, such as fault trees or event trees, the uncertainty is propagated analytically based on the operational rules of uncertain variables. For nonmonotone risk models, we propose a simulationbased method for uncertainty propagation. Three indexes, i.e., average risk, valueatrisk and bounded valueatrisk, are defined for riskinformed decision making in the level2 uncertainty setting. Two numerical studies and an application on a real example from literature are worked out to illustrate the developed method. A comparison is made to some commonly used uncertainty analysis methods, e.g., the ones based on probability theory and evidence theory.
Keywords
Uncertainty theory Uncertainty analysis Epistemic uncertainty1 Introduction
Uncertainty modeling and analysis is an essential part of probabilistic risk assessment (PRA) and has drawn numerous attentions since 1980s (Apostolakis 1990; Parry and Winter 1981). Two types of uncertainty are usually distinguished: aleatory uncertainty, which refers to the uncertainty inherent in the physical behavior of a system, and epistemic uncertainty, which refers to the uncertainty in the modeling caused by lack of knowledge on the system behavior (Kiureghian and Ditlevsen 2009). In practice, uncertainty modeling and analysis involving both aleatory and epistemic uncertainty is often formulated in a level2 setting: aleatory uncertainty is considered by developing probabilistic models for risk assessment, while the parameters in the probabilistic models might subject to epistemic uncertainty (Aven et al. 2014).
In general, it has been well acknowledged that aleatory uncertainty should be modeled using probability theory. However, there appears to be no consensus on which mathematical framework should be used to describe epistemic uncertainty, since its modeling usually involves subjective information from human judgements. Indeed, various mathematical frameworks have been proposed in the literature to model the epistemically uncertain variables, e.g., probability theory (subjective interpretation), evidence theory, possibility theory (Aven 2013; Aven and Zio 2011; Helton et al. 2010). As a result, different methods for level2 uncertainty analysis are developed. Aven et al. (2014) systematically elaborate on level2 uncertainty analysis methods and developed a purely probabilistic for level2 uncertainty analysis. Limbourg and Rocquigny (2010) apply evidence theory to both level1 and level2 uncertainty modeling and analysis, and the two settings were compared through a benchmark problem. Some explanations of the results are discussed in the context of evidence theory. Considering the large calculation cost for level2 uncertainty analysis, Limbourg et al. (2010) develop an accelerated method for monotonous problems using the monotonous reliability method (MRM). Pedroni et al. (2013) and Pedroni and Zio (2012) model the epistemic uncertainty using possibility distributions and develop a level2 Monte Carlo simulation for uncertainty analysis, which is then compared to a purely probabilistic approach and an evidence theorybased (ETB) approach. Pasanisi et al. (2012) reinterpret the level2 purely probabilistic frameworks in the light of Bayesian decision theory and apply the approach to risk analysis. Hybrid methods based on probability theory and evidence theory are also presented (Aven et al. 2014). Baraldi et al. (2013) introduce the hybrid level2 uncertainty models to consider maintenance policy performance assessment.
In this paper, we enrich the research of level2 uncertainty analysis by introducing a new mathematical framework, the uncertainty theory, to model the epistemically uncertain variables. Uncertainty theory has been founded in 2007 by Liu (2007) as an axiomatic mathematical framework to model subjective belief degrees. It is viewed as a reasonable and effective approach to describe epistemic uncertainty (Kang et al. 2016). To simulate the evolution of an uncertain phenomenon with time, concepts of uncertain process (Liu 2015) and uncertain random process (Gao and Yao 2015) are proposed. The uncertain differential equation is also developed as an effective tool to model events affected by epistemic uncertainty (Yang and Yao 2016). After these years of development, uncertainty theory has been applied in various areas, including finance (Chen and Gao 2013; Guo and Gao 2017), decision making under uncertain environment (Wen et al. 2015a, b), game theory (Yang and Gao 2013, 2016; Gao et al. 2017; Yang and Gao 2014). There are also considerable real applications in reliability analysis and risk assessment considering epistemic uncertainties. For example, Zeng et al. (2013) propose a new concept of belief reliability based on uncertainty theory accounting for both aleatory and epistemic uncertainties. Wen et al. (2017) develop an uncertain optimization model of spare parts inventory for equipment system, where the subjective belief degree is adopted to compensate the data deficiency. Ke and Yao (2016) apply uncertainty theory to optimize scheduled replacement time under block replacement policy considering human uncertainty. Wen and Kang (2016) model the reliability of systems with both random components and uncertain components. Wang et al. (2017) develop a new structural reliability index based on uncertainty theory.
To the best of our knowledge, in this paper, it is the first time that uncertainty theory is applied to level2 uncertainty analysis. Through comparisons to some commonly used level2 uncertainty analysis methods, new insights are brought with respect to strength and limitations of the developed method.
The remainder of the paper is structured as follows. Section 2 recalls some basic concepts of uncertainty theory. Level2 uncertainty analysis method is developed in Sect. 3, for monotone and nonmonotone risk models. Numerical case studies and applications are presented in Sect. 4. The paper is concluded in Sect. 5.
2 Preliminaries
In this section, we briefly review some basic knowledge on uncertainty theory. Uncertainty theory is a new branch of axiomatic mathematics built on four axioms, i.e., normality, duality, subadditivity and product axioms. Founded by Liu (2007) and refined by Liu (2010) , uncertainty theory has been widely applied as a new tool for modeling subjective (especially human) uncertainties. In uncertainty theory, belief degrees of events are quantified by defining uncertain measures:
Definition 1
(Uncertain measure Liu 2007) Let \(\varGamma \) be a nonempty set, and Open image in new window be a \(\sigma \)algebra over \(\varGamma \). A set function Open image in new window is called an uncertain measure if it satisfies the following three axioms,
Axiom 1
(Normality Axiom) Open image in new window for the universal set \(\varGamma \).
Axiom 2
(Duality Axiom) Open image in new window for any event Open image in new window .
Axiom 3
Uncertain measures of product events are calculated following the product axiom (Liu 2009):
Axiom 4
Definition 2
(Uncertain variable Liu 2007) An uncertain variable is a function \(\xi \) from an uncertainty space Open image in new window to the set of real numbers such that \(\left\{ \xi \in {\mathcal {B}} \right\} \) is an event for any Borel set \({\mathcal {B}}\) of real numbers.
Definition 3
(Uncertainty distribution Liu 2007) The uncertainty distribution \(\varPhi \) of an uncertain variable \(\xi \) is defined by Open image in new window for any real number x.
Theorem 1
Definition 4
3 Level2 uncertainty analysis based on uncertainty theory
In this section, a new method for level2 uncertainty analysis is presented based on uncertainty theory. Sect. 3.1 formally defines the problem of level2 uncertainty analysis. Then, the uncertainty analysis method is introduced for monotone and nonmonotone models in Sects. 3.2 and 3.3, respectively.
3.1 Problem definition
Uncertainty in (10) is assumed to come from the input parameters \({\mathbf {x}} \), i.e., model uncertainty (e.g., see Nilsen and Aven (2003)) is not considered in the present paper. Aleatory and epistemic uncertainty are considered separately. Depending on the ways the uncertainty in the model parameters is handled, level1 and level2 uncertainty models are distinguished.
Level1 uncertainty models separate the input vector into \({\mathbf {x}}=({\mathbf {a}}, {\mathbf {e}}) \), where \({\mathbf {a}}=(x_1,x_2,\ldots ,x_m) \) represents the parameters affected by aleatory uncertainty while \({\mathbf {e}}=(x_{m+1},x_{m+2},\ldots , x_{n}) \) represents the parameters that are affected by epistemic uncertainty (Limbourg and Rocquigny 2010). In level1 uncertainty models, probability theory is used to model the aleatory uncertainty in \({\mathbf {a}}=(x_1,x_2,\ldots ,x_m) \) by identifying their probability density functions (PDF) \(f(x_i\theta _i) \). These PDFs are assumed to be known, i.e., the parameters in the PDFs, denoted by \({\varvec{\Theta }}=(\theta _1,\theta _2,\ldots ,\theta _n)\), are assumed to have precise values. In practice, however, \({\varvec{\Theta }}=(\theta _1,\theta _2,\ldots ,\theta _n) \), are subject to epistemic uncertainty, and the corresponding uncertainty model is called level2 uncertainty model.
 (1)
The aleatory uncertainty in the input parameters is described by the PDFs \(f(x_i\theta _i), \quad i=1,2,\ldots ,n \).
 (2)
\({\varvec{\Theta }}=(\theta _1,\theta _2,\ldots ,\theta _n) \) are modeled as independent uncertain variables with regular uncertainty distributions \(\varPhi _1,\varPhi _2,\ldots ,\varPhi _n \).
3.2 Monotone risk model
3.2.1 Uncertainty analysis using operational laws
Two risk indexes are defined for riskinformed decision making, considering the level2 uncertainty settings presented.
Definition 5
3.2.2 Numerical case study
Time threshold and distributions for level2 uncertain parameters
\(\lambda _1 (10^{5}/h^{1}) \)  \(\lambda _2 (10^{5}/h^{1}) \)  \(t_0 \)  \(\gamma \)  

UTB method  \(10^{4}h\)  0.9  
PB method  U(0.8, 1.2)  U(0.5, 0.8) 
Risk indexes of the monotone risk model
Method  \(\overline{p} \)  VaR(0.9) 

UTB method  0.1519  0.1755 
PB method  0.1520  0.1685 
3.3 Nonmonotone risk model
3.3.1 Uncertainty analysis using uncertain simulation
In many practical situations, the risk index of interest cannot be expressed as a strictly monotone function of the level2 uncertain parameters. For such cases, we cannot obtain the exact uncertainty distributions for p by directly applying the operational laws. Rather, the maximum uncertainty principle needs to be used to derive the upper and lower bounds for the uncertainty distribution based on an uncertain simulation method developed by (Zhu 2012). The uncertain simulation can provide a reasonable uncertainty distribution of a function of uncertain variables and does not require the monotonicity of the function with respect to the variables. In this section, the method is extended to calculate the upper and lower bounds of an uncertainty distribution for risk assessment.
Definition 6
(Zhu 2012) An uncertain variable \(\xi \) is common if it is from the uncertain space Open image in new window to \(\mathfrak {R}\) defined by \(\xi (\gamma )=\gamma \), where \({\mathcal {B}} \) is the Borel algebra over \(\mathfrak {R}\). An uncertain vector \(\varvec{\xi }=(\xi _1,\xi _2,\ldots ,\xi _n) \) is common if all the elements of \(\varvec{\xi }\) are common.
Theorem 2
From Theorem 2, it can be seen that (27) gives a theoretical bound of each Open image in new window in (26). Let Open image in new window , Open image in new window . It is clear that any values within m and \(1n\) is a reasonable value for Open image in new window . Hence, we use m as the upper bound and \(1n\) as the lower bound of Open image in new window and develop a numerical algorithm for level2 uncertainty analysis.
 step 1

Set \(m_1(i)=0 \) and \(m_2(i)=0 \), \(i=1,2,\ldots ,n \).
 step 2

Randomly generate \(u_k=\left( \gamma _k^{(1)},\gamma _k^{(2)},\ldots ,\gamma _k^{(n)} \right) \) with \(0<\varPhi _i\left( \gamma _k^{(i)}\right) <1 \), \(i=1,2,\ldots ,n \), \(k=1,2,\ldots ,N \).
 step 3

From \(k=1\) to \(k=N\), if \(f(u_k)\le c \), \(m_1(i)=m_1(i)+1 \), denote \(x_{m_1(i)}^{(i)}=\gamma _k^{(i)} \);
otherwise, \(m_2(i)=m_2(i)+1 \), denote \(y_{m_2(i)}^{(i)}=\gamma _k^{(i)} \), \(i=1,2,\ldots ,n \).
 step 4

Rank \(x_{m_1}^{(i)} \) and \(y_{m_2}^{(i)} \) from small to large, respectively.
 step 5
 Set$$\begin{aligned} a^{(i)}=&\,\,\varPhi \left( x_{m_1(i)}^{(i)}\right) \wedge \left( 1\varPhi \left( x_1^{(i)}\right) \right) \wedge \\&\left( \varPhi \left( x_1^{(i)}\right) +1\varPhi \left( x_2^{(i)}\right) \right) \wedge \\&\left( \varPhi \left( x_{m_1(i)1}^{(i)}\right) +1 \varPhi \left( x_{m_1(i)}^{(i)}\right) \right) ;\\ b^{(i)}=&\,\,\varPhi \left( y_{m_2(i)}^{(i)}\right) \wedge \left( 1\varPhi \left( y_1^{(i)}\right) \right) \wedge \\&\left( \varPhi \left( y_1^{(i)}\right) +1\varPhi \left( y_2^{(i)}\right) \right) \wedge \\&\left( \varPhi \left( y_{m_2(i)1}^{(i)}\right) +1 \varPhi \left( y_{m_2(i)}^{(i)}\right) \right) . \end{aligned}$$
 step 6

\(L_{1U}^{(i)}=a^{(i)},L_{1L}^{(i)}=1b^{(i)},L_{2U}^{(i)}=b^{(i)},L_{2L}^{(i)}=1a^{(i)}\).
 step 7

If \(a_U=L_{1U}^{(1)}\wedge L_{1U}^{(2)}\wedge \cdots \wedge L_{1U}^{(n)}>0.5 \), \(L_U=a_U \); if \(b_U=L_{2L}^{(1)}\wedge L_{2L}^{(2)} \wedge \cdots \wedge L_{2L}^{(n)}>0.5 \), \(L_U=1b_U \); otherwise, \(L_U=0.5 \).
If \(a_L=L_{1L}^{(1)}\wedge L_{1L}^{(2)}\wedge \cdots \wedge L_{1L}^{(n)}>0.5 \), \(L_L=a_L \); if \(b_L=L_{2U}^{(1)}\wedge L_{2U}^{(2)} \wedge \cdots \wedge L_{2U}^{(n)}>0.5 \), \(L_L=1b_L \); otherwise, \(L_L=0.5 \).
Through this algorithm, the upper and lower bounds for the uncertainty distribution of p can be constructed, denoted by \(\left[ \varPsi _L(p),\varPsi _U(p)\right] \). Similar to the monotone case, we define two risk indexes considering the level2 uncertainty:
Definition 7
The defined average risk is a reflection of the average belief degree of the risk index p, and a greater value of \(\overline{p} \) means more severe risk that we believe we will suffer. The meaning of the bounded valueatrisk is that, with belief degree \(\gamma \), we believe that the value of risk index is within the interval \(\left[ \text {VaR}_L,\text {VaR}_U\right] (\gamma ) \). Obviously, if we fix the value of \(\gamma \), a wider bounded valueatrisk means a more conservative assessment result. Meanwhile, we believe a greater \(\text {VaR}_U(\gamma ) \) reflects that the risk is more severe.
3.3.2 Numerical case study
Distributions of level1 and level2 parameters
Parameter  Level1  Level2  

UTB method  ETB method  
\(x_1\)  \(N(\mu _1,5)\)  \(\mu _1\sim U(9,11) \)  
\(x_1\)  \(N(\mu _2,5)\)  \(\mu 2\sim {\mathcal {N}}(10,0.3) \)  \(\mu _2\sim N(10,0.3) \) 
Since the developed method offers a bounded uncertainty distribution of \(p_f\), it is then compared with an evidence theorybased (ETB) method, in which the belief degree of \(p_f\) is also given as upper and lower distributions called plausibility (Pl) and belief (Bel) function, respectively. In this paper, the ETB method models the belief degrees of \(\mu _1\) and \(\mu _2\) using probability distributions (see Table 3). A double loop Monte Carlo simulation combined with a discretization method for getting basic probability assignments (BPAs) is used to obtain \(Bel\left( p_f\right) \) and \(Pl\left( p_f\right) \) (Limbourg and Rocquigny 2010; Tonon 2004). In Fig. 3, the dotted line and dot–dash line represent Bel and Pl, respectively. It should be noted that although we use Bel and Pl as mathematical constructs, they are not strictly the concepts of belief and plausibility defined by Shafer (i.e., the degree of truth of a proposition Shafer 1976). The two functions only represent bounds on a true quantity. To illustrate this, a cumulative density function (CDF) of \(p_f\) is calculated via a double loop MC simulation method, shown as the crossed line in Fig. 3. It is seen that the CDF is covered by the area enclosed by Bel and Pl. In this sense, the CDF obtained in PB method is a special case of the ETB model, and the \(Bel\left( p_f\right) \) and \(Pl\left( p_f\right) \) give a reasonable bound of the probability distribution of \(p_f\).
Risk indexes of the nonmonotone risk model
Method  \(\overline{p} \)  \(\left[ \text {VaR}_L,\text {VaR}_U\right] (0.9) \) 

UTB method  0.001980  [0.001689, 0.003548] 
ETB method  0.002012  [0.002440, 0.003140] 
Figure 3 shows a comparison of the distributions of belief degrees on \(p_f\) in UTB method and ETB method. The distributions have the same supports, whereas the upper and lower uncertainty distributions fully cover the CDF and the area enclosed by Bel and Pl, which indicates that the developed method is more conservative. This is because the subjective belief described by uncertainty distributions usually tends to be more conservative and is more easily affected by epistemic uncertainty. This phenomenon is also reflected by the two defined risk indexes: the average risk \(\overline{p}\)s are nearly the same on different theory basis, while the bounded valueatrisk of ETB method is within that of the UTB method.
We also find that the bounded valueatrisk obtained by the developed UTB method may be too wide for some decision makers. This may be a shortcoming of the proposed method. Therefore, when choosing a method for risk analysis from the PB method, ETB method and UTB method, we need to consider the attitude of decision maker. For a conservative decision maker, the bounded uncertainty distribution is an alternative choice.
4 Application
Uncertainty description of level1 and level2 parameters
Parameter  Probability distribution  Level2 uncertainty distribution  Theoretical bounds  

Q  \(Gum(\alpha ,\beta )\)  \(\alpha \)  \({\mathcal {N}}_\alpha (1013, 48) \)  [10, 10000] 
\(\beta \)  \({\mathcal {N}}_\beta (558,36) \)  
\(K_s\)  \(N(\mu _{K_s},\sigma _{K_s}^2)\)  \(\mu _{K_s}\)  [5, 60]  
\(\sigma _{K_s}\)  
\(Z_m\)  \(N(\mu _{Z_m},\sigma _{Z_m}^2)\)  \(\mu _{Z_m}\)  [53.5, 57]  
\(\sigma _{Z_m}\)  
\(Z_v\)  \(N(\mu _{Z_v},\sigma _{Z_v}^2)\)  \(\mu _{Z_v}\)  [48, 51]  
\(\sigma _{Z_v}\)  
l  5000 (constant)  –  
b  30 (constant)  – 
4.1 System description
4.2 Parameter setting
The input variables in 34 are assumed to be random variables and the form of their PDFs are assumed to be known, as shown in Table 5 (Limbourg and Rocquigny 2010). However, due to limited statistical data, the distribution parameters of these PDFs cannot be precisely estimated using statistic methods and, therefore, are affected by epistemic uncertainty, which should be evaluated based on experts knowledge. In this paper, the experts knowledge on the distribution of these parameters is obtained by asking the experts to give the uncertainty distributions of the parameters, as shown in Table 5. For example, the yearly maximal water flow, denoted by Q, follows a Gumbel distribution \(Gum(\alpha ,\beta ) \), and according to expert judgements, \(\alpha \) and \(\beta \) follow normal uncertainty distributions \({\mathcal {N}}_\alpha (1013,48) \) and \({\mathcal {N}}_\beta (558,36) \), respectively. In addition, considering some physical constraints, the input quantities also have theoretical bounds, as given in Table 5.
4.3 Results and discussion
Risk indexes for the flood system
Index  Value 

\(\overline{p_\text {flood}}\)  0.0161 
\(\left[ \text {VaR}_L,\text {VaR}_U\right] (0.9)\)  [0.0073, 0.0476] 
It follows that the average yearly risk is \(p_{\text {flood}}\), which corresponds to an average return period of 62 years. This is unacceptable in practice, because it is too short when compared to a commonly required 100yearreturn period. To solve this problem, one measure is to heighten the dike for a more reliable protection. Another solution might be increasing the friction coefficient of the riverbed \(K_s\), noting from 34 that \(Z_c\) decreases with \(K_s\).
The bounded valueatrisk is relatively wide, which indicates that due to the presence of epistemic uncertainty, we cannot be too confirmed on the calculated risk index. The same conclusion is also drawn from Fig. 5: the difference between the upper and lower bounds of the uncertainty distributions are large, indicating great epistemic uncertainty. To reduce the effect of epistemic uncertainty, more historical data need to be collected to support a more precise estimation of the distribution parameters in the level1 probability distributions.
5 Conclusions
In this paper, a new level2 uncertainty analysis method is developed based on uncertainty theory. The method is discussed in two respects: for monotone risk models, where the risk index of interest is expressed as an explicit monotone function of the uncertain parameters, and level2 uncertainty analysis is conducted based on operational laws of uncertainty variables; for nonmonotone risk models, an uncertain simulationbased method is developed for level2 uncertainty analysis. Three indexes, i.e., average risk, valueatrisk and bounded valueatrisk, are defined for riskinformed decision making in the level2 uncertainty setting. Two numerical studies and an application on a real example from literature are worked out to illustrate the developed method. The developed method is also compared to some commonly used level2 uncertainty analysis methods, e.g., PB method and ETB method. The comparisons show that, in general, the UTB method is more conservative than the methods based on probability theory and evidence theory.
Notes
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 61573043, 71671009).
Compliance with ethical standards
Conflicts of interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Human and animal rights
This article does not contain any studies with human participants performed by any of the authors.
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