Sensitivity estimation of failure probability applying line sampling

Reliability Engineering and System Safety 171 (2018) 99–111

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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress

Sensitivity estimation of failure probability applying line sampling M.A. Valdebenito a, H.A. Jensen a,∗, H.B. Hernández a, L. Mehrez b a

Departmento de Obras Civiles, Universidad Tecnica Federico Santa Maria, Av. España 1680, Valparaiso, Chile Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, 3620 S. Vermont Avenue, KAP 210, Los Angeles, CA 90089-2531, USA b

a r t i c l e

i n f o

Keywords: Probability sensitivity Line sampling Failure probability Post-processing

a b s t r a c t This contribution presents a framework for calculating a sensitivity measure for problems of computational stochastic mechanics. More specifically, the sensitivity measure considered is the derivative of the failure probability with respect to parameters of the probability distributions (e.g. mean value, standard deviation) associated with the random input quantities of a system’s model. The proposed framework is formulated as a post-processing step of Line Sampling, which is a simulation-based method for estimating small failure probabilities. In particular, the proposed framework comprises two different approaches for estimating the sought sensitivity. The application of the proposed framework and comparison of the two aforementioned approaches is discussed through a number of numerical examples. The results obtained indicate that both approaches allow estimating the sought sensitivity measure. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction The application of probability theory has been widely accepted as a means for quantifying the unavoidable effects of uncertainty on the performance of mechanical and structural systems [54]. Thus, the safety of a system can be measured in terms of, e.g. a failure probability. It should be noted that the failure probability can be highly sensitive to the characterization of the uncertainty in the input variables of a system. That is, the failure probability may vary considerably in case the numerical value of a distribution parameter (such as mean value or standard deviation) varies [7,9,39]. Undoubtedly, evaluating the sensitivity of the probability with respect to such parameters is of paramount importance. For example, it can allow pinpointing the most influential parameters of a model [26,29] or perform reliability-based optimization [2,63]. In this context, the objective of this contribution is proposing a framework for evaluating the sensitivity of the failure probability. The sensitivity measure considered herein is the derivative of the failure probability with respect to distribution parameters that describe the uncertainty in the input variables of a model. Most of the approaches for probability sensitivity estimation with respect to distribution parameters developed so far have been formulated as a post-processing step of an existing strategy for estimating failure probabilities. For example, the estimation of probability sensitivity applying the First- and Second-Order Reliability Methods [17] has been addressed in, e.g. [8,19,20,33]. In these contributions, the sensitivity

∗

analysis is closely linked with the gradient of the so-called design point [17] with respect to the distribution parameters. The estimation of the probability sensitivity applying simulation methods such as Monte Carlo [44], Importance Sampling [42,57] and Subset Simulation (which was originally introduced in [5] and further extended in [3,65]) has also been addressed in, e.g. [29,30,43,58,62]. A common feature found in the latter contributions is that the samples generated for estimating the probability are post-processed in order to obtain the sensitivity estimates, thus requiring no additional system (structural) analyses. The framework for probability sensitivity estimation proposed in this contribution follows a similar scheme when compared to the approaches described above. In particular, the proposed framework is developed as a post-processing of Line Sampling (LS), which is a simulation method introduced in [34] and further extended in [14,15]. It should be noted that LS produces accurate probability estimates for problems which involve a linear, weakly nonlinear or even mildly nonlinear behavior while exhibiting high efficiency when compared to other simulation strategies [52]. The main idea behind LS is estimating the failure probability by assessing the response of the system along lines (which are generated randomly in the space of the uncertain input variables). The proposed framework for probability sensitivity estimation is implemented considering two different approaches. The first approach involves calculating the gradient of the function describing the performance of the system at a specific point for each of the lines associated with LS. The second approach involves the estimation of a one-dimensional integral along each of the lines generated when applying LS. Although the two approaches

Corresponding author. E-mail address: [email protected] (H.A. Jensen).

https://doi.org/10.1016/j.ress.2017.11.010 Received 3 July 2017; Received in revised form 29 September 2017; Accepted 18 November 2017 Available online 21 November 2017 0951-8320/© 2017 Elsevier Ltd. All rights reserved.

M.A. Valdebenito et al.

Reliability Engineering and System Safety 171 (2018) 99–111

are conceptually different, they produce similar sensitivity estimates. The performance of both approaches is discussed using several numerical examples. It is important to note that the application of LS for probability sensitivity estimation has already been explored in [37,47]. However, the work reported herein possesses substantial differences when compared with those contributions. First, the objective of the current work is estimating probability sensitivity with respect to distribution parameters while [47] focuses on estimating sensitivity with respect to deterministic parameters of the model. Second, the results obtained in the current work generalize the results on sensitivity analysis along each line associated with LS which were presented in [37]. Third, the contributions [37,47] are developed following the first approach of the framework reported herein, while the second approach has not been explored as yet in context with LS. The range of application of the proposed approach for probability sensitivity estimation based on LS is similar to that of LS applied for probability estimation, i.e. reliability problems that involve weakly to mildly nonlinear behavior. In fact, in such class of problems, LS may exhibit a high numerical efficiency when compared to other approaches for probability estimation, as discussed in [55]. While the proposed approach is also applicable to more general reliability problems, it is expected that its efficiency can decrease. Thus, on one hand, for problems that exhibit a highly nonlinear behavior, the application of Subset Simulation can provide a more efficient means for estimating probability sensitivity, as discussed in [30,58]. On the other hand, for those problems that exhibit a linear or close to linear behavior, the application of the First Order Reliability method may be appropriate [8,19,20,33]. This paper is organized as follows. The formulation of the problem studied in this paper (i.e. sensitivity of the failure probability) is presented in Section 2. Section 3 contains a brief overview on Line Sampling (LS). Two approaches for estimating probability sensitivity applying LS are presented in Sections 4 and 5, respectively. These two approaches are compared in Section 6 while its application to a number of examples is investigated in Section 7. The contribution closes with some conclusions and outlook in Section 8.

In the above equation, pF denotes failure probability and gx (x) is the so-called performance function [11,17], which assumes a value equal or smaller than zero whenever a realization x of the random input variables causes an undesirable structural response. Throughout this work, it is assumed that the performance function is differentiable with respect to x. The numerical evaluation of the failure probability integral is usually a challenging task. This stems from two issues. First, the number of random variables involved in a problem may be large, thus precluding the application of numerical quadrature. Second, for most cases of practical interest, the performance function gx (x) is not known analytically. In fact, its evaluation must be performed often in a point-wise manner for particular values of x, which implies performing a deterministic system (structural) analysis. The two aforementioned issues favor the application of approximate methods (such as the First- and Second-order Reliability Methods [17]) and simulation techniques (such as Monte Carlo and its more advance variants [4]) for evaluating the failure probability. 2.2. Sensitivity of failure probability The structure of Eq. (1) indicates that the value of the failure probability is affected by the vector 𝜽 that groups the distribution parameters. A possible means for quantifying the sensitivity of the failure probability with respect to these distribution parameters is calculating the partial derivative of pF with respect to each entry in 𝜽 [62], i.e.: 𝜕𝑝𝐹 = ℎ (𝑥 |𝜽 )𝑓 (𝒙|𝜽)𝑑 𝒙, 𝑙 = 1, … , 𝑛𝑖 , 𝑖 = 1, … , 𝑛 𝜕𝜃𝑙,𝑖 ∫𝑔𝒙 (𝒙)≤0 𝑥𝑖 ,𝜃𝑙,𝑖 𝑖 𝑖 𝑿

(2)

where 𝜃 l,i represents the lth distribution parameter associated with the ith random variable, ni is the number of distribution parameters associated with the ith random variable and ℎ𝑥𝑖 ,𝜃𝑙,𝑖 (𝑥𝑖 |𝜽𝑖 ) is the following function. ℎ𝑥𝑖 ,𝜃𝑙,𝑖 (𝑥𝑖 |𝜽𝑖 ) =

𝜕𝑓𝑋𝑖 (𝑥𝑖 |𝜽𝑖 ) 1 𝑓𝑋𝑖 (𝑥𝑖 |𝜽𝑖 ) 𝜕𝜃𝑙,𝑖

(3)

The challenges associated with the calculation of the partial derivative of the probability in Eq. (2) are – in principle– similar to those associated with the failure probability integral in Eq. (1).

2. Formulation of the problem 2.1. Failure probability

2.3. Transformation into standard normal space

Assume that a computational model of a mechanical or structural system of interest is available, which has been generated using an appropriate technique such as, e.g. the finite element method [6]. A total of n input variables of this model are uncertain and are characterized as random variables 𝑋𝑖 , 𝑖 = 1, … , 𝑛. The physical values that these input variables may assume are denoted as 𝑥𝑖 , 𝑖 = 1, … , 𝑛. For the sake of simplicity and without loss of generality, it is assumed in the remaining part of this contribution that these random variables are independent. However, possible dependencies between these random variables could be accounted for considering appropriate models, see e.g. [36,45]. The probability density function (pdf) associated with each input random variable is denoted as 𝑓𝑋𝑖 (𝑥𝑖 |𝜽𝑖 ), where 𝜽i is a vector that collects the distribution parameters of Xi such as mean, standard deviation, etc. The joint pdf is denoted as fX (x|𝜽), where 𝒙 = [𝑥1 , … , 𝑥𝑛 ]𝑇 , 𝜽 = [𝜽𝑇1 , … , 𝜽𝑇𝑛 ]𝑇 and (·)T represents transpose. Due to the independence between random ∏ variables, it is evident that 𝑓𝑿 (𝒙|𝜽) = 𝑛𝑖=1 𝑓𝑋𝑖 (𝑥𝑖 |𝜽𝑖 ). As some of the input variables of the model are random, the response of the model is random as well. Moreover, some particular realizations of the input variables may lead to an undesirable response of the system, such as loss of serviceability or structural collapse. The chances that such undesirable response occur can be measured in terms of a failure probability.

A common practice in structural reliability is expressing the failure probability integral in the standard normal space. Thus, each of the input random variables of the model (which are denoted as physical random variables) is mapped into a standard normal random variable. In view of the assumption of independence between physical random variables, such projection is performed by equating the cumulative density function associated with the ith physical random variable with the cumulative density function of the ith standard normal random variable [16]. That is, 𝐹𝑋𝑖 (𝑥𝑖 |𝜽𝑖 ) = Φ(𝑧𝑖 ), where 𝐹𝑋𝑖 (⋅) and Φ(·) are the cumulative distribution functions of the physical random variable Xi and standard normal random variable Zi , respectively. In this way, it is possible to define a transformation function such that 𝑧𝑖 = 𝑡𝑖 (𝑥𝑖 |𝜽𝑖 ), 𝑖 = 1, … , 𝑛, where ( ( )) 𝑡𝑖 (𝑥𝑖 |𝜽𝑖 ) = Φ−1 𝐹𝑋𝑖 𝑥𝑖 |𝜽𝑖 and Φ(⋅)−1 represents the standard normal

𝑝𝐹 =

∫𝑔𝒙 (𝒙)≤0

𝑓𝑿 (𝒙|𝜽)𝑑 𝒙

inverse cumulative distribution function. The collection of the n transformation functions (which is actually a vector-valued function) is denoted as 𝒛 = 𝒕(𝒙|𝜽). In view of the definitions discussed above and applying a change of variables, Eqs. (1) and (2) can be recast in the standard normal space as: 𝑝𝐹 =

(1)

100

𝜙𝑛 (𝒛)𝑑 𝒛

(4)

𝜕𝑝𝐹 = ℎ (𝑧 |𝜽 )𝜙 (𝒛)𝑑 𝒛 𝜕𝜃𝑙,𝑖 ∫𝑔𝒛 (𝒛|𝜽)≤0 𝑧𝑖 ,𝜃𝑙,𝑖 𝑖 𝑖 𝑛

(5)

∫𝑔𝒛 (𝒛|𝜽)≤0

M.A. Valdebenito et al.

Reliability Engineering and System Safety 171 (2018) 99–111

each line L(j) intersects only once the limit state function (defined as 𝑔𝒛 (𝒛|𝜽) = 0), the estimator for the failure probability simplifies to: 𝑝𝐹 ≈ 𝑝̂𝐹 =

where 𝑔𝒛 (𝒛|𝜽) = 𝑔𝒙 (𝒕−1 (𝒛|𝜽)) = 𝑔𝒙 (𝒙), ℎ𝑧𝑖 ,𝜃𝑙,𝑖 (𝑧𝑖 |𝜽𝑖 ) = ℎ𝑥𝑖 ,𝜃𝑙,𝑖 (𝑡−1 𝑖 (𝑧𝑖 |𝜽𝑖 )|𝜽𝑖 ) = ℎ𝑥𝑖 ,𝜃𝑙,𝑖 (𝑥𝑖 |𝜽𝑖 ) and where 𝜙n (·) is the n-dimensional independent standard normal pdf; furthermore, note that 𝒕−1 (⋅) and 𝑡−1 𝑖 (⋅) represent the inverse vector-valued and inverse scalar-valued transformation functions, respectively.

The above description of LS assumes a weakly or mildly nonlinear performance function. However, it should be noted that LS can be applied for reliability problems involving several and/or nonlinear performance functions as well. For such cases, more than one important direction may be required [34] or the important direction may have to be selected adaptively [15]. For the sake of simplicity and without loss of generality, the remaining part of this work focuses on the application of LS for cases of weakly or mildly nonlinear performance functions, where a single important direction suffices for estimating the failure probability. Furthermore, when applying LS, it is assumed that each line L(j) intersects only once the limit state function.

3. Estimation of failure probability applying line sampling Line Sampling (LS) is a simulation technique introduced in [34] and is closely related with another reliability method known as Axis Orthogonal Sampling [25,59]. LS operates in the standard normal space and can be highly efficient for estimating small failure probabilities whenever the associated performance function gz (z|𝜽) is weakly or mildly non-linear. As LS is well documented in the literature (see, e.g. [64]), only essential aspects of this technique are described herein. The basis of LS is the identification of a so-called important direction, which is denoted as 𝜶. This important direction is a unit vector which points towards the region in the standard normal space such that gz (z|𝜽) ≤ 0. Fig. 1 presents a schematic representation of 𝜶 for 𝑛 = 2. Several criteria have been proposed for selecting 𝜶, see e.g. [15,34,50]. In this contribution, 𝜶 is selected as the negative of the gradient of the performance function at the origin of the standard normal space, i.e. 𝜶 = −∇𝒛 𝑔𝒛 (𝒛 = 𝟎|𝜽)∕||∇𝒛 𝑔𝒛 (𝒛 = 𝟎|𝜽)||, where || · || denotes Euclidean norm and ∇z denotes nabla operator, i.e. ∇𝒛 = [𝜕 ∕𝜕 𝑧1 , … , 𝜕 ∕𝜕 𝑧𝑛 ]𝑇 . The next step of LS consists in drawing samples of the standard normal random variables that lie in the hyperplane orthogonal to 𝜶. These samples are denoted as 𝒛⟂,(𝑗) , 𝑗 = 1, … , 𝑁 and are generated as 𝒛⟂,(𝑗) = 𝒛(𝑗) − (𝜶 𝑇 𝒛(𝑗) )𝒛(𝑗) , where 𝒛(𝑗) , 𝑗 = 1, … , 𝑁 are independently and identically distributed samples drawn according to 𝜙n (z). Fig. 1 illustrates the samples z⊥, (j) for 𝑁 = 2. The step described above can be interpreted as a Monte Carlo simulation in a (𝑛 − 1)-dimensional space, as all samples lie in the hyperplane orthogonal to 𝜶. Taking into account these samples, the failure probability is estimated as: 𝑁 1 ∑ 𝜙(𝑦)𝑑𝑦 𝑁 𝑗=1 ∫𝑔𝒛 (𝜶𝑦+𝒛⟂,(𝑗) |𝜽)≤0

(7)

where c(j) fulfills the equation 𝑔𝒛 (𝜶𝑐 (𝑗) + 𝒛⟂,(𝑗) |𝜽) = 0. In other words, c(j) is the (Euclidean) distance between z⊥, (j) and the intersection with the limit state function, measured along the line L(j) . Clearly, this distance depends on the sample z⊥, (j) and the distribution parameters 𝜽, i.e. 𝑐 (𝑗) = ) ( 𝑐 (𝑗) 𝒛⟂,(𝑗) |𝜽 . A practical means for determining c(j) (as suggested in [56]) is to evaluate the performance function over a discrete grid of points along line L(j) . Then, c(j) is calculated by numerical interpolation. Under the assumption of a mildly nonlinear performance function, few grid points should be required for interpolating c(j) with sufficient accuracy. The accuracy of the estimator in Eq. (7) is quantified in terms of its coefficient of variation 𝛿𝑝̂𝐹 , which is equal to [56]: √ ((∑ ) ) ) ( 𝑁 1 (𝑗) 2 − 𝑁 𝑝̂2 𝑗=1 Φ −𝑐 𝐹 𝑁 (𝑁 −1) 𝛿𝑝̂𝐹 = (8) 𝑝̂𝐹

Fig. 1. Schematic illustration of Line Sampling and important direction 𝜶.

𝑝𝐹 ≈ 𝑝̂𝐹 =

𝑁 1 ∑ ( (𝑗) ) Φ −𝑐 𝑁 𝑗=1

4. Estimation of failure probability sensitivity applying line sampling – first approach In order to formulate an expression for estimating the probability sensitivity applying LS, consider the following observations. •

•

Recall that the estimator in Eq. (6) is the average of a series of onedimensional integrals. The argument of each integral is the standard normal pdf and the integration interval is [c(j) , ∞[. Hence, each of these integrals could be interpreted as the probability content associated with a given line L(j) . Assume that the lth distribution parameter of the ith random variable undergoes a small perturbation Δ𝜃 l,i . The perturbation Δ𝜃 l,i induces a perturbation in the distance between z⊥, (j) and the intersection with the limit state function (measured along the line L(j) ), which is equal to Δc(j) . This occurs because the limit state functions 𝑔𝒛 (𝒛|𝜽) = 0 and 𝑔𝒛 (𝒛|𝜽 + 𝒆𝑙,𝑖 Δ𝜃𝑙,𝑖 ) = 0 may differ from each other (note that el,i is a ∑ vector whose entries are zeros, except for its ( 𝑖𝑘−1 =1 𝑛𝑘 + 𝑙)th entry, which is equal to one). This is represented schematically in Fig. 2.

Taking into account these two observations, it becomes clear that estimating the probability sensitivity applying LS reduces to calculating the change of the probability content associated with the jth line (which is denoted as Δ𝑝(𝐹𝑗) ) with respect to the perturbation Δ𝜃 l,i and then taking

(6)

where 𝑝̂𝐹 is the estimator of pF , 𝜙(·) denotes the standard normal pdf and y is an auxiliary coordinate in the direction of 𝜶 (see Fig. 1). It is important to note two characteristics of this estimator. First, it comprises a one-dimensional integral, while the probability integral (see Eq. (1)) comprises a total n dimensions. While this may seem contradictory, the difference in the number of dimensions is justified as the application of Monte Carlo simulation sweeps the (𝑛 − 1) dimensions orthogonal to 𝜶. Second, the estimator involves an integral of the standard normal pdf over the portion of the line 𝐿(𝑗) = {𝒛 ∈ ℝ𝑛 ∧ 𝑦 ∈ ℝ ∶ 𝒛 = 𝜶𝑦 + 𝒛⟂,(𝑗) } that lies in the failure domain (see Fig. 1). Under the assumption that

the average of the values of Δ𝑝(𝐹𝑗) associated with each line 𝑗 = 1, … , 𝑁. The change in the probability content of the jth line can be easily estimated by examining Fig. 2. Under the assumption that Δc(j) is suffi( ) ciently small, Δ𝑝(𝐹𝑗) = −𝜙 −𝑐 (𝑗) Δ𝑐 (𝑗) , i.e. the change in the probability content associated with the jth line is equal to the product between the ( ) probability mass 𝜙 −𝑐 (𝑗) and the portion of the line which no longer belongs to the failure domain (Δc(j) ); the negative sign reflects that a positive value of Δc(j) implies that a smaller portion of the jth line is contained within the failure domain. Hence, the rate of change of the failure 101

M.A. Valdebenito et al.

Reliability Engineering and System Safety 171 (2018) 99–111

Fig. 3. Schematic representation of function associated with calculation of probability integral. Fig. 2. Perturbation of limit state function due to a change in a distribution parameter.

( ) ing Eq. (11) at 𝒛0,(𝑗) + 𝜶Δ𝑐 (𝑗) |𝜽 + 𝒆𝑙,𝑖 Δ𝜃𝑙,𝑖 and letting Δ𝜃 l,i → 0 yields: ( )−1 𝜕𝑔𝒙

) ( probability content along the line is Δ𝑝(𝐹𝑗) ∕Δ𝜃𝑙,𝑖 = −𝜙 −𝑐 (𝑗) Δ𝑐 (𝑗) ∕Δ𝜃𝑙,𝑖 ; as Δ𝜃 l,i → 0, the change rate of the failure probability for the jth line is ( ) 𝜕 𝑝(𝐹𝑗) ∕𝜕 𝜃𝑙,𝑖 = −𝜙 −𝑐 (𝑗) 𝜕 𝑐 (𝑗) ∕𝜕 𝜃𝑙,𝑖 . Thus, the sensitivity of the failure probability applying LS is estimated as shown below. ) (𝑗) 𝑁 𝑁 ( ( ) 𝜕𝑐 (𝑗) 𝜕 𝑝̂ 𝜕𝑝𝐹 1 ∑ 𝜕𝑝𝐹 1 ∑ ≈ 𝐹 = = −𝜙 −𝑐 (𝑗) 𝜕𝜃𝑙,𝑖 𝜕𝜃𝑙,𝑖 𝑁 𝑗=1 𝜕𝜃𝑙,𝑖 𝑁 𝑗=1 𝜕𝜃𝑙,𝑖

𝜕𝑥𝑖 𝜕𝑐 (𝑗) = ∑𝑛 𝜕𝜃𝑙,𝑖

𝜕𝑔𝒙 𝑖=1 𝛼𝑖 𝜕𝑥𝑖

1 𝑁 (𝑁 −1)

𝛿𝜕 𝑝̂𝐹 ∕𝜕𝜃𝑙,𝑖 =

(9)

(( ) ( ) ) ∑𝑁 ( ( (𝑗) ) 𝜕𝑐 (𝑗 ) )2 𝜕 𝑝̂𝐹 2 𝜙 − 𝑐 − 𝑁 − 𝑗=1 𝜕𝜃 𝜕𝜃 𝑙,𝑖

|𝜕 𝑝̂𝐹 ∕𝜕𝜃𝑙,𝑖 |

𝑙,𝑖

(10)

where | · | denotes absolute value. This absolute value is introduced in the definition of 𝛿𝜕 𝑝̂𝐹 ∕𝜕𝜃𝑙,𝑖 as 𝜕 𝑝̂𝐹 ∕𝜕𝜃𝑙,𝑖 may assume negative values. Note that the proposed formula for 𝛿𝜕 𝑝̂𝐹 ∕𝜕𝜃𝑙,𝑖 is valid for 𝜕 𝑝̂𝐹 ∕𝜕𝜃𝑙,𝑖 ≠ 0. In case 𝜕 𝑝̂𝐹 ∕𝜕𝜃𝑙,𝑖 = 0, it is more convenient to assess the accuracy of the estimator for sensitivity of Eq. (9) in terms of its standard deviation, which is the numerator in Eq. (10). The practical application of the estimator in Eq. (9) demands evaluating 𝜕 c(j) /𝜕 𝜃 l,i . An analytical expression for this derivative can be deduced by constructing a Taylor expansion of the performance function about the point where the line L(j) intersects with the limit state function 𝑔𝒛 (𝒛|𝜽) = 0. This expansion is denoted as 𝑔̃𝒛(𝑗) (𝒛|𝜃) and is equal to: 𝑛 ) ( ) ∑ 𝜕𝑔𝒛 ( 𝑔̃𝒛(𝑗) (𝒛|𝜽) = 𝑔𝒛 𝒛0,(𝑗) |𝜽0 + 𝑧𝑖 − 𝑧0𝑖 ,(𝑗) + 𝜕𝑧𝑖 𝑖=1 𝑛𝑖 𝑛 ∑ ) ∑ 𝜕𝑔𝒛 ( 0 𝜃𝑙,𝑖 − 𝜃𝑙,𝑖 +… 𝜕𝜃𝑙,𝑖 𝑖=1 𝑙=1

(

𝜕𝑧𝑖 𝜕𝜃𝑙,𝑖

𝜕𝑧𝑖 𝜕𝑥𝑖

)−1

(12)

Thus, the calculation of 𝜕 c(j) /𝜕 𝜃 l,i demands evaluating 𝜕 zi /𝜕 xi , 𝜕 zi /𝜕 𝜃 l,i and 𝜕 gx /𝜕 xi . The evaluation of these three partial derivatives is discussed in the following. The derivative 𝜕 zi /𝜕 xi measures the rate of change of zi due to a perturbation on the physical variable xi for a fixed value of 𝜽i while 𝜕 zi /𝜕 𝜃 l,i measures the rate of change of zi due to a perturbation on the distribution parameter 𝜃 l,i for a fixed value of xi . Undoubtedly, these derivatives depend on the specific probability distribution associated with Xi . It is interesting to note that analytical expressions for these derivatives can be determined for several types of probability distributions. Appendix B presents these expressions for the case where Xi follows either a Gaussian or a lognormal distribution. The calculation of the partial derivative 𝜕 gx /𝜕 xi may be more involved than the calculation of 𝜕 zi /𝜕 xi or 𝜕 zi /𝜕 𝜃 l,i , as usually no analytical expressions for gx (x) are available. In some cases, this gradient can be calculated as a byproduct of a structural analysis, see e.g. [61]. In other cases, it may be necessary to evaluate this gradient using semi-analytical or numerical strategies, see e.g. [12,24,31,32,46]. In remaining part of this contribution, it is assumed that the gradient of the performance function is available either as an analytical formula or as a byproduct of structural analysis. Finally, it should be noted that an expression for calculating 𝜕 c(j) /𝜕 𝜃 l,i has already been reported in [47]. However, in that contribution, 𝜃 l,i represents a deterministic parameter that influences the system’s behavior and not a distribution parameter as considered herein. Thus, the expression for 𝜕 c(j) /𝜕 𝜃 l,i as shown in Eq. (12) and the one reported in [47] differ in the term 𝜕 zi /𝜕 𝜃 l,i . Moreover, in [37], another expression for 𝜕 c(j) /𝜕 𝜃 l,i is proposed, which is different from Eq. (12). The expression proposed herein involves four quantities: the important direction 𝜶 and the partial derivatives 𝜕 zi /𝜕 xi , 𝜕 zi /𝜕 𝜃 l,i and 𝜕 gx /𝜕 xi . Nonetheless, the formula suggested in [37] omits the last of these four quantities.

Note that this expression could also be deduced by differentiating Eq. (7) with respect to 𝜃 l,i [37]. It can be shown that the coefficient of variation of the estimator in Eq. (9) (denoted as 𝛿𝜕 𝑝̂𝐹 ∕𝜕𝜃𝑙,𝑖 ) is equal to [4,56]: √

𝜕𝑧𝑖 𝜕𝑥𝑖

(11)

5. Estimation of failure probability sensitivity applying line sampling – second approach

where the expansion point is (z0, (j) |𝜽0 ); note that z0, (j) represents the point where the jth line intersects the limit state function, i.e. 𝒛0,(𝑗) = 𝜶𝑐 (𝑗) + 𝒛⟂,(𝑗) , while 𝜽0 represents the nominal value of the distribution parameters. The derivatives involved in Eq. (11) are cal( )−1 culated as 𝜕 𝑔𝒛 ∕𝜕 𝑧𝑖 = 𝜕 𝑧𝑖 ∕𝜕 𝑥𝑖 (𝜕 𝑔𝒙 ∕𝜕 𝑥𝑖 ), 𝑖 = 1, … , 𝑛 and 𝜕 𝑔𝒛 ∕𝜕 𝜃𝑙,𝑖 = −(𝜕 𝑔𝒛 ∕𝜕 𝑧𝑖 )(𝜕 𝑧𝑖 ∕𝜕 𝜃𝑙,𝑖 ), 𝑙 = 1, … , 𝑛𝑖 , 𝑖 = 1, … , 𝑛 (see Appendix A) and ( ) are evaluated at the expansion point. Note that 𝑔𝒛 𝒛0,(𝑗) |𝜽0 = 0 ) (𝑗) ( 0,(𝑗) and 𝑔̃𝒛 𝒛 + 𝜶Δ𝑐 (𝑗) |𝜽 + 𝒆𝑙,𝑖 Δ𝜃𝑙,𝑖 = 0, as the points (z0, (j) |𝜽0 ) and ( 0,(𝑗) ) + 𝜶Δ𝑐 (𝑗) |𝜽 + 𝒆𝑙,𝑖 Δ𝜃𝑙,𝑖 are located over the nominal and perturbed 𝒛 limit state functions, respectively, as indicated in Fig. 2. Thus, evaluat-

A comparison between Eqs. (4) and (5) suggests that the integrals associated with the calculation of the probability and its sensitivity are quite similar. In fact, their integration domain is identical (i.e. the failure domain gz (z|𝜽) ≤ 0) while the difference lies in the function being integrated. This idea is illustrated in Figs. 3 and 4, which show the functions associated with the integrals for pF and 𝜕 pF /𝜕 𝜃 l,i for a particular case in which the limit state function is linear. Note that these two Figures also include a suitable important direction for applying LS. 102

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While the two approaches described above may seem as completely independent, they are actually linked through the divergence theorem, as discussed in [10,38]. On one hand, the first approach can be seen as the calculation of the probability flux through the limit state surface. The probability flux represents the probability density function at each point in the standard normal space over the limit state surface times the speed of that point. This speed is given by the velocity of the point (i.e. 𝜕 z/𝜕 𝜃 l,i ) and its dot product with the vector normal to the limit state surface (i.e. ∇z gz /||∇z gz ||). On the other hand, the second approach just calculates the divergence of the probability flux over the failure domain. Hence, the approaches presented in Sections 4 and 5 are actually two different means for estimating the same quantity. It is important to remark that the two approaches for probability sensitivity estimation discussed above have been formulated within the framework of LS as proposed in [34]. Hence, these two approaches are not directly applicable within the framework of Adaptive Line Sampling (ALS), which was proposed in [14,15]. Such issue stems from the adaptive selection of the important direction performed in ALS, which may bring substantial efficiency improvements for estimating small failure probabilities in reliability problems. Extending the proposed approaches for probability sensitivity estimation such that they are applicable within the framework of ALS is outside the scope of this contribution.

Fig. 4. Schematic representation of function associated with calculation of probability sensitivity integral.

In view of the similarities between the integrals associated with the probability and its sensitivity, it becomes evident that the latter can be estimated applying LS. In fact, the evaluation of the probability sensitivity comprises the same two basic steps of LS as applied for probability estimation: (1) generation of random samples 𝒛⟂,(𝑗) , 𝑗 = 1, … , 𝑁 in the hyperplane orthogonal to the important direction 𝜶 and (2) onedimensional integration along each line 𝐿(𝑗) , 𝑗 = 1, … , 𝑁 over the failure domain. The only difference is that the function integrated at the second step is in this case equal to the product between ℎ𝑧𝑖 ,𝜃𝑙,𝑖 (𝑧𝑖 |𝜽𝑖 )

7. Examples 7.1. General remarks

L(j) )

(evaluated along and the standard normal pdf. Thus, the sought estimator for probability sensitivity is: 𝑁 ( ⟂,(𝑗) ) 𝜕 𝑝̂ 𝜕𝑝𝐹 1 ∑ ≈ 𝐹 = 𝜑 |𝜽𝑖 𝒛 𝜕𝜃𝑙,𝑖 𝜕𝜃𝑙,𝑖 𝑁 𝑗=1 𝑧𝑖 ,𝜃𝑙,𝑖

In the following, three numerical examples are presented in order to illustrate the application of the two approaches for estimating probability sensitivity applying LS. In all of these examples, the results obtained using these approaches are compared with the results obtained applying Monte Carlo simulation (MCS), which is considered to be a reference. The estimators for the probability and the sensitivity when applying MCS are (see, e.g. [21,53]):

(13)

where: ( ) 𝜑𝑧𝑖 ,𝜃𝑙,𝑖 𝒛⟂,(𝑗) |𝜽𝑖 =

∞

∫𝑐 (𝑗 )

(𝑗) ℎ𝑧𝑖 ,𝜃𝑙,𝑖 (𝛼𝑖 𝑦 + 𝑧⟂, |𝜽𝑖 )𝜙(𝑦)𝑑𝑦 𝑖

(14) 𝑝̂𝐹 =

Analytical expressions for the integral in Eq. (14) can be obtained for certain types of probability distributions associated with Xi . Appendix C presents such expressions for the case where Xi follows either a Gaussian or lognormal distribution. The formula for calculating the coefficient of variation of the estimator in Eq. (13) is analogous to the one presented in Eq. (10), except ( ) ( ) that −𝜙 −𝑐 (𝑗) 𝜕𝑐 (𝑗) ∕𝜕𝜃𝑙𝑖 must be replaced by 𝜑𝑧𝑖 ,𝜃𝑙,𝑖 𝒛⟂,(𝑗) |𝜽𝑖 .

𝑁 1 ∑ ( (𝑗) ) 𝐼 𝒛 |𝜽 𝑁 𝑗=1

𝑁 𝜕 𝑝̂𝐹 1 ∑ ( (𝑗) ) = 𝐼 𝒛 |𝜽 ℎ𝑧𝑖 ,𝜃𝑙,𝑖 (𝑧(𝑖𝑗) |𝜽𝑖 ) 𝜕𝜃𝑙,𝑖 𝑁 𝑗=1

(15)

(16)

where 𝒛(𝑗) , 𝑗 = 1, … , 𝑁 are independent, identically distributed samples that follow a n-th dimensional standard normal distribution and I(z|𝜽) is the indicator function, which is equal to one in case gz (z|𝜽) ≤ 0 and zero otherwise. Details on the deduction of these estimators and their corresponding coefficient of variations can be found in, e.g. [21,53]. For implementing LS in the three numerical examples, the important direction 𝜶 is selected as the unit vector whose direction is the negative of the gradient of the performance function evaluated at the origin of the standard normal space, as already discussed in Section 3. Such criterion for selecting the important direction has been adopted due to its simplicity. However, it should be noted that there are other more advanced criteria for selecting the important direction based on e.g. Markov Chains [34], Monte Carlo gradient estimation [50] or adaptive selection [14,15]. Furthermore, whenever the performance function is evaluated along each line of LS, the evaluation points are chosen on a evenly spaced grid in the standard normal space; the specific construction of this grid is problem dependent [34]. Once again, the criterion applied for evaluating the performance function along a line is chosen due to its simplicity. However, in [14,15], a more efficient criterion has been proposed for selecting the evaluation points in an adaptive way.

6. Comparison of proposed approaches The two approaches for estimating probability sensitivity described above can be interpreted as a post-processing step of LS. Indeed, both approaches are applied once the distances 𝑐 (𝑗) , 𝑗 = 1, … , 𝑁 have been determined for each line; recall that these distances are required for probability estimation. The application of the first approach (see Eq. (9)) involves the sensitivity of the probability model (i.e. 𝜕 zi /𝜕 xi and 𝜕 𝑧𝑖 ∕𝜕 𝜃𝑙𝑖 ) and the sensitivity of the response of the system (i.e. 𝜕 gx /𝜕 xi ). It is interesting to note that all of these quantities are evaluated at the point where the jth line intersects the limit state function (also termed as limit state surface). Hence, the first approach can be seen as an integration over the limit state surface, as all information required to estimate the probability sensitivity is gathered at the locus {𝒛 ∈ ℝ𝑛 ∶ 𝑔𝒛 (𝒛|𝜽) = 0}. The second approach for estimating 𝜕 pF /𝜕 𝜃 l,i (see Eq. (13)) demands knowledge on the sensitivity of the probability model only and does not require calculating the sensitivity of the response of the system. However, in this case, the function to be evaluated is not calculated at a point (as in the first approach) but along each of the lines of LS. Hence, this second approach can be seen as an integration over the failure domain.

7.2. Example 1 The first example is taken from [37] and involves the performance function 𝑔𝒙 (𝒙) = 𝑒0.4𝑥1 +7 − 𝑒0.3𝑥2 +5 − 200, where X1 and X2 are inde103

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Fig. 5. Limit state function and important directions 𝜶 and 𝜸 in the standard normal space – Example 1. Fig. 6. Limit state function in the standard normal space evaluated considering nominal and perturbed values for the distribution parameters – Example 1.

Table 1 Probability and probability sensitivity estimates (and their coefficient of variation 𝛿) obtained applying Line Sampling and Monte Carlo Simulation – Example 1. Quantity

𝑝̂𝐹 (𝛿) 𝜕 𝑝̂𝐹 ∕𝜕 𝜇1 𝜕 𝑝̂𝐹 ∕𝜕 𝜇2 𝜕 𝑝̂𝐹 ∕𝜕 𝜎1 𝜕 𝑝̂𝐹 ∕𝜕 𝜎2

Simulation Method

(𝛿) (𝛿) (𝛿) (𝛿)

LS – First Approach

LS – Second Approach

MCS

3.6 × 10−3 −1 × 10−2 ( < 0.1%) 3.8 × 10−3 ( < 0.1%) 2.6 × 10−2 ( < 0.1%) 4.0 × 10−3 (0.1%)

( < 0.1%) −1 × 10−2 (0.1%) 3.8 × 10−3 (0.1%) 2.6 × 10−2 ( < 0.1%) 4.0 × 10−3 (0.3%)

3.6 × 10−3 (0.2%) −1 × 10−2 (0.2%) 3.8 × 10−3 (0.2%) 2.6 × 10−2 (0.2%) 4.0 × 10−3 (0.4%)

approach tends to produce estimators with slightly less variability than those produced with the second approach. Furthermore, all probability and sensitivity estimators produced with LS are comparable with the results obtained applying MCS. It is interesting to note that the probability sensitivity assumes different values depending on the distribution parameter under consideration. For example, a perturbation on 𝜇1 produces a decrease in the failure probability, i.e. negative sensitivity value. On the contrary, a perturbation on 𝜎 1 , 𝜇 2 or 𝜎 2 produces an increase in the failure probability, i.e. positive sensitivity value. Such behavior is explained because different distribution parameters have different effect on the limit state function in the standard normal space. Such effect is represented schematically in Fig. 6, where the limit state function is represented in the standard normal space for nominal and perturbed values of the distribution parameters. As noted from this Figure, a perturbation in the value of 𝜇 1 induces a shift of the limit state function towards the left, i.e. the safe region increases and the failure probability decreases. In case either 𝜎 1 , 𝜇 2 or 𝜎 2 is perturbed, the safe region contracts, thus causing an increase in the failure probability. In a second stage, the two approaches for estimating probability sensitivity with LS are applied considering an important direction 𝜸 which is rotated 15° in counterclockwise direction with respect to the important direction 𝜶, i.e. 𝜸 = [−0.997, 0.081]𝑇 (see Fig. 5). This second important direction 𝜸 is chosen according to the suggestion in [37], in order to assess its effect on the efficiency and accuracy of LS for estimating probability sensitivity. In this particular case, LS is applied considering a total of 104 lines. The results obtained are shown in Fig. 7. This Figure presents the evolution of the estimator for 𝜕 𝑝̂𝐹 ∕𝜕𝜎2 according to the number of lines drawn for each of the approaches for calculating sensitivity. Note that the estimators for the other probability sensitivities are not reported, as they present a similar behavior. For comparison purposes, Fig. 7(a) presents the results associated with the important direction 𝜶 while Fig. 7(b) presents the results associated with the important direction 𝜸. The results of Fig. 7 allow drawing three important conclusions. First, the accuracy of any of the proposed approaches is not affected by the considered important directions 𝜶 and 𝜸. In fact, for both important directions, the estimators converge to similar sensitivity values (provided that a sufficient number of lines is considered). This is a most important conclusion, which differs from the observations in [37]. In the latter, it is indicated that the probability sensitivity estimates are affected by the important direction. The difference between that conclusion and the one reported herein stems in the way the quantity 𝜕 c(j) /𝜕 𝜃 l,i is calculated, as

pendent Gaussian random variables with distribution parameters 𝜽𝑖 = [𝜇𝑖 , 𝜎𝑖 ]𝑇 = [0, 1]𝑇 , 𝑖 = 1, 2. A schematic representation of the limit state function associated with this performance function is shown in Fig. 5. It is noted that this performance function is weakly nonlinear. In addition, the Figure also illustrates two important directions 𝜶 and 𝜸, which are considered for performing two different simulation runs of LS, as suggested in [37] (details on these two simulation runs are discussed later on). This example is considered here for two reasons. First, as it involves an analytic performance function, the probability and its sensitivity can be evaluated considering a large number of samples with negligible numerical costs. Second, it is possible to establish a comparison between the results produced by the first approach proposed in this contribution and the results reported in [37]. In a first stage, the failure probability and its sensitivity with respect to 𝜃𝑙,𝑖 , 𝑙, 𝑖 = 1, 2 are calculated applying LS and the two approaches reported in Sections 4 and 5, respectively. LS is applied considering the important direction 𝜶 = [−0.942, 0.336]𝑇 (see [37]). A total of 106 lines are considered for applying LS and the performance function is evaluated at five points along each of these lines for interpolating the distance 𝑐 (𝑗) , 𝑗 = 1, … , 106 ; thus, for each line L(j) , the performance function is evaluated at points 𝜶𝑦 + 𝒛⟂,(𝑗) , 𝑦 ∈ {0, 1, 2, 3, 4}, ensuring that distance c(j) can be appropriately interpolated (see Fig. 5). Note that the number of lines considered is quite large. However, this number of lines is chosen in order to verify that the proposed approaches for estimating the sensitivity of the probability converge to the correct values. The results obtained are presented in Table 1. The second and third columns of this Table present the estimators for probability and its sensitivity calculated using LS and their coefficient of variation (𝛿). In addition, the fourth column presents the estimators obtained applying MCS considering a total of 108 samples. The results presented in Table 1 indicate that the two approaches for estimating probability sensitivity applying LS produce similar results. However, the values of the coefficient of variation suggest that the first 104

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Reliability Engineering and System Safety 171 (2018) 99–111 Table 2 Distribution parameters of lognormal random variables – Example 2. Variable

Mean (𝜇i )

Standard deviation (𝜎 i )

P [N] EB [N/mm2 ] ES [N/mm2 ] EC [N/mm2 ] rBO [mm] L [mm]

2388 20,300 210,000 2634 15 100

406 2300 7000 150 1.4 10

exceeding a prescribed threshold due to axial loading, as such event is related with implant loosening. Undoubtedly, the occurrence of this failure event is subjected to a large degree of uncertainty, due to inter (patient to patient) and intra (within one patient) patient variability. Thus, the aim of this example is studying the sensitivity of the probability of occurrence of this failure event with respect to the different parameters that affect the response. In order to calculate the stem’s distal displacement uS, D due to an axial loading P, a simplified model proposed in [27] and applied in [41] is considered here. In this model, both the bone and stem are idealized as elastic beams of length L connected through elastic springs that simulate the effect of the cement layer, as depicted in Fig. 8(c). The axial stiffness of the bone and stem are equal to EB AB and ES AS ; EB and ES represent the Young’s modulus of the bone and stem, respectively, and AB and AS are the cross section areas of the bone and stem, respectively. In view of the assumption of cylindrical shape for the bone and stem, 𝐴𝐵 = 𝜋(𝑟2𝐵𝑂 − 𝑟2𝐵𝐼 ) and 𝐴𝑆 = 𝜋𝑟2𝑆 , where rBO and rBI are the outer and inner radii of the bone and rS is the radius of the stem. The stiffness per unit length of the elastic springs that represent the effect of the cement layer is denoted as Ca and is calculated as [27]:

Fig. 7. Evolution of estimator for 𝜕 𝑝̂𝐹 ∕𝜕𝜎2 with respect to the number of lines of LS. The important direction is 𝜶 in (a) and 𝜸 in (b) – Example 1.

𝐶𝑎 =

𝜋𝐸𝐶 (1 + 𝜈𝐶 ) ln(𝑟𝐵𝐼 ∕𝑟𝑆 )

(17)

where Ec and 𝜈 c are the Young’s modulus and Poisson ratio of the cement. Taking into account all parameters described previously, it can be shown [27] that the distal displacement uS, D is equal to: ( ) 𝑃 𝑒2𝐿𝜓 + 1 2𝑃 𝑒𝐿𝜓 𝑢𝑆,𝐷 = (18) ( )+ ( ) 𝐸𝐵 𝐴𝐵 𝜓 𝑒2𝐿𝜓 − 1 𝐸𝑆 𝐴𝑆 𝜓 𝑒2𝐿𝜓 − 1

Fig. 8. Schematic representation of example 2. (a) Hip replacement. (b) Idealized uniform axisymmetric femoral part of hip replacement between proximal and distal ends. (c) Simplified one-dimensional model of bone-cement-stem interaction between proximal and distal end.

where: √ 𝜓=

already indicated at the end of Section 4. Second, the efficiency of the estimator for the probability sensitivity is affected by the important direction. In fact, in Fig. 7(a), the estimator converges to an stable value after 100 lines have been drawn, while in Fig. 7(b), the estimator converges only after drawing 1000 lines. Third, visual inspection of Fig. 7(a) and (b) confirms an observation already drawn from Table 1: the estimator associated with the first approach presents slightly less variability than the estimator associated with the second approach.

( 𝐶𝑎

1 1 + 𝐸𝑆 𝐴𝑆 𝐸𝐵 𝐴𝐵

) (19)

Several quantities that affect the stem’s distal displacement uS, D are modeled as lognormal random variables, as discussed in [40]. Table 2 summarizes the expected value and standard deviation of these random variables. Furthermore, it is assumed that the Poisson ratio of the cement, the radius of the stem and the inner radius of the bone can be modeled as deterministic quantities which are equal to 𝜈𝐶 = 0.3, 𝑟𝑆 = 4.5 [mm] and 𝑟𝐵𝐼 = 7 [mm], respectively. The performance function associated with the problem involves the stem’s distal displacement exceeding a prescribed threshold of 10 [μm]. The probability of failure associated with this performance function and the probability sensitivity with respect to the mean values of the different random variables involved are calculated applying LS. A total of 2000 lines are considered, involving 5 evaluations of the performance function along each line; thus, for each line L(j) , the performance function is evaluated at points 𝜶𝑦 + 𝒛⟂,(𝑗) , 𝑦 ∈ {0, 1, 2, 3, 4} (see Fig. 9). In addition, the results produced with LS are compared with those obtained applying MCS considering a total of 108 samples. Before presenting the results obtained from the probability sensitivity analysis, it is of interest assessing the degree of nonlinearity of the performance function associated with the reliability problem under study. In order to perform such analysis, Fig. 9 presents the value of the

7.3. Example 2 This example involves a simplified model of a hip replacement, as investigated in [41]. The problem is represented schematically in Fig. 8(a), where the hip joint is replaced by a metallic prosthetic implant. A cement layer attaches the stem of the implant and the femur. Fig. 8(b) illustrates an idealization of the bone-cement-stem structure between the proximal and distal ends of the femur, where the bone is assumed to be a hollow cylinder that contains the cement layer and the stem of the implant (which is also assumed to be of cylindrical shape). One of the possible failure mechanisms of the hip replacement involves the longitudinal displacement of the stem at the distal end (uS, D , see Fig. 8(c)) 105

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Reliability Engineering and System Safety 171 (2018) 99–111 Table 3 Probability and probability sensitivity estimates (and their coefficient of variation 𝛿) obtained applying Line Sampling and Monte Carlo Simulation – Example 2. Quantity

Simulation Method LS – First Approach

𝑝̂𝐹 (𝛿) 𝜕 𝑝̂𝐹 ∕𝜕 𝜇𝑃 (𝛿) 𝜕 𝑝̂𝐹 ∕𝜕 𝜇𝐸𝐵 (𝛿) 𝜕 𝑝̂𝐹 ∕𝜕 𝜇𝐸𝑆 (𝛿) 𝜕 𝑝̂𝐹 ∕𝜕 𝜇𝐸𝐶 (𝛿) 𝜕 𝑝̂𝐹 ∕𝜕 𝜇𝑟𝐵𝑂 (𝛿) 𝜕 𝑝̂𝐹 ∕𝜕 𝜇𝐿 (𝛿)

7.1 × 10 2.7 × 10−6 (0.7%) −13 −3.3 × 10 (0.3%) 6.7 × 10−15 (0.4%) −1.9 × 10−12 (0.4%) −1.5 (0.6%) −1.7 × 10−3 (1.8%)

LS – Second Approach −4

(0.5%) 2.7 × 10−6 (0.4%) −3.3 × 10−13 (2.3%) 7 × 10−15 (33.1%) −1.9 × 10−12 (6.0%) −1.5 (1.3%) −8.9 × 10−4 (185.4%)

MCS 7.1 × 10−4 (0.4%) 2.7 × 10−6 (0.4%) −3.3 × 10−13 (0.6%) 6.1 × 10−15 (6.2%) −1.9 × 10−12 (1.1%) −1.5 (0.4%) −1.7 × 10−3 (16.2%)

Fig. 9. Value of performance function along each line of LS – Example 2.

performance function along each of the 2000 lines sampled using LS. Each different color indicates the set of values associated with a single line. As noted from this Figure, the performance function along each line behaves in a nonlinear fashion. However, the low dispersion between different curves suggests that the limit state function is weakly nonlinear [51,60]. The results of the probability and probability sensitivity analysis are shown in Table 3. Two observations can be drawn from this Table. First, the coefficient of variation of the probability sensitivity estimates produced with LS and the second approach are larger than those produced with the first approach. Nevertheless, they are comparable for most of the parameters. Second, the estimates produced with LS and the two proposed approaches for probability sensitivity possess excellent agreement when compared with the estimates produced applying MCS. However, some differences are seen with the estimates of the probability sensitivity with respect to the mean value of the Young’s modulus of the stem (𝜇𝐸𝑆 ) and the mean value of length of the model (𝜇L ). In order to investigate these differences and compare the different probability sensitivity results, the so-called elasticities are calculated [18,30,35]. The elasticity 𝜖 of the sensitivity of the failure probability with respect to the distribution parameter 𝜃 l,i is defined as follows. 𝜖𝜃𝑙,𝑖 =

𝜕 𝑝̂𝐹 𝜃𝑙,𝑖 𝜕𝜃𝑙,𝑖 𝑝̂𝐹

Fig. 10. Elasticity of probability sensitivity – Example 2.

these parameters have a negligible effect over the failure probability, it is more challenging to quantify precisely their sensitivity numerically. In conclusion, both approaches give accurate estimates with small coefficients of variation for the important parameters that control the failure probability. The results presented in Fig. 10 allow gaining valuable insight on the behavior of the hip replacement model. For example, it is seen that the distribution parameter having the largest effect on the probability sensitivity is the mean value of the outer radius of the bone (𝜇𝑟𝐵𝑂 ). In fact, an increase in 𝜇𝑟𝐵𝑂 would cause an increase in the overall stiffness of the model, thus reducing the distal displacement uS, D and hence, causing a decrease in the value of the failure probability. It is also seen from Fig. 10 that an increase the mean values of the Young’s modulii of the bone and cement (𝜇𝐸𝐵 and 𝜇𝐸𝐶 , respectively) would have an effect similar as that of 𝜇𝑟𝐵𝑂 , because they increase the stiffness of the model. On the contrary, an increase in the mean value of the axial loading 𝜇P would induce larger values of distal displacement uS, D and an increase in the failure probability. In addition to analyzing the effect of variations of the different expected values on the failure probability, it is also of interest assessing the variability of the probability sensitivity estimates. Thus, Fig. 11 illustrates the coefficient of variation (𝛿) associated with each of the probability sensitivity estimates as a function of the number of lines drawn for each of the two proposed approaches. This Figure includes the four most important parameters identified in Fig. 10, i.e. the expected values of the load (𝜇 P ), Young’s modulus of the bone (𝜇𝐸𝐵 ), Young’s modulus of the cement (𝜇𝐸𝐶 ) and outer radius of the bone (𝜇𝑅𝐵𝑂 ). It is seen from this Figure that in order to produce probability sensitivity estimates with less than 10% of coefficient of variation, between 100 and 400 lines are required. This number is similar to the number of lines that has been

(20)

The elasticity 𝜖𝜃𝑙,𝑖 provides a means for comparing the sensitivity of the failure probability with respect to different distribution parameters which may possess different physical meaning. Fig. 10 shows the elasticity associated with each of the sensitivity measures of Table 3. From this Figure, it is seen that the smallest elasticities are associated with 𝜇𝐸𝑆 and 𝜇 L , thus revealing that the sensitivity of the probability with respect to these two distribution parameters is negligible. In addition, this explains the relatively high coefficient of variation of the estimators associated with these distribution parameters shown in Table 3, i.e. as 106

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Fig. 11. Evolution of the coefficient of variation (𝛿) of the probability sensitivity estimate with respect to the number of lines applying LS (A1: first approach; A2: second approach) – Example 2.

gravel. The permeability is modeled as anisotropic and the uncertainty on its magnitude is characterized through lognormal random variables. The expected value and standard deviation of the vertical and horizontal permeabilities of the ith soil layer (kxx, i and kxx, i , respectively) are shown in Table 4. It should be noted that the coefficient of variation associated with each permeability is equal to 100%, reflecting that a high degree of uncertainty can be encountered when estimating these parameters in the context of engineering applications (see, e.g. [13]). The governing partial differential equation associated with the hydraulic head of a seepage problem is (see, e.g. [23]): 𝑘𝑥𝑥,𝑖

Fig. 12. Schematic representation of Example 3.

𝜕 2 ℎ𝑊 𝜕𝑥2

+ 𝑘𝑦𝑦,𝑖

𝜕 2 ℎ𝑊 𝜕𝑦2

= 0, 𝑖 = 1, 2

(21)

where x and y denote horizontal and vertical coordinates, respectively. The boundary conditions for this equation are the hydraulic head over segments AB and CD and null flow over the other boundaries, as already described above. This equation is solved numerically applying the finite element (FE) method (see, e.g. [6]). In particular, a FE mesh comprising 3413 nodes and 1628 quadratic triangular elements is considered. Once the hydraulic head has been computed, the seepage q can be calculated, e.g. at the downstream side of the dam.

reported in the literature for estimating failure probabilities using LS, see e.g. [49]. 7.4. Example 3 The final example involves the study of a steady state confined seepage below a dam. Fig. 12 illustrates the elevation of the dam. The water height hD and permeabilities of the soil layers are modeled as random variables. The failure event involves the seepage discharge below the dam exceeding a threshold level of 30 [L/h/m]. The objective is estimating the probability of occurrence of this event as well as its sensitivity with respect to variations in the expected values of the soil permeabilities. The dam rests over a soil composed of two permeable layers and one impermeable layer. A cutoff wall is included in the bottom of the dam for preventing excessive seepage (see Fig. 12). The upstream side of the dam retains a water column of height hD [m], where hD is modeled as a random variable with uniform distribution between 7 [m] and 10 [m]. Thus, the hydraulic head hW over segment AB in Fig. 12 is equal to ℎ𝑊 = 20 + ℎ𝐷 [m], where the elevation head is measured with respect to the impermeable soil layer. The water flows through two permeable soil layers towards the downstream side of the dam (segment CD of Fig. 12), where the hydraulic head is equal to 20 [m]. It is assumed that there is no water flow on any of the boundaries of the problem except for the aforementioned segments AB and CD. The first layer of soil is composed of silty sand, while the second layer is composed of silty

𝜕ℎ𝑊 𝑘 𝑑𝑥 (22) ∫CD 𝑦𝑦,2 𝜕𝑦 It should be noted that q is measured in units of volume over time over distance, i.e. the seepage discharge is calculated in terms of a unit width of the dam. Before calculating the failure probability and its sensitivity, it is of interest visualizing the characteristics of the water flow through the soil in terms of equipotentials and streamlines [22,23,28]. Equipotentials are curves that indicate the locus of points that share the same value of hydraulic head. Streamlines illustrate the path followed by water through the soil. The area between streamlines is called a streamtube and represents a fraction of the total seepage discharge q. Fig. 13 illustrates the equipotentials and streamlines for the problem under study, considering expected values for the permeabilities and a water height ℎ𝐷 = 10 [m]. This Figure includes a total of three streamtubes, each containing one third of the total seepage discharge 𝑞0 = 12.5 [L/h/m]. It should be noted that the streamlines present a sudden change of slope at the vertical coordinate 𝑦 = 15 [m]; this is caused by the different permeability values of each soil layer. Moreover, note that the streamtube between stream𝑞=−

107

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Reliability Engineering and System Safety 171 (2018) 99–111 Table 4 Expected value and standard deviation of permeability – Example 3. Soil layer

Permeability

Expected value (𝜇 [m/s])

Standard deviation (𝜎 [m/s])

Silty sand

horizontal (kxx, 1 ) vertical (kyy, 1 ) horizontal (kxx, 2 ) vertical (kyy, 2 )

5 × 10−7 2 × 10−7 5 × 10−6 2 × 10−6

5 × 10−7 2 × 10−7 5 × 10−6 2 × 10−6

Silty gravel

Fig. 13. Equipotentials (hW [m]) and streamlines (q [L/h/m]) considering expected values of permeabilities and water height ℎ𝐷 = 10 [m] – Example 3.

Table 5 Probability and probability sensitivity estimates (and their coefficient of variation 𝛿) obtained applying Line Sampling and Monte Carlo Simulation – Example 3. Quantity

Simulation Method LS – First Approach

𝑝̂𝐹 (𝛿) 𝜕 𝑝̂𝐹 ∕𝜕 𝜇𝑘𝑥𝑥,1 𝜕 𝑝̂𝐹 ∕𝜕 𝜇𝑘𝑦𝑦,1 𝜕 𝑝̂𝐹 ∕𝜕 𝜇𝑘𝑥𝑥,2 𝜕 𝑝̂𝐹 ∕𝜕 𝜇𝑘𝑦𝑦,2

(𝛿) (𝛿) (𝛿) (𝛿)

8.2 × 102 1.1 × 104 3.1 × 102 9.3 × 102

1.5 × 10 (9.2%) (1.9%) (1.7%) (1.5%)

LS – Second Approach −3

(1.3%) 8.3 × 102 1.1 × 104 3.2 × 102 9.6 × 102

(9.6%) (1.8%) (3.4%) (2.2%)

MCS 1.5 × 10−3 (1.8%) 8 × 102 (8.7%) 1.1 × 104 (2.0%) 3.2 × 102 (2.6%) 9.2 × 102 (2.2%)

Table 5 summarizes the probability and probability sensitivity estimates obtained applying LS and the two approaches for sensitivities reported herein. In addition and for comparison purposes, the fourth column of this table includes the results obtained applying MCS considering a total of 2 × 106 samples. The analysis of this Table indicates that the probability sensitivity estimates obtained using different approaches are in good agreement. Moreover, the tendency already observed in the two previous examples is repeated here: the first approach for calculating probability sensitivity applying LS yields estimates with a smaller coefficient of variation than those produced with the second approach. Nonetheless, small coefficients of variation are obtained by both approaches. The analysis of the results presented in Table 5 allows drawing the following observations. First, it is noted that all sensitivities are positive. This is consistent with the physics of the problem: an increase in the value of the mean value of the permeability should result in a larger seepage discharge, thus leading to an increase in the value of the failure probability. Second, the sensitivity associated with the mean value of the permeability of the silty sand layer 𝜇𝑘𝑦𝑦,1 is the largest one. This is due to the fact that 𝜇𝑘𝑦𝑦,1 is much smaller than the other mean permeability values and hence, it controls the total seepage discharge. Thus, an increase of 𝜇𝑘𝑦𝑦,1 causes a large impact on the total seepage and on the failure probability. In order to further analyze results reported in Table 5, Fig. 15 shows the elasticity associated with each probability sensitivity value. This Figure confirms the observation drawn previously that 𝜇𝑘𝑦𝑦,1 is the most influential parameter over the failure probability;

Fig. 14. Value of performance function along each line of LS – Example 3.

lines 𝑞 = 𝑞0 and 𝑞 = 2𝑞0 ∕3 (which contains one third of the total seepage discharge) is located predominantly on the silty gravel layer. This indicates that a considerable share of the total seepage discharge flows through each of the soil layers, suggesting that all permeabilities (horizontal and vertical of both soil layers) affect the total water flow q below the dam. The failure event associated with the problem involves the seepage discharge q exceeding a prescribed threshold of 30 [L/h/m], i.e. the performance function is 𝑔𝒙 (𝒙) = 30 − 𝑞(𝒙), where x is a vector that collects the uncertain variables of the problem, namely the four permeabilities and the water height hD . In order to estimate the failure probability and its sensitivity with respect to the mean values of the permeabilities applying LS, a total of 2000 lines are considered. The performance function is evaluated over a grid of eight points along each line; thus, for each line L(j) , the performance function is evaluated at points 𝜶𝑦 + 𝒛⟂,(𝑗) , 𝑦 ∈ {0, 1, 2, 3, 4, 5, 6, 7} (see Fig. 14). The value of the performance function along each of the aforementioned lines is shown in Fig. 14. It can be seen from this Figure that performance function presents a nonlinear behavior along each line and that there is significant dispersion in the values of the performance function for different lines. This suggests that the performance function associated with the failure event is mildly nonlinear. 108

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The numerical examples presented in this contribution show that the variability (measured in terms of the coefficient of variation) of the probability sensitivity estimates produced with the two proposed approaches is similar. The exception to this observation are the estimators that are associated with distribution parameters that possess small elasticity (i.e. a small impact on the failure probability). Although the numerical implementation of any of the two approaches is relatively straightforward, the second approach may be preferable over the first one. This is due to the fact that the first approach demands the evaluation of the gradient of the performance function at a specific point. Such calculation may be demanding, particularly in those cases where the gradient is not available either in analytical or semianalytical form. On the contrary, the second approach does not require any additional information on the structural response and just demands solving a one-dimensional integral. Such integral can be solved either analytically (for some particular cases) or numerically. Future research efforts on the application of Line Sampling for probability sensitivity estimation involve the application of the proposed framework to reliability problems involving dependent random variables or random fields. It is expected that in these type of problems, both the number of random variables and their correlation structure may pose a challenge for estimating probability sensitivity. Two additional research challenges that should be investigated are: (a) the integration of the proposed approaches with Adaptive Line Sampling [14,15], which is an advanced, more efficient development of the original LS simulation method; and (b) the application of the two approaches simultaneously for each line of Line Sampling. Addressing both challenges may allow estimating the probability sensitivity with greater efficiency.

Fig. 15. Elasticity of probability sensitivity – Example 3.

in addition this Figure indicates that 𝜇𝑘𝑥𝑥,1 is the least influential parameter among the four considered in this example. Fig. 16 shows the evolution of the coefficient of variation (𝛿) of the probability sensitivity estimates with respect to the number of lines drawn using the two approaches for probability sensitivity estimation proposed herein. It is observed that between 100 and 200 lines suffice for generating estimates with a coefficient of variation less than 10%, except for the case of 𝜇𝑘𝑥𝑥,1 . For this last parameter, evaluating the sensitivity is more challenging, as it possesses a small impact on the probability, as seen in Fig. 15. Note that the study of the probability sensitivity with respect to distribution parameters associated with the water height hD is not considered here. This is due to the fact that for a uniform distribution, the bounds of the integration range for probability depend on its statistical moments, thus precluding the calculation of the probability sensitivity as discussed here (see [30]).

Acknowledgments This research is partially supported by CONICYT (National Commission for Scientific and Technological Research) under grant number 1150009. This support is gratefully acknowledged by the authors.

7.5. Final remarks

Appendix A. Calculation of partial derivatives of gz (z|𝜽) with respect to z and 𝜽

The results presented in the previous examples reinforce the ideas discussed in Section 6, i.e. the two approaches for calculating probability sensitivity provide different means for estimating the same quantity. The first approach tends to produce probability sensitivity estimates which possess a smaller coefficient of variation than those produced with the second approach, even though small coefficient of variations are obtained by both approaches for the parameters that affect the most the failure probability. In fact, as noted from Examples 2 and 3, the proposed approaches can produce estimates of the sought probability sensitivity with a coefficient of variation of 10% (which is considered to be an accurate estimate) with about 100 lines for the most important parameters. For those parameters that have a small impact on the failure probability, a large number of lines (e.g. 1000 or more) may be required in order to produce probability sensitivity estimates with sufficient accuracy. In fact, such issue also occurs when applying either Monte Carlo Simulation or Subset Simulation, as discussed in [30]. In conclusion, the performance of the proposed approaches in terms of the variability of the sensitivity estimates depends on the importance of the corresponding parameters on the reliability problem at hand.

Consider the equality between the performance function in the physical and standard normal space, i.e. 𝑔𝒛 (𝒛|𝜽) = 𝑔𝒙 (𝒙). On one hand, differentiating this equality with respect to xi yields [17]: 𝜕𝑔𝒛 𝜕𝑧𝑖 𝜕𝑔 = 𝒙 , 𝑖 = 1, … , 𝑛 𝜕𝑧𝑖 𝜕𝑥𝑖 𝜕𝑥𝑖

(A.1)

In the above equation, recall that zi depends on xi alone due to the independence between the random variables of the problem. Thus: ( ) 𝜕𝑔𝒛 𝜕𝑔 𝜕𝑧𝑖 −1 = 𝒙 , 𝑖 = 1, … , 𝑛 (A.2) 𝜕𝑧𝑖 𝜕𝑥𝑖 𝜕𝑥𝑖 On the other hand, the differentiation of the equality 𝑔𝒛 (𝒛|𝜽) = 𝑔𝒙 (𝒙) with respect to 𝜃 l,i results in [17]: 𝜕𝑔𝒛 𝜕𝑔 𝜕𝑧𝑖 + 𝒛 = 0, 𝑙 = 1, … , 𝑛𝑖 , 𝑖 = 1, … , 𝑛 𝜕𝜃𝑙,𝑖 𝜕𝑧𝑖 𝜕𝜃𝑙,𝑖

(A.3)

In the last equation, the right hand side is equal to zero as xi does not depend on 𝜃 l,i . Therefore, the partial derivative of gz (z|𝜽) with respect to 𝜃 l,i is equal to:

8. Conclusions and outlook

𝜕𝑔𝒛 𝜕𝑔 𝜕𝑧𝑖 =− 𝒛 , 𝑙 = 1, … , 𝑛𝑖 , 𝑖 = 1, … , 𝑛 𝜕𝜃𝑙,𝑖 𝜕𝑧𝑖 𝜕𝜃𝑙,𝑖

This contribution has presented a framework for estimating the sensitivity of the probability of failure with respect to distribution parameters. The proposed framework is cast as a post-processing step of Line Sampling and is implemented through two different approaches. Although both approaches produce similar probability sensitivity estimates, their basis is different: the first approach involves an integral over the limit state function, while the second one comprises an integral over the failure domain.

(A.4)

Appendix B. Analytic expressions for 𝝏 zi /𝝏 𝜽l,i – Gaussian and lognormal random variables Consider that Xi is characterized either as a Gaussian or a lognormal random variable. In both cases, the distribution parameters are 𝜇i and 𝜎 i , which represent the mean value and standard deviation, respectively. 109

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Fig. 16. Evolution of the coefficient of variation (𝛿) of the probability sensitivity estimate with respect to the number of lines applying LS (A1: first approach; A2: second approach) – Example 3.

Both parameters are collected in the vector 𝜽i such that 𝜽𝑖 = [𝜇𝑖 , 𝜎𝑖 ]𝑇 . For the case where Xi follows a Gaussian distribution, the transformation function is 𝑧𝑖 = 𝑡𝑖 (𝑥𝑖 |𝜽𝑖 ) = (𝑥𝑖 − 𝜇𝑖 )∕𝜎𝑖 [1]. Differentiating this function, it is straightforward to determine that:

where: 𝜅𝑖 =

( ) −𝜎 2 1 −𝜇𝐺𝑖 −𝜎𝐺2 𝑖 ∕2 𝑒 2 − 𝑒 𝐺𝑖 𝜎𝐺 𝑖

(B.9)

𝜆𝑖 =

( ) 1 −𝜇𝐺𝑖 −𝜎𝐺2 𝑖 ∕2 −𝜎𝐺2 𝑖 𝑒 𝑒 −1 2 𝜎𝐺

(B.10)

𝜕𝑧𝑖 1 = 𝜕𝑥𝑖 𝜎𝑖

(B.1)

𝜕𝑧𝑖 1 =− 𝜕𝜃1,𝑖 𝜎𝑖

(B.2)

1 −𝜇𝐺𝑖 −3𝜎𝐺2 𝑖 ∕2 𝜁𝑖 = − 𝑒 𝜎𝐺 𝑖

𝜕𝑧𝑖 𝑥 − 𝜇𝑖 𝑧 =− 𝑖 =− 𝑖 𝜕𝜃2,𝑖 𝜎𝑖 𝜎2

(B.3)

𝜂𝑖 = −

𝑖

𝑖

√ 𝑒

2 𝜎𝐺

𝑖

−1

𝜁𝑖 𝜎𝐺 𝑖

(B.11)

(B.12)

In case Xi follows a lognormal distribution, the transformation function is 𝑧𝑖 = 𝑡𝑖 (𝑥𝑖 |𝜽𝑖 ) = (ln(𝑥𝑖 ) − 𝜇𝐺𝑖 )∕𝜎𝐺𝑖 , where 𝜇𝐺𝑖 and 𝜎𝐺𝑖 are equal to [1]:

Appendix C. Analytic solution of one-dimensional integral – Gaussian and lognormal random variables

⎛ ⎞ 𝜇2 ⎜ ⎟ 𝜇𝐺𝑖 = ln ⎜ √ 𝑖 ⎟ ⎜ 𝜇2 + 𝜎 2 ⎟ 𝑖 𝑖 ⎠ ⎝

Assume that Xi follows a Gaussian distribution. In such case, it can be shown that the function ℎ𝑧𝑖 ,𝜃𝑙,𝑖 (𝑧𝑖 |𝜽𝑖 ) is equal to [30,53]:

𝜎𝐺 𝑖

√ ( ) √ √ 𝜎𝑖2 = √ln +1 𝜇𝑖2

(B.4)

ℎ𝑧𝑖 ,𝜃1,𝑖 (𝑧𝑖 |𝜽𝑖 ) = ℎ𝑧𝑖 ,𝜇𝑖 (𝑧𝑖 |𝜽𝑖 ) = (B.5)

ℎ𝑧𝑖 ,𝜃2,𝑖 (𝑧𝑖 |𝜽𝑖 ) = ℎ𝑧𝑖 ,𝜎𝑖 (𝑧𝑖 |𝜽𝑖 ) = −

Note that 𝜇𝐺𝑖 and 𝜎𝐺𝑖 represent the expected value and standard deviation of ln (Xi ). Differentiation of the transformation function associated with the lognormal distribution described above allows determining the following expressions for the partial derivatives of zi with respect to xi and 𝜃 l,i : 𝜕𝑧𝑖 1 𝑒 = = 𝜕𝑥𝑖 𝑥 𝑖 𝜎𝐺 𝑖

− 𝜇𝐺 𝑖 − 𝜎𝐺 𝑖 𝑧 𝑖

𝜎𝐺 𝑖

(ln(𝑥𝑖 ) − 𝜇𝐺𝑖 ) 𝜕𝑧𝑖 = −𝜅𝑖 − 𝜆𝑖 = −𝜅𝑖 − 𝜆𝑖 𝑧𝑖 𝜕𝜃1,𝑖 𝜎𝐺 𝑖 (ln(𝑥𝑖 ) − 𝜇𝐺𝑖 ) 𝜕𝑧𝑖 = −𝜁𝑖 − 𝜂𝑖 = −𝜁𝑖 − 𝜂𝑖 𝑧𝑖 𝜕𝜃2,𝑖 𝜎𝐺 𝑖

𝑧𝑖 𝜎𝑖 𝑧2 1 + 𝑖 𝜎𝑖 𝜎𝑖

(C.1)

(C.2)

Note that the above functions ℎ𝑧𝑖 ,𝜃𝑙,𝑖 (𝑧𝑖 |𝜽𝑖 ), 𝑙 = 1, 2 are polynomials of zi . Thus, for these two cases, the integral in Eq. (14) possesses the following analytical solutions (see e.g. [48]). ⟂,(𝑗) ( ( ) 𝑧 ) 𝛼 ( ) 𝜑𝑧𝑖 ,𝜃1,𝑖 𝒛⟂,(𝑗) |𝜽𝑖 = 𝑖 Φ −𝑐 (𝑗) + 𝑖 𝜙 𝑐 (𝑗) 𝜎𝑖 𝜎𝑖

(B.6)

(( ) )2 ( ( ) ) 1 (𝑗) 𝑧⟂, 𝜑𝑧𝑖 ,𝜃2,𝑖 𝒛⟂,(𝑗) |𝜽𝑖 = + 𝛼𝑖2 − 1 Φ −𝑐 (𝑗) + … 𝑖 𝜎𝑖 ) ( ) 𝛼𝑖 ( ⟂,(𝑗) + 𝛼𝑖 𝑐 (𝑗) 𝜙 𝑐 (𝑗) 2𝑧𝑖 𝜎𝑖

(B.7)

(C.3)

(C.4)

In case Xi follows a lognormal distribution, it can be shown that the function ℎ𝑧𝑖 ,𝜃𝑙,𝑖 (𝑧𝑖 |𝜽𝑖 ) is equal to [30]:

(B.8)

ℎ𝑧𝑖 ,𝜃1,𝑖 (𝑧𝑖 |𝜽𝑖 ) = ℎ𝑧𝑖 ,𝜇𝑖 (𝑧𝑖 |𝜽𝑖 ) = −𝜆𝑖 + 𝜅𝑖 𝑧𝑖 + 𝜆𝑖 𝑧2𝑖 110

(C.5)

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ℎ𝑧𝑖 ,𝜃2,𝑖 (𝑧𝑖 |𝜽𝑖 ) = ℎ𝑧𝑖 ,𝜎𝑖 (𝑧𝑖 |𝜽𝑖 ) = −𝜂𝑖 + 𝜁𝑖 𝑧𝑖 + 𝜂𝑖 𝑧2𝑖

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(C.6)

while the integration of these functions as indicated in Eq. (14) yields the following results. ( (( ) ) )2 ( ( ) ) (𝑗) (𝑗) 𝜑𝑧𝑖 ,𝜃1,𝑖 𝒛⟂,(𝑗) |𝜽𝑖 = 𝜆𝑖 𝑧⟂, + 𝛼𝑖2 − 1 + 𝜅𝑖 𝑧⟂, Φ −𝑐 (𝑗) + … 𝑖 𝑖 (

( ) ) ( ) (𝑗) 𝜆𝑖 𝛼𝑖 2𝑧⟂, + 𝛼𝑖 𝑐 (𝑗) + 𝜅𝑖 𝛼𝑖 𝜙 𝑐 (𝑗) 𝑖

( ) 𝜑𝑧𝑖 ,𝜃2,𝑖 𝒛⟂,(𝑗) |𝜽𝑖 = (

(C.7)

( (( ) ) )2 ( ) (𝑗) (𝑗) 𝜂𝑖 𝑧⟂, + 𝛼𝑖2 − 1 + 𝜁𝑖 𝑧⟂, Φ −𝑐 (𝑗) + … 𝑖 𝑖

( ) ) ( ) (𝑗) + 𝛼𝑖 𝑐 (𝑗) + 𝜁𝑖 𝛼𝑖 𝜙 𝑐 (𝑗) 𝜂𝑖 𝛼𝑖 2𝑧⟂, 𝑖

(C.8)

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