System reliability under prescribed marginals and correlations: Are we correct about the effect of correlations?

Accepted Manuscript

System reliability under prescribed marginals and correlations: Are we correct about the effect of correlations? Fan Wang , Heng Li PII: DOI: Reference:

S0951-8320(17)30679-8 10.1016/j.ress.2017.12.018 RESS 6038

To appear in:

Reliability Engineering and System Safety

Received date: Revised date: Accepted date:

8 June 2017 17 November 2017 28 December 2017

Please cite this article as: Fan Wang , Heng Li , System reliability under prescribed marginals and correlations: Are we correct about the effect of correlations?, Reliability Engineering and System Safety (2017), doi: 10.1016/j.ress.2017.12.018

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Highlights   

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Probability distribution model is built on prescribed marginals and correlations A sequential search strategy (S3) is proposed to retrieve pair-copula parameters Both Gaussian and Non-Gaussian dependency can be incorporated into system reliability Reliability can be evaluated using qualitative dependency and quantitative statistics

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Title page

System reliability under prescribed marginals and correlations:

Fan Wanga,b,c, Heng Lia

a

Department of Building and Real Estate, The Hong Kong Polytechnic University,

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Kowloon, Hong Kong

b

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Are we correct about the effect of correlations?

Department of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan, 430074, PR China

Department of Resource and Civil Engineering, Wuhan Institute of Technology,

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c

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Wuhan, 430073, PR China

[email protected] (H. Li)

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[email protected] (F. Wang)

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System reliability under prescribed marginals and correlations: Are we correct about the effect of correlations?

Abstract

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Many reliability problems involve correlated random variables. However, the probabilistic specification of random variables is commonly given in terms of marginals and correlations, which is actually incomplete because the data dependency needed for distribution modeling is not characterized. The implicitly assumed

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Gaussian dependence structure is not necessarily true and may bias the reliability result. To investigate the effect of correlations on system reliability under non-Gaussian dependence structures, a general approach to the probability distribution model construction based on the pair-copula decomposition is proposed. Numerical examples have highlighted the importance of dependence modeling in

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system reliability since large deviation in failure probabilities under different

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dependencies is observed. The method for identifying the best fit data dependency from data is later provided and illustrated with a retaining wall. It is demonstrated that

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the reliability result can be accurately estimated if the qualitative dependence

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structure is complemented to the available quantitative statistical information.

Keywords:

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System reliability; Marginals and correlations; Transformation method; Dependence structure; Probability distribution model; Pair-copula decomposition

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1 Introduction Engineering systems are associated with various uncertainties including variable material properties and random environmental conditions, which necessitate the use of reliability concept for engineering design within the constraint of economy [1]. A critical step to this target is to capture the joint probability distribution of the uncertain

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parameters. Unfortunately, due to the high cost or the limited conditions for in situ experiments, the data needed to characterize the joint probability distribution is frequently insufficient [2]. In contrast, the univariate data analysis can be readily performed to obtain the marginal distribution of each parameter. If the random variables representing the parameters are independent, the joint probability

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distribution will be a simple multiplication of its marginal distributions. However, correlations may exist between different stochastic parameters or in the variance of a stochastic parameter over time or space [36]. Many articles and books have illustrated how to consider the correlations in reliability analysis (e.g., [7–8]). The basic idea is

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to transform the correlated random variables into uncorrelated standard ones to simplify the subsequent reliability evaluation. When the correlated random variables

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are normal, the orthogonal transformation can be used for this goal. If the correlated non-normals are involved, the Nataf transformation [9] is commonly adopted.

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It is known that the marginals and correlations can be readily derived from the joint probability distribution; however, the reverse of the process is difficult because

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the probabilistic description in terms of the first two statistical moments (including the correlation or covariance matrix) is actually incomplete [9]. The missing information

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is the dependence structure that is usually characterized by a copula function [10–11]. From the copula perspective, the orthogonal and the Nataf transformation arbitrarily assumes a Gaussian dependence structure characterized by a Gaussian copula [12]. Although this arbitrary selection of distribution model is done quite often, recent investigation has shown that the data dependency is not always Gaussian [13–14], which implies that the reliability results based on the Gaussian dependence assumption could be biased. -2-

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In this paper, the system reliability problems involving correlated random variables are revisited. The aim is to investigate the effect of correlations between random variables on the system reliability under both Gaussian and non-Gaussian dependence structures, and tries to provide insights into uncertainty modeling in reliability analysis. To this end, this paper proposes a general method for constructing the multivariate distribution that is consistent with the given marginal distributions

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and linear correlations. The pair-copula decomposed model is adopted to represent the multivariate distribution while the model parameters are retrieved from given probabilistic information via a sequential search strategy (S3). The constructed distribution model is flexible as the correlated random variables are not necessarily

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Gaussian dependent. Then the reliability results under the same probabilistic description of input random variables but different dependence structures are compared, and the cause of large deviation in results is discussed. Finally, with a given dataset, we show how to choose an appropriate distribution model in reliability

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reliability-based design.

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analysis and demonstrate the impact of inaccurate dependence modeling on

2 Reliability analysis considering correlations

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2.1 A brief review on the formulation of reliability

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Let x   x1 ,, xn  be a random vector representing uncertain parameters in a system and f  x  be its joint probability density function (PDF). In practice, an

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engineering system often has multiple components or subsystems. The failure probability of a system can be generally defined as:

P  Esys   

Esys

f  x  dx

(1)

where Esys denotes the failure event of the system. For a cut-set system, Esys can be expressed as: N

N



Esys   Ck    Ei k 1

k 1 iCk

-3-



(2)

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where Ck , k  1,, N is the kth cut-set event that the system fails if any cut-set event occurs; Ei   Gi  x   0  is the ith component failure event and Gi  x  defines its performance function. Note Eq. (2) is also valid for a link-set system, which is defined as the intersection of the unions of component failure events, if the complementary event of Esys (i.e., survival event of the link-set system) is of

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interest.

Various methods can be used to approximate the system reliability (e.g., [15–17]). Generally, these methods are accurate and efficient; yet, it is usually assumed that the random variables are statistically independent. If the random variables are correlated,

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they are commonly mapped into the independent standard normal variables u through a transformation of the joint probability distribution model of x , namely

u  T  x  , and the reliability evaluation can be equivalently implemented in the standard normal space as:

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P  Esys   

Esys

f  u  du

(3)





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where the component failure event in Eq. (2) is re-written as Ei  Gi T 1  u    0

and T 1   is the inverse of the transformation T   ; f  u  is the PDF of an

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independent standard multivariate normal distribution.

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Theoretically, f  x  is needed to perform the above nonlinear transformation. However, this requirement is not fulfilled at all times. Instead, the marginal

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distributions of x and the correlation matrix is commonly given, which motivates the use of copula theory [10] to bridge the gap between the incomplete probability information and a probability distribution model.

2.2 A copula perspective on dealing with correlations Copula theory allows the analysis of a joint probability distribution by investigating its dependence structure and marginal distributions respectively [10]. According to Sklar’s theorem, the joint bivariate cumulative distribution function -4-

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(CDF) can be formulated as:

F ( x1 , x2 )  C (u1 , u2 ; )

(4)

where u1  F1  x1  and u2  F2  x2  are the marginal CDFs of x1 and x2 respectively, C is a bivariate copula function with a parameter  indicating the strength of dependence. A number of copula functions can be found in the literature

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[10], and some widely used copulas are listed in Table 1. The corresponding joint bivariate PDF can be written as:

f ( x1 , x2 ) 

(5)

 2C (u1 , u2 ; ) is a copula density function, f1 ( x1 ) and f 2 ( x2 ) u1u2

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where c(u1 , u2 ; ) 

 2C (u1 , u2 ; )  c(u1 , u2 ; ) f1 ( x1 ) f 2 ( x2 ) x1x2

are the marginal PDFs of x1 and x2 , respectively. Table 1. Examples of bivariate copulas

1

 ( (u1 ),  (u2 ))

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1

*

Frank

 (e u1  1)(e u2  1)   ln 1     e  1 

Clayton

(u1  u2  1)1/

 1 2  21 2   2 2  exp    2 2 2 1    1    e

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

1    u1u2  

u1  u2  1  (1  u1 )  (1  u2 )  1 *



 u1 u2 

 1

2

(, ) \{0}

(u1  u2   1)21/

1    (1  u1 )(1  u2 )

 1

1/

[1, 1]



(e  1)

(e  1)  (e u1  1)(e u2  1) 



[(1  u1 )  (1  u2 )   1]21/

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CClayton

1

Range of 

1

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Gaussian

c(u1 , u2 ; )

C (u1 , u2 ; )

Copula

 is the bivariate standard normal distribution function with a correlation coefficient  ,

1   1  u1  , 2  1  u2  , where  1 is the inverse function of the CDF of a univariate

standard normal distribution.

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(0, ) (0, )

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-2

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2

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(d)

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x2

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x2

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(a) 3

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Fig. 1 joint PDF isoline with the same standard normal marginal distributions, a correlation

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coefficient of 0.5, and the different copulas: (a) Gaussian copula; (b) Frank copula; (c) Clayton copula; and (d) CClayton copula

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Given a certain copula, the distribution model can be uniquely established by linking the copula parameter to the correlation coefficient. Fig. 1 shows the

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iso-density contours of f  x1 , x2  when different copulas are adopted. It is obvious that the joint PDFs differ significantly despite that they share the same standard

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normal marginal distributions and correlation coefficient. Several studies have reported that the reliability can be differed significantly due to a different copula selection [11, 13, 14]. However, most probability distribution models considering non-Gaussian dependency are illustrated in two-dimensions [11, 13, 14]. If more than two random variables are mutually correlated, the above copula-based approach to the multivariate distribution modeling usually fails except for the elliptical copula (e.g., Gaussian copula). For the widely used n-variate Archimedean copulas (e.g., Frank, Clayton, and CClayton copulas), the maximum number of copula parameters is n-1 -6-

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while the n-variate probability distribution has n(n-1)/2 pair-wise correlation coefficients. Consequently, the number of copula parameters is less than the number of

correlation

coefficients

unless

in

two-dimensions.

This

inequality

in

multi-dimensions prohibits a one-to-one mapping between the copula parameters and the correlation matrix that is required to rebuild a probability distribution model from the prescribed marginals and correlations [18]. In other words, the simple extension of

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bivariate copulas to its multivariate form generally has limited flexibility that cannot be used to build a distribution in consistency with arbitrary correlations. Unfortunately, there are many engineering systems involving incomplete probabilistic description of correlated multivariates. To provide insights into the effect of copulas on the system

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reliability, it is necessary to develop a general approach to probability distribution model construction under prescribed marginals and correlations.

3 Construction of multivariate distribution model

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3.1 Pair-copula decomposed model for multivariate distributions The joint PDF of x   x1 ,, xn  can be decomposed as:

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f  x   f  x1  f  x2 | x1  f  xn | x1 ,, xn1 

(6)

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wherein the marginal conditional PDFs in the general form of f ( x | v) can be

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expressed as [19]:





f  x | v   c F  x | v  j  , F  v j | v  j  ; f  x | v  j 

(7)

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where v j is an arbitrarily excluded element from vector v and v  j denotes the vector v after excluding v j ; c is a bivariate copula density function defined in Eq. (5); the subscript of  indicates the conditioning variables to be drawn. Using Eq. (7) recursively, f (x) can be written as a product of bivariate copula density functions with marginal conditional CDFs in the form of F ( x | v) , where F ( x | v) can also be represented recursively [20]: -7-

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F  x | v 



C F  x | v  j  , F  v j | v  j  ; F  v j | v  j 



(8)

If v is a univariate, Eq. (8) simplifies to:

F  x2 | x1  

C  F1  x1  , F2  x2  ;  F1  x1 

 h  u2 , u1 ; 

(9)

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where h(u2 , u1; ) is the derivative of C (u1 , u2 ; ) with respect to u1 . Note the second variable in the h-function is always the conditioning variable (i.e.,

F  x2 | x1   h(u2 , u1; ) and F  x1 | x2   h(u1 , u2 ; ) ). For reference, the h-functions corresponding to the bivariate copulas in Table 1 are listed in Table 2.

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Table 2 The h-functions of the example copulas

h(u2 , u1; )

Copula

Frank

e u1 (e u2  1) (e  1)  (e u1  1)(e u2  1)

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Gaussian

  1 (u2 )   1 (u1 )    1 2  

u1 1 (u1  u2  1)11/

CClayton

1  (1  u1 ) 1[(1  u1 )  (1  u2 )  1]11/

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Clayton

The n-dimensional joint PDF f (x) can thus be expressed in terms of a series of

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bivariate copulas (i.e., pair-copula decomposition). However, there are many different

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choices of v j during the decomposition of the joint PDF, leading to many different ways of representing f (x) . Bedford and Cooke [21–22] proposed a graphical tool known as vines to help categorize different pair-copula decompositions. The canonical vine (C-vine) and the drawable vine (D-vine) are the two major types of regular vines. The joint PDF f (x) corresponding to a C-vine is [19]: n

n 1 n  j

k 1

j 1 i 1

f (x)   f k ( xk ) c F ( x j | x1 , , x j 1 ), F ( x j i | x1 , , x j 1 ); j , j i|1,, j 1

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(10)

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while f (x) corresponding to a D-vine is given as [19]: n

n 1 n  j

k 1

j 1 i 1

f ( X)   f k ( xk ) c F ( xi | xi 1 , , xi  j 1 ), F ( xi  j | xi 1 , , xi  j 1 ); i ,i  j|i 1,,i  j 1 (11) where the subscript indices indicate the conditional random variables to be drawn. For an n-dimensional distribution, there are n!/2 different C-vines and n!/2

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different D-vines. Each vine structure contains a specific set of pair-copula parameters that need to be estimated. The critical step to a general PDF construction thus lies in the retrieval of these parameters, which will be presented below.

3.2 A sequential search strategy for retrieving the pair-copula parameters from marginals and correlations

i , j  





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The pair-wise correlation coefficient, by definition, has the following expression:  x    i  i i 

 xj   j     j

  f ( xi , x j )dxi dx j 

(12)

where i (  j ) and  i (  j ) are respectively the mean and the standard deviation of

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x1 ( x2 ); f  x1 , x2  is the bi-dimensional marginal PDF which can be written as:

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f ( xi , x j ) 

f ( x) f (x i , j | xi , x j )

(13)

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where x i , j denotes the random vector excluding the i-th and j-th variables. By

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analogy with Eq. (6), the conditional PDF f (xi , j | xi , x j ) can also be expressed as a product of marginal conditional PDFs such as: (14)

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f ( xi , j | xi , x j )  f ( x1 | xi , x j ) f ( x2 | x1 , xi , x j )  f ( xn | xi , j ,n , xi , x j )

Therefore, both f (x) and f (xi , j | xi , x j ) in Eq. (13) can be expressed by a series of pair-copulas. However, in multi-dimensions, both f (x) and f (xi , j | xi , x j ) have more than one pair-copula-based expressions, which leads to various formulations of f ( xi , x j ) . Unfortunately, the pair-copula parameters cannot be directly determined from a single correlation coefficient using Eq. (12) except those associated to the unconditional pair-copulas. To retrieve the pair-copula parameters in -9-

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higher hierarchical levels, additional parameters irrelevant to the desired construction of f (x) have to be estimated, which can result in unacceptable computational cost [18]. Bedford and Cooke [22] use vines to demonstrate a one-to-one mapping relation between the pair-copula parameters and correlation coefficients. This mapping

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relation is further visualized in Fig. 2, in which the pair-copula parameters associated to the edges in a C-vine structure are separately related to the pair-wise correlation coefficients in a correlation matrix. Moreover, the nodes at a particular tree level are only necessary for determining the edges in the next level. In other words, the conditional pair-copula parameters at a specific level are dependent on the lower-level

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pair-copula parameters. Bearing this in mind, a sequential search strategy shown in Algorithm 1 is proposed to approximate the pair-copula parameters, where each pair-copula parameter is retrieved from its corresponding correlation coefficient by a simulation-based bisection search method shown in Algorithm 2. Appendix A

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illustrates that the pair-copula parameter at higher levels can be uniquely determined

...

from the corresponding coefficient if the solution exists.

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i-1,i|1,…,i-2

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Tree i

i+j |1, …, i-1

θi,i+

j+1|1

,…,i-1

...

... i-1,i+j|1,…,i-2

  Correlation  matrix   

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 i ,i  j i 1,i  j

i-1,i+j+1|1,…,i-2

i ,i  j 1 i 1,i  j 1

     

i,i+1|1,…,i-1

Tree i+1

θi+

...

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θi,

1,i +

j|1 ,… ,i

θi+1

,i+j+ 1|1,…

,i

...

... i,i+j|1,…,i-1

i,i+j+1|1,…,i-1

Fig. 2 Illustration of the one-to-one relation in a C-vine structure.

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Algorithm 1 Sequential search strategy for retrieving pair-copula parameters Input: Vine structure: V correlation matrix: ρ Output: pair-copula parameters: Θ

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Initialize Θ (e.g., Θ ← zero matrix) Case V=C-vine For i=1 to n

θi,j|1,2,…,i-1←ρi,j by Algorithm 2 End End Case V=D-vine

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For i=1 to n-1

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For j=i+1 to n

For j=1 to n-i

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θj,j+i|j+1,j+2,…,j+i-1←ρj,j+i by Algorithm 2 End

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End

Algorithm 2 Approximating pair-copula parameter from marginals and correlations

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Input:

A set of pair-copulas: C

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Marginal CDFs: Fd(xd) d=1,…,n Targeted correlation coefficient: ρij Search interval: ij|... ,ij|...  Current pair-copula parameters: Θold Tolerance of approximation error: ε Sample population: m Output: Updated pair-copula parameters: Θnew - 11 -

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%Fix seed for generating random numbers While ij|...  ij|...   Θtrial ← replace ij|... in Θold by ijtrial |...  (ij|...  ij|... ) / 2 um×n←{V, C, Θtrial} % Sampling m copula random numbers from the vine xm×n←F-1(um×n) % obtain the random variables by the inverse of marginal CDFs

 trial ←[x( : , i), x( : , j)] % obtain the i-th and j-th column of xm×n ij

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If  ijtrial  ij

ij|...  ijtrial |... Else

ij|...  ijtrial |... End

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End Θnew=Θtrial

4 MCS approach to system reliability

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Due to the conceptual simplicity and algorithm robustness, the MCS is adopted

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in this paper for system reliability analysis. In general, the system failure probability

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can be approximated by:

Pf 

nt

nf nt



 I x  i 1

F

i

(15)

nt

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where nf is the number of failure samples and nt is the total number of generated

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random variables; I F  xi  is the indicator function, i.e., I F  xi   1 if the system failure event Esys shown in Eq. (2) occurs and I F  xi   0 otherwise. As mentioned above, the generation of correlated normal or non-normal random

variables based on the orthogonal transformation or the Nataf transformation inherently adopts a Gaussian dependence structure. Therefore, strictly speaking, the Rosenblatt transformation U k  1  F ( xk | x1 , xk 1 ) , k  1,, n [23] should be used for transforming the uncorrelated random variables into the correlated ones. - 12 -

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The use of Rosenblatt transformation requires the joint probability distribution to be available. By rebuilding the distribution, the Rosenblatt transformation can be re-written in terms of pair-copulas:

w1   U1   F1 ( x1 ) w2   U 2   h  F2 ( x2 ), F1 ( x1 );1,2 



w3   U 3   h h  F3 ( x3 ), F1 ( x1 );1,3  , h  F2 ( x2 ), F1 ( x1 );12  ; 2,3|1

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

(16)



where w1 ,, wn are independent random variables uniformly distributed on [0,1] , and U1 ,,U n are independent standard Gaussian random variables. Therefore, the

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random variables x1 ,, xn can be generated from U1 ,,U n or w1 ,, wn by applying the inverse of the Rosenblatt transformation. For example: x1  F1-1 ( w1 )



x2  F2-1 h 1  w2 , F1 ( x1 );1,2 

 



(17)



M

x3  F3-1 h 1 h 1  w3 , h  F2 ( x2 ), F1 ( x1 );12  ; 2,3|1  , F1 ( x1 );1,3 

ED

where F 1 and h 1 are the inverse functions of marginal CDFs and h-functions, respectively. Note Eqs. (16) and (17) are true for a C-vine structured model, in which

PT

the conditioning variable is always excluded as:





(18)

CE

F  x j | x1 ,, x j 1   h F  x j | x1 ,, x j 2  , F  x j 1 | x1 ,, x j 2  ; j 1, j|1,2 j 2

The Rosenblatt transformation corresponds to a D-vine structured model can be

AC

derived in a similar manner. Note the Nataf transformation is actually a special case of the Rosenblatt transformation if all pair-copulas are Gaussian [18].

5 Numerical examples Two examples are used to illustrate the proposed method for reliability under prescribed marginals and correlations. Gaussian and Non-Gaussian data dependencies are respectively assumed to examine the possible deviation in failure probability. For simplicity, we use the C-vine structured decomposed model with canonical - 13 -

ACCEPTED MANUSCRIPT

conditioning order to represent the multivariate probability distribution, as shown in Eq. (10). The MCS population is set to 106. 5.1 A five-variate example Consider the following series system which has five performance functions.

2 x3  x5 2 2 g 2  x   x1  x3  x5  x4 2 g3  x   2 x1  x5 g 4  x   2 x3  x5 g5  x   x1  x2  x5  x4

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g1  x   x2 

(19)

parameters:

x1 ~ N 137.2, 20.58 ,

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All the random variables conform to normal distribution with the following

x2 ~ N  98.0,14.70  ,

x3 ~ N 196.0, 29.40 ,

x4 ~ N  29.4, 4.41 , x5 ~ N 127.4,38.22  . When all the random variables are independent, the system failure probability is 1.5e-2 regardless of the dependence

M

structure. If all the pair-wise correlation coefficients are 0.5, the failure probability

ED

from the Nataf transformation is 6.6e-4; however, the failure probability can increase up to 2.6e-3 if all pair-copulas are Frank. Fig. 3 shows the PDFs and CDFs of the

PT

minimum response values of the series system corresponding to different copulas. Fig. 4 illustrates the conditional bivariate distribution of

x1 and x2 provided that the

CE

remaining variables are set to their mean values. The assumed dependence structure underlying the random variables obviously impacts the distribution of the uncertain

AC

parameters and thus the system reliability, even if the marginals and correlations remain the same.

- 14 -

ACCEPTED MANUSCRIPT

(a) 0.014

10 normal Frank

0.012

10

0.01 0.008

CDF

PDF

10

0.006

10

10

(b)

0

normal Frank

-1

-2

-3

-4

0.004

0 -50

0

50

100

150

200

10

250

-5

-6

-50

0

50

100

150

200

250

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10

0.002

min(g1,g2,g3,g4,g5)

min(g1,g2,g3,g4,g5)

Fig. 3 (a) PDFs and (b) CDFs of the minimum response values of the series system under different copulas. 120 0.1

0. 1

110

0.3

0.1

0.1 0.2 0.2

0.1

x2

0. 2

0.3

100 95

0.1

0.2 0.2 0.3

0.2 0.3

105

0.1

0.1

0.1

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normal Frank

115

3 0.

2 0.

M

90 85

0.2

0.3 .2 0

0.1 0. 1

80

ED

75 100

110

120

130

0.11 0.

140

150

160

170

x1

PT

Fig. 4 Contour of f  x1 , x2 | x3  196.0, x4  29.4, x5  127.4  .

5.2 A sewer system example

CE

Consider a sewer system consisting of three pipes (denoted as ac, bc, and cd,

respectively) as shown in Fig. 5. The system performance is considered unsatisfactory

AC

if overflow occurs in any pipe. According to Ang and Tang [24], the performance function for each pipe is given as:

Gac  8.3Y  4.3542C1W1

(20a)

Gbc  8.3Y  4.3542C2W2

(20b)

Gcd  25.835Y  4.3542W3  C3  KC1  KC2 

(20c)

where Y is proportional to the pipe flow capacity; C1 , C2 , C3 are respectively the - 15 -

ACCEPTED MANUSCRIPT

runoff coefficients at basin areas a, b, and c; W1 ,W2 ,W3 is proportional to the precipitation rate at each basin area; and K is an attenuation factor. The statistics of the random variables are summarized in Table 3 and they are all assumed to follow a normal distribution. Since basin areas a, b, and c are adjacent, it is reasonable to assume that the precipitation rates at the three different areas are positively correlated.

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Generally, the value of correlation coefficient is inversely related to the relative distance between the areas (i.e., precipitation rates at two locations with shorter relative distance are more likely to have similar values). For simplicity, we assume the three correlation coefficients between W1 and W2 , W1 and W3 , W2 and W3 have

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the same value and the other random variables are statistically independent.

b

c

d

ED

M

a

Fig. 5 The sewer system example.

AC

CE

PT

Table 3. Statistics of the eight random variables in the sewer system example. Variable

Distribution



COV

C1 C2 C3 W1 W2 W3 K Y

normal normal normal normal normal normal normal normal

0.825 0.825 0.9 8.59 8.59 8.59 0.9 7.01

0.07 0.07 0.05 0.233 0.233 0.233 0.05 0.123

The Gaussian copula, Clayton copula, and CClayton copula are respectively examined. Fig. 6 shows the effect of dependence assumption on the sewer system CClayton Gaussian Clayton  Psys  Psys failure probability. It can be observed that Psys and the failure

- 16 -

ACCEPTED MANUSCRIPT

probabilities are exactly the same only if W1 ,W2 ,W3 are fully correlated or uncorrelated. This consistency is expected because W1 ,W2 ,W3 are equivalent to a single random variable if they are fully correlated and the joint PDF of W1 ,W2 ,W3 is a simple multiplication of its marginals if they are independent, which leads to a unique joint PDF. However, for a medium strength of correlation between random

12

x 10

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variables, the effect of dependence assumptions is not negligible. -3

Gaussian Clayton CClayton

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P(Esys )

10

8

0

0.2

0.4

0.6

12=13=23

0.8

1

M

6

ED

Fig. 6 Effect of dependence structure on the sewer system reliability as the correlation coefficients increase from 0 to 1.

PT

The effect of dependence assumption is illustrated in a series system. It is of interest if the effect remains similar for other systems. Suppose the sewer system is

CE

defined satisfactory if water can be drained from basin area a to d or b to d without occurrence of overflow. Thus the sewer system becomes to a series-parallel combined

AC

system Fig. 7 shows the effect of dependence assumption on the system reliability and the CIMs of components, respectively. The system failure probability now increases as

the

correlation

coefficients

between

W1 ,W2 ,W3

increase

and

CClayton Gaussian Clayton Psys  Psys  Psys . However, the pattern of influence of dependence

assumptions on system reliability in the series-parallel combined system is basically the same to that in the series system.

- 17 -

ACCEPTED MANUSCRIPT

3

x 10

-3

Gaussian Clayton CClayton

P(Esys )

2.5

1.5

0

0.2

0.4

0.6

0.8

12=13=23

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2

1

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Fig. 7 Effect of dependence assumption on reliability by taking the sewer pipes as a series-parallel combined system.

6 A retaining wall case

The above examples have demonstrated the effect of pair-copulas on system reliability under prescribed marginal distributions and correlations. Therefore, how to

M

choose the appropriate copula function is of great value for an unbiased estimate of reliability. A retaining wall subjected to backfill soil pressure (Fig. 8) is investigated

ED

here for this purpose. The retaining wall system has three failure modes, which are

H

 soil , csoil , soil

W2

Soil

L2 W1

AC

CE

PT

defined with respect to the factor of safety (FS) [24].

L1

Pa H0

O

b

a Foundation

Fig. 8 schematic diagram of the retaining wall.

(1) Sliding of the wall along its base - 18 -

ACCEPTED MANUSCRIPT

The FS against sliding failure mode is defined as:

FS1 

  a  b   W1  W2  tan  2 / 3base   1/ 3cbase Pa

(21)

 and base  respectively the cohesion and friction angle of foundation soil; where cbase

Pa is the active earth thrust calculated by:

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W1  1/ 2 wall bH and W2   wall aH ;  wall is the unit weight of the retaining wall;

2 1     2coh 2 Pa   H K a  2cohH tan     2   4 2

(22)

where coh,  ,  are respectively the cohesion, friction angle, and unit weight of the

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backfill soil. a, b, and H are geometry parameters shown in Fig. 8. (2) Overturing of the wall with respect to its toe

The FS against overturning failure mode is defined as: M r W1L1  W2 L2  Mo Pa H 0

(23)

M

FS2 

ED

1 1  2 a      where L1  b , L2  b  , and H 0   H  2coh   tan      . 3  3 2  4 2    

(3) Ultimate bearing capacity of the foundation soil exceeded

PT

The FS against bearing capacity failure mode is defined as:

CE

FS3 

qu qu  6e  qmax W1  W2  1   ab  ab

AC

where qu is the bearing capacity of the foundation and e 

(24)

a  b Mr  Mo  . 2 W1  W2

The retaining wall system fails if any failure mode occurs. The corresponding

performance functions are as follows:

G1  FS1  1

(25a)

G2  FS2  1

(25b)

G3  FS3  1

(25c)

- 19 -

ACCEPTED MANUSCRIPT

Here coh ,  , and  are random variables. The other parameters are taken as

 =100kPa, base  =30°, qu =400kPa, a=0.5m, deterministic:  wall =24kN/m3, cbase b=1.45m, H=6.0m. 42 samples of the soil properties, namely coh,  ,  , are given as shown in

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Appendix B. The correlation coefficients are coh,   0.458 , coh,   0.270 , and

 ,   0.033 . The trivariate distribution best fitted to the above 42 data samples can be identified by investigating its marginal distributions and the copula separately. The univariate data analysis results are given in Appendix B. The set of best fit

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pair-copulas can be selected based on the minimum value of Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), which are calculated as follows:

AIC  2L  2k

(26)

BIC  2L  k ln M

M

(27)

ED

where k is the number of pair-copula parameters and M is the sample size; L is the pseudo-likelihood given by:







(m) (m) L  coh, , coh, , , |coh    log c  scoh , s( m ) ; coh,    log c  scoh , s( m ) ; coh, 

PT

M

m 1



M

m 1

  log c h  s , s ;  coh,  , h  s , s ;  coh,  ;  , |coh M

(m) coh

(m)

( m) coh



(28)

CE

m 1

(m)



where s( m)  r( m) /  M  1 ,  coh,  ,   are the pseudo-observations and r( m )

AC

are the respective ranks associated to the m-th sample. Table 4. AIC and BIC values of different pair-copula decomposed models. Pairs of

Gaussian

Frank

Clayton

CClayton

variables

(AIC, BIC)

(AIC, BIC)

(AIC, BIC)

(AIC, BIC)

coh  

-6.81, -5.07

-7.13, -5.39

2.00, 3.74

2.00, 3.74

coh  

-0.98, 0.76

-2.75, -1.02

-3.77, -2.03

0.44, 2.17

 

1.77, 3.51

1.81, 3.54

1.71, 3.45

1.98, 3.72

Note: the underlined AIC and BIC values indicate the preferred pair-copula. The best fit - 20 -

ACCEPTED MANUSCRIPT

pair-copula in tree 2 (    data) is selected by adopting the preferred pair-copulas in tree 1 (i.e., Frank for coh   data and Clayton for coh   ).

Four bivariate copulas, namely the Gaussian, Frank, Clayton, and CClayton copula, are taken as the candidate pair-copulas. Table 4 summarizes the result of model selection. Note this identification of best fit pair-copulas is independent of the

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univariate data analysis. The CDF of the trivariate distribution can thus be calculated by:

1 F  coh,  ,    C  u1 , u2 , u3    C h  u2 , u;coh,  , h  u3 , u;coh,  ; , |coh du 0

u

(29)

where u1  Fcoh  coh  , u2  F   , u3  F   are the fitted marginal CDFs of the

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soil properties. Fig. 9 compares the empirical bivariate CDF with the fitted bivariate CDF, which are generally in good agreement. Note the fitted bivariate CDF can be conveniently obtained from the trivariate distribution (e.g., F  coh,    C  u1 , u2 ,1 ). (a)

M  (kN/m3)

 (°)

ED

PT 24

6 0.

19

0.3

16

0.2

34

14 12

39

4

9

14

CE

19

24

(c) 0.3

0.8

0.3

0.6

0.1

0.4

22

0.2

20

0.6 0.5 0.4

0.3 0. 2

0.1

18

0.7

0.5 0.4

 (kN/m3)

9 0.

0.7

0.2

0.5

28

24

0.8

0.6

30

0.3 0.2

0.1

16

0.1

14 12

empirical best fit 15

20

 (°)

25

30

Fig. 9 Comparison of the empirical and the best fit bivariate CDFs. - 21 -

0.1

0.1

empirical best fit

c (kPa)

26

0.3

0.2

0.1

29

0.4

0.3

0.2

c (kPa)

AC

7 0.

14

0.8

9

0.6

4

0.6 0.5

0.4

18

0.1

empirical best fit

0.8

0.7

0.5

0.4

12

20

0.3

0.1

14

0.5

0.2

16

0.4

22 0.1

0.1

0. 4

0.6

0.3

0.2

18

0.7

0.5

0.3

0.3

0.1

20

0.6

0.9

24 0.2

0.4

0.2

22

26

0.8

0.7

0.4

0.5

24

0.1

0.8

0.3

0.1

26

28

0.9

(b)

0.2

9 0.

30

0.6

0.4

28

0.7

0.2

0.5

30

29

34

39

ACCEPTED MANUSCRIPT

The data dependencies show no Gaussian characteristics. Table 5 presents the failure probabilities by different methods. The result from the best fit trivariate distribution can be taken as the actual failure probability. If marginals and correlations are given, the transformation method based on Gaussian dependence assumption apparently gives a biased estimate. In contrast, by adopting the qualitative best fit

Table 5. Failure probabilities approximated by different methods. Available information

Best fit from data

Pf Relative error (%)

0.1473 /

Marginal distributions and correlations 0.1598a 8.49

0.1487b 0.95

Assuming the Gaussian dependence structure.

b

Assuming the best fit non-Gaussian dependence structure.

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a

10

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copula functions, the failure probability from the transformation method is corrected.

0

best fit non-Gaussian Gaussian (Nataf)

-1

ED

p(Esys )

M

10

-2

=2.5 pf =0.0062

PT

10

CE

10

-3

1

2

3

4

 =b/a

5

6

7

8

Fig. 10 System failure probability as a function of λ.

The dependence assumption can further impact the reliability-based system

AC

design. Suppose the normalized width  =b / a of the retaining wall is a design parameter ( a  0.5m ). Fig. 10 shows the system failure probability with respect to  . If the targeted system reliability is   2.5 ( p f       6.2e-3 ), the  value that optimized towards this targeted reliability should be approximately 6.98 based on the best fit distribution. The  value from the transformation method with the best fit non-Gaussian dependence structure is 7.12 (the actual failure probability is 6.0e-3), which is slightly biased towards the safety side. However, if the Nataf transformation - 22 -

ACCEPTED MANUSCRIPT is used, the optimized  value is only 6.29 (the actual failure probability is 8.3e-3) and the targeted reliability is actually not reached.

7 Concluding remarks System reliability has gained extensive attentions in the past few decades. A

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major challenge in system reliability analysis is how to deal with correlated random variables. The orthogonal transformation and the Nataf transformation are commonly used to deal with correlations in the reliability analysis; however, these transformations actually assume a Gaussian dependence structure for random variables. This study investigates the effect of the dependence assumption, an

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unawarely ignored factor in the transformation method, on the system reliability, and tries to shed some light on reliability analysis considering correlations. To investigate the reliability under the assumption of non-Gaussian dependence structure, a general method for probability distribution reconstruction from prescribed marginals and

M

correlations is needed. The pair-copula decomposition approach with a sequential search strategy for retrieving the model parameters is proposed to construct various

ED

joint multivariate PDFs that can share the same marginals and correlations. The Rosenblatt transformation in terms of pair-copulas is presented for transforming

PT

non-Gaussian dependent random variables. The effect of correlations under different dependence structures are first shown

CE

in two numerical examples. Given the same marginal distributions and correlations, different multivariate distributions are demonstrated and the deviation in failure

AC

probabilities is non-trivial except that the random variables tend to be uncorrelated or fully correlated. The result from the Gaussian dependence structure can be either smaller or larger than the results from other dependence structures. This result runs counter to the intuition that the reliability is unique under specific marginal distributions and correlations. Note the result also has implications for time-dependent reliability, in which the time-variance is commonly realized by a set of correlated random variables. - 23 -

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Nevertheless, the actual system failure probability remains unique. It is just the incomplete probabilistic specification in terms of marginals and correlations that causes the deviation. The identification of the appropriate probability distribution model is thus critical to an unbiased estimate of system reliability. The retaining wall example has shown that the best fit pair-copulas can be determined independently, and the probabilistic result can be sufficiently accurate as long as the qualitative

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pair-copulas are complemented to the quantitative statistical information regarding marginals and correlations. The widely used Nataf transformation, however, failed to provide an accurate estimate and the subsequent reliability-based design drifted towards the unconservative side because it adopts a Gaussian dependence structure

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without proper validation.

Acknowledgments

This work is supported by the National Natural Science Foundation of China

M

(NSFC) under Grant No. 51608399 and the Research Grants Council of Hong Kong

ED

under Grant No. PolyU 152093/14E.

PT

Appendix A. Monotonicity of Gaussian pair-copula parameters at higher levels

CE

Fig. A1 illustrates the monotonicity of Gaussian pair-copula parameters with a trivariate example. As the correlation coefficients corresponding to the first level tree

AC

(i.e., 1,2 and 1,3 ) increases, the relation between  2,3 and  2,3|1 changes such that the interval of viable values of  2,3 becomes narrow. This variation is explained below. Suppose 1,2 =1,3 =0.5 and 2,3  0.5 , the corresponding unconditional Gaussian

pair-copula

2,3  0.5237

.

parameters

Thus

the

are

respectively

conditional

- 24 -

1,2 =1,3 =0.5076 ,

pair-copula

parameter

and is

ACCEPTED MANUSCRIPT

 2,3|1 

 2,3  1,21,3 2 2 1  1,2 1  1,3

 1.0525 , which obviously cannot be reached since the range

of Gaussian pair-copula parameters are bounded in  1, 1 . The reason is rooted in 1,2 1,3   in the equivalent trivariate Gaussian copula 1  2,3   2,3 1 

 1

that the correlation matrix 

 1,2 1,3

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is not positive definite. 1 0.8 0.6 0.4

1,2=y if 1,2=x

0

or

1,3=y if 1,3=x

-0.2 -0.4

2,3|1=y if 2,3=x

-0.6

2,3|1=y if 2,3=x and 1,2= 1,3=0.2

-0.8

-0.8

-0.6

-0.4

2,3|1=y if 2,3=x and 1,2=1,3=0.5

-0.2

0 x

0.2

and

1,2=1,3=0.8

0.4

0.6

0.8

1

M

-1 -1

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y

0.2

Fig. A1 C-vine structured Gaussian pair-copula parameters as a function of the correlation

ED

coefficient in a trivariate distribution. Lognormal marginals with mean of 40 and standard deviation of 10 are assumed for all three random variables. Black line indicates the variation of

PT

pair-copula parameters in tree 1. Blue dashed lines indicate the variation of pair-copula parameter in tree 2 if the pair-copula parameters in tree 1 are determined by the corresponding correlation

AC

CE

coefficients ρ1,2=ρ1,3=0.2, 0.5, or 0.8.

Appendix B. Best fit marginal distributions for the 42 data samples The 42 data samples of soil mass properties are listed in Table B1. The values of soil properties have a wide range because the samples are measured at different depths. Some samples at deep depth are weathered rock.

- 25 -

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Table B1. Statistics of the random variables used in the retaining wall example.

coh (kPa)

 (°)

 (kN/m3)

No.

coh (kPa)

 (°)

 (kN/m3)

1

11.3

24.37

16.9

22

14.4

19.33

27.0

2

9.8

24.98

23.3

23

16.2

20.51

20.0

3

9.0

23.34

17.3

24

5.1

22.65

17.9

4

20.3

30.48

24.5

25

15.0

19.33

19.8

5

9.0

27.98

17.3

26

19.2

18.07

27.5

6

12.8

27.98

18.9

27

11.1

17.50

16.4

7

6.8

28.59

25.9

28

11.9

23.34

19.2

8

25.5

21.62

16.5

29

10.5

19.33

18.5

9

26.6

17.11

17.2

30

1.9

19.16

20.0

10

27.0

18.98

23.0

31

11.4

16.91

14.7

11

14.4

27.61

25.2

32

10.8

23.87

21.5

12

2.0

26.22

13.0

13

5.7

18.62

14.3

14

18.5

22.50

22.7

15

15.6

20.68

20.7

16

22.2

15.06

22.5

17

21.6

20.51

25.3

18

11.6

24.12

19

19.8

15.27

20

31.5

21

29.4

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CR IP T

No.

22.5

15.48

21.3

34

6.3

22.79

12.0

35

12.8

16.51

24.2

36

9.0

13.74

15.9

37

1.5

20.18

30.7

38

26.3

10.41

21.2

30.2

39

24.0

19.51

18.5

26.0

40

24.8

10.41

21.5

ED

M

33

13.51

29.8

41

9.8

15.48

18.4

16.11

17.6

42

39.0

16.11

24.7

PT

The best fit marginal distribution is selected based on the Kolmogorov–Smirnov

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statistic. Table B2 summarizes the test results and Fig. B1 shows the comparison of empirical CDFs of the soil properties with their respective best fitted CDFs.

AC

Table B2. Results of the Kolmogorov–Smirnov test for the marginal distributions (α=0.05). The best fit distribution is underlined. Soil property

coh

Distribution

Parameters

p-value

Test statistic

Normal

15.55, 8.58

0.46

0.1279

Lognormal

2.54, 0.73

0.26

0.1524

Gamma

2.60, 5.98

0.65

0.1103

GEV*

-0.09, 7.42, 11.85

0.88

0.0874

Weibull

17.47, 1.86

0.89

0.0865

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

Normal

20.15, 4.78

0.95

0.0772

Lognormal

2.97, 0.25

0.97

0.0708

Gamma

16.98, 1.19

0.99

0.0640

-0.27, 4.68, 18.43

0.98

0.0698

Weibull

22.03, 4.65

0.82

0.0937

Normal

20.93, 4.57

0.91

0.0833

Lognormal

3.02, 0.22

0.99

0.0642

Gamma

20.77, 1.01

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GEV



GEV

-0.20, 4.28, 19.16

Weibull

22.78, 4.95

* GEV: generalized extreme value (a) 0.9

0.0684

0.84

0.0915

empirical best fit

0.8

0.7

0.7 0.6

CDF

0.6

CDF

0.98

AN US

1

empirical best fit

0.8

0.5 0.4

0.5 0.4 0.3

0.2 0.1 0

5

10

15

20

25

30

M

0.3

0

0.0630

(b)

1 0.9

0.99

35

40

0.2 0.1

0 10

15

25

30

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c (kPa)

20

 (°)

25

(c)

1

PT

0.9

empirical best fit

0.8 0.7

AC

CDF

CE

0.6 0.5 0.4 0.3 0.2 0.1 0

15

20

 (kN/m3)

Fig. B1 Comparison of the empirical and the best fit CDFs for each soil property.

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30

ACCEPTED MANUSCRIPT

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