Structural Safety. Impact of copulas for modeling bivariate distributions on system reliability

Transcription

1 Structural Safety 44 (2013) Contents lists available at SciVerse ScienceDirect Structural Safety j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / s t r u s a f e Impact of copulas for modeling bivariate distributions on system reliability Xiao-Song Tang a, Dian-Qing Li a, *, Chuang-Bing Zhou a, Kok-Kwang Phoon b, Li-Min Zhang c a State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 8 Donghu South Road, Wuhan , PR China b Department of Civil and Environmental Engineering, National University of Singapore, Blk E1A, #07 03, 1 Engineering Drive 2, Singapore , Singapore c Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong a r t i c l e i n f o a b s t r a c t Article history: Received 12 April 2013 Received in revised form 24 June 2013 Accepted 25 June 2013 Keywords: Joint probability distribution Correlation coefficient Copulas Parallel system System reliability Reliability analysis A copula-based method is presented to investigate the impact of copulas for modeling bivariate distributions on system reliability under incomplete probability information. First, the copula theory for modeling bivariate distributions as well as the tail dependence of copulas are briefly introduced. Then, a general parallel system reliability problem is formulated. Thereafter, the system reliability bounds of the parallel systems are generalized in the copula framework. Finally, an illustrative example is presented to demonstrate the proposed method. The results indicate that the system probability of failure of a parallel system under incomplete probability information cannot be determined uniquely. The system probabilities of failure produced by different copulas differ considerably. Such a relative difference in the system probabilities of failure associated with different copulas increases greatly with decreasing component probability of failure. The maximum ratio of the system probabilities of failure for the other copulas to those for the Gaussian copula can happen at an intermediate correlation. The tail dependence of copulas has a significant influence on parallel system reliability. The copula approach provides new insight into the system reliability bounds in a general way. The Gaussian copula, commonly used to describe the dependence structure among variables in practice, produces only one of the many possible solutions of the system reliability and the calculated probability of failure may be severely biased. c 2013 Elsevier Ltd. All rights reserved. 1. Introduction In reliability analyses, quantities such as loads, material properties and structural dimensions are typically treated as random variables or random fields (e.g., [ 1 3 ]). Furthermore, these quantities are usually correlated with each other. For example, flood peak discharge and volume are interrelated in a flood frequency analysis [ 4 ]; peak and permanent displacements of a system subjected to earthquake loading [ 5 ], curve-fitting parameters underlying a soil water characteristic curve [ 6 ], and curve-fitting parameters underlying load displacement curves of piles [ 7 ] are correlated with each other. It is well known that the joint cumulative distribution function (CDF) or probability density function (PDF) of these parameters should be available to evaluate the reliability accurately. In engineering practice, unfortunately, the available data are only sufficient for determining appropriate marginal distributions as well as covariances, which poses a problem of incomplete probability information [ 8 11 ]. Under this condition, the modeling and simulation of correlated non-normal variables are challenging problems [ 12, 13 ], and a rigorous evaluation of reliability is impossible [ 14, 15 ]. * Corresponding author. Tel.: ; fax: address: dianqing@whu.edu.cn (D.-Q. Li). Conventionally, the Nataf model [ 8, 14, 16, 17 ] is used to construct the joint CDF or PDF based on incomplete probability information. As pointed out by Lebrun and Dutfoy [ 18 ] and Li et al. [ 19 ], the Nataf model essentially adopts a Gaussian copula for modeling the dependence structure among the variables. That is to say, there is an implicit assumption that the Gaussian copula is adequate for characterizing the dependence structure. Unfortunately, this commonly used assumption is not validated in a rigorous way. In this situation, it is natural to question if the less than best-fit Gaussian copula will lead to unacceptable errors in probability of failure when the Gaussian copula cannot rigorously characterize the dependence structure but is still used to construct the joint probability distribution of correlated non-normal variables associated with the reliability analyses. To evaluate the accuracy of Gaussian copula for modeling the dependence structure between two random variables, Li et al. [ 20 ] investigated the performance of two commonly used translation approaches, in which the dependence structure between two random variables is modeled by the Gaussian copula based on their abilities to match the high order joint moments, joint PDFs, and component probabilities of failure. Li et al. [ 21 ] further explored the impact of the two translation approaches on parallel system reliability. Essentially, Li et al. [ 20, 21 ] only evaluated the performance of the Gaussian copula. As for the dependence structures among variables characterized / $ - see front matter c 2013 Elsevier Ltd. All rights reserved.

2 Xiao-Song Tang et al. / Structural Safety 44 (2013) by other copulas such as t, Clayton, Frank, and Plackett copulas [ 22 ], the differences in the probabilities of failure associated with different copulas have not been investigated. For this reason, Tang et al. [ 10, 11 ] studied the effect of copulas for modeling the bivariate distributions on component reliability. However, structural systems usually consist of more than one structural component or failure mode (e.g., [ 1, 23 ]). It is evident that the reliability of an individual component cannot represent the reliability of the entire structural system. It is of practical interest to distinguish between the reliability of each component and the reliability of the entire structural system. In addition, the effect of tail dependence of copulas on system reliability has not been investigated. With these in mind, the effect of copulas for modeling the joint distributions on the system reliability should be explored, and is the topic of the present research. This paper aims to investigate the impact of copulas for constructing bivariate distributions on parallel system reliability under incomplete probability information. To achieve this goal, this article is organized as follows. In Section 2, the copula theory for modeling the joint probability distributions of multiple correlated variables is introduced briefly. Six copulas, namely Gaussian, t, Frank, Plackett, Clayton and CClayton copulas are selected to model the dependence structure between two variables. In Section 3, a general parallel system reliability problem is formulated, and the corresponding bounds of system probability of failure are derived from the copula viewpoint. The ratios of the system probabilities of failure for the other copulas to those for the Gaussian copula and the effect of tail dependence of copulas on system reliability are presented in Section A copula-based method for modeling the bivariate distribution of two variables According to Sklar s theorem (e.g., [ 22 ]), the bivariate joint CDF of two random variables X 1 and X 2 is given by F ( x 1, x 2 ) = C ( F 1 ( x 1 ), F 2 ( x 2 ) ; θ) = C ( u 1, u 2 ; θ) (1) where F ( x 1, x 2 ) is the joint CDF of X 1 and X 2 ; u 1 = F 1 ( x 1 ) and u 2 = F 2 ( x 2 ) are the marginal distributions of X 1 and X 2, respectively; C ( u 1, u 2 ; θ) is the copula function in which θ is a copula parameter describing the dependency between X 1 and X 2. In mathematical terms, a bivariate copula function C ( u 1, u 2 ; θ) is a two-dimensional probability distribution on [0, 1] 2 with uniform marginal distributions on [0, 1]. From Eq. (1), the bivariate PDF of X 1 and X 2, f ( x 1, x 2 ), can be obtained as (e.g., [ 22 ]) f ( x 1, x 2 ) = f 1 ( x 1 ) f 2 ( x 2 ) c ( F 1 ( x 1 ), F 2 ( x 2 ) ; θ) (2) where f 1 ( x 1 ) and f 2 ( x 2 ) are the marginal PDFs of X 1 and X 2, respectively; c ( F ( x 1 ), F 2 ( x 2 ); θ) is the copula density function, which is given by c ( F 1 ( x 1 ), F 2 ( x 2 ) ; θ) = c ( u 1, u 2 ; θ) = 2 C ( u 1, u 2 ; θ) / u 1 u 2 (3) Theoretically, the joint CDF and PDF of X 1 and X 2 can be determined by Eqs. (1) and (2) if the marginal distributions of X 1 and X 2, and the copula function are known. Many copulas can be used to describe the dependence between two random variables. Generally, the dependence structures underlying different copulas differ significantly. To clearly capture the differences in various copulas, the Gaussian copula, t copula, Plackett copula, Frank copula, Clayton copula and CClayton copula [ 22 ] are examined in this study. The aforementioned six copulas, along with the ranges of the θ parameter are listed in Table 1. The six copulas are selected due to the following three reasons. First, they are commonly used copulas associated with several typical copula families. Among them, the Gaussian and t copulas are elliptical copulas. The Plackett copula is a member of the Plackett copula family. The Frank, Clayton and CClayton copulas are commonly used Archimedean copulas. In addition, the Gaussian, t, Plackett, and Frank copulas are symmetric copulas. Second, the aforementioned six copulas can describe positive dependences, and the values of the correlation coefficients can approach 1. Finally, the t, Clayton and CClayton copulas have tail dependence (e.g., [ 22 ]), which can account for the effect of tail dependence on reliability. These three features are suitable for investigating the effect of copulas on system reliability. It should be noted that most copula applications are concerned with bivariate data. One possible reason for this is that relatively few copula families have practical N -dimensional generalization (e.g., [ 22 ]). Among the selected six copulas, only the Gaussian and t copulas belonging to the family of elliptical copulas can be readily generalized to multivariate case. For this reason, the Gaussian and t copulas are widely used to characterize the dependence structure among multivariables in practical application. Unlike the Gaussian and t copulas, the Plackett, Frank, Clayton and CClayton copulas have only a single parameter. Hence, they cannot describe general dependence among more than two random variables. The Archimedean copulas such as Frank, Clayton and CClayton copulas have two generalizations, but both of them are afflicted by some shortcomings. The first generalization, termed symmetric [ 22 ], uses the same generator as the bivariate case. Thus, all variables are described by the same dependence structure, which is extremely simple for most practical applications. The second generalization, termed asymmetric [ 24 ], uses ( N 1) different generators. Although ( N 1) different dependence structures are allowed, existence conditions must be satisfied for the generalized multivariate copula to be valid. These conditions impose restrictions on the copula parameters, and only a limited range of correlations can be taken into consideration. Hence, the commonly used approach is to analyze the multivariate data pair by pair using bivariate copulas when confronted with a multivariate data. To facilitate the understanding of the subsequent analyses, the concept of tail dependence is introduced briefly. The tail dependence relates to the amount of dependence at the upper-quadrant tail or lower-quadrant tail of a bivariate distribution. The coefficients of upper and lower tail dependence, λ U and λ L, are defined as [ ] λ U = lim P X 2 > F 1 X q 1 2 ( q ) 1 > F 1 1 ( q ) (4) [ ] λ L = lim P X 2 < F 1 X q ( q ) 1 < F 1 1 ( q ) given that these limits λ U [0,1] and λ L [0,1] exist. In Eqs. (4) and (5), F 1 1 ( ) and F 1 2 ( ) are the inverse CDFs of X 1 and X 2, respectively. If λ U (0,1] or λ L (0,1], X 1 and X 2 are said to be asymptotically dependent at the upper tail or lower tail; if λ U = 0 or λ L = 0, X 1 and X 2 are said to be asymptotically independent at the upper tail or lower tail. It can be seen from the definition of tail dependence that the coefficient of upper tail dependence is the probability that the random variable X 2 exceeds its quantile of order q, knowing that X 1 exceeds its quantile of the same order when order q approaches 1. The coefficient of lower tail dependence is the probability that X 2 is smaller than its quantile of order q, knowing that X 1 is smaller than its quantile of the same order when order q approaches 0. As mentioned, the Gaussian, Plackett and Frank copulas do not have tail dependence. Unlike the aforementioned three copulas, the t copula, Clayton copula and CClayton copula have tail dependence. The t copula with v degrees of freedom has both lower and upper tail dependences, which are given by [ ( )] ( v + 1 ) ( 1 θ) λ U = λ L = 2 1 t v+ 1 (6) ( 1 + θ) in which t v + 1 is the CDF of the one-dimensional Student distribution with v + 1 degrees of freedom. (5)

3 82 Xiao-Song Tang et al. / Structural Safety 44 (2013) Table 1 Summary of the adopted bivariate copula functions in this study. Copula Copula function, C ( u 1, u 2 ; θ) Limiting cases of C ( u 1, u 2 ; θ) Range of θ Gaussian θ ( 1 ( u 1 ), 1 ( u 2 )) C 1 = W, C 0 =, C 1 = M [ 1, 1] t t θ,v ( t 1 v ( u 1 ), t 1 v ( u 2 )) C 1 = W, C 0, C 1 = M [ 1, 1] S S Plackett 4 u 1 u 2 θ( θ 1), S = 1 + ( θ 1)( u 2( θ 1) 1 + u 2 ) C 0 = W, C 1 =, C = M (0, )\{ 1 } Frank 1 θ ln [1 + ( e θu 1 1)( e θu 2 1) ] C e θ 1 = W, C 0 =, C = M (, )\{ 0 } Clayton ( u θ 1 + u θ 2 1) 1 θ C 0 =, C = M (0, ) CClayton u 1 + u [(1 u 1 ) θ + (1 u 2 ) θ 1] 1 θ C 0 =, C = M (0, ) Note: The subscript on C denotes the value of copula parameter θ. Unlike the t copula, the Clayton copula only has lower tail dependence. The coefficient of lower tail dependence is given by λ L = 2 1 /θ (7) Since the CClayton copula is the survival copula of the Clayton copula, it only has upper tail dependence. The coefficient of upper tail dependence is given by λ U = 2 1 /θ (8) It can be observed from Eqs. (7) and (8) that the coefficients of tail dependence for the Clayton and CClayton copulas remain the same. Based on the above discussions, the copulas and their coefficients of tail dependence are closely related to the copula parameter θ. The value of θ can be determined through the linear correlation coefficient or rank correlation coefficients such as the Pearson linear correlation coefficient, Spearman and Kendall rank correlation coefficients (e.g., [ 25 ]). In this work, the Kendall correlation coefficient is adopted because it is easy to estimate in a robust way and invariant under strictly increasing transformations in comparison with the linear correlation coefficient. Note that the Spearman rank correlation coefficient can also be used for such a purpose. The Kendall rank correlation coefficient between two random variables X 1 and X 2, τ, is expressed as 1 1 τ = 4 C ( u 1, u 2 ; θ) dc ( u 1, u 2 ; θ) 1 (9) 0 0 For the Gaussian and t copulas, Eq. (9) can be further simplified as τ = 2 arcsin θ π (10) Similarly, for the Clayton and CClayton copulas, Eq. (9) can be simplified as τ = θ 2 + θ (11) Hence, with the known Kendall rank correlation coefficient τ, the copula parameter θ for the selected six copulas can be easily obtained through Eqs. (9)-(11). 3. Parallel system reliability associated with different copulas 3.1. Formulation of the system reliability problem for a parallel system It is well known that a system can be classified as a series system or a parallel system, or combinations thereof. Among them, the series and parallel systems are two basic systems. In comparison with the series system reliability problem, the parallel system reliability problem captures the difference in system probability of failure effectively. Hence, only the parallel system reliability is investigated. For simplicity, the system reliability of a parallel system consisting of two components is adopted. The failure of the considered parallel system with two components connected in parallel requires that all the two components fail. The system probability of failure, p fs, is expressed as p fs = P [ g 1 ( x ) < 0 g 2 ( x ) < 0 ] (12) where x represents the random vector; g 1 ( x ) and g 2 ( x ) are the performance functions for the two components, respectively. Two cases of g 1 ( x ) are considered as below: { g 1 ( x ) = S 1 x 1 g 1 ( x ) = x 1 S 3 (13) Similarly, two cases of g 2 ( x ) are considered as below: { g 2 ( x ) = S 2 x 2 g 2 ( x ) = x 2 S 4 (14) in which S 1 S 4 are four constants. The reliability levels can be varied when the four constants take different values. Based on Eqs. (12)-(14), there exist four combinations of g 1 ( x ) and g 2 ( x ). The resulting system probabilities of failure are expressed as p fs 1 = P [ S 1 x 1 < 0 S 2 x 2 < 0 ] p fs 2 = P [ x 1 S 3 < 0 x 2 S 4 < 0 ] p fs 3 = P [ x 1 S 3 < 0 S 2 x 2 < 0 ] p fs 4 = P [ S 1 x 1 < 0 x 2 S 4 < 0 ] (15a) (15b) (15c) (15d) Theoretically, the four parallel systems are able to scan the entire area of the joint PDF surface provided that the four constants are varied over a wide range. For brevity, the parallel systems represented by Eqs. (15a) (15d) are hereafter referred to as systems I, II, III, and IV, respectively. Fig. 1 illustrates the failure domains for the considered four parallel systems. Note that all the failure domains are in the semiinfinite space. Furthermore, the diagonals of the failure domains are parallel or perpendicular to the symmetric planes of the selected six copula functions although the copula functions are not plotted in Fig. 1. These features are very suitable for identifying the differences in the probabilities of failure produced by different copulas Formulae for calculating system probability of failure Since the failure domains shown in Fig. 1 are simple, the system probabilities of failure in Eq. (15) can be calculated readily. Let p fc 1 and p fc 2 be the probabilities of failure for components 1 and 2, respectively. They are defined as { p fc 1 = P ( g 1 ( x ) < 0 ) (16) p fc 2 = P ( g 2 ( x ) < 0 ) For simplicity, the probability of failure of component 1 is assumed the same as that of component 2, p fc 1 = p fc 2 (17) According to Eqs. (16) and (17), the constants S 1 S 4 can be derived as { F 1 ( S 1 ) = F 2 ( S 2 ) = 1 p fc F 1 ( S 3 ) = F 2 ( S 4 ) = p fc (18)

4 Xiao-Song Tang et al. / Structural Safety 44 (2013) Fig. 1. Illustration of the analyzed parallel systems and their failure domains. Substituting Eqs. (16) and (18) into Eq. (15) yields p fs 1 = 1 F 1 ( S 1 ) F 2 ( S 2 ) + F ( S 1, S ( 2 ) = 2 p fc 1 + C 1 p fc, 1 p fc ; θ ) (19a) p fs 2 = F ( S 3, S 4 ) = C ( p fc, p fc ; θ ) (19b) p fs 3 = F 1 ( S 3 ) F ( S 3, S 2 ) = p fc C ( p fc, 1 p fc ; θ ) (19c) p fs 4 = F 2 ( S 4 ) F ( S 1, S 4 ) = p fc C ( 1 p fc, p fc ; θ ) (19d) It can be seen from Eq. (19) that the system probabilities of failure for the four parallel systems only depend on the component probabilities of failure p fc and the copula parameters θ that are directly related to τ as shown in Eq. (9). A change in p fc or τ leads to a change in the system probability of failure. The system probability of failure does not relate to the marginal distributions directly. Therefore, there is no need to make any assumptions on the marginal distributions of the random variables. For simplicity, the random variables are assumed to be standard normally distributed System reliability bounds from the copula viewpoint In system reliability analyses, system reliability bounds are often used to validate the system probabilities of failure. The general bounds for probabilities of failure of a parallel system are available in the literature (e.g., [ 2, 26 ]). For the general reliability bounds, the dependence among variables is represented by the Pearson linear correlation coefficient instead of the rank correlation coefficients. In this study, the bounds of system probabilities of failure for the considered parallel systems are derived from the copula viewpoint. To the best

5 84 Xiao-Song Tang et al. / Structural Safety 44 (2013) of our knowledge, this approach extends the conventional system reliability bounds that are based on the Pearson linear correlation coefficient (e.g., [ 2, 26 ]). According to the Fr échet Hoeffding bounds inequality (e.g., [ 22 ]), for every copula C and every ( u 1, u 2 ) in I 2, one can obtain W ( u 1, u 2 ) C ( u 1, u 2 ) M ( u 1, u 2 ) (20) in which W ( u 1, u 2 ) and M ( u 1, u 2 ) are the Fr échet Hoeffding lower and upper bounds, respectively, and are given by W ( u 1, u 2 ) = max ( u 1 + u 2 1, 0 ) (21) M ( u 1, u 2 ) = min ( u 1, u 2 ) (22) It should be noted that W ( u 1, u 2 ) and M ( u 1, u 2 ) are also copulas. For brevity, they are referred to as W and M, respectively. Besides these copulas, a third important copula that is frequently encountered is the product copula ( u 1, u 2 ) = u 1 u 2. In general, when τ approaches zero, copula C converges to. When τ approaches 1 and 1, copula C converges to W and M, respectively. Table 1 also summarizes the limiting cases for the selected six copulas when τ takes the above values. Note that all the copulas except the t copula converge to when τ approaches zero. When the number of degrees of freedom diverges, the t copula converges to, which is also equivalent to the Gaussian copula. If the t copula is excluded from the aforementioned six copulas, for τ [0, 1], the Fr échet Hoeffding bounds associated with the Gaussian, Plackett, Frank, Clayton, and CClayton copulas can be expressed as ( u 1, u 2 ) C ( u 1, u 2 ) M ( u 1, u 2 ) (23) By substituting Eq. (23) into Eq. (19), the system reliability bounds for the considered four parallel systems are derived as p 2 fc p fs 1 p fc p 2 fc p fs 2 p fc ( ) max 2 p fc 1, 0 p fs 3 p 2 fc ( ) max 2 p fc 1, 0 p fs 4 p 2 fc (24a) (24b) (24c) (24d) It can be seen from Eq. (24) that the reliability bounds for system I are the same as those for system II. The reliability bounds for systems III and IV remain the same. The system probabilities of failure for systems I and II are significantly higher than those for systems III and IV. When the t copula is used for systems I and II, the upper bounds of the system probability of failure are the same as those in Eq. (24), and the lower bounds of the system probability of failure are above the lower bounds in Eq. (24). When the t copula is used for systems III and IV, the lower bounds are the same as those in Eq. (24), and the upper bounds also exceed the upper bounds in Eq. (24). In addition, the system reliability bounds increase with decreasing component probability of failure. 4. Reliability analysis results for the parallel systems The system probabilities of failure for different copulas can be obtained using Eq. (19). The lower and upper bounds of the system probability of failure are calculated by Eq. (24). For illustrative purposes, only positive Kendall rank correlation coefficients [0, 1] are analyzed in this study because the Clayton and CClayton copulas can only account for the positive correlation between two correlated random variables. Since the two component probabilities of failure are assumed the same and the selected six copulas are symmetrical about the 45 diagonal line, the system probabilities of failure for systems III and IV remain the same. Therefore, only the system probabilities of failure for systems I, II and III are studied step by step in the following. The reliability results associated with the Gaussian copula are highlighted because the Gaussian copula is commonly used to describe the dependence structure among variables in practice when the available data are not enough to select one copula among a set of candidate copulas [ 6, 15 ]. In addition, since the Gaussian copula is a limiting case of the t copula when the degrees of freedom v become infinity, a t copula with v = 2 is used in this study in order to distinguish the difference in dependence modeling between the Gaussian and t copulas System probabilities of failure for parallel system I Fig. 2 compares the system probabilities of failure of parallel system I associated with the selected six copulas for various component probabilities of failure. For comparison, the bounds of system probability of failure are also presented. The system probabilities of failure produced by different copulas differ considerably. Such a relative difference in the system probabilities of failure associated with different copulas increases with decreasing component probability of failure. The results imply that the system probability of failure under incomplete probability information cannot be determined uniquely. When τ is small (see Fig. 2 (a)), the system probabilities of failure associated with the six copulas approach the lower bound. When τ is very large (see Fig. 2 (c)), they appear to converge to the upper bound. These observations further demonstrate the validity of the results. Fig. 3 shows the system probabilities of failure for various Kendall rank correlation coefficients. In Fig. 3, a value of p fc = 1.0E 03 is used for illustration. The system probabilities of failure increase with increasing Kendall rank correlation coefficient. When τ approaches 1, the system probabilities of failure associated with the six copulas converge to the upper bound value of 1.0E 03. When τ approaches 0, the system probabilities of failure associated with the six copulas except the t copula converge to the lower bound value of 1.0E 06, which has been explained in Section 3. The Clayton copula leads to the smallest system probability of failure, whereas the t and CClayton copulas produce higher system probabilities of failure. These results indicate that the tail dependence underlying the t, Clayton, and CClayton copulas has a significant influence on the reliability results, which will be explained later. For engineers, the relative errors in system probability of failure produced by the commonly used Gaussian copula for a prescribed target system reliability level may be of greatest interest. For this reason, the ratios of p fs for the other five copulas to p fs for the Gaussian copula are plotted against various target system probabilities of failure in Fig. 4. The ratios increase with decreasing target system probability of failure. The ratios can be significant especially for a large τ. For τ = 0, the ratios of p fs for the t copula to p fs for the Gaussian copula are 34, 331, 3302, and 33,017 for p E = 1.0E 03, 1.0E 04, 1.0E 05, and 1.0E 06, fs respectively. The maximum ratio is about 30,000, which implies that the Gaussian copula will produce unacceptable system probabilities of failure if the t copula is assumed to be the correct one among the six copulas, and vice versa. For the t copula, the ratios monotonically decrease as τ increases. As expected, when τ approaches 1, the t copula and the Gaussian copula produce the same results because both system probabilities of failure approach the upper bound in Eq. (24a). For the other four copulas, the maximum ratio happens at an intermediate τ System probabilities of failure for parallel system II For system II, the plots are identical to those for system I with the curves for the Clayton and CClayton copulas switched. The reason is that both the Clayton and CClayton copulas are symmetrical with respect to the 45 diagonal line of a unit square (i.e., a domain defined by [0, 1] 2 ), and the CClayton copula is the survival copula of the Clayton copula. Hence, the probabilities of failure produced by the Clayton copula for system II are identical to those produced by the CClayton

6 Xiao-Song Tang et al. / Structural Safety 44 (2013) Fig. 3. Effect of correlation coefficients on probabilities of failure produced by different copulas for parallel system I ( p fc = 0.001). Fig. 2. Probabilities of failure produced by different copulas for parallel system I. copula for system I, and vice versa. Additionally, the results produced by the t, Gaussian, Plackett, and Frank copulas remain unchanged because these copulas are all symmetrical with respect to the 45 and 135 diagonal lines of a unit square. For brevity, the results for system II are not presented again System probabilities of failure for parallel system III Fig. 5 compares the system probabilities of failure of parallel system III associated with the six copulas for various component probabilities of failure. The system reliability bounds are calculated by Eq. (24c). In comparison with systems I and II, the system probabilities of failure for system III decrease significantly even though the same component probability of failure and Kendall rank correlation coefficient are used for systems I, II, and III. The Clayton and CClayton copulas produce the same system probabilities of failure. This is because the CClayton copula is the survival copula of the Clayton copula. The probabilities of failure associated with the t copula (see Fig. 5 (a)) exceed the upper bound of the system probability of failure significantly. The reason is that when τ approaches zero, the t copula does not converge to the independent copula. The resulting system probability of failure is not constrained by the upper bound in Eq. (24c). Fig. 6 shows the system probabilities of failure for various Kendall rank correlation coefficients. Unlike systems I and II, a value of p fc = 0.1 is used herein due to computational accuracy. The reason is that the commonly used computers can only compute a system probability of failure above 1.0E 17 with a sufficient accuracy. A value of p fc below 0.1 would result in system probabilities of failure significantly below 1.0E 17, which is beyond the precision of the commonly used computers. In comparison with systems I and II, the system probabilities of failure decrease with increasing τ, which decrease dramatically when τ exceeds 0.6. The system probabilities of failure associated with the six copulas except the t copula approach the upper bound value of 0.01 when τ approaches zero. For the six copulas, the system probabilities of failure approach the lower bound, 0, when τ approaches 1. Similarly, Fig. 7 shows the ratios of p fs for the other five copulas to p fs for the Gaussian copula with various target system probabilities of failure. The target system probability of failure associated with the t and Plackett copulas is only taken as the level of 1.0E 03 (see Fig. 7 (a) and (b)) because the ratios significantly exceed 1.0E05 for p E fs below 1.0E 03. The ratios associated with the t and Plackett copulas are very sensitive to the target system reliability level, which implies that there is a large difference in dependence modeling among the t, Plackett and Gaussian copulas for system III. For the t copula, when p E fs increases only from 1E 03 to 4E 03, the maximum ratio changes from 96,976 to 54 (see Fig. 7 (a)). For the analyzed five copulas, the maximum ratio occurs at an intermediate Kendall rank correlation coefficient. In general, the ratios associated with different copulas could be very large if the copulas for modeling the dependence structure among variables are not selected properly.

7 86 Xiao-Song Tang et al. / Structural Safety 44 (2013) Fig. 4. Ratios of the system probabilities of failure of t ( v = 2), Plackett, Frank, Clayton and CClayton copulas to those of Gaussian copula for parallel system I.

8 Xiao-Song Tang et al. / Structural Safety 44 (2013) Fig. 6. Effect of correlation coefficients on probabilities of failure produced by different copulas for parallel system III ( p fc = 0.1) Effect of tail dependence of copulas on system reliability In this section, we further investigate the effect of tail dependence of copulas on system reliability. Fig. 8 shows the probabilities of failure associated with parallel systems I, II and III for various coefficients of tail dependence. The tail dependence of copulas has a significant influence on the parallel system reliability, especially for system III. For instance, when the coefficient of tail dependence, λ, varies from zero to 1, the probabilities of failure for systems I and II increase from 1E 06 to 1E 03. The latter is 1000 times of the former. For systems I and II, the probabilities of failure produced by different copulas increase with increasing coefficient of tail dependence. However, the probabilities of failure for system III decrease as the coefficient of tail dependence increases. As expected, the probabilities of failure produced by the Clayton copula in Fig. 8 (a) are equal to those produced by the CClayton copula in Fig. 8 (b), and vice versa. This is because the CClayton copula is the survival copula of the Clayton copula as mentioned previously. 5. Discussion Fig. 5. Probabilities of failure produced by different copulas for parallel system III. Based on the aforementioned study, it can be observed that the system probability of failure of a parallel system cannot be determined uniquely under incomplete probability information. The system probabilities of failure produced by different copula models differ considerably. Hence, evaluation of the system reliability under incomplete probability information is still a challenging problem. This section will explore some solutions to deal with practical problems based on the above observations. To estimate the system reliability under incomplete probability information, three approaches are discussed in this section. The first approach is to select a copula corresponding to the highest estimate of probability of failure. In this approach, one may select a copula from a set of candidate copulas, for example the set { Gaussian, t, Plackett, Frank, Clayton and CClayton copulas } studied in this work, which results in the highest estimate of probability of failure. The rationale behind this is that a conservative estimate of reliability is generally accepted by engineers when limited information is available [ 27 ]. Applying this approach, for parallel system I in the example studied in Section 4, the t copula or the CClayton copula should be selected because it results in the highest probability of failure. The second approach can be derived from the tail dependence underlying copulas. For instance, if the available data associated with the practical problems do not exhibit tail dependence, then the set of candidate copulas { Gaussian, t, Plackett, Frank, Clayton and CClayton

9 88 Xiao-Song Tang et al. / Structural Safety 44 (2013) Fig. 7. Ratios of the system probabilities of failure of t ( v = 2), Plackett, Frank, Clayton and CClayton copulas to those of Gaussian copula for parallel system III.

10 Xiao-Song Tang et al. / Structural Safety 44 (2013) The third approach is to collect more data for practical problems. If the available data from field or laboratory tests associated with a specific project are sufficient to identify the best-fit copula among the set of candidate copulas, then the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) can be used for identifying the best-fit copula, as illustrated by Li et al. [ 19 ]. The system reliability produced by the best-fit copula can be taken as the reliability of the considered system. It should be noted that, in most cases, the available data associated with a specific project are very limited. Therefore, the data from similar projects should be explored. The rationale behind this approach is that the determination of a copula only relies on the ranks underlying the observed data rather than the real values of the data. With the relatively enough data at hand, a cautious identification of the best-fit copula to the dependence structure underlying the empirical data from the set of candidate copulas can be conducted using the AIC or BIC. 6. Summary and conclusions Copulas have been applied to construct bivariate distributions with given marginal distributions and a correlation coefficient for two correlated variables. A general parallel system reliability problem is formulated. The effect of the bivariate distribution models using copulas on the parallel system reliability is investigated. Several conclusions can be drawn from this study: Fig. 8. Effect of tail dependence of copulas on system probabilities of failure for parallel systems I, II, and III, respectively. (1) The system probability of failure of a parallel system under incomplete probability information is not unique. The copulas characterizing the dependence structures among the random variables can significantly influence the probability of failure of the system. The relative difference in system probabilities of failure associated with different copulas increases tremendously with decreasing component probability of failure. (2) The ratios of the system probabilities of failure for the other copulas to those for the Gaussian copula increase with increasing target system reliability level. The maximum ratio may not be associated with a large correlation. It can happen at an intermediate correlation level. (3) The tail dependence of copulas has a significant influence on the parallel system reliability. This finding highlights the importance of tail dependence of copulas that should be paid more attention in system reliability analyses. (4) The bounds of system probabilities of failure for the considered parallel systems are derived from the copula viewpoint. To the best of our knowledge, this approach extends the conventional system reliability bounds that are based on the Pearson linear correlation coefficient, and provides new insight into the system reliability bounds. (5) It is emphasized that the Gaussian copula, commonly used to model the dependence structure among variables in practice, produces only one of the many possible solutions of the parallel system reliability problem and the calculated probability of failure may be severely biased, which has never been explained so far. This finding should be noted in practical system reliability analyses. copulas } can be reduced to { Gaussian, Plackett and Frank copulas }. Again, if the available data have the upper tail dependence or the lower tail dependence, the set of candidate copulas can be reduced to { t and CClayton copulas } or { t and Clayton copulas }. With the reduced set of candidate copulas, the bound of system probability of failure can be improved. Acknowledgments This work was supported by the National Science Fund for Distinguished Young Scholars (Project No ), the National Natural Science Foundation of China (Project No ) and the Doctoral Program Fund of Ministry of Education of China (Project No ).

11 90 Xiao-Song Tang et al. / Structural Safety 44 (2013) References [1] Ang AH-S, Tang WH. Probability concepts in engineering planning and design, vol. II: decision, risk, and reliability. New York: John Wiley and Sons; [2] Melchers RE. Structural reliability analysis and prediction. 2nd ed. Chichester: John Wiley and Sons; [3] Wang Y, Rosowsky DV. Joint distribution model for prediction of hurricane wind speed and size. Struct Saf 2012;35: [4] De Michele C, Salvadori G, Canossi M, Petaccia A, Rosso R. Bivariate statistical approach to check adequacy of dam spillway. J Hydrol Eng 2005;10(1):50 7. [5] Goda K. Statistical modeling of joint probability distribution using copula: application to peak and permanent displacement seismic demands. Struct Saf 2010;32(2): [6] Phoon KK, Santoso A, Quek ST. Probabilistic analysis of soil water characteristic curves. J Geotech Geoenviron Eng 2010;136(3): [7] Uzielli M, Mayne PW. Load displacement uncertainty of vertically loaded shallow footings on sands and effects on probabilistic settlement. Georisk 2012;6(1): [8] Liu PL, Der Kiureghian A. Multivariate distribution models with prescribed marginals and covariances. Probab Eng Mech 1986;1(2): [9] Lebrun R, Dutfoy A. Do Rosenblatt and Nataf isoprobabilistic transformations really differ? Probab Eng Mech 2009;24(4): [10] Tang XS, Li DQ, Rong G, Phoon KK, Zhou CB. Impact of copula selection on geotechnical reliability under incomplete probability information. Comput Geotech 2013;49: [11] Tang XS, Li DQ, Zhou CB, Zhang LM. Bivariate distribution models using copulas for reliability analysis. J Risk Reliab 2013, http: // dx.doi.org / / X [12] Grigoriu M. Existence and construction of translation models for stationary non- Gaussian processes. Probab Eng Mech 2009;24(4): [13] Beer M, Zhang Y, Quek ST, Phoon KK. Reliability analysis with scarce information: comparing alternative approaches in a geotechnical engineering context. Struct Saf 2013;41:1 10. [14] Der Kiureghian A, Liu PL. Structural reliability under incomplete probability information. J Eng Mech 1986;112(1): [15] Der Kiureghian A, Ditlevsen O. Aleatory or epistemic? Does it matter? Struct Saf 2009;31(2): [16] Nataf A. D étermination des distributions de probabilit és dont les marges sont donn ées. C R Acad Sci 1962;225:42 3. [17] Noh Y, Choi KK, Du L. Reliability-based design optimization of problems with correlated input variables using a Gaussian Copula. Struct Multidiscip Optim 2009;38(1):1 16. [18] Lebrun R, Dutfoy A. A generalization of the Nataf transformation to distributions with elliptical copula. Probab Eng Mech 2009;24(2): [19] Li DQ, Tang XS, Phoon KK, Chen YF, Zhou CB. Bivariate simulation using copula and its application to probabilistic pile settlement analysis. Int J Numer Anal Methods Geomech 2013;37(6): [20] Li DQ, Wu SB, Zhou CB, Phoon KK. Performance of translation approach for modeling correlated non-normal variables. Struct Saf 2012;39: [21] Li DQ, Phoon KK, Wu SB, Chen YF, Zhou CB. Impact of translation approach for modelling correlated non-normal variables on parallel system reliability. Struct Infrastruct Eng 2013;9(10): [22] Nelsen RB. An introduction to copulas. 2nd ed. New York: Springer; [23] Lehar M, Zimmermann M. An inexpensive estimate of failure probability for high-dimensional systems with uncertainty. Struct Saf 2012(36 37):32 8. [24] Whelan N. Sampling from Archimedean copulas. Quant Finance 2004;4(3): [25] Mari DD, Kozt S. Correlation and dependence. UK: Imperial College Press; [26] Ditlevsen O, Madsen HO. Structural reliability methods. New York: John Wiley and Sons; [27] Ditlevsen O. Generalized second moment reliability index. J Struct Mech 1979;7(4):