proc imeche part o: j risk and reliability bivariate...

Original Article

Proc IMechE Part O:J Risk and Reliability0(0) 1–14� IMechE 2013Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1748006X13481928pio.sagepub.com

Bivariate distribution models usingcopulas for reliability analysis

Xiao-Song Tang1, Dian-Qing Li1, Chuang-Bing Zhou1

and Li-Min Zhang2

AbstractThe modeling of joint probability distributions of correlated variables and the evaluation of reliability under incompleteprobability information remain a challenge that has not been studied extensively. This article aims to investigate the effectof copulas for modeling dependence structures between variables on reliability under incomplete probability informa-tion. First, a copula-based method is proposed to model the joint probability distributions of multiple correlated vari-ables with given marginal distributions and correlation coefficients. Second, a reliability problem is formulated and adirect integration method for calculating probability of failure is presented. Finally, the reliability is investigated to demon-strate the effect of copulas on reliability. The joint probability distribution of multiple variables, with given marginal distri-butions and correlation coefficients, can be constructed using copulas in a general and flexible way. The probabilities offailure produced by different copulas can differ considerably. Such a difference increases with decreasing probability offailure. The reliability index defined by the mean and standard deviation of a performance function cannot capture thedifference in the probabilities of failure produced by different copulas. In addition, the Gaussian copula, often adoptedout of expedience without proper validation, produces only one of the various possible solutions of the probability offailure and such a probability of failure may be biased towards the non-conservative side. The tail dependence of copulashas a significant influence on reliability.

KeywordsJoint probability distribution, correlation, copula, reliability analysis

Date received: 11 September 2012; accepted: 13 February 2013

Introduction

The engineering literature is replete with correlationsbetween two parameters. For example, internal frictionangle and cohesion of soils,1 seismic peak and perma-nent displacements,2 two stress components underlyingweld fatigue damage,3 curve-fitting parameters of soil–water characteristic curves,4 and two hyperbolic curve-fitting parameters underlying a pile load–displacementcurve.5 When these correlated parameters are involvedin a reliability analysis, the joint cumulative distribu-tion function (CDF) or joint probability density func-tion (PDF) of these correlated parameters has to beestablished in order to properly evaluate reliability.However, owing to limited data from field tests, labora-tory tests, or other resources, the joint CDF or PDF isoften unknown.6 In most cases, only the marginal dis-tributions and correlation coefficients, referred to asincomplete probability information in this study, areknown.7,8 Based on the incomplete probability infor-mation, the joint CDF or PDF of random variables

may not be determined uniquely and the reliability ofproblems concerned may not be evaluated exactly.Therefore, the modeling of multivariate distributionsbased on incomplete probability information and itseffect on reliability remain an outstanding challenge.9

In engineering practice, the multivariate normal dis-tribution, often adopted out of expedience withoutproper validation, is used to deal with the joint prob-ability distribution of multiple random variables.10 Fora multivariate non-normal distribution, the Natafmodel11,12 is used in FORM(First-order reliability

1State Key Laboratory of Water Resources and Hydropower Engineering

Science, Wuhan University, Wuhan, China2Department of Civil and Environmental Engineering, The Hong Kong

University of Science and Technology, Kowloon, Hong Kong

Corresponding author:

Dian-Qing Li, State Key Laboratory of Water Resources and

Hydropower Engineering Science, Wuhan University, 8 Donghu South

Road, Wuhan 430072, China.

Email: [email protected]

at NATIONAL UNIV SINGAPORE on March 27, 2013pio.sagepub.comDownloaded from

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method)/SORM(Second-order reliability method)13 tomap a multivariate non-normal distribution to a multi-variate normal distribution. For instance, DerKiureghian and Liu7 constructed a bivariate distribu-tion of two random variables using the Nataf modeland investigated the reliability under incomplete prob-ability information. Dithinde et al.14 used the Natafmodel to construct the bivariate lognormal distributionof two parameters underlying load–displacement curvesof piles. Leira3 applied the Nataf model to constructthe joint distribution of correlated stress componentsunderlying weld fatigue damage. Li et al.15 investigatedthe performance of the multivariate normal distribu-tions based on their abilities to match the exact prob-abilities of failure from one benchmark bivariateexample where the joint distribution is known.

Recently, a copula approach16,17 has found wideapplications in constructing joint probability distribu-tions of multivariate random data. In the copulaapproach, the dependence structure of multivariate ran-dom data is described by mathematical functions thatsatisfy requirements as a copula function,17 such as theelliptical copulas and the Archimedean copulas. Thecopula approach provides a general and flexible way ofdescribing nonlinear dependence among multivariatedata in isolation from their marginal probability distri-butions. From the copula viewpoint, the Nataf modelessentially uses the Gaussian copula to construct a multi-variate distribution.8,18–20 The copula approach has beenextensively used in financial and hydrological applica-tions.16,21,22 In structural engineering and geotechnicalengineering, Goda2 used the copula approach to con-struct the bivariate distribution of seismic peak and per-manent displacements. Lebrun and Dutfoy8,23 discussedthe difference between the Nataf transformation and theRosenblatt transformation from the copula viewpoint.Li et al.5 used the copula approach to construct thebivariate distribution of two curve-fitting parametersunderlying load–displacement curves of piles. Uzielli andMayne24 used the copula approach to investigate thedependence among load–displacement parameters forvertically loaded shallow footings on sands.

Although the copula approach has been widely usedfor modeling joint probability distributions of multiplecorrelated variables, the effect of copulas for modelingdependence structures between variables on calculatedreliability has not been investigated. Particularly, thedifference in the probabilities of failure associated withthe commonly used Gaussian copula and other copu-las, has not been identified. The objective of this studyis to investigate the effect of the type of copulas on cal-culated reliability with incomplete probability informa-tion, where only the marginal distributions andcorrelation coefficients are known. To achieve the goal,this article is organized as follows. In ‘Modeling thebivariate distribution of two variables using copulas’, acopula-based method is presented for modeling jointprobability distributions of correlated random vari-ables. Six copulas, namely independent, Gaussian,

Plackett, Frank, Clayton and CClayton copulas arecompared. In ‘Probabilities of failure using direct inte-gration method’, a reliability problem is formulatedand a direct integration method for calculating compo-nent probabilities of failure is proposed. Also, newinsights into the reliability index defined by the meanand standard deviation of a performance function aregiven from the copula viewpoint. The effect of copulason reliability is demonstrated in ‘Results and discus-sion’. Furthermore, the pros and cons associated withthe commonly used Nataf model and the copulaapproach to estimate reliability under incomplete prob-ability information are distinguished.

Modeling the bivariate distribution of twovariables using copulas

Copulas

Sklar’s theorem.17 Let F(x1, x2, ��, xn) be the joint CDFwith marginal CDFs F1(x1), F2(x2), ��, Fn(xn). Thenthere exists an n-dimensional copula C such that for allreal x1, x2, ��, xn

F x1, x2, � � � , xnð Þ=C F1 x1ð Þ,F2 x2ð Þ, � � � ,Fn xnð Þ½ �ð1Þ

If C is a copula and F1(x1), F2(x2), ��, Fn(xn) are CDFs,then the function F(x1, x2, ��, xn) defined by equation(1) is a joint distribution function with marginalsF1(x1), F2(x2), ��, Fn(xn). Sklar’s theorem essentiallystates that the joint probability distribution of randomvariables can be expressed in terms of a copula functionand their marginal distributions.

According to Sklar’s theorem, the bivariate jointCDF of two random variables X1 and X2 can be givenby

F x1, x2ð Þ=C F1 x1ð Þ,F2 x2ð Þ; uð Þ=C u1, u2; uð Þ ð2Þ

in which F(x1, x2) is the joint CDF of X1 and X2;u1=F1(x1) and u2=F2(x2) are the corresponding mar-ginal distributions of X1 and X2, respectively; C (u1, u2;u) is the copula function in which u is the copula para-meter describing the dependency between X1 and X2. Inmathematical terms, a bivariate copula function C (u1,u2; u) is a two-dimensional probability distribution on[0, 1]2 with uniform marginal distributions on [0, 1].From equation (2), the bivariate PDF f (x1, x2) can beobtained as16

f x1, x2ð Þ= f1 x1ð Þf2 x2ð Þc F1 x1ð Þ,F2 x2ð Þ; uð Þ ð3Þ

where f1(x1) and f2(x2) are the marginal PDFs of X1

and X2, respectively; c (F(x1), F2(x2); u) is the copuladensity function, which is given by

c F1 x1ð Þ,F2 x2ð Þ; uð Þ= c u1, u2; uð Þ= ∂2C u1, u2; uð Þ=∂u1∂u2ð4Þ

2 Proc IMechE Part O: J Risk and Reliability 0(0)



Theoretically, the joint CDF and PDF of X1 and X2

can be determined by equations (2) and (3) if the mar-ginal distributions of X1 and X2, and the copula func-tion are known.

The copula parameter, u, can be determined throughthe linear correlation coefficient or rank correlationcoefficient, such as the Spearman and Kendall correla-tion coefficients.25 In this work, the linear correlationcoefficient is adopted to determine u owing to its famil-iarity among practitioners. According to the definitionof linear correlation coefficient,26 the following integralrelation between u and correlation coefficient, r, can beobtained

r=Cov x1, x2ð Þ

s1s2=

ð+‘

�‘

ð+‘

�‘

x1 � m1

s1

� �x2 � m2

s2

� �f1 x1ð Þf2 x2ð Þc F1 x1ð Þ,F2 x2ð Þ; uð Þdx1dx2

ð5Þ

where m1 and m2 are the means of X1 and X2, respec-tively; s1 and s2 are the standard deviations of X1 andX2, respectively; Cov(x1, x2) is the covariance betweenX1 and X2. For given marginal distributions of X1 andX2, and correlation coefficient r between X1 and X2, thepreceding integral equation can be solved iteratively.For example, Li et al.27 developed a two-dimensionalGaussian-Hermite integral technique to solve the aboveintegral equation, which is adopted in this study.

Tail dependence

To facilitate the understanding of the subsequent anal-yses, the concept of tail dependence17 is introducedbriefly. The tail dependence relates to the amount ofdependence at the upper-quadrant tail or lower-quadrant tail of a bivariate distribution. The coefficientof upper tail dependence of X1 and X2, lU, is definedas

lU = limq!1�

P X2 .F�12 qð Þ X1 .F�11 qð Þ��

ð6Þ

given that the limit lU2[0,1] exists. In equation (6),F1

21(�) and F221(�) are the inverse CDFs of X1 and

X2, respectively. If lU2(0,1], X1 and X2 are said to beasymptotically dependent at the upper tail; if lU=0,X1 and X2 are said to be asymptotically independent atthe upper tail.

Similarly, the coefficient of the lower tail dependenceof X1 and X2, lL, is defined as

lL = limq!0+

P X2 \F�12 qð Þ X1 \F�11 qð Þ��

ð7Þ

provided that the limit lL2[0,1] exists. If lL2(0,1], X1

and X2 are said to be asymptotically dependent at thelower tail; if lL=0, X1 and X2 are said to be asympto-tically independent at the lower tail.

It can be seen from the definition of tail dependencethat the coefficient of upper tail dependence is the prob-ability that the random variable X2 exceeds its quantileof order q, knowing that X1 exceeds its quantile of thesame order when order q approaches 1. The coefficientof lower tail dependence is the probability that X2 issmaller than its quantile of order q, knowing that X1 issmaller than its quantile of the same order when orderq approaches 0. With regard to the commonly usedGaussian copula, lU=lL=0. Hence, the Gaussiancopula is asymptotically independent at both tails.

Comparison among selected copulas

There exist many copulas to describe the dependencebetween variables. The dependence structures underly-ing different copulas can differ considerably. For illus-trative purposes, the independent copula, Gaussiancopula, Plackett copula, Frank copula, Clayton copulaand CClayton copula17 are examined in this study. Theabove six copulas, along with the range of the u para-meter are listed in Table 1. The independent copula isused to account for the effect of independent variableson reliability. The Gaussian copula is an elliptical andsymmetric copula. The Frank, Clayton and CClaytoncopulas are commonly used Archimedean copulas. ThePlackett copula is a member of Plackett family of

Table 1. Summary of the adopted bivariate copula functions in this study.

Copula Copula function, C(u1, u2) First derivative of C(u1, u2) withrespect to u1, M(u1, u2)

Range of u

Independent u1u2 u2 —Gaussian Fu F�1 u1ð Þ, F�1 u2ð Þ

� �F

F�1 u2ð Þ�uF�1 u1ð Þffiffiffiffiffiffiffiffi1�u2p

� �[21, 1]

Plackett S�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS2�4u1u2u u�1ð Þp

2 u�1ð Þ , S = 1 + u� 1ð Þ u1 + u2ð Þ 12�

1 + u�1ð Þu1� u + 1ð Þu2

2 1 + u�1ð Þ u1 + u2ð Þ½ �2�4u1u2u u�1ð Þf g12

(0, + N)\{1}

Frank � 1uln 1 +

e�uu1�1ð Þ e�uu2�1ð Þe�u�1

�e�uu1 e�uu2�1ð Þ

e�u�1ð Þ+ e�uu1�1ð Þ e�uu2�1ð Þ(2N, + N)\{0}

Clayton u�u1 + u�u

2 � 1� ��1

u u�u�11 u�u

1 + u�u2 � 1

� ��1u�1 (0, + N)

CClaytonu1 + u2 � 1 + 1� u1ð Þ�u + 1� u2ð Þ�u � 1

h i�1u

1� 1� u1ð Þ�u�1 1� u1ð Þ�u + 1� u2ð Þ�u � 1h i�1

u�1 (0, + N)

The symbol ‘‘—’’ denotes that the range of u is not available.

Tang et al. 3



copulas, which was proposed by Plackett.28 In addition,the independent, Gaussian, Plackett, and Frank copu-las do not have tail dependence. Unlike the aforemen-tioned four copulas, the Clayton copula and CClaytoncopula have lower and upper tail dependence, respec-tively. The coefficient of lower tail dependence for theClayton copula is given by

lL =2�1=u ð8Þ

and the coefficient of upper tail dependence for theCClayton copula is given by

lU =2�1=u ð9Þ

It can be observed from equations (8) and (9) that thecoefficient of tail dependence increases as the para-meter u increases. Note that the aforementioned sixcopulas, except the independent copula, can describepositive dependence. The absolute values of the correla-tion coefficients can approach 1. Such features are suit-able for examining the effect of copulas on reliability.

Figure 1 presents the joint PDF isolines produced bythe six copulas to clearly visualize the differencesamong these six selected copulas. In Figure 1, the jointPDFs of X1 and X2 are calculated by equation (3) inwhich the two random variables X1 and X2 are standardnormal distributions and a linear correlation coefficientr=0.5 between X1 and X2 is assumed. There is a signif-icant difference in the joint PDF isolines produced bydifferent copulas even though the same marginal distri-butions and correlation coefficient are followed. In par-ticular, the joint PDF isolines produced by the Claytoncopula and CClayton copula with tail dependence aresignificantly different from those produced by the otherfour copulas. Such a difference may lead to significantdifferences in reliability because the tail of the jointPDF can influence the reliability greatly,29 which willbe discussed in the following sections.

Probabilities of failure using directintegration method

Performance functions for reliability analysis

It is widely accepted to characterize reliability in termsof a probability of failure. In a general reliability prob-lem with a performance function g(X), the probabilityof failure, pf, can be calculated as29

pf =P½g(X)40�=ðg(X)40

fx(x)dx ð10Þ

where fX(x) denotes the joint PDF for the n-dimen-sional vector X of basic variables; g(X) 4 0 is the fail-ure domain denoting the space of limit state violation.Except for some special cases, the integration of equa-tion (10) over the failure domain g(X) 4 0 cannot beperformed analytically. Notwithstanding this, a direct

integration method is used to calculate the probabilityof failure in this study, as explained further.

To examine the difference in the probabilities of fail-ure more extensively, the following four performancefunctions are considered

g1(x)=C1 � x1 � x2 ð11aÞg2(x)=C2 + x1 + x2 ð11bÞg3(x)=C3 + x1 � x2 ð11cÞg4(x)=C4 � x1 + x2 ð11dÞ

in which C1–C4 are constants. The reliability levels canbe varied when the four constants C1–C4 take differentvalues. Theoretically, the four performance functionsare able to scan the entire area of the joint PDF sur-face provided that the four constants are varied overa wide range. For convenience, the aforementionedfour performance functions are hereafter referred toas performance functions I, II, III, and IV, respec-tively. For visualization, Figure 2 shows the probabil-ities of failure for the considered performancefunctions. Note that all the failure domains are semi-infinite spaces. Furthermore, the limit state surfacesare parallel or perpendicular to the symmetricalplanes of the considered six copula density functions,as illustrated in Figure 1. These features are useful tocapture the difference in the probabilities of failureassociated with different copulas. Since the mainobjective of this study is to examine the effect ofcopulas on reliability, the type of marginal distribu-tions of random variables is not a constraint. Forsimplicity and discussions in ‘Discussions’, the stan-dard normal distributions are adopted for describingthe distributions of random variables X1 and X2. Itshould be noted that there exists a symmetric planefor the bivariate distributions produced by the sixcopulas selected, as shown in Figure 1. In this situa-tion, the probabilities of failure on the two sides ofthe symmetric plane remain the same. In other words,the reliability results associated with performancefunctions III and IV are the same. Hence, only one ofthem needs to be evaluated. Thus, the probabilities offailure for performance functions I, II, and III arepresented in the subsequent analyses.

Direct integration for calculating probability of failure

As can be seen from equations (11a)–(11d), the consid-ered performance functions are relatively simple. Theresulting component probabilities of failure associatedwith equation (10) can be solved by direct integration,which is described further. The component probabilityof failure, pf, can be expressed as

pf =

ðg40

f x1, x2ð Þdx1dx2 ð12Þ

where f(x1, x2) is the joint PDF of X1 and X2.Substituting equation (3) into equation (12) yields




pf =

ðg40

f1 x1ð Þf2 x2ð Þc F1 x1ð Þ,F2 x2ð Þ; uð Þdx1dx2

ð13Þ

It is evident from equation (13) that the double integralmay be time-consuming. For this reason, the first deriva-tive of a copula function is employed, which is given by

M u1, u2; uð Þ= ∂C u1, u2; uð Þ=∂u1 ð14Þ

By substituting equation (14) into equation (13), thedouble integral in equation (13) can be reduced to a sin-gle integral

pf =

ðg40

f1 x1ð ÞM F1 x1ð Þ,F2 x2ð Þ; uð Þdx1 ð15Þ

Based on the four performance functions shown inequation (11) and the corresponding failure domains

Figure 1. Comparison of PDF isolines of correlated variables associated with the six copulas selected: (a) Independent copula; (b)Gaussian copula; (c) Plackett copula; (d) Frank copula; (e) Clayton copula; (f) CClayton copula.

Tang et al. 5



shown in Figure 2, the formulae for calculating theprobabilities of failure can be derived as

pf1 =1�ð+‘

�‘

f1 x1ð ÞM F1 x1ð Þ,F2 C1 � x1ð Þ; uð Þdx1

ð16aÞ

pf2 =

ð+‘

�‘

f1 x1ð ÞM F1 x1ð Þ,F2 �C2 � x1ð Þ; uð Þdx1

ð16bÞ

pf3 =1�ð+‘

�‘

f1 x1ð ÞM F1 x1ð Þ,F2 C3 + x1ð Þ; uð Þdx1

ð16cÞ

pf4 =

ð+‘

�‘

f1 x1ð ÞM F1 x1ð Þ,F2 �C4 + x1ð Þ; uð Þdx1

ð16dÞ

The probabilities of failure shown in equations (16a)–(16d) are relatively simple if the first derivative of acopula is available. For convenience, Table 1 shows thefirst derivatives of the six copulas considered. When thecopula parameters u are known, the probabilities offailure can be evaluated efficiently using equation (16).Note that the probabilities of failure calculated by anyone of equations (16a)–(16d) for different copulas are

different. These probabilities of failure can account forthe effect of copulas on reliability because the actualjoint distributions of X1 and X2 constructed using dif-ferent copulas are used. In other words, the probabil-ities of failure calculated using equation (16) representthe actual reliability levels of the structures.

Reliability index defined by mean and standarddeviation of a performance function

It is well known that the reliability index is an alterna-tive index for characterizing reliability. When the per-formance function g(x) is normally distributed, thereliability index, b, is defined as

b=mg

sgð17Þ

where mg and sg are the mean and standard deviationof g(x), respectively. For normally distributed perfor-mance functions, it can be shown that the reliabilityindex is related to the probability of failure by

pf =1�F bð Þ ð18Þ

in which F(�) is the standard normal CDF.

Figure 2. Illustration of the adopted performance functions and their failure domains: (a) Performance function I; (b) Performancefunction II; (c) Performance function III; (d) Performance function IV.




It should be pointed out that equations (17) and (18)are approximate for non-normally distributed perfor-mance functions. The reliability indexes defined byequation (17) do not account for the effect of type ofcopulas on calculated reliability, because the reliabilityindexes are evaluated only based on the first twomoments of the performance function, namely themean and standard deviation. The mean and standarddeviation of a performance function can completelycharacterize the statistics of a normally distributed per-formance function.

Applying the concepts of expected value and var-iance of a function,26 it can be shown that the means ofperformance functions gi (i=1;4) shown in equations(11a)–(11d) are equal to constants Ci (i=1;4), respec-tively; and the corresponding standard deviationssgi(i=1;4) can be obtained as

sgi =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∂gi∂x1

� �2

s21 +2

∂gi∂x1

∂gi∂x2

Cov x1, x2ð Þ+ ∂gi∂x2

� �2

s22

s

ð19Þ

where ∂gi=∂x1 and ∂gi=∂x2 are the partial derivatives ofgi to X1 and X2, respectively. Based on equation (5) andequation (19), one can obtain

sgi =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∂gi∂x1

� �2

s21 +2

∂gi∂x1

∂gi∂x2

rs1s2 +∂gi∂x2

� �2

s22

s

ð20Þ

Having obtained the means and standard deviationsof the performance functions, the reliability indexes canbe calculated using equation (17) under the assumptionthat the performance functions follow normal distribu-tions. The four reliability indexes b1;b4 for the consid-ered four performance functions in equation (11) can beobtained as

b1 =C1

. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1+ rð Þ

pð21aÞ

b2 =C2

. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1+ rð Þ

pð21bÞ

b3 =C3

. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1� rð Þ

pð21cÞ

b4 =C4

. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1� rð Þ

pð21dÞ

It can be observed that the reliability indexes bi

(i=1;4) only depend on constants Ci(i=1;4) andthe correlation coefficient r between X1 and X2.Changes in Ci and r are presented in a uniform way aschanges in the reliability indexes. The reliability indexescalculated by equation (21) only depend on the con-stants and the linear correlation coefficients. The relia-bility indexes calculated by any one of equations (21a)–(21d) are the same for different copulas because differ-ent copula parameters are obtained based on the samer using equation (5). Consequently, the effect of thetype of copulas on calculated reliability cannot betaken into consideration using bi(i=1;4). Since the

main objective of this study is to investigate the effectof the type of copulas on reliability, equation (16) isused to calculate the probabilities of failure in the sub-sequent analyses. The reliability indexes bi(i=1;4)evaluated using equation (21) are only presented in thehorizontal axes to relate the probabilities of failure tothe more familiar reliability indexes among engineers.

Results and discussion

Applying the proposed direct integration method pre-sented in ‘Direct integration for calculating probabilityof failure’, the probabilities of failure can be obtainedfor different copulas. The probabilities of failure forperformance functions I, II, and III are studied step bystep in the following. Additionally, in the subsequentanalyses, the reliability indexes are calculated usingequation (17) unless stated otherwise. It should benoted that the reliability results associated with theGaussian copula are highlighted because the Gaussiancopula, often adopted out of expedience without propervalidation, is commonly assumed adequate to describethe dependence structure between random variables inpractice.

Reliability results for the three performance functions

Probabilities of failure for performance function I. Figure 3compares the probabilities of failure on log scale forperformance function I (equation (11a)) produced bydifferent copulas. It can be observed that the probabil-ities of failure associated with different copulas differconsiderably. These results imply that the probabilitiesof failure under incomplete probability informationcannot be determined uniquely. For the consideredcopulas, the probability of failure associated with theCClayton copula with upper tail dependence is signifi-cantly larger than those associated with the other fivecopulas, whereas the independent copula produces thesmallest probability of failure. Such results indicate thatthe CClayton copula may significantly overestimate theprobability of failure if any of the other five copulas isadequate to model the dependence structure betweenX1 and X2, which will be conservative for structuralsafety assessment. In contrast, the independent copulamay underestimate the probability of failure if it is notadequate to represent the dependence structure betweenX1 and X2, which will lead to unsafe structural designs.In addition, the difference in the probabilities of failurebetween the Gaussian copula and the other five copulasincreases with increasing reliability index or decreasingprobability of failure. For r=0.5 and b1=0, theratios of the probabilities of failure for the independent,Clayton, Frank, Plackett, and CClayton copulas to thatfor the Gaussian copula are 1, 0.948, 1, 1, and 0.946,respectively. For r=0.5 and b1=5, the correspondingratios associated with the independent, Clayton, Frank,Plackett, and CClayton copulas are 6.273 102,

Tang et al. 7



3.143 102, 1.833 102, 1.343 102, and 2.023 101,respectively. These results indicate that if any one ofthe other five copulas is adequate to describe the depen-dency between X1 and X2, the error in probability offailure produced by the commonly used Gaussiancopula will be unacceptable. This conclusion can alsobe applied to any one of the other five copulas.

To account for the effect of correlation between X1

and X2 on reliability, Figure 4 shows the probabilitiesof failure on log scale for various correlation coeffi-cients between X1 and X2. Constant C1 is assumed tobe 5

ffiffiffi2p

, which will result in b1=5 for r=0 usingequation (21a). As expected, the probabilities of failureproduced by the independent copula for various corre-lation coefficients remain the same. Unlike the indepen-dent copula, the probabilities of failure produced bythe other five copulas increase as the correlationbetween two variables becomes stronger. Except for theindependent copula, the difference in the probabilitiesof failure between the Gaussian copula and the otherfour copulas is at its maximum value at an intermediatecorrelation coefficient. When the two variables X1 andX2 are perfectly correlated, there is no difference in theprobabilities of failure produced by the consideredcopulas except for the independent copula.

Probabilities of failure for performance function II. Similarly,Figure 5 compares the probabilities of failure on logscale for performance function II (equation (11b)) pro-duced by different copulas. Similar to the results shownin Figure 3, the probabilities of failure produced by

Figure 4. Effect of correlation coefficients on probabilities offailure produced by different copulas for performance function I.

Figure 3. Probabilities of failure produced by different copulas for performance function I: (a) r = 0.25, C1 = [0, 7.91]; (b) r = 0.50,C1 = [0, 8.66]; (c) r = 0.75, C1 = [0, 9.35].




different copulas differ greatly. The independent copulaleads to the smallest probability of failure again andthe Clayton copula produces the largest probability offailure. These results indicate that the Clayton copulawith lower tail dependence may significantly overesti-mate the probability of failure like the CClayton copulain performance function I, which will be conservativefor structural safety assessment.

Figure 6 shows the probabilities of failure on logscale for various Pearson correlation coefficientsbetween X1 and X2. C2 is also assumed to be 5

ffiffiffi2p

toproduce a b2 equal to 5 when r=0 using equation(21b). Except for the independent copula, the differencein the probabilities of failure between the Gaussiancopula and the other four copulas does not increasemonotonically as an increasing correlation coefficient.In comparison with the results shown in Figure 4, theprobabilities of failure produced by the Clayton copulashown in Figure 6 are equal to those produced by theCClayton copula shown in Figure 4, and vice versa.The reason is that both the Clayton and CClaytoncopulas are symmetrical with respect to the 45� diago-nal line of a unit square, as shown in Figure 1.Furthermore, the CClayton copula is the survival

copula17 of the Clayton copula and the upper taildependence coefficient for the CClayton copula is equalto the lower tail dependence coefficient for the Claytoncopula. The results produced by the Gaussian,

Figure 5. Probabilities of failure produced by different copulas for performance function II: (a) r = 0.25, C2 = [0, 7.91]; (b) r = 0.50,C2 = [0, 8.66]; (c) r = 0.75, C2 = [0, 9.35].

Figure 6. Effect of correlation coefficients on probabilities offailure produced by different copulas for performance function II.

Tang et al. 9



Plackett, Frank, and independent copulas remainunchanged because these copulas are symmetrical withrespect to the 45� and 135� diagonal lines of a unitsquare simultaneously in Figure 1.

Probabilities of failure for performance function III. Withregard to performance function III (equation (11c)), theprobabilities of failure on log scale produced by differ-ent copulas are plotted in Figure 7. In comparison withperformance functions I and II, the Gaussian copulaleads to the smallest probability of failure, and the inde-pendent copula results in the largest probability of fail-ure. These results mean that the Gaussian copula mayoverestimate the component reliability in the case of per-formance function III. Unlike performance functions Iand II, the CClayton and Clayton copulas produce thesame probabilities of failure owing to the former beingthe survival copula of the latter and the special form ofthe performance function as shown in Figure 2(c).

Similarly, the probabilities of failure on log scale forvarious Pearson correlation coefficients between X1 andX2 are plotted in Figure 8. Constant C3 is here assumedto be

ffiffiffi2p

, which will lead to b3=1 for r=0 usingequation (21c). Unlike performance functions I and II,

the probabilities of failure produced by the Gaussiancopula are significantly lower than those produced by theother five copulas, especially for a strong positive correla-tion. The difference in the probabilities of failure betweenthe Gaussian copula and the other five copulas stillincreases, although the correlation coefficient is set as0.999. Theoretically, when the correlation coefficientapproaches 1.0, the joint PDFs of X1 and X2 produced bydifferent copulas tend to be the same.17 The resultantprobabilities of failure will thus converge to the samevalue 0. These results are very different from those associ-ated with performance functions I and II. At r=0.99, theratios of the probabilities of failure for the Clayton,CClayton, Frank, Plackett, and independent copulas tothat for the Gaussian copula are 1.913 1012, 1.913 1012,6.623 1011, 1.463 1012, and 2.043 1014, respectively.This further indicates that if one of the other five copulasis adequate to describe the dependency between X1 andX2, the commonly used Gaussian copula may lead toresults of unacceptable errors.

Discussion

As mentioned previously, the effect of copulas on thereliability cannot be uniquely represented by the

Figure 7. Probabilities of failure produced by different copulas for performance function III: (a) r = 0.25, C3 = [0, 6.12];(b) r = 0.50, C3 = [0, 5.00]; (c) r = 0.75, C3 = [0, 3.54].




reliability index defined by the mean and standarddeviation of a performance function as shown in equa-tion (17). This interesting finding will be furtherexplained. For convenience, the exact probabilities offailure for performance functions I, II, and III aredenoted by pf1, pf2 and pf3, respectively, as shown inequations (16a)–(16c). When the three performancefunctions (equations (11a)–(11c)) are assumed to benormal distributions, the corresponding probabilities offailure, denoted by pf01, pf02, and pf03, are calculated byequation (18) in which the reliability indexes are calcu-lated by equations (21a)–(21c), respectively. Takingr=0.5 as an example, Figure 9 shows the probabilitiesof failure pfi(i=1;3) for various values of pf0i(i=1;3). For most copulas, the probabilities of failure pfifor the same performance functions are significantlydifferent from pf0i. pfi are the same as pf0i only when theGaussian copula is used to model the joint probabilitydistribution of X1 and X2. These results indicate thatthe performance functions shown in equation (11) arenormally distributed only when the Gaussian copula isadopted to construct the joint probability distributionbetween the two variables. Whereas the performancefunctions will follow non-normal distributions whenthe non-Gaussian copulas are employed to model thejoint probability distribution of X1 and X2.

In engineering practice, the available observations ofrandom variables such as X1 and X2 are often sufficientfor determining the marginal distributions and covar-iance. However, the available data are often insufficientfor determining the joint probability distribution. Inother words, the dependence structure between X1 andX2 cannot be determined uniquely. As a result, only themarginal distributions and covariance, namely theincomplete probability information in this study, areavailable. In this situation, a large number of copulasin the literature are available to describe the depen-dence structure between X1 and X2 being consistentwith the known marginals and covariance of X1 and

X2. Consequently, a robust estimation of the probabil-ity of failure is not possible. As can be seen from thereliability results in ‘Reliability results for the three per-formance functions’, the probabilities of failure pro-duced by different copulas differ greatly. These resultshighlight the need for engineers to collect more data inorder to improve the dependence information. Ifobtaining data from field or laboratory tests associatedwith a specific project is cost-prohibitive, the data fromsimilar projects should be explored. The rationalebehind this approach is that the dependence structureonly relies on the ranks underlying the variables ratherthan the real values of the variables.17 With theimproved dependence information, the candidate copu-las are reduced and the estimation of probability offailure is improved.

In the case that enough observations of randomvariables are available, one may first rule out somecopulas that are obviously different from the depen-dence structure of the empirical data at hand by simplevisual inspection. For instance, if the empirical dataexhibit neither the upper tail dependence nor the lowertail dependence, then the set of candidate copulas{Independent, Gaussian, Plackett, Frank, Clayton, andCClayton copulas} can be reduced to {Independent,Gaussian, Plackett, and Frank copulas} in which theIndependent copula is used for comparison eventhough it cannot account for the correlation betweenvariables. Furthermore, a cautious identification of thebest-fit copula to the dependence structure underlyingthe empirical data from the reduced set of copulas{Independent, Gaussian, Plackett, and Frank copulas}can also be conducted. The Akaike InformationCriterion (AIC) and the Bayesian InformationCriterion (BIC) can be used for such purposes, as illu-strated by Li et al.5 It should be noted that the identifi-cation of the best-fit copula needs a large number ofempirical data. Otherwise, the identified copulas maylead to misleading results.

It is well known that, in practical applications, theNataf model or the multivariate normal distribution isoften employed to model the joint probability distribu-tion of random variables. As pointed out by Lebrunand Dutfoy8 and Nelsen,17 the Nataf model or the mul-tivariate normal distribution essentially adopts aGaussian copula for modeling the dependence structurebetween variables. The tradeoff is that the probabilityof failure estimated from the Nataf model or the multi-variate normal distribution is only one of the variouspossible solutions that is evaluated using the Gaussiancopula and may be biased towards the non-conservativeside, e.g. the performance function III investigated inthis study. As mentioned, various copulas in the litera-ture are available to describe the dependence structurebetween variables being consistent with the known mar-ginals and covariance. This study proposes a copula-based method to model the joint probability distribu-tions of variables. The copula approach embraces thereliability methods that use the Nataf model or the

Figure 8. Effect of correlation coefficients on probabilities offailure produced by different copulas for performance function III.

Tang et al. 11



multivariate normal distribution as an approximationof the actual joint probability distribution of randomvariables. It provides a tool for modeling the multivari-ate distribution in a more general and rigorous way.Compared with the other copulas, such as the Frank,Clayton, CClayton, and Plackett copulas, the Gaussiancopula can be easily extended to high-dimensional case.Furthermore, it is slightly simpler than the other copu-las. Again, the Gaussian copula has been widely used inpractical applications. In order to obtain realistic relia-bility results, the copulas for modeling the multivariatedistribution should be selected carefully when onlyincomplete probability information is available.

Summary and conclusions

Copulas have been applied to construct bivariate distri-butions for reliability analysis. The effect of the bivari-ate distribution models using copulas on reliability isevaluated considering three performance functions.Several conclusions can be drawn.

1. The joint probability distribution of multiple ran-dom variables with given marginal distributions

and correlation coefficients can be constructedusing copulas in a general and flexible way. Thecopula approach embraces the reliability methodsthat use the Nataf transformation in dealing withnon-normal variables.

2. The probability of failure of a structural compo-nent with given marginal distributions and correla-tion coefficient of two variables cannot bedetermined uniquely. The probabilities of failureassociated with different copulas differ consider-ably. The difference in probability of failureincreases as the probability of failure decreases.The Gaussian copula, often adopted out ofexpedience without proper validation, producesonly one of the various possible solutions of theprobability of failure and such a probability of fail-ure may be biased towards the non-conservativeside. These results are often not realized in practi-cal reliability analysis.

3. The tail dependence of copulas has a significantinfluence on reliability. When the failure domainof a performance function coincides with the tailof a copula, the probability of failure produced bythe copula with tail dependence is significantly

Figure 9. Comparison between probabilities of failure pf1~ pf3 associated with three performance functions for various copulas andprobabilities of failure pf01~ pf03: (a) Performance function I; (b) Performance function II; (c) Performance function III.




larger than that associated with the copula withouttail dependence. On the contrary, when the failuredomain of a performance function does not fallwithin the tail of copulas, the component probabil-ities of failure are not sensitive to the tail depen-dence of copulas.

4. Except for the independent copula, the largest dif-ference between the probabilities of failure for theGaussian copula and the other four copulas stud-ied herein may not be associated with a large cor-relation. The largest difference can happen at anintermediate correlation.

5. The reliability index defined by the mean and stan-dard deviation of a performance function cannotcapture the differences in probabilities of failureassociated with different copulas. Such differ-ences can be considered by the probabilities offailure evaluated based on the actual distributionof a performance function. For the linear perfor-mance functions involving multiple random vari-ables, they are normally distributed only whenthe Gaussian copula is adopted to construct thejoint probability distribution of these multiplevariables.

6. It should be noted that the unacceptable error ofGaussian copula and the importance of tail depen-dence underlying copulas, such as Clayton andCClayton copulas, cause difficulties in establishingan appropriate way to select the appropriate bivari-ate model with respect to a given reliability prob-lem with incomplete probability information. Oneeffective way is to collect more data to improve thedependence information.

Funding

This work was supported by the National ScienceFund for Distinguished Young Scholars (Project No.51225903), the National Natural Science Foundationof China (Project No. 51079112) and Doctoral Fund ofMinistry of Education of China (Project No.20120141110009).

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