impact of copulas for modeling bivariate distributions on...

Structural Safety 44 (2013) 80–90

Contents lists available at SciVerse ScienceDirect

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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

mpact of copulas for modeling bivariate distributions on system

eliability

iao-Song Tang a , Dian-Qing Li a , * , Chuang-Bing Zhou a , Kok-Kwang Phoon b , Li-Min Zhang c

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 8 Donghu South Road, Wuhan 430072, PR China Department of Civil and Environmental Engineering, National University of Singapore, Blk E1A, #07–03, 1 Engineering Drive 2, Singapore 117576, Singapore Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

r t i c l e i n f o

rticle history:

eceived 12 April 2013

eceived in revised form 24 June 2013

ccepted 25 June 2013

eywords:

oint probability distribution

orrelation coefficient

opulas

arallel system

ystem reliability

eliability analysis

a b s t r a c t

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 gen-

eralized 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 dif-

ferent 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.

. Introduction

In reliability analyses, quantities such as loads, material properties

nd structural dimensions are typically treated as random variables

r random fields (e.g., [ 1 –3 ]). Furthermore, these quantities are usu-

lly correlated with each other. For example, flood peak discharge

nd volume are interrelated in a flood frequency analysis [ 4 ]; peak

nd permanent displacements of a system subjected to earthquake

oading [ 5 ], curve-fitting parameters underlying a soil–water char-

cteristic curve [ 6 ], and curve-fitting parameters underlying load–

isplacement curves of piles [ 7 ] are correlated with each other. It

s well known that the joint cumulative distribution function (CDF)

r probability density function (PDF) of these parameters should be

vailable to evaluate the reliability accurately. In engineering prac-

ice, unfortunately, the available data are only sufficient for determin-

ng appropriate marginal distributions as well as covariances, which

oses a problem of incomplete probability information [ 8 –11 ]. Under

his condition, the modeling and simulation of correlated non-normal

ariables are challenging problems [ 12 , 13 ], and a rigorous evaluation

f reliability is impossible [ 14 , 15 ].

* Corresponding author. Tel.: + 86 27 6877 2496; fax: + 86 27 6877 4295. E-mail address: [email protected] (D.-Q. Li).

167-4730/ $ - see front matter c © 2013 Elsevier Ltd. All rights reserved. ttp://dx.doi.org/10.1016/j.strusafe.2013.06.004

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 depen-

dence 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 as-

sumption 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 de-

pendence structure between two random variables, Li et al. [ 20 ] in-

vestigated the performance of two commonly used translation ap-

proaches, 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 cop-

ula. As for the dependence structures among variables characterized

http://dx.doi.org/10.1016/j.strusafe.2013.06.004http://www.sciencedirect.com/science/journal/01674730http://www.elsevier.com/locate/strusafehttp://crossmark.dyndns.org/dialog/?doi=10.1016/j.strusafe.2013.06.004&domain=pdfmailto:[email protected]://dx.doi.org/10.1016/j.strusafe.2013.06.004

Xiao-Song Tang et al. / Structural Safety 44 (2013) 80–90 81

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 practi-

cal interest to distinguish between the reliability of each component

and the reliability of the entire structural system. In addition, the ef-

fect 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 construct-

ing bivariate distributions on parallel system reliability under incom-

plete probability information. To achieve this goal, this article is or-

ganized as follows. In Section 2 , the copula theory for modeling the

joint probability distributions of multiple correlated variables is intro-

duced 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 reliabil-

ity problem is formulated, and the corresponding bounds of system

probability of failure are derived from the copula viewpoint. The ra-

tios 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 4 .

2. A copula-based method for modeling the bivariate

distribution of two variables

According to Sklar’s theorem (e.g., [ 22 ]), the bivariate joint CDF oftwo 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 distri-

bution 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 , respec-

tively; 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 under-

lying different copulas differ significantly. To clearly capture the dif-

ferences 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 posi-

tive dependences, and the values of the correlation coefficients can

approach 1. Finally, the t , Clayton and CClayton copulas have tail de-

pendence (e.g., [ 22 ]), which can account for the effect of tail depen-

dence 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 multi-

variables 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 sim-

ple 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 up-

per and lower tail dependence, λU and λL , are defined as

λU = lim q→ 1 −

P [

X 2 > F −1 2 ( q )

∣∣∣X 1 > F −1 1 ( q ) ] (4)

λL = lim q→ 0 +

P [

X 2 < F −1 2 ( q )

∣∣∣X 1 < F −1 1 ( q ) ] (5)given that these limits λU ∈ [0,1] and λL ∈ [0,1] exist. In Eqs. (4) and(5) , F 1

−1 ( ·) and F 2 −1 ( ·) are the inverse CDFs of X 1 and X 2 , respectively.If λU ∈ (0,1] or λL ∈ (0,1], X 1 and X 2 are said to be asymptotically de-pendent at the upper tail or lower tail; if λU = 0 or λL = 0, X 1 and X 2are 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

λU = λL = 2 [

1 − t v+ 1 ( √

( v + 1 ) ( 1 − θ) ( 1 + θ)

) ] (6)

in which t v + 1 is the CDF of the one-dimensional Student distributionwith v + 1 degrees of freedom.

82 Xiao-Song Tang et al. / Structural Safety 44 (2013) 80–90

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 v −1 ( u 1 ) , t v −1 ( u 2 )) C −1 = W , C 0 �= �, C 1 = M [ −1, 1]

Plackett S −

√ S 2 −4 u 1 u 2 θ( θ−1)

2( θ−1) , S = 1 + ( θ − 1)( u 1 + u 2 ) C 0 = W , C 1 = �, C ∞ = M (0, ∞ ) \{ 1 } Frank − 1

θln [1 + ( e −θu 1 −1)( e −θu 2 −1)

e −θ −1 ] C −∞ = W , C 0 = �, C ∞ = M ( −∞ , ∞ ) \{ 0 } Clayton ( u 1

−θ + u 2 −θ − 1) −1 θ C 0 = �, C ∞ = M (0, ∞ )

CClayton u 1 + u 2 − 1 + [ (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 .

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Unlike the t copula, the Clayton copula only has lower tail depen-
ence. The coefficient of lower tail dependence is given by

L = 2 −1 /θ (7) ince the CClayton copula is the survival copula of the Clayton cop-

la, it only has upper tail dependence. The coefficient of upper tail

ependence is given by

U = 2 −1 /θ (8) t can be observed from Eqs. (7) and (8) that the coefficients of tail

ependence for the Clayton and CClayton copulas remain the same.

Based on the above discussions, the copulas and their coefficients

f tail dependence are closely related to the copula parameter θ . The

alue of θ can be determined through the linear correlation coefficient

r rank correlation coefficients such as the Pearson linear correlation

oefficient, Spearman and Kendall rank correlation coefficients (e.g.,

25 ]). In this work, the Kendall correlation coefficient is adopted be-

ause it is easy to estimate in a robust way and invariant under strictly

ncreasing transformations in comparison with the linear correlation

oefficient. Note that the Spearman rank correlation coefficient can

lso be used for such a purpose. The Kendall rank correlation coeffi-

ient between two random variables X 1 and X 2 , τ , is expressed as

= 4 ∫ 1

0

∫ 1 0 C ( u 1 , u 2 ; θ) dC ( u 1 , u 2 ; θ) − 1 (9)

or the Gaussian and t copulas, Eq. (9) can be further simplified as

= 2 arcsin θπ

(10)

imilarly, for the Clayton and CClayton copulas, Eq. (9) can be simpli-

ed as

= θ2 + θ (11)

ence, with the known Kendall rank correlation coefficient τ , the

opula parameter θ for the selected six copulas can be easily obtained

hrough Eqs. (9)-(11) .

. Parallel system reliability associated with different copulas

.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

r a parallel system, or combinations thereof. Among them, the se-

ies and parallel systems are two basic systems. In comparison with

he series system reliability problem, the parallel system reliability

roblem captures the difference in system probability of failure effec-

ively. Hence, only the parallel system reliability is investigated. For

implicity, the system reliability of a parallel system consisting of two

omponents is adopted. The failure of the considered parallel system

ith two components connected in parallel requires that all the two

omponents 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 per-

formance 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 ] (15a)

p fs 2 = P [ x 1 − S 3 < 0 ∩ x 2 − S 4 < 0 ] (15b)

p fs 3 = P [ x 1 − S 3 < 0 ∩ S 2 − x 2 < 0 ] (15c)

p fs 4 = P [ S 1 − x 1 < 0 ∩ x 2 − S 4 < 0 ] (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 semi-

infinite 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.

3.2. 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, re-

spectively. They are defined as {p fc 1 = P ( g 1 ( x ) < 0 ) p fc 2 = P ( g 2 ( x ) < 0 )

(16)

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)


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.

3.3. System reliability bounds from the copula viewpoint

In system reliability analyses, system reliability bounds are of-

ten 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 cor-

relation 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


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f our knowledge, this approach extends the conventional system

eliability bounds that are based on the Pearson linear correlation

oefficient (e.g., [ 2 , 26 ]).

According to the Fr ́echet–Hoeffding bounds inequality (e.g., [ 22 ]),

or every copula C and every ( u 1 , u 2 ) in I 2 , one can obtain

( u 1 , u 2 ) ≤ C ( u 1 , u 2 ) ≤ M ( u 1 , u 2 ) (20) n which W ( u 1 , u 2 ) and M ( u 1 , u 2 ) are the Fr ́echet–Hoeffding lower

nd upper bounds, respectively, and are given by

( u 1 , u 2 ) = max ( u 1 + u 2 − 1 , 0 ) (21)

M ( u 1 , u 2 ) = min ( u 1 , u 2 ) (22) t should be noted that W ( u 1 , u 2 ) and M ( u 1 , u 2 ) are also copulas. For

revity, they are referred to as W and M , respectively. Besides these

opulas, a third important copula that is frequently encountered is

he product copula �( u 1 , u 2 ) = u 1 u 2 . In general, when τ approaches ero, copula C converges to �. When τ approaches −1 and 1, copula converges to W and M , respectively. Table 1 also summarizes the

imiting cases for the selected six copulas when τ takes the above

alues. Note that all the copulas except the t copula converge to �

hen τ approaches zero. When the number of degrees of freedom

iverges, the t copula converges to �, which is also equivalent to the

aussian copula.

If the t copula is excluded from the aforementioned six copulas, for

∈ [0, 1], the Fr ́echet–Hoeffding bounds associated with the Gaussian, lackett, Frank, Clayton, and CClayton copulas can be expressed as

( u 1 , u 2 ) ≤ C ( u 1 , u 2 ) ≤ M ( u 1 , u 2 ) (23) y substituting Eq. (23) into Eq. (19) , the system reliability bounds for

he considered four parallel systems are derived as

p 2 fc ≤ p fs 1 ≤ p fc (24a)

p 2 fc ≤ p fs 2 ≤ p fc (24b)

ax (2 p fc − 1 , 0

) ≤ p fs 3 ≤ p 2 fc (24c) ax

(2 p fc − 1 , 0

) ≤ p fs 4 ≤ p 2 fc (24d) t can be seen from Eq. (24) that the reliability bounds for system I

re the same as those for system II. The reliability bounds for systems

II and IV remain the same. The system probabilities of failure for

ystems I and II are significantly higher than those for systems III and

V. When the t copula is used for systems I and II, the upper bounds

f the system probability of failure are the same as those in Eq. (24) ,

nd the lower bounds of the system probability of failure are above

he lower bounds in Eq. (24) . When the t copula is used for systems

II and IV, the lower bounds are the same as those in Eq. (24) , and the

pper bounds also exceed the upper bounds in Eq. (24) . In addition,

he system reliability bounds increase with decreasing component

robability of failure.

. Reliability analysis results for the parallel systems

The system probabilities of failure for different copulas can be

btained using Eq. (19) . The lower and upper bounds of the system

robability of failure are calculated by Eq. (24) . For illustrative pur-

oses, only positive Kendall rank correlation coefficients [0, 1] are

nalyzed in this study because the Clayton and CClayton copulas can

nly account for the positive correlation between two correlated ran-

om variables. Since the two component probabilities of failure are

ssumed the same and the selected six copulas are symmetrical about

he 45 ◦ diagonal line, the system probabilities of failure for systems II and IV remain the same. Therefore, only the system probabilities

f failure for systems I, II and III are studied step by step in the fol-

owing. 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.

4.1. System probabilities of failure for parallel system I

Fig. 2 compares the system probabilities of failure of parallel sys-

tem I associated with the selected six copulas for various component

probabilities of failure. For comparison, the bounds of system proba-

bility of failure are also presented. The system probabilities of failure

produced by different copulas differ considerably. Such a relative dif-

ference 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 incom-

plete 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 in-

creasing Kendall rank correlation coefficient. When τ approaches 1,

the system probabilities of failure associated with the six copulas con-

verge 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 small-

est 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 cop-

ulas 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 tar-

get 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 fail-

ure. 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 fs

= 1.0E −03, 1.0E −04, 1.0E −05, and 1.0E −06, 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 monotoni-

cally 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 .

4.2. 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 Clay-

ton copula. Hence, the probabilities of failure produced by the Clayton

copula for system II are identical to those produced by the CClayton


Fig. 2. Probabilities of failure produced by different copulas for parallel system I.

Fig. 3. Effect of correlation coefficients on probabilities of failure produced by different

copulas for parallel system I ( p fc = 0.001).

copula for system I, and vice versa. Additionally, the results produced

by the t , Gaussian, Plackett, and Frank copulas remain unchanged be-

cause these copulas are all symmetrical with respect to the 45 ◦ and135 ◦ diagonal lines of a unit square. For brevity, the results for systemII are not presented again.

4.3. System probabilities of failure for parallel system III

Fig. 5 compares the system probabilities of failure of parallel sys-

tem III associated with the six copulas for various component prob-

abilities of failure. The system reliability bounds are calculated by

Eq. (24c) . In comparison with systems I and II, the system probabil-

ities 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 CClay-

ton 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 = 0.1is 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 below0.1 would result in system probabilities of failure significantly below

1.0E −17, which is beyond the precision of the commonly used com-puters. 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 prob-

abilities 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 verysensitive 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 96,976to 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.


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.


Fig. 5. Probabilities of failure produced by different copulas for parallel system III.

Fig. 6. Effect of correlation coefficients on probabilities of failure produced by different

copulas for parallel system III ( p fc = 0.1).

4.4. 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 fail-

ure 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 systemsI and II, the probabilities of failure produced by different copulas in-

crease 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 pro-

duced 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

Based on the aforementioned study, it can be observed that the

system probability of failure of a parallel system cannot be deter-

mined uniquely under incomplete probability information. The sys-

tem probabilities of failure produced by different copula models differ

considerably. Hence, evaluation of the system reliability under in-

complete 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, whichresults 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


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.


Fig. 8. Effect of tail dependence of copulas on system probabilities of failure for parallel

systems I, II, and III, respectively.

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 reducedset of candidate copulas, the bound of system probability of failure

can be improved.

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 prob-

lem is formulated. The effect of the bivariate distribution models us-

ing copulas on the parallel system reliability is investigated. Several

conclusions can be drawn from this study:

(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 tremen-

dously 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 increas-

ing target system reliability level. The maximum ratio may not

be associated with a large correlation. It can happen at an in-

termediate correlation level.

(3) The tail dependence of copulas has a significant influence on

the parallel system reliability. This finding highlights the im-

portance 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 lin-

ear 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 par-

allel system reliability problem and the calculated probability

of failure may be severely biased, which has never been ex-

plained so far. This finding should be noted in practical system

reliability analyses.

Acknowledgments

This work was supported by the National Science Fund for Distin-

guished Young Scholars (Project No. 51225903 ), the National Natural

Science Foundation of China (Project No. 51079112 ) and the Doc-

toral Program Fund of Ministry of Education of China (Project No.

20120141110009 ).


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http://dx.doi.org/10.1177/1748006X13481928http://dx.doi.org/10.1177/1748006X13481928

Impact of copulas for modeling bivariate distributions on system reliability1 Introduction2 A copula-based method for modeling the bivariate distribution of two variables3 Parallel system reliability associated with different copulas3.1 Formulation of the system reliability problem for a parallel system3.2 Formulae for calculating system probability of failure3.3 System reliability bounds from the copula viewpoint

4 Reliability analysis results for the parallel systems4.1 System probabilities of failure for parallel system I4.2 System probabilities of failure for parallel system II4.3 System probabilities of failure for parallel system III4.4 Effect of tail dependence of copulas on system reliability

5 Discussion6 Summary and conclusionsAcknowledgmentsReferences

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