Environmental Engineering Reference
In-Depth Information
distributions are constructed by substituting the CDFs of
X
1
and
X
2
into the corresponding
copula functions in
Table 2.1
. The copula parameters θ are obtained from Kendall rank corre-
lation coefficients τ using
Equation 2.13
. It is clear that the bivariate distributions constructed
from different copulas are different, indicating that each copula has its own dependence struc-
ture. It is well known that the dependence structure underlying a copula has a significant
impact on reliability (Tang et al. 2013a,b,c). To represent a dependence structure, the concepts
of symmetry and tail dependence can be used. These two concepts are introduced below.
There are two forms of symmetry for a copula function. One is the exchangeability. The
other is the radial symmetry. For the former, a copula function
C
(
u
1
,
u
2
; θ) is said to be sym-
metric when
C
(
u
1
,
u
2
; θ) =
C
(
u
2
,
u
1
; θ) for all (
u
1
,
u
2
) in [0, 1]
2
. Alternatively,
C
(
u
1
,
u
2
; θ) is
symmetric when
u
1
and
u
2
are exchangeable (e.g., Nelsen 2006). The exchangeability can
only guarantee that a copula function is symmetrical with respect to the 45° diagonal line of
a unit square (i.e., a domain defined by [0, 1]
2
). It is clear from
Table 2.1
that
u
1
and
u
2
are
exchangeable for the Gaussian, Plackett, Frank, and No.16 copulas. Therefore, the Gaussian,
Plackett, Frank, and No.16 copulas are symmetric copulas. If the variables
X
1
and
X
2
have
the same marginal CDFs, the bivariate distributions of
X
1
and
X
2
using the symmetric cop-
ulas are also symmetric. For example, the bivariate distributions of
X
1
and
X
2
using the
Gaussian, Plackett, Frank, and No.16 copulas in
Figures 2.1
and
2.2
a
re symmetrical with
respect to the 45° diagonal line of a square (i.e., a domain defined by [−3, 3]
2
in this example).
Besides the changeability, the other symmetry is the radial symmetry. A copula func-
tion
C
(
u
1
,
u
2
; θ) is said to be radially symmetric when
C
(
u
1
,
u
2
; θ) =
u
1
+
u
2
− 1 +
C
(1 −
u
1
,
1 −
u
2
; θ) for all (
u
1
,
u
2
) in [0, 1]
2
(e.g., Nelsen 2006). The radial symmetry implies that
a copula function is symmetrical with respect to the center (i.e., a point defined by [0.5,
0.5]) of a unit square. With the above requirement, it can be concluded from
Table 2.1
that
the Gaussian, Plackett, and Frank copulas are radially symmetric. On the other hand, the
No.16 copula is not radially symmetric. The reason is that
C
(
u
1
,
u
2
; θ) is not identical to
u
1
+
u
2
− 1 +
C
(1 −
u
1
, 1 −
u
2
; θ) for the No.16 copula. If
X
1
and
X
2
have the same marginal
CDFs, the bivariate distributions of
X
1
and
X
2
using the radially symmetric copulas are
also radially symmetric. For example, the bivariate distributions of
X
1
and
X
2
using the
Gaussian, Plackett, and Frank copulas in
Figures 2.1
and
2.2
are symmetrical with respect
to the center (i.e., a point defined by [0, 0] in this example) of a square. On the contrary, the
cal with respect to the center of a square. The common feature of radial symmetry underly-
ing the Gaussian, Plackett, and Frank copulas will lead to relatively similar reliability results
as demonstrated in Section 2.5.
Tail dependence, as its name says, measures the dependence between
X
1
and
X
2
in the
upper-right quadrant or in the lower-left quadrant of a bivariate distribution. Recall that
F
1
(
x
1
) and
F
2
(
x
2
) are the marginal CDFs of two continuous random variables
X
1
and
X
2
,
respectively. Let
Fx
1
−
() be the inverse CDFs of
X
1
and
X
2
, respectively. Then,
the coefficient of upper tail dependence, λ
U
, is defined as the limit (if it exists) of the condi-
tional probability that
X
2
is greater than the 100
t
-th percentile of
F
2
given that
X
1
is greater
than the 100
t
-th percentile of
F
1
as
t
approaches 1 (e.g., Nelsen 2006)
1
−
1
() and
Fx
2
1
2
2
1
−
−
1
λ
U
=
lim
PX
[
>
FtXFt
( )
>
( )]
(2.16)
2
1
1
−
t
→
1
where
Ft
1
−
() are the 100
t
-th percentiles of
F
1
and
F
2
, respectively. If λ
U
∈ (0,1],
X
1
and
X
2
are said to be asymptotically dependent the upper tail; if λ
U
= 0,
X
1
and
X
2
are
said to be asymptotically independent the upper tail. Similarly, the coefficient of lower tail
dependence, λ
L
, is defined as the limit (if it exists) of the conditional probability that
X
2
is
−
() and
Ft
2
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