Environmental Engineering Reference
In-Depth Information
Fx x
(, ,
…
,
x
)
=
CF xFx
(
( ,
(),
…
,();
F x
θ
)
(2.2)
12
n
11
22
n
n
where θ is a vector of copula parameters describing the dependence among
X
1
,
X
2
, …,
X
n
.
The dimension of this vector varies with copula types from 1 × 1 to 1 × 0.5
n
(
n
− 1). For the
Gaussian and
t
copulas, the number of copula parameters is 0.5
n
(
n
− 1), which is the same
as the number of the correlation coefficient pairs in a multivariate normal distribution dis-
cussed in
Chapter 1
. If
C
is an
n
-dimensional copula and
F
1
(
x
1
),
F
2
(
x
2
), …,
F
n
(
x
n
) are mar-
marginal distributions
F
1
(
x
1
),
F
2
(
x
2
), …,
F
n
(
x
n
).
Sklar's theorem essentially states that the joint CDF of
X
can be expressed in terms of a
copula function and its marginal CDFs. In other words, fitting a joint CDF to measured data
using copulas involves two steps: (1) determining the best-fit marginal distributions for all
individual random variables, and (2) identifying the copula that provides the best fit to the
measured dependence structure from a set of candidate copulas. The above two steps can be
carried out separately, allowing different marginal distributions and dependence structures
to be incorporated into a multivariate distribution. This advantage underlying the copula
approach is critical since the processes of determining the marginal distributions and the
correlation structure in a multivariate distribution are decoupled.
The joint PDF of
X
can be derived from the joint CDF of
X
. Basically, the joint PDF is the
derivative of the joint CDF (or the joint CDF is the integration function of the joint PDF).
By taking derivatives of
Equation 2.2
,
the joint PDF of
X
,
f
(
x
1
,
x
2
, …,
x
n
), can be obtained
as (e.g., McNeil et al. 2005)
n
∂
n
CF xFx
(( ),
( ,
…
,
F x
( ;)
θ
∂
Fx
x
()
∏
…
11 22
11
nn
nn
i
i
fx x
(, ,
,
x
)
=
12
n
∂
Fx
()
…
∂
F
(
x
)
∂
i
i
=
1
n
∏
=
DF xFx
(( ),
( ,
…
,
F x
( ;)
θ
f x
()
(2.3)
11 22
nn
i
i
i
=
1
where
DCFx Fx
is a copula density function,
=∂
n
(( ),
( ,
…
,
Fx
( ;)
θ
∂
Fx
()
…∂
Fx
()
11 22
nn
11
nn
which is the derivative of a copula function
C
.
fx
=∂ ∂
is the marginal PDF of
X
i
for
i
= 1, 2, …,
n
. By introducing
u
i
=
F
i
(
x
i
), the copula function
C
(
F
1
(
x
1
),
F
2
(
x
2
), …,
F
n
(
x
n
); θ)
and the copula density function
D
(
F
1
(
x
1
),
F
2
(
x
2
), …,
F
n
(
x
n
); θ) can be rewritten as
C
(
u
1
,
u
2
,
…,
u
n
; θ) and
D
(
u
1
,
u
2
, …,
u
n
; θ), respectively. Note that
U
i
=
F
i
(
X
i
) is a standard uniform
random variable. This is because the CDF of
U
i
can be expressed as
()
Fx
()
x
i
i
i
i
i
PU
(
≤= ≤= ≤
uPFX
)
(( )
uPXFu
)
(
−
1
(
))
=
F Fu
(
−
1
(
))
=
u
(2.4)
i
i
i
i
i
i
i
i
i
i
i
i
dard uniform random variable that is bounded on the interval of [0, 1].
Since the focus of this chapter is the modeling of the bivariate distribution of shear
strength parameters, the copula theory involving two variables (i.e.,
n
= 2) is introduced in
detail from hereon. According to Sklar's theorem, the bivariate CDF of two random vari-
ables
X
1
and
X
2
,
F
(
x
1
,
x
2
), can be given by
Fx
(, )
xCFx
=
(
( ,
Fx
(); )
θ
=
Cu
(
,
u
; )
θ
(2.5)
12
11
22
12
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