Environmental Engineering Reference
In-Depth Information
4. Set
U
1
=
V
1
and
U
2
= [
c
− (1 − 2
V
2
)
d
]/2
b
. Then, two correlated standard uniform vectors
are obtained as
u
m
×2
= [
U
1
,
U
2
] belonging to the Plackett copula (e.g., Nelsen 2006).
2.4.1.3 Frank and No.16 copulas
1. Simulate two independent standard normal vectors
Z
m
×2
= [
Z
1
,
Z
2
]. This can be obtained
using MATLAB:
z = randn (m,2)
and MATLAB function
ra n d n('state',1)
is
used to fix the initial seed. If the sample size
m
is small, the MATLAB command
z*inv(chol(cov(z)))
is further adopted to eliminate the sampling correlations
underlying the simulated
Z
m
×2
.
2. Set
v
= Φ(
Z
). Then two independent standard uniform vectors
v
m
×2
= [
V
1
,
V
2
] are
obtained. This can be realized from MATLAB using
v
=
normcdf
(
Z
).
3. Set
U
1
=
V
1
.
4. Set
V
2
=
C
2
(
U
2
|
U
1
) in which
C
2
(
U
2
|
U
1
) is the conditional distribution of
U
2
given the
values of
U
1
. It can be calculated by (e.g., Nelsen 2006)
ϕϕ ϕ
ϕϕ
−
11
()
(( )
u
+
(
u
))
1
2
Cuu
(
|
)
=
(2.30)
221
−
11
()
(( ))
u
1
in which φ(
.
) is the generator function of an Archimedean copula. The generator functions
the equation
V
2
=
C
2
(
U
2
|
U
1
) using the bisection method. Two correlated standard uniform
vectors are obtained as
u
m
×2
= [
U
1
,
U
2
] belonging to the Frank copula or No.16 copula (e.g.,
Nelsen 2006).
After simulating the correlated standard uniform samples
u
m
×2
= [
U
1
,
U
2
] from the four
copulas, the physical samples of shear strength parameters
X
m
×2
= [
X
1
,
X
2
] = [
c
, ϕ] can be
easily obtained using the usual CDF transform method. Set
U
1
=
F
1
(
X
1
) and
U
2
=
F
2
(
X
2
),
then
X
m
−
(. and
F
2
−
(. are the inverse CDFs
of
X
1
and
X
2
, respectively (e.g., Ang and Tang 1984). The inverse CDFs for the four distri-
between shear strength parameters only, whereas
X
m
×2
relies on both the correlation and
marginal distributions underlying the shear strength parameters.
−
1
12
1
−
1
=
[
XX
,
]
=
[
FUFU
(
)
,
(
)]
in which
F
1
×
2
1
2
1
2
2.4.2 Simulation of copulas and bivariate distribution
The copulas are simulated through obtaining their correlated standard uniform samples
u
m
×2
= [
U
1
,
U
2
] using the simulation algorithms in Section 2.4.1. The samples of
U
1
and
U
2
Table 2.8
Transformations of
U
to
X
for the selected four distributions
Distribution
X
=
F
−
1
(U; p, q)
TruncNormal
XpqU
−
1
{[
1
(
pq
)]
(
pq
)}
=+ −− +−
Φ
Φ
Φ
Lognormal
X
exp[
q
Φ
1
−
()
Up
]
=
+
1
{
}
(
)
+−
TruncGumbel
Xp
q
ln
ln
U
1
exp(
exp(
pq
))
exp(
exp(
pq
))
=− −
−
−
Weibull
Xp
[ n(
1
U
)]
(/ )
1
q
=− −
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