Environmental Engineering Reference
In-Depth Information
(a)
(b)
35
35
Measured data
Simulated data
Measured data
Simulated data
30
30
25
25
20
20
15
15
10
10
0
30
60
90
120
150
0
30
60
90
120
150
c
(kPa)
c
(kPa)
(c)
(d)
35
35
Measured data
Simulated data
Measured data
Simulated data
30
30
25
25
20
20
15
15
10
10
0
30
60
90
120
150
0
30
60
90
120
150
c
(kPa)
c
(kPa)
Figure 2.6
Scatter plots of measured and simulated
c
and
ϕ
for CU dataset. (a) Gaussian copula, (b) Plackett
copula, (c) Frank copula, and (d) No.16 copula.
marginal distributions and copula goodness of fits. Similarly, the MATLAB codes for simu-
lating copulas and bivariate distribution are appended.
On the basis of simulated samples
u
m
×2
= [
U
1
,
U
2
], Pearson's rho, ρ, and Kendall's tau, τ,
the similar approach, the ρ and τ between
X
1
and
X
2
can be obtained using the simulated
and τ of CU dataset are also listed. Note that the ρ between
X
1
and
X
2
are significantly dif-
ferent from those between
U
1
and
U
2
. The reason is that a nonlinear monotonic transforma-
tion underlying the CDF transformation is employed to transform
u
into
X
, and Pearson's
rho is not invariant under such nonlinear transformations as mentioned previously. Unlike
Pearson's rho, the τ between
X
1
and
X
2
are the same as those between
U
1
and
U
2
as shown
monotone transformations. In addition, the τ between
X
1
and
X
2
associated with various
copulas agree well with the prescribed τ between the measured data and the maximum
relative error in τ is only 2%, while the resulting ρ between
X
1
and
X
2
may differ consider-
ably from the prescribed ρ between the measured data. The reason is that there is no non-
Gaussian simulation technique available to date that can match rank and product-moment
correlation simultaneously (Grigoriu 1998; Phoon et al. 2004).
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