Environmental Engineering Reference
In-Depth Information
(a)
1.0
(b)
1.0
Measured data
Simulated data
Measured data
Simulated data
0.5
0.5
0.0
0.0
0.0
0.5
1.0
0.0
0.5
1.0
u 1
u 1
(c)
1.0
(d)
1.0
Measured data
Simulated data
Measured data
Simulated data
0.5
0.5
0.0
0.0
0.0
0.5
1.0
0.0
0.5
1.0
u 1
u 1
Figure 2.5 Scatter plots of measured and simulated u 1 and u 2 for CU dataset. (a) Gaussian copula, (b)
Plackett copula, (c) Frank copula, and (d) No.16 copula.
for measured datasets of shear strength parameters are given by Equation 2.29 . To examine
the fitness of the identified copulas to dependence structures underlying the measured data,
300 samples (i.e., m = 300) are drawn from the four selected copulas. As an example, Figure
2.5 s hows the simulated (300 samples) and measured (64 samples) data of U 1 and U 2 for
CU dataset. Note that all the copulas selected can capture the symmetry of the measured
data about the 135° diagonal line of a unit square. From visual inspection, it is very difficult
to distinguish the best-fit copula among them due to the limited number of measured data.
Generally, the Frank copula, having been identified as the best-fit copula using the AIC and
BIC criteria, can match the measured data adequately.
Similarly, the bivariate distribution of shear strength parameters is simulated by obtaining
its physical samples X m ×2 = [ c , ϕ] using the algorithms presented previously. In this simu-
lation, the best-fit marginal distributions (TruncNormal distribution for both c and ϕ in
CU dataset) are adopted to convert u m ×2 into X m ×2 . To further examine the fitness of the
constructed bivariate distributions using various copulas, Figure 2.6 compares the simu-
lated (300 samples) and measured (64 samples) data of c and ϕ for CU dataset. The bivari-
ate distribution constructed using the Frank copula agrees well with the measured data in
comparison with the other three copulas. Compared with the results in Figure 2.5 that are
affected by the copula goodness of fit alone, the results in Figure 2.6 are affected by both
 
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