Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 96

Table 2.7 Copula parameters, AIC, and BIC values for the Gaussian, Plackett, Frank, and No.16 copulas
Dataset

Gaussian [ θ , AIC, BIC] Plackett [ θ , AIC, BIC] Frank [ θ , AIC, BIC] No.16 [ θ , AIC, BIC]

CD

− 0.567, − 19.63, − 17.49

0.164, − 18.00, − 15.85

− 3.946, − 20.18, − 18.04

0.027, − 15.31, − 13.17

CU − 0.754, − 43.50, − 41.35 0.068, − 43.00, − 40.84 − 6.590, − 44.10, − 41.95 0.011, − 32.39, − 30.23

UU

− 0.645, − 26.40, − 24.29 0.118, − 27.38, − 25.27 − 4.841, − 27.23, − 25.11 0.020, − 23.28, − 21.17

Note: The AIC and BIC values are bold if the corresponding copula is preferred.

Table 2.7 summarizes the AIC and BIC values associated with the four copulas for various
datasets of ( c , ϕ). Note that both the AIC and BIC values indicate that the Plackett copula is
the best-fit copula for the UU dataset. For the CD and CU datasets, the Frank copula is the
best-fit copula. All of them are better fits to measured data than the Gaussian copula based
on AIC and BIC. These results indicate that the Gaussian copula may not provide the best fit
to the dependence between c and ϕ. In short, the marginal distributions and copula to con-
struct the bivariate distribution of shear strength parameters should be determined based
on the measured data in some sense such as the AIC and BIC. It is recommended that the
marginal distributions of c and ϕ and the copula for characterizing the dependence structure
between c and ϕ must be selected carefully. This is because they have a significant impact

(a) 1.0

1.0

(b)

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

0.5

0.0

0.0 0.5 1.0

u 1

Figure 2.4 Scatter plots of u 1 and u 2 for measured cohesions and friction angles. (a) CD dataset, (b) CU
dataset, and (c) UU dataset.