Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 95

As stated previously, the Gaussian, Plackett, Frank, and No.16 copulas are selected as can-
didate copulas to fit the measured dependence structure between c and ϕ. Similarly, the AIC
and BIC can be used to identify the best-fit copula underlying the measured data. A copula
producing the smallest AIC and BIC values is considered to be the best-fit copula. The cor-
responding AIC and BIC are, respectively, defined as

N

∑

AIC =− 2 ln Du (, ;) u θ + 2 k (2.27)

12 i i 2

i = 1

and

N

∑

BIC =− 2 ln Du (, ;) u θ + k ln N (2.28)

12 i i 2

i = 1

ln (, ;θ is
the logarithm of the likelihood function for a specified copula; k 2 is the number of copula
parameters; and {( u 1 i , u 2 i ), i = 1, 2, …, N } are the empirical distribution values of measured
( c , ϕ), which are defined as

where D ( u 1 i , u 2 i ; θ) is the copula density function shown in Table 2.1 ; ∑ =

N

Du u

i 1 1 i 2 i

‡ rank () c

i

u =

‰

1 i

N + 1

= 1, 2, … ,

i N (2.29)

ˆ

rank () φ

‰

u =

i

2 i

Š N + 1

in which rank( c i ) [or rank(ϕ i )] denotes the rank of c i (or ϕ i ) among the list { c 1 , …, c N } (or
{ϕ 1 , …, ϕ N }) in an ascending order. Note that Equation 2.29 is another estimator of the
empirical CDF, which is similar to the empirical CDFs in Equations 1.11 and 1.12 of
Chapter 1 . Hence, ( u 1 i , u 2 i ) are realizations of standard uniform variables. For the four
selected copulas, all of them are single-parameter copulas. Therefore, k 2 = 1 is used in
Equations 2.27 and 2.28 . Similarly, identification of the best-fit copula using AIC and
BIC includes the following five steps:

1. Compute Kendall's tau, τ, of the measured data of shear strength parameters using
Equations 2.14 and 2.15 . The corresponding results are listed in Table 2.2 .

2. Estimate the copula parameters θ underlying the four candidate copulas using Equation
2.13 or Equation 2.20 for the Gaussian copula or Equation 2.21 for the Frank and
No.16 copulas based on the measured τ in Step 1. The results are listed in Table 2.7 .

3. Calculate the empirical distributions u = ( U 1 , U 2 ) of measured ( c , ϕ) using Equation
2.29 . Figure 2.4 shows the scatter plots of U 1 and U 2 . There is a strong negative depen-
dence between U 1 and U 2 . Furthermore, the samples of U 1 and U 2 for the three data-
sets are basically symmetrical with respect to the 135° diagonal line of a unit square,
which graphically demonstrates that the selected four copulas are capable of capturing
the dependence structure underlying the measured data.

4. Evaluate the AIC and BIC values for the four candidate copulas using Equations 2.27
and 2.28 , respectively.

5. The copula producing the smallest AIC and BIC values is identified to be the best-fit
copula to the measured data.