Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 79

where C ( u 1 , u 2 ; θ) is a bivariate copula function in which θ is a copula parameter measur-
ing the dependence between X 1 and X 2 . It is worthwhile to point out that the vector θ in
Equation 2.2 is reduced to a single parameter θ when n = 2. In other words, all bivariate
copulas have only one copula parameter θ. By taking derivatives of Equation 2.5 , the bivari-
ate PDF of X 1 and X 2 , f ( x 1 , x 2 ), is obtained as

fx xDFx Fx (, ) = ( ( ,(); )( )( ) θ fxfx = Du uf ( ,;)( ) θ xf

22 ()

x (2.6)

12 11 22 1122 12 11

where D ( u 1 , u 2 ; θ) is a bivariate copula density function, which is given by

2

Du u (, ;) θ =∂ C uu (, ;) θ / ∂∂ uu (2.7)

12 12 12

Theoretically, the joint CDF and PDF of X 1 and X 2 can be determined by Equations
2.5 and 2.6 if the marginal CDFs of X 1 and X 2 , and the copula function are known. For
example, when both X 1 and X 2 are standard normal random variables (the CDF and PDF
of a standard normal random variable are given in Chapter 1 ), substituting the CDFs of X 1
and X 2 into the Gaussian copula in Table 2.1 leads to the well-known bivariate standard
normal distribution. It is noted that this is not the case for other copulas. If the CDFs of
X 1 and X 2 are substituted into the Plackett copula, a bivariate standard normal distribu-
tion will not be expected. In other words, the dependence structure underlying a bivariate
standard normal distribution is uniquely characterized by a Gaussian copula (Lebrun and
Dutfoy 2009a, b). Hence, the Pearson correlation coefficient ρ (see Chapter 1 ) underlying
the bivariate standard normal distribution is incidentally equal to the copula parameter θ
underlying the Gaussian copula.
Figure 2.1 shows the contour plots for the bivariate distributions of two standard nor-
mal random variables X 1 and X 2 using a Gaussian copula with θ equal to −0.5, 0, and 0.5.
Note that these copula parameters θ are obtained from the Kendall rank correlation coef-
ficients τ (see Section 2.2.2) equal to −1/3, 0, and 1/3, respectively. It is clear that there is a
strong negative correlation between X 1 and X 2 when θ = −0.5 (τ = −1/3), whereas a strong
positive correlation exists between X 1 and X 2 when θ = 0.5 (τ = 1/3). For θ = 0 (τ = 0), X 1
and X 2 are uncorrelated. It is well known that an important step in copula modeling is
the determination of copula parameters θ. Since various copulas have their own param-
eters, it is desirable to have a common dependence measure such as the Kendall rank cor-
relation coefficient adopted above to obtain the copula parameters θ from the measured
data. In the next section, two commonly used dependence measures are introduced for
this purpose.

2.2.2 Dependence measures
This section focuses on two kinds of dependence measure: (1) the usual Pearson linear
correlation coefficient (Pearson's rho, ρ), and (2) the Kendall rank correlation coefficient
(Kendall's tau, τ). Their relations to the copula parameter θ are highlighted.

2.2.2.1 Pearson's rho
The Pearson linear correlation coefficient has been defined in Chapter 1 . In this chapter, the
Pearson linear correlation coefficient and its relation to the copula parameter θ are further
explained within the framework of the copula theory. Pearson's rho, which is also called a