Environmental Engineering Reference
In-Depth Information
2.2.3 Four selected copulas
There are many copulas in the literature belonging to various classes of copula families. The
difference among various copulas is mainly characterized by dependency characteristics such as
symmetry, tail dependence, and range of correlation coefficients (e.g., Nelsen 2006). Owing to
the strongly negative correlation between
c
and ϕ, the copulas allowing a wide range of negative
correlation coefficients are selected to fit the dependence structure between
c
and ϕ. A literature
review reveals that the Gaussian copula, Plackett copula, Frank copula, and No.16 copula (e.g.,
Nelsen 2006) are appropriate for describing such dependence structure between
c
and ϕ. The
Among the four copulas, the Gaussian copula is an elliptical copula. The Plackett copula is a
member of the Plackett copula family. The Frank and No.16 copulas are Archimedean copulas.
All the four copulas can describe negative dependences, and the values of the Kendall rank
correlation coefficients between
c
and ϕ can approach −1. For a more general rule of choosing
copulas, the interested readers are referred to Nelsen (2006) and McNeil et al. (2005).
Figure 2.2
s
hows the contour plots for the bivariate distributions of two standard normal
variables
X
1
and
X
2
using Plackett, Frank, and No.16 copulas. The bivariate distributions
(a)
3
(b)
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
x
1
x
1
(c)
3
(d)
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
x
1
x
1
Figure 2.2
Contour plots for the bivariate distributions of
X
1
and
X
2
using Plackett, Frank and No.16 copu-
las. (a) Plackett copula (
θ
=
0.213,
τ
=
−
1/3), (b) Frank copula (
θ
=
−
3.306,
τ
=
−
1/3), (c) No.16
copula (
θ
=
0.035,
τ
=
−
1/3), and (d) No.16 copula (
θ
=
0.067,
θ
=
−
1/5).
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