Civil Engineering Reference
In-Depth Information
by trying several candidates. For quantitative model selection purpose, the
AIC is useful; a model with the smallest AIC is the preferred copula func-
tion. The commonly used copula functions include the elliptical copulas (e.g.
normal and
t
copulas) and Archimedean copulas (McNeil
et al.
, 2005). For
modelling seismic loss data, an investigation by Goda and Ren (2010)
showed that the Gumbel and heavy right tail (HRT) copulas are adequate,
because of signifi cant upper tail dependence.
The bivariate normal copula and
t
copula are defi ned by:
[
]
N
ρ
(
)
=
−
1
()
−
1
()
Cuu
,
ΦΦ
u
,
Φ
u
,
[28.7]
12
ρ
1
2
and
[
]
(
)
=
() ()
C uu
t
νρ
,
t
t
−
1
ut
,
−
1
u
,
[28.8]
,
12
νρ ν
,
1
ν
2
where
Φ
ρ
(•) is the bivariate normal distribution function with zero mean
and the linear correlation coeffi cient
Φ
−1
(•) is the inverse standard normal
distribution;
t
ν
,
ρ
(•) is the bivariate
t
distribution function with zero mean,
degree-of-freedom parameter
ρ
;
; and
t
−1
(•) is the inverse standard
t
distribution. Both normal and
t
copulas are symmetrical, and the normal
copula is a limiting case of the
t
copula, when
ν
, and
ρ
becomes infi nity.
On the other hand, the Gumbel and HRT copulas are given by:
ν
{
}
1
θ
(
)
=
(
)
+−
θ
(
)
θ
Cuu
Gu
,
exp
− −
⎣
ln
u
ln
u
⎦
,
[28.9]
12
1
2
and
−
1
θ
(
)
=+−+ −
(
)
+−
−
θ
(
)
−
−
θ
Cuuuu
HRT
,
11
⎣
u
1
u
1
⎦
,
[28.10]
12
1
2
1
2
where
is the copula model parameter. It is noteworthy that the Gumbel
copula has been studied extensively in bivariate extreme value theory, and
is known as the logistic dependence model (Coles, 2001). The logistic model
parameter
θ
α
is related to
θ
in Equation [28.9] as
α
=
1/
θ
;
α
=
0 indicates
independence, whereas
1 indicates perfect dependence.
Regarding the estimation of the dependence parameter, one more remark
is in order. In bivariate extreme value theory, estimation of both marginal
GP parameters and dependence parameter is carried out based on the
maximum likelihood method by treating data pairs as censored samples
(Coles, 2001; McNeil
et al.
, 2005). This is a sensible approach by utilising all
available information of POT data (not only data pairs with joint exceed-
ance but also those with exceedance of one of the thresholds only). Another
more simplifi ed approach is to focus only on data pairs that jointly exceed
respective threshold values (Dupuis and Jones, 2006). Both approaches are
valid, but emphasise different aspects of the bivariate POT data; the former
provides an overall picture of data pairs that exceed threshold values at
α
=
Search WWH ::
Custom Search