Civil Engineering Reference
In-Depth Information
Pickands-Balkema-de Haan theorem states that the limiting distribution
of POT data
Y
that exceeds a suffi ciently high threshold
μ
, converges to the
GP distribution (note:
Y
=
X
−
μ
> 0). The distribution function of the GP
model is given by:
(
1
ξβ
)
−
ξ
⎧
⎩
11
−+
y
ξ
βξ
≠
0
()
=
(
)
=
Gy
PY
≤
yY
>
0
,
[28.2]
(
)
1
−
exp
−
y
=
0
where
is the shape parameter and determines the upper tail behaviour of
the data, and
ξ
is the scale parameter. Note that Equation [28.2] is given in
terms of the shifted variable
Y
, rather than the original variable
X
. Given
data
Y
, it is straightforward to estimate the parameters of the GP model
based on the maximum likelihood method. The conditional representation
of the POT data shown in Equation [28.2] can also be expressed in terms
of the original data
X
as follows:
β
(
)
=
(
)
(
)
=
(
)
+−
[
(
)
]
−
1
ξ
PX
>
x
PX
>
μ
PX
>
xX
>
μ
PX
>
μ
1
ξ
x
μ β
,
[28.3]
where
P
(
X
>
, and this value can be
estimated from the data directly. Furthermore, the fractile value
x
p
that
corresponds to the exceedance probability
p
can be obtained as:
μ
) is the probability of exceedance of
μ
β
ξ
(
(
)
)
−
ξ
x
=+
μ
⎣
P X
>
μ
p
1.
⎦
[28.4]
p
The confi dence interval of the fractile value
x
p
can also be calculated rela-
tively easily (Coles, 2001; McNeil
et al.
, 2005).
A key step in modelling POT data using the GP model is to fi nd a rea-
sonable value of
value that is too low results in poor approximation of data using the GP
model, whereas one that is too high reduces the number of data above the
threshold level. Several diagnostic methods are available, such as the empir-
ical mean excess plot and the shape parameter plot (Cebrian
et al.
, 2003;
McNeil
et al.
, 2005). The mean excess plot examines a trend of
E
[
X
μ
. A trade-off exists for the estimation of
μ
, because a
μ
−
μ
|
X
>
: a linear relationship is indicative of a suffi ciently
high threshold level. The shape parameter plot uses the stability of
μ
] as a function of
μ
for
upper tail data. These two methods will be employed in Section 28.4. In
summary, with given data at hand, an adequate threshold parameter
ξ
needs
to be selected based on diagnostic methods, and then, the other parameters
P
(
X
>
μ
are obtained.
The extreme value distributions, such as GEV and GP models, have sta-
tistical foundation in dealing with extreme data. Nevertheless, when actual
data are considered, it is recommended to explore a wider class of probabil-
ity distributions, particularly with heavy right tails, because other popular
distributions, such as the lognormal and gamma models, and more elabo-
μ
),
ξ
and
β
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