Civil Engineering Reference
In-Depth Information
(formulated in Sections 26.3.1 and 26.3.2) for deformation, shear and
moment demands on offshore wind turbine support structures due to
seismic loading in addition to environmental (wind, wave, and current) and
operational loadings. To develop parsimonious probabilistic demand models
(i.e., with only the explanatory functions that are strictly needed) a model
selection process is used to identify the important explanatory functions
among the candidates presented in Table 26.1.
Model selection
A step-wise addition method is used for model selection. The fi rst step of
this process is to choose
1 and
estimate the model parameters using the Bayesian approach described
next. In the second step, the residuals (i.e., the difference between the
virtual data and the median prediction from the probabilistic model) are
plotted versus all the remaining explanatory functions. Finally, in the third
step, an additional term
γ
k
(
x
,
w
,
θ
k
)
=
θ
k
1
h
k
1
(
x
,
w
), where
h
k
1
(
x
,
w
)
=
γ
k
if there is evi-
dence of a trend in the plots of the residuals. In particular, the term that
corresponds to the explanatory function with the clearest trend (here
h
kj
(
x
,
w
)) should be added. Steps 2 and 3 are then repeated until no clear trend
remains.
θ
kj
h
kj
(
x
,
w
) might be added to
Bayesian updating
The unknown model parameters
Θ
k
are estimated using a Bayesian approach
using the following updating rule (Box and Tiao, 1992):
(
)
=
(
) (
)
f
Q
κ
L
Q
p
Q
[26.7]
k
k
k
where
f
(
Θ
k
)
=
the posterior distribution of
Θ
k
that represents the updated
state of knowledge,
L
(
Θ
k
)
=
the likelihood function that represents the
objective information on
Θ
k
that comes from this study's virtual experi-
ments,
p
(
Θ
k
)
=
the prior distribution of
Θ
k
that refl ects the state of knowl-
edge about
a nor-
malizing factor. The likelihood is a function that is proportional to the
conditional probability of observing the results from the virtual experi-
ments for a given value of
Θ
k
prior to conducting the virtual experiments, and
κ
=
Θ
k
. Following Gardoni
et al.
(2002),
L
(
Θ
k
) is
written as:
{
}
×
()
()
1
σ
r
q
r
q
⎡
⎢
⎤
⎥
⎡
⎢
⎤
⎦
∏
∏
lower bound data
(
)
∝
ik
k
ik
k
L
Q
ϕ
Φ
−
[26.8]
⎥
k
σ
σ
equality data
k
k
k
ˆ
k
(
x
i
,
w
i
)
where
r
ik
(
θ
k
)
=
D
ik
−
−
γ
k
(
x
i
,
w
i
,
θ
k
),
D
ik
=
observed value for the
k
th demand for a given
x
i
and
W
i
.
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