Environmental Engineering Reference
In-Depth Information
Table 7.1
Summary of hypothesis test results for the
polynomial function example
Uncertain parameter
X
1
X
2
1.00
1.00
Unconditional mean
μ
0.05
0.10
Standard deviation
σ
Failure sample mean
μ
f
1.01
1.20
Absolute value of
Z
H
15.76
157.65
Ranking
2
1
values of
Z
H
are 15.76 and 157.65 for
X
1
and
X
2
, respectively. Based on the absolute values
of
Z
H
, the uncertain parameter
X
2
has more significant effects on the failure probability
than
X
1
. The effect of the important uncertain parameters (e.g.,
X
2
in the polynomial func-
tion example) on failure probability can be further quantified using a Bayesian analysis
described in the next section.
7.4.2 bayesian analysis
For the important uncertain parameter identified from hypothesis tests, Bayesian analysis
can be performed to explore how the failure probability varies as the important parameter
changes. Let
X
k
be an important uncertain parameter identified from the hypothesis test. In
the context of the Bayes' theorem (e.g., Ang and Tang, 2007):
px FPF
px
(
|
) ()
()
P
(| )
Fx
=
k
(7.1 2)
k
k
where
P
(
F
|
x
k
) is the conditional failure probability at
X
k
=
x
k
,
p
(
x
k
|
F
) is the conditional PDF
of
x
k
given that failure occurs,
P
(
F
) is the failure probability, and
p
(
x
k
) is the unconditional
PDF of
x
k
that is given before simulation and can be determined analytically. Both
P
(
F
)
and
p
(
x
k
|
F
) are estimated from failure samples of MCS. Consider an MCS run with a total
sample number of
n
t
.
P
(
F
) is estimated using the following equation:
n
n
f
t
P
(
F
=
(7.13)
in which
n
f
is the number of failure samples in the simulation.
p
(
x
k
|
F
) is estimated from an
x
k
histogram in which
x
k
is divided into a number of bins (e.g.,
n
b
bins) and
p
(
x
k
|
F
) for
x
k
within a bin
j
(i.e.,
Px
(
∈
binF
|
)
, for
j
= 1, 2, …,
n
b
) is estimated using
k
j
n
n
j
f
px FPx in F
(
|
)
= ∈ =
(
|
)
(7.14)
k
k
j
in which
n
j
is the number of simulation samples where failure occurs and the
x
k
value falls
into bin
j
. In this way, the “interval” probability
Px
(
∈
binF
| )
(instead of
p
(
xF
|
)
) for
k
j
k
x in
k
∈
is obtained, yielding an interval analog of
Equation 7.12
:
j
Px
(
binFPF
Px
∈
|
)()
k
j
PF x in
(|
∈=
)
,
for
j
= …
12
,
,
,
n
(7.15)
k
j
b
(
∈
bin
)
k
j
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