Environmental Engineering Reference
In-Depth Information
where
∫
Px
(
∈=
bin
p
()
x
dx
k
j
k
k
(7.16)
bin
j
Since
p
(
x
k
) is known before the simulation,
Px
(
∈
bin
)
can be determined analytically
k
j
without any information from MCS.
Combining
Equations 7.13
and
7.14
leads to
n
n
j
t
(7.17 )
Px
(
∈
binFPF
|
) ()
=
,
for
j
= …
12
,
,,
n
k
j
b
Since both
n
f
and
n
j
, for
j
= 1, 2, …,
n
b
, are obtained directly from a single simulation run
by simply counting the numbers of failure samples,
PF x in
k
| ∈ , for
j
= 1, 2, …,
n
b
, from
Equations 7.15
and
7.17
can be estimated directly from a single simulation run.
Note that counting the sample numbers for
Equations 7.15
and
7.17
can be considered
to contain two steps: first, grouping the simulation samples in accordance with the bins of
x
k
and second, counting the sample number for each bin of
x
k
. From this perspective, the
Bayesian analysis herein is equivalent to grouping all simulation samples into
n
b
subsets
(i.e.,
n
b
bins) in accordance with the
x
k
value and to determining the
P
(
F
) in each subset
(i.e.,
PF
(
)
j
(| ∈ ). When the bin interval is small, the
x
k
value is virtually deterministic
and constant. Note that
PF
x in
k
j
(| ∈ , for
j
= 1, 2, …,
n
b
, is a variation of the failure
probability
P
(
F
) as a function of the value of the
x
k
bin. It can be considered as results of
the
n
b
repeated simulation runs in which the
x
k
value adopted in each simulation run is
deterministic but different (i.e., the value of
bin
j
, for
j
= 1, 2, …,
n
b
), and the other uncertain
parameters (i.e.,
X
1
,
X
2
, …,
X
k
−1
,
X
k
+1
, …,
X
n
) remain random. In other words, the Bayesian
analysis approach, which makes use of the failure samples generated in a single simulation
run for assessment of failure probability, provides results that are equivalent to those from
a sensitivity study, which frequently includes many repeated simulation runs with differ-
ent given values of
x
k
in each run. Additional computational times and efforts for repeated
simulation runs in the sensitivity study can be avoided when using the Bayesian analysis
described herein. In addition, it is also worthwhile to point out that the Bayesian analysis
results can be used to further provide the sensitivity on failure probability or PDF of the
operator output to
X
k
through sample reassembling when
X
k
is considered as random, but
with statistical parameters or the distribution type which are different from those adopted
in the nominal case. Interested readers are referred to Wang (2012) for details on sampling
reassembling in MCS.
unconditional one
p
(
x
k
) provides an indication of the effect of the uncertain parameter
X
k
on failure probability. Generally speaking,
P
(
F
|
x
k
) changes as the values of the uncertain
parameter
x
k
change. However, when
p
(
x
k
|
F
) is similar to
p
(
x
k
),
P
(
F
|
x
k
) remains more or
less constant regardless of the values of
x
k
. This implies that the effect of
X
k
on the failure
probability is minimal. Such implication can be used to validate the results obtained from
hypothesis tests.
Consider again the polynomial function example and the direct Monte Carlo run with
100,000 samples in Sections 7.2.2 and 7.4.1. The hypothesis tests in the previous section
have shown that the uncertain parameter
X
2
has the most significant effect on the failure
probability. Bayesian analysis is therefore performed to further quantify the variation
of
P
(
F
) as the
X
2
value changes. The 6213 failure samples are collected to analyze their
x in
k
j
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