Environmental Engineering Reference
In-Depth Information
The above expectations of products can be given analytical expressions only when
Hermite polynomials are used (see Sudret et al. (2006, Appendix I)). Otherwise, they may
be computed numerically by quadrature.
6.4.2 Distribution analysis and confidence intervals
surface. Thus, the output PDF of
ˆ
Y
=∑
∈
y
ˆ
Ψ
X
can be obtained by merely sampling
()
α
A
α
α
the input random vector
X
, say X
MCS
and evaluating the PC expansion onto
=…
{
x
,
,
x
}
1
n
MCS
this sample, that is,
Y M M
MCS
PC
= {
PC
(
x
1
…
),
,
PC
(
x
)}.
(6.52)
n
MCS
Using a sufficiently large sample set (e.g.,
n
MCS
= 10
5−6
), one can then compute and plot the
almost-exact PDF of
Ŷ
by using a kernel density estimator (Wand and Jones, 1995):
n
MCS
1
y
−
M
x
PC
()
.
ˆ
∑
i
ˆ
fy
()
=
nh
K
(6.53)
h
Y
MCS
i
=
1
In this equation, the
kernel function K
is a positive definite function integrating to one
(e.g., the standard normal PDF φ()
ye
y
=
−
2
2
2π ) and
h
is the
bandwidth.
The latter can be
taken, for instance, from Silverman's equation:
−
15
·
h
=
09
.
n
in(,(
σ
·
QQ
−
)
134
. ,
(6.54)
MCS
y
075
.
0 25
.
where σ
y
(resp.
Q
0.25
,
Q
0.75
) is the empirical standard deviation of Y
MCS
PC
(resp. the first and
PC
). Note that these quantiles as well as any other can be obtained from
the large sample set in
Equation 6.52
.
Having first reordered it in ascending order, say
{,
third quartile of Y
MCS
y
·
1
…
,
y
n
·
},
the empirical
p
-quantile
Q
p
%
, 0 <
p
< 1 is the
pn
% ⋅
MCS
-th point in the
()
(
)
MCS
ordered sample set, that is,
·
Qy
p
=
,
(6.55)
(
)
%
pn
%
⋅
MCS
is the largest integer that is smaller than
u
. This allows one to compute confidence
intervals (CIs) on the QoI
Ŷ
. For instance, the 95% centered CI, whose bounds are defined by
the 2.5% and 97.5% quantile, is
where
u
=
95
%
ˆ
,
ˆ
CI
y
y
.
(6.56)
(
)
(
)
ˆ
25
.%
⋅
n
97 5
.%
⋅
n
Y
MCS
MCS
Note that all the above post-processing may be carried out on large Monte Carlo samples
since the function to evaluate in
Equation 6.52
is the polynomial surrogate model and not
the original model
M
. Such an evaluation is nowadays a matter of seconds on standard
computers, even with
n
MCS
= 10
5−6
.
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