Environmental Engineering Reference
In-Depth Information
et al., 2000, 2008). This variability being well described by the variance of
Y
, the ques-
tion reduces to apportioning Var [
Y
] to each input parameter {
X
1
, …,
X
M
}, second-order
interactions
X
i
X
j
, and so on. For this purpose, variance decomposition techniques have
gained interest since the mid-1990s. The Sobol' decomposition (Sobol', 1993) states that any
square-integrable function
M
with respect to a probability measure associated with a PDF
f
M
X
()
=∏
=1
f
()
x
(independent components) may be cast as
i
Xi
i
M
∑
∑
MMM
()
x
=
+
()
x
+
M
(, )
x
x
+
+
M
( ),
x
(6.61)
0
i
i
ij
i
j
12
,,
…
M
i
=
1
1
≤<≤
ijM
that is, as a sum of a constant, univariate functions {
M
i
(
x
i
), 1 ≤
i
≤
M
}, bivariate functions
{
M
ij
(
x
i
,
x
j
), 1 ≤
i
<
j
≤
M
}, and so on. Using the
set notation
for indices
(6.62)
def
{, ,}
…
…
u
=
i
i
⊂
{,
1
,
M
,
1
s
the Sobol' decomposition in
Equation 6.61
reads:
∑
MM M
()
x
=
+
( ,
x
0
uu
(6.63)
u
⊂
{1,,}
0
…
M
u
≠
where
x
u
is a subvector of
x
that only contains the components that belong to the index set
u
. It can be proven that the Sobol' decomposition is unique when the orthogonality between
summands is required, namely:
E[
MM
u
(
x
)
(
x
)]
=∀ ⊂… ≠
0
uv
,
{ ,
1
,
M
},
u
v
.
(6.64)
u
v
v
A recursive construction is obtained by the following recurrence relationship:
MM
MM M
=
=
E
E
E
[()].
X
X
X
0
()
x
[()
|
X
=
x
]
−
.
(6.65)
i
i
i
i
0
M
(,)
xx
=
[
M
( |
Xx
=
,
Xx
=− −
]
MMM .
()
x
()
x
−
ij
i
j
i
i
j
j
i
i
j
j
0
The latter equation is of little interest in practice since the integrals required to compute
the various conditional expectations are cumbersome. Nevertheless, the existence and unic-
ity of
Equation 6.61
together with the orthogonality property in
Equation 6.64
now allow
one to decompose the variance of
Y
as follows:
∑
∑
0
def
[
]
DY
=
Var
[]
=
ar
M
()
x
=
Var
M
()
X
,
(6.66)
uu
uu
u
⊂
{1,,}
0
…
M
u
⊂
{{, ,}
1
…
M
u
≠
u
≠
where the
partial variances
read:
def
Var[
(6.67)
D
u
=
M
( ]
X
=
E
[
M
2
(
X
)].
u
u
u
u
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