Environmental Engineering Reference
In-Depth Information
pose a general problem of finding linear combination which minimizes
msqe
. The
solution is given in
Table 1.
Table 1.
Weights and optimal variances
Model un-correlated case
Model correlated case
1
−
1
K
l
1
V
(
x
)
α
=
,
V
(
X
)
=
1
k
opt
α
=
,
V
(
X
)
=
−
1
−
1
(
K
l
,
l
)
(
K
l
,
l
)
k
opt
1
1
∑
∑
T
l
=
(
,...,
1
,
(
⋅
,
⋅
)
=
dot
product
V
(
x
)
V
(
x
)
j
j
j
j
K
represents covariance matrix i.e. its
ij
-th element is the covariance of
x
i
and
x
j
. From these formulas one can conclude some basic properties of multi-model
ensemble shown i
n Table 2. I
n case of correlated models eigenvalues of the
covariance matrix (
s
1
,…,
s
m
) play the role of variances
V(x
j
) - the basic properties
are analogous. It is easy to find an example showing that the ensemble mean can
produce worse results than the best individual model, while optimal combination
always overperforms any single model [1]. The last implications in
Table 2
show
which conditions ensure that the ensemble mean produces better
msqe
than any
single model.
Table 2.
Basic properties of multi-model ensembles
Model un-correlated case
Model correlated case
V
(
x
)
V
(
x
)
s
s
1
m
1
m
≤
V
(
X
)
≤
,
V
(
x
)
≤
...
≤
V
(
x
)
≤
V
(
X
)
≤
,
s
≤
...
≤
s
opt
1
m
opt
1
m
m
m
m
m
V
(
x
)
s
m
m
V
(
X
)
≤
min{
V
(
x
),
}
V
(
X
)
≤
min{
V
(
x
),
}
opt
1
opt
1
m
m
V
(
x
)
s
m
m
≤
m
+
1
⇒
V
(
X
)
≤
V
(
x
)
≤
m
⇒
V
(
X
)
≤
s
mean
1
mean
1
V
(
x
)
s
1
1
3. Conclusions
First of all by choosing appropriate combination of model results we can find an
optimal representative of the ensemble that after bias correction minimizes
msqe
and ensemble will not be deteriorated even by a model with a big variance. Secondly
general estimations for the optimal variance and msqe have been obtained showing
that multi-model ensemble oveperforms any single model. Thirdly we have shown
which condition guarantees that the ensemble mean gives more accurate results
than any individual model. This is expressed in the terms of the ratio between
