Environmental Engineering Reference
In-Depth Information
structure can lead to insensitive parameters and param-
eter interdependence.
11.3.3.2 Bayesian Approaches.
Bayes equation yields
the (posterior) pdf,
p
(
θ|
X Y
, of the model parameters,
θ
, conditioned on the observed input data,
X
, and the
observed output variables,
Y
, according to the relation
,
)
11.3.3.1 Multiobjective Optimization.
The tradi-
tional multiobjective approach is to select several dif-
ferent model performance measures and then merge
them together into a single function for optimization.
However, there is significant advantage to maintaining
the independence of the various performance measures,
and that a multiobjective optimization should be per-
formed to identify the entire set of optimal solutions.
Formally, the problem of multiobjective optimization
can be expressed in the form
p
(
Y
|
q
X
Y X
,
) (
p
q
)
|
p
(
q
X Y
,
)
=
(11.39)
p
(
|
)
X
|
ensures that the pdf
integrates to 1. The posterior pdf
p
(
where the denominator
p
(
)
q|
X Y
of the model
parameters is the basis for identifying the most likely
(i.e., highest probability) values of the model parame-
ters, and the posterior pdf can be combined with the
model function to estimate the uncertainty in the model
output that is derived from parameter uncertainty. In
applying Equation (11.39), it is typically not necessary
to explicitly evaluate
p
(
,
)
=
1
…
minimize
F
( )
q
minimize {
f
( ),
q
f
( ),
q
,
f
( )}
q
2
M
(11.38)
Y
|
and any factors indepen-
dent of
θ
in the likelihood function*
p
(
)
where
f
m
(
θ
) is a performance measure associated with
the parameter set
θ
. The solution to this problem is
P
(
Θ
), which is called a
Pareto optimum set
of solutions
in the feasible parameter space, which defines the
minimum parameter uncertainty that can be achieved
without stating a subjective relative preference for mini-
mizing one specific component of
F
(
θ
) at the expense
of another. The Pareto set is defined such that any
member
θ
i
of the set has the following properties:
(1) For all nonmembers
θ
j
, there exists at least one
member
θ
i
such that
F
(
θ
i
) is strictly less than
F
(
θ
j
), and
(2) it is not possible to find
θ
j
within the Pareto set such
that
F
(
θ
j
) is strictly less than
F
(
θ
i
), where by “strictly
less,” it is meant that
Y
|
can be
absorbed into the proportionality constant, giving
)
p
(
q
|
X Y
,
)
=
L
(
Y X
|
q
,
) (
p
q
)
(11.40)
If a noninformative prior is imposed, Equation
(11.40) becomes
p
(
q
|
X Y
,
)
=
L
(
Y X
|
q
,
)
(11.41)
Although the prior pdf has an important role in
classic Bayesian analysis, hydrological applications tend
to use noninformative priors, with justification that this
“let the data speak for itself” (Kavetski et al., 2003). The
likelihood function,
L
(
f
(
q
)
<
f
(
q
)
∀ ∈ 1
m
[ ,
M
]
|q, )
, in Equations (11.40) and
(11.41) represents the likelihood of observing the data
Y
given the model parameters
θ
, the observed data
X
,
and the assumed model structure. The form of the likeli-
hood function reflects the way error and uncertainty
enter an propagate through the system.
In most cases, model errors and errors in forcing
variables cannot be separated and are lumped together.
In an attempt to separate these errors, a method called
Bayesian total error analysis (BATEA) has been pro-
posed in which an error model is specified explicitly and
hence the adequacy of the model and errors can be dealt
with separately and explicitly (Kavetski et al., 2006a,b).
Y X
m
j
m
i
The multiobjective formulation results in the parti-
tioning of the feasible parameter space into “good”
solutions (Pareto solutions) and “bad” solutions. In the
absence of additional information, it is not possible to
distinguish any of the good (Pareto) solutions as being
objectively better than any of the other good solutions
(i.e., there is no uniquely best solution). Further, every
member
θ
i
of the Pareto set will match some character-
istics of the system behavior better than every other
member of the Pareto set, but the tradeoff will be that
some other characteristics of the system behavior will
not be as well-matched. A powerful advantage of this
approach is that it includes the best solution for each
error component of the vector
F
(
θ
), meaning that the
classical single objective optimum value for each sepa-
rate function is an element of the Pareto set. An example
of an algorithm for identifying the Pareto solution can
be found in gupta et al. (2003b).
The Pareto set can be used to quantify the uncertainty
in model predictions by determining the range of model
output as the parameters vary within the Pareto set.
Homoscedastic Model Error.
Errors in model predic-
tions are sometimes expressed explicitly in the form
y
=
f
(
x
q
,
)
+
ε
(11.42)
i
i
i
*
The concept of a likelihood function was first introduced by Fisher
(1922).
Search WWH ::
Custom Search