Environmental Engineering Reference
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chain Monte Carlo (MCMC) approach based on the
Metropolis-Hastings algorithm (e.g., Arhonditsis et al.,
2008; Hastings, 1970; Metropolis, 1953; Metropolis and
Ulam, 1949; Samanta et al., 2007). However, care should
be taken in using conventional MCMC approaches to
determine the parameter probability distributions, since
such approaches can lead to an inadequate sample of
the parameter space and hence incomplete parameter
probability distributions. Improved approaches that
address some of these limitations in the MCMC approach
have been proposed (e.g., Smith and Marshall, 2008).
where y i is the observed value of the predicted variable
y at time t i ( i = 1, . . . , N ), f is the model function, x i is
the observed or specified vector of forcing variables at
time t i , θ is the parameter vector, and ε i is the error in
model prediction of y i . The model given by Equation
(11.42) is also consistent with the assumption that the
model, f ( x i , θ ), is exact, and the errors, ε i , are associated
with observation and measurement inaccuracies
(Arhonditsis et al., 2008). It is usually assumed that ε i is
a set of normally distributed independent variables with
a mean of zero and a standard deviation of σ ; the fact
that the errors at each prediction point have the same
variance makes the errors homoscedastic . With this
assumption, the likelihood function for the entire series
of observations y can be estimated using the relation
11.3.3.3  Generalized  Likelihood  Uncertainty  Esti-
mation.  The gLUE methodology, initially proposed by
Binley and Beven (1991) and Beven and Binley (1992),
is a Bayesian uncertainty estimation method that uses
random samples of the parameter set to calculate the
corresponding model output, and uses the level of
agreement between the model output and observations
to estimate both the probability distribution of param-
eters and the probability distribution of model output.
The gLUE method allows for the concept of equifinal-
ity of parameter sets, which means that different param-
eter sets can produce model predictions that agree with
observations equally well. Using the gLUE approach,
there is no requirement to minimize (or maximize) any
measurement statistic or objective function, but the per-
formance of individual parameter sets are characterized
by a likelihood measure, such as the Nash-Sutcliffe
measure. In a typical application, parameter ranges are
selected, Monte Carlo simulations (MCSs) are gener-
ated from these ranges, and a likelihood plot for each
parameter is viewed to identify optimal sets and param-
eter insensitivities.
To illustrate the application of the gLUE method,
consider a generic model that is used to predict a set of
values of a single variable, y , based on a set of input
forcing variables, x , and model parameters, θ ; such a
model can be expressed in the form
{
}
N
1
1
[
]
2
2
L
(
y
|q
,
x
,
σ
)
exp
y
f
(
x
,
q
)
i
i
i
σ
N
2
σ
2
i
=
1
(11.43)
Assuming that θ is distributed uniformly within a
specified interval and the prior distribution of σ is
uniform over log σ , the noninformative prior is given by
1
p (
q,
σ
2
)
(11.44)
σ
2
Using this prior distribution and combining Equa-
tions (11.43) and (11.44) in accordance with Bayes equa-
tion (Eq. 11.40) yields the joint posterior distribution
{
}
N
1
1
[
]
2
p
(
θ σ
,
2
|
y x
,
)
exp
y
f
(
x
,
q
)
i
i
σ
N
+
2
2
σ
2
i
=
1
(11.45)
This equation gives the probability, or likelihood, dis-
tribution of the model parameters and error variance,
and is basis on which inferences about the parameters
are made.
The posterior probability distribution of parameters
given by Equation (11.45) is the basis for estimating
the uncertainty bounds of the predicted variables
ˆ
y
= f ( ,
x
q
(11.46)
Observed values of ( x , y ) have some errors included
and are denoted by ( x , y ), and for each realization of a
feasible parameter set, a likelihood function L (
q|
x y
for θ based on the Nash Sutcliffe efficiency (NSE) cri-
terion is:
,
)
i = q . The procedure is to equate the frequency of
occurrence of each value of the output variable to the
frequency of occurrence of the parameters set causing
that output variable. This is usually implemented
numerically by sampling the parameter set from the
posterior distribution.
The posterior probability distribution of model
parameters, given by Equation (11.45) for homoscedas-
tic model errors, can be obtained using a Markov
y
f xi
(
,
)
σ
σ
2
ε
q|
(11.47)
L (
x y
,
) =
1
2
0
where σ 2 is the error variance between model simula-
tions and observed data, and σ 2 is the variance of the
observed data. A negative value of L (
q|
x y indicates
,
)
 
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