Geoscience Reference
In-Depth Information
f
(
#|z
1
,…,
z
n
) "
f
(
#|z
1
,…,
z
n
!1
)
L(#|z
n
)
(3.28)
The predictive distribution of the model responses (water levels) is determined with the
formulation of a model of prediction errors. Romanowicz et al. (1996) and Romanowicz
and Beven (1998) assumed an additive error model (
$
t
=
y
!
y
t
,
where
y
and
y
t
are
realizations of simulated and observed water levels, respectively) and a Gaussian
distribution of errors with first-order correlation. For details on the GLUE method, the
readers are referred to Beven and Binley (1992) and Romanowicz et al. (1996). Further
discussion on GLUE is found in Aronica et al. (1998), Beven and Freer (2001) and
McIntyre et al. (2002).
3.4
Hybrid techniques in uncertainty modelling
Uncertainty in a given system or a parameter may constitute components of the major
types of uncertainty: randomness and vagueness. The former is dealt with using the
theory of probability and the latter using the theory of fuzzy sets. This suggests that
uncertainty assessment using only one of the approaches may be incomplete. Various
concepts, which we refer as hybrid techniques, have emerged whereby the combined use
of both the probabilistic and fuzzy approaches are proposed. In-depth coverage of this
topic is beyond the scope of this thesis. Kaufmann and Gupta (1991) presented more
discussion on this issue. A brief description of the two of such concepts, which have been
used in some applications, is presented here. These are the concept of
fuzzy probability
by
Zadeh (1968 & 1984) and the concept of
fuzzy-random variable
by Ayyub and Chao
(1998). The readers are referred to these references for more details of these concepts.
3.4.1
The concept of fuzzy probability
The concept of fuzzy probability, that is, the probability of a fuzzy event was first
proposed by Zadeh (1968) and later elaborated by Zadeh (1984). The concept is also
addressed by Dubois and Prade (1988), Ross (1995), Terano et al. (1992), Tsoukalas and
Uhrig (1997) and Zimmermann (1991). The fuzzy probability concept considers the
probability (uncertainty) of an event that is fuzzy. The probability of a fuzzy event, also
known as fuzzy probability, is given by
(3.29)
where
is the fuzzy event on the universe
X, x
X,
is the probability of the
fuzzy event
is the membership function of the fuzzy event and
p
X
(x)
is a
probability distribution.
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