Geoscience Reference
In-Depth Information
(3.32)
Then a joint PDF
defined by the multiplication of
and
as
(3.33)
Then the fuzzy-random PDF f x (x) of the fuzzy-random variable is defined as the
marginal PDF of the joint PDF
i.e.
(3.34)
3.5 Methods for probability-possibility transformation
Establishing relationship between probabilistic and fuzzy (or more widely possibilistic)
representation of uncertainty has gained significant attention since Zadeh (1978). Such a
relationship or the transformation between probabilistic and possibilistic uncertainty is a
key issue in handling both types of uncertainty in a given system. Various situations that
give rise to the existence of probabilistic and possibilistic uncertainties in an uncertain
parameter and/or in a system of modelling or forecasting are discussed in Chapter 4
(Section 4.4). Different methods of uncertainty modelling based on these theories have
their own advantages and relevancies. The challenge is therefore to utilise the advantages
and relevancies of both theories opportunistically. To achieve this we need the capability
to move from one theory to the other as appropriate. That is, we need to integrate the
theories in the sense that the uncertainty represented in one theory can be converted by a
justifiable transformation into an equivalent representation in the other theory (Klir,
1992).
Various transformation methods have been suggested in the literature. Klir and
Wierman (1998) presented discussions on various transformation methods based upon
various principles reported in the literature (see, e.g., Leung, 1982; Kaufmann and Gupta,
1991; Klir, 1992; Wonneberger, 1994), such as based on maximum entropy, insufficient
reason, maximum specificity and different forms of the principle of possibility-
probability consistency. Based on experience, an ad-hoc-type conversion was also
proposed by Bardossy and Duckstein (1995). Klir (1992) proposed a method based upon
uncertainty invariance principle arguing that the transformation must preserve the amount
of uncertainty contained in the information. Zimmermann (1991) presented discussion on
 
Search WWH ::




Custom Search