CONTRIBUTION OF PRESENT RESEARCH TO UNCERTAINTY ASSESSMENT METHODS - Modelling Uncertainty in Flood Forecasting Systems - page 69

4.4.1 Type I and Type II Problems
In many situations, an uncertain parameter may contain components of two major types of
uncertainty: one due to randomness and the other due to imprecision and vagueness (or
fuzziness). Probability theory and fuzzy set theory (or possibility theory) are used to rep-
resent these types of uncertainty, respectively. In the presence of both types of
uncertainty, representing uncertainty by either one of the two approaches alone is
considered incomplete. This type of problem is referred to here as a “Type I Problem”.
Various concepts have been emerged to deal with this type of problem. In-depth
coverage of this topic is beyond the scope of this thesis. Kaufmann and Gupta (1991)
presented some discussions on the issue. Two particular concepts have been presented in
Chapter 3. They are the concept of fuzzy probability by Zadeh (1968 and 1984) and the
concept of fuzzy-random variable by Ayyub and Chao (1998).
For a given system, there may exist some parameters for which enough data are available
to characterise their uncertainty probabilistically (using PDFs) by statistical methods. In
contrast there may remain some uncertain parameters with very little or no data to char-
acterise their uncertainty probabilistically. This is a very common problem in uncertainty
analysis in practice. What is usually done is to exclude such parameters (with no data for
uncertainty characterisation) from the analysis, or assume their PDFs based on judgements.
This latter set of parameters may be more suitably characterised for uncertainty by means
of the possibilistic approach using the possibility distributions or fuzzy membership functions.
This then gives rise to a system with two sets of uncertain parameters, one represented by
PDFs and the other represented by MFs. This type of situation is referred to here as “Type
II Problem”. The Type II Problem is more practical but, ironically, more difficult to deal with.
An ideal solution to the Type II Problem could be to separate the uncertain parameters
into probabilistic (defined by PDFs) and possibilistic (defined by MFs), and transform
from one type to another as appropriate. Some of the methods for such transformations
are discussed in Chapter 3. These and other transformation methods reported in the
literature are neither fully rational nor are free from criticism.
4.4.2 Operations on random and fuzzy variables
Uncertain variables represented by PDFs and MFs are commonly referred to as random
and fuzzy variables, respectively. Note that random and fuzzy variables are sometimes
also referred to as random and fuzzy numbers (Kaufmann and Gupta, 1991). The
methods of operations between random-random (R-R) and fuzzy-fuzzy (F-F) variables
are very different. The major obstacle in modelling the uncertainty of a system consisting
of random and fuzzy variables comes from these differences.
The Monte Carlo (MC) method and the fuzzy Extension Principle (EP) are the two
standard methods for the operations on random variables and fuzzy variables,
respectively. Some publications that address the issue of differences in MC and EP are
Chen et al. (1998), Dubois and Prade (1991), Ferson and Kuhn (1994), Ferson and
Ginzburg (1995), Fishwick, (1991), Guyonnet et al. (1999), Kaufmann and Gupta (1991)
and Schulz and Huwe (1997). Some discussion on the issue is also presented in
Subsection 3.6.1 of this thesis. Two notable differences between the MC and the EP arise
from (i) the correlation between the variables, and (ii) the nature of the function
(monotonic or non-monotonic). In the MC method, the correlations can be incorporated.