FLOOD FORECASTING MODELS AND UNCERTAINY REPRESENTATION - Modelling Uncertainty in Flood Forecasting Systems - page 16

structure (Cullen and Frey, 1999). Model uncertainty may also be assessed by comparing
results from different models (Radwan et al., 2002). Such assessments of model
uncertainty however, remain largely subjective.
The methods and methodologies developed and applied in this research are primarily
focused on the treatment of input and parameter uncertainty. An exception is the
framework of the expert judgement based qualitative method (Subsection 3.2.2), in which
all identified sources of uncertainty can be represented qualitatively.
2.3 Theories of uncertainty representation
Historically, probability theory has been the primary tool for representing uncertainty in
mathematical models (Ross, 1995). With the rapid development of computer technology
and its use in mathematical modelling, the need of error representation in digital
computation was increasingly realised in the 1950s and early 1960s, leading to the
invention of interval arithmetic (Moore, 1962 & 1966). However, it was not until mid
1960s when Zadeh (1965) developed the fuzzy set theory that the representation of
uncertainty by a non-probabilistic approach began to increase in pace rapidly. Further,
Zadeh (1978) developed a broader framework for uncertainty representation called
possibility theory, which is also known as a fuzzy measure . He interpreted a normal fuzzy
membership function (Appendix I) as a possibility measure. Another broad theory of
uncertainty representation was advanced by Shafer (1976) in the name of the theory of
evidence . Shafer's theory has its origins in the work of Dempster (1969) on upper and
lower probabilities. Therefore the theory of evidence is more commonly known as
Dempster-Shafer theory of evidence . These theories are very rich in content, and
therefore a detailed coverage of these cannot be presented within the scope of the present
thesis. Some details on probability theory, fuzzy set theory and possibility theory (fuzzy
measures) are presented in the following Subsections (2.3.1 to 2.3.3).
2.3.1 Probability theory
Of all the methods for handling uncertainty, probability theory has by far the longest
tradition and is the best understood. This of course does not imply that it should be
beyond criticism as a method of handling uncertainty. It does, however, mean that it is
relatively well tested and well developed and can act as a standard against which other
more recent approaches may be measured (Hall, 1999). There are two broad views on
probability theory for representing uncertainty: frequentist view and subjective or
Bayesian view .
Frequentist view of probability
The frequentist view of probability relates to the situation where an experiment can be
repeated indefinitely under identical conditions, but the observed outcome is random.
Empirical evidence suggests that the relative occurrence of any particular event, i.e. its
relative frequency, converges to a limit as the number of repetitions of the experiment
increases. This limit is what is called the probability of the event. The frequentist
approach requires that we base statistical inferences on data that are collected preferably
at random from a defined population. The mathematics of probability rests on three basic