Description

One feature of uncertainty is the degree of precision with which a statement is made. Statements that are less precise contain less information. This notion of uncertainty is quite different to probabilistic uncertainty, which (according to the frequentist interpretation) expresses the relative frequency with which specified events will occur. Imprecision can be represented in mathematical terms using set-theoretic methods. For example, an imprecise statement about the outcome of the next throw of a six-sided dice would be that it will result on one of the outcomes in the set {1,2,3}. Fuzzy set theory extends this type of approach to the situation where it is not precisely defined whether a given outcome is in the set of interest or not. For example an assessment of the outcome of a given computer experiment being in the set of “behavioural runs” can be expressed with a fuzzy membership function between 0 and 1.

A further extension of this approach is to deal with the situation where probabilities are not precisely known, which in practice is almost universally the case. The theory of imprecise probabilities (Walley, 1991) deals with sets of probability measures, or, more generally, sets of gambles. Indeed, Klir and Smith (Klir and Smith, 2001) illustrate how the theory of imprecise probabilities generalises classical probability theory and some interpretations of fuzzy set theory. A well justified theory of decision-making forms the basis of this approach.

In practice using fuzzy and imprecise methods in flood modelling involve propagating sets of intervals through a numerical model. A practical approach is described by Hall and Anderson (2002) and the references therein. The computational expense of this approach can be greatly reduced if there is some prior knowledge about the behaviour of the numerical model (whether, for example, its response is monotonic with given variables).

Software

Fuzzy Analysis Framework

Advantages

• Able to represent the imprecision that is inherent in some (arguably most) information.

• Enables the exploration of robustness of decision options to imprecision in available knowledge.

Disadvantages

• A proliferation of alternative approaches and interpretations (particularly for fuzzy methods) some of which lack sound theoretical basis.

References and Further Reading

Klir, G.J. and Folger, T.A. Fuzzy Sets, uncertainty and Information, Prentice Hall , New Jersey, 1988

Ross, T, Fuzzy Logic with Engineering Applications, Wiley, 2004

Hall, J. and Anderson, M., 2002. Handling uncertainty in extreme or unrepeatable hydrological processes - the need for an alternative paradigm. Hydrological Processes, 16(9): 1867-1870.

Klir, G.J. and Smith, R.M., 2001. On measuring uncertainty and uncertainty-based information: recent developments. Annals of Mathematics and Artificial Intelligence, 32: 5-33.

Walley, P., 1991. Statistical Reasoning with Imprecise Probability. Chapman and Hall.

Möller, B. et al., Uncertainty in Engineering