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fuzzy sets

  Fuzzy sets are sets (categories, classes, types) for which the definitions of set membership are vague or \'fuzzy\', and contrast with the sharp, clearly-defined definitions used by standard logic and set theory. Examples include \'cold\', \'warm\' and \'hot\' for the weather: \'cold\' is not a precise category, and will overlap with \'warm\' and there is no precise boundary between them (e.g. 10 °C), unless we artificially impose one. Similarly \'poor\', \'middle-class\' and \'rich\' are fuzzy categories in everyday life and usage, only converted into precise categories by government or other statistical definitions.

Fuzziness should be distinguished from randomness, and it attempts to represent and model a different type of uncertainty to that dealt with by probability — Kosko (1992, p. 265) says that:

Fuzziness describes event ambiguity. It measures the degree to which an event occurs, not whether an event occurs. Randomness describes the uncertainty of event occurrence. An event occurs or not, and you can bet on it. The issue concerns the occurring event: is it uncertain in any way. … Whether an event occurs is \'random\'. to what degree it occurs is fuzzy.There may be a 40 per cent chance of a US citizen being in the \'middle-class\' category (if we can define the bounds of the category and \'defuzzy\' it), for example, and that would be a probability, whereas the ambiguous, uncertain nature of \'middle-class\' is a fuzzy set issue.

Most standard quantitative and mathematical modelling works with non-fuzzy sets and logic, yet much of social life and the environment is represented by fuzzy categories. Engineers such as Zadeh developed \'fuzzy logic\' as a new group of methods to allow modelling of such fuzzy problems. The methods require further assumptions (and precision) — ordering the categories; estimating membership shape; and centring (e.g. \'middle-class\' centred on $30,000 income with a triangular membership function, with overlap at either side with the other two categories) — but one can test sensitivity to the assumptions. In human geography, the main applications so far have been in the arenas of artificial intelligence (AI) and non-linear systems modelling, such as neural nets. Openshaw and Openshaw (1997) provide examples to spatial interaction modelling using fuzzy distances (\'short\', \'average\', \'big\', \'long\') and fuzzy trip-frequencies (\'some\', \'lots\', \'massive\', etc.), and show how fuzzy systems modelling can perform as well (or better) than many traditional interaction methods.

Fuzzy sets undoubtedly have potential to extend the limits of what can be modelled in socio-economic and spatial systems. Advocates such as Openshaw (1996) see fuzzy logic as the key which will enable \'soft human geography\' — all the aspects currently studied by qualitative and discursive methods — to be made \'scientific\', but most would paint a much more modest picture. It is notable that most examples come from engineering, systems modelling and AI; the social sciences (including economics) have not made great use of fuzzy sets. Much of human geography is not asking questions of the type that fuzzy logic can help answer, and fuzzy models are likely to be most useful within specialist contexts. (LWH)

References Kosko, B. 1992: Neural networks and fuzzy systems. Englewood Cliffs, NJ: Prentice-Hall. Openshaw, S. 1996: Fuzzy logic as a new scientific paradigm for doing geography. Environment and Planning A 28: 761-8. Openshaw, S. and Openshaw C. 1997: Artificial intelligence in geography. Chichester: John Wiley.



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