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people with more and more hair are less and less bald, until someone with a small thinning of hair
on the top of their head might be described as balding showing only a minor affinity with the set.
Any person may to some extent be tall and to some extent be bald. The extent may be so small that
they are effectively not in the set, but if we define the sets tall and not tall, many people will belong
to both sets to varying degrees. It is this degree of belonging that Zadeh suggests is described by
fuzzy set theory. Essentially, the concepts of bald and tall are sorites susceptible. In an era of smok-
ing bans inside buildings, it is possible to see people's varied interpretations of inside and outside a
building. When the weather is warm, smokers will congregate some distance from the doors, but if
it is raining, smokers may be very close to the doors and there may even be a discussion to be had
as to whether a porch or foyer is within the building. People can easily exploit the vagueness of the
definition of the extent of the building.
In geography, we can distinguish a large number of concepts that can comfortably be conceived
of as having a degree of belonging, but do not fit so well with a concept of a hard crisp set. The
degree of belonging may relate to the categorisation of that property of either the geographical indi-
vidual or a location irrespective of the individual. In more complex formulations, membership may
relate to the interaction between individual and locational attributes.
Fuzzy set theory has encountered some opposition due in large part to the similarity of the scales
of measurement for fuzzy set membership and probability: in both approaches, an object acquires
a value between 0 and 1 (Haack, 1974; Zadeh, 1980; Kosko, 1990; Laviolette and Seaman, 1994).
Fisher (1994) has contributed a geographical example to illustrating this problem arguing that in
determining the visibility from a position in the landscape (a viewshed operation), the line of sight
to a location may be uninterrupted making the location visible (an issue of probability), but it may
not be possible to clearly discern an object which may be at that location (fuzziness).
Dale (1988), Moraczewski (1993a,b) and Roberts (1986, 1989) all advocate the use of fuzzy
sets as being more appropriate than applying Boolean concepts for the analysis of vegetation com-
munities, and Dale (1988) and Roberts (1986) argue that the traditional method of ordination used
in the analysis of phytosociology is a tacit admission that the vegetation at any particular location
possesses partial characteristics of some central concept of the community, and ordination analysis
(through principal component or factor analysis) is one method of describing the strength of any
location belonging to these concepts. The same argument with respect to ordination has been made
by soil researchers (Powell et al . , 1991).
12.5 FUZZY MEMBERSHIPS
At the heart of fuzzy set theory is the concept of a fuzzy membership. A crisp set is defined by the
binary coding {0,1}, whereby an object that is in the set has code 1 and an object that is not has
code 0. A fuzzy set membership, on the other hand, is defined by any real number in the interval
[0,1] (noting switch from curly brackets to square brackets, indicating that a series of potential num-
bers can exist ranging from 0 to 1). If the value is closer to 0, then the object is less like the concept
being described; if closer to 1, then it is more like it. A person 2 m high might have a value 0.9 in the
set of tall people whereas one 1.8 m high might have a value of 0.2. The fuzzy membership is com-
monly described in formulae by μ and, particularly, an object x has fuzzy membership of the set A ,
μ( x ) A . The difference between crisp and fuzzy set memberships is shown in Figure 12.3. The binary
crisp sets have memberships 0 and 1. There is no grade of belonging. Fuzzy sets have continuous
membership values that can range between 0 and 1.
Historically, authors have referred to two major approaches to defining the fuzzy membership
of an object: either the semantic import model or the similarity relation model (Robinson, 1988;
Burrough, 1989). This nomenclature is based on how fuzzy memberships are used to represent fuzz-
iness in a geographic database, not the method whereby the membership values are derived. For this
reason, Fisher (2000) suggested that an experimental model also needed to be acknowledged. As
suggested by Robinson (2009), it is now more useful to consider that fuzzy memberships are usually
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