Geoscience Reference
In-Depth Information
1
0.8
0.6
0.4
Crisp
Fuzzy
0.2
0
FIGURE 12.3 The difference between a crisp set and a fuzzy set illustrated by cross sections (Figure 12.2).
(From Fisher, P.F., Fuzzy modelling, in: GeoComputation , Openshaw, S. and Abrahart, R., eds., Taylor &
Francis Group, London, U.K., 2000, pp. 161-186.)
a function of a direct assignment (DA), an indirect assignment (IA) or an assignment by transforma-
tion (AT) methodology (Verkuilen, 2005), and these are illustrated in examples that follow.
Until now, we have discussed what are known as Type-1 fuzzy sets. These are the most perva-
sive type of fuzzy set used in GC endeavours. For some time, it has been recognised that there are
several potential sources of uncertainty inherent in the membership functions of Type-1 fuzzy sets
(Zadeh, 1975; Fisher et al., 2007a). In typical geospatial applications, such sources of uncertainty
may be (1) the meaning of words used in fuzzy rule-based systems, (2) measurements upon which
Type-1 fuzzy sets are based and/or (3) the data used to tune the parameters of a Type-1 fuzzy set
membership function. Type-2 fuzzy sets allow for the explicit incorporation of uncertainty about the
membership function into fuzzy set theory. In Type-1 fuzzy sets, the memberships are by definition
crisp, whereas in Type-2 fuzzy sets, memberships are themselves fuzzy. Following Mendel and John
(2002), a Type-2 fuzzy set, denoted A , is characterised by a Type-2 membership function µ A xu
(,)
01 A can be expressed as
where x X and ∀∈ ⊆
uJ x
[,].
∫∫ µ (,)
xu
xu
A
A
=
J
[,]
01
(12.4)
x
(,)
xXuJ
∈∈
x
in which A B symbolises a subset, having fewer or equal elements to that of the main set.
Note that the restriction ∀∈ ⊆
01 is consistent with the Type-1 constraint that
uJ x
[,]
µ A () . Thus, when uncertainties about the membership function disappear, a Type-2
membership function reduces to a Type-1 membership function. It is beyond the scope of this
chapter to provide a concise tutorial on Type-2 fuzzy sets. Suffice it to say that the difficulty of
working with their 3D nature poses a challenge for their application. However, some examples of
this type of analysis will be given at the end of this review.
0
1
12.5.1 d irect a SSignMent
Either experts or standard functions assigning memberships to entities of interest characterise the
DA methodology. Any function which relates the fuzzy membership to some measureable property,
d , can be used but most common are functions such as the linear, triangular, trapezoidal and trigo-
nometric (Figure 12.4; Robinson, 2003). The first is based on two critical values associated with
the inflection of the membership line ( d 1 , d 2 ) and the second on four inflection values ( d 1 , d 2 , d 3 , d 4 ).
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