Fuzzy Modelling - GeoComputation - page 293

The DA approach is not without its issues. First, its interpretation can be difficult because rarely
is there anything other than opinion underlying the number. Second, the assignment may be too
challenging for the expert(s) to do reliably, especially if they are not well versed in fuzzy set theory.
Third, there can be a systematic bias towards the end points (Thole et al., 1979). Fourth, since the
assignments from multiple experts can be at extreme variance from one another, the process of DA
is often difficult when trying to combine their opinions (Verkuilen, 2005). In spite of these issues,
DA remains a commonly used strategy for defining membership values, and the use of standard
membership functions is customary.
12.5.2 i indirect a SSignMent
Rather than have membership values directly assigned by experts, IA first elicits responses of some
kind from experts , then applies a model to those responses in order to generate membership values.
Robinson (2000) showed how an intelligent, interactive question/answer system could be used to
generate fuzzy representations of a basic spatial relation such as near . The expert need only provide
a yes/no answer to a question posed by the software. From crisp answers, the system generates a
fuzzy representation of a spatial concept. This approach can also be used to generate fuzzy repre-
sentations of individual concepts - spatial or non-spatial. One of the reasons IA is less often used
is the difficulty of the knowledge elicitation process. Zhu (1999) used personal construct theory to
formulate a rigorous methodology for eliciting expert knowledge about soils. Part of the process
included the expert interacting with a graphical user interface to assist in formalising the relations.
Nevertheless, this proved difficult and very time-consuming. Kuo et al. (2003) used questionnaires
to acquire data on store location decisions from business experts . Questionnaire data were then used
to estimate the weight assigned to each factor in their model of location decisions. Other studies that
have reported using questionnaires in the process of constructing fuzzy memberships include Fritz
et al. (2000) and Lin et al. (2006).
12.5.3 a SSignMent By t ranSforMation
In AT, numerical variables are taken and mapped into membership values by some transformation.
Although there are many different approaches that assign fuzzy membership using some version of
AT, by far one of the most common is the use of a fuzzy clustering algorithm. The most common
fuzzy clustering algorithm for assigning membership is the fuzzy c -means (FCM) algorithm origi-
nally developed by Dunn (1973) and later generalised by Bezdek (1973, 1981). It is also commonly
referred to as the fuzzy k -means algorithm. It takes as input a multivariate data set of p variables
by q cases. Then, minimising an objective function optimises the identification of a predetermined
number of groups, c . For each of the q cases, the degree to which any case resembles the properties
of each of the c groups is reported as a fuzzy membership value in the range [0,1]. Thus, the mem-
bership value is a function of the classification process and of the input data. Two primary variants
are possible. The supervised method is where information on prototypical groups is used a priori to,
in essence, train the algorithm. The unsupervised method is where a search is executed to determine
the division into some number of predetermined classes.
Since the algorithm was published (Bezdek et al., 1984), FCM has been applied to a diverse set
of situations in geographic computing. It has been most prominent in studies of soils mapping and
geomorphometry (Odeh et al., 1990; Irvin et al., 1997; McBratney and Odeh, 1997; Burrough et al.,
2001; Triantafilis et al., 2001; Bragato, 2004; Amini et al., 2005; Deng et al., 2006; Arrell et al.,
2007). There are several other areas where FCM has been used in the study of geographic phenom-
ena. In plant ecology, Brown (1998) mapped historical forest types as fuzzy sets. Spatial subpopu-
lations of woodland caribou were identified using FCM (Schaefer et al., 2001). As an example of
FCM use in a business GIS environment, Wanek (2003) illustrated the utility of fuzzy clustering
in identifying spatial customer potentials for prefabricated housing markets. Fuzzy clustering has