Fuzzy Modelling - GeoComputation - page 294

served as an attractive method of analysis for ordering world cities since they will differ in both the
nature and degree of world cityness (Taylor and Derudder, 2004).
Since the objective function does not take into consideration spatial dependence between obser-
vations, noisy spatial data can adversely affect the performance of the algorithm. Few attempts
to incorporate spatial information in an FCM algorithm have been published outside the image
analysis community. Liew et al. (2000) presented a modification of the FCM whereby the normed
distance computed at each pixel within an image is replaced with the weighted sum of distances
from within the neighbourhood of a pixel. Pham (2001) followed with a more general solution that
uses penalty functions to constrain the membership value of a class, so as to be negatively correlated
with the membership values of the other classes for neighbouring pixels. Both approaches produced
promising results. It remains to be seen if, or when, these adaptations of FCM will develop and be
applied outside the image analysis community.
Another problem is that the number of classes needs to be specified a priori . Where the number
of classes is not known, FCM is used as an exploratory data analysis technique. In order to deter-
mine which number of classes provides a useful partitioning of the data set, measures of cluster
validity are used such as compactness and separation. There are a large number of indices for mea-
suring cluster validity in an FCM analysis. Many indices of cluster validity are detailed and evalu-
ated by Wang and Zhang (2007). After extensive comparison on both artificial and well-known data
sets, they found that none of the indices correctly partitions all data sets into the predetermined
number of clusters.
Another approach that has been used to map data to fuzzy membership functions is the adap-
tive neuro-fuzzy inference system (Jang, 1993). Using a given input/output data set, the objective
is to construct a fuzzy inference system where the input membership functions best suit that
particular data set. Using a hybrid backpropagation algorithm/least-squares method, the member-
ship parameters are tuned in a training exercise similar to that of a neural network. The method
has been used to identify map revision (Teng and Fairbairn, 2002) and land cover classification
(Peschel et al., 2002).
Neural approaches have the advantage of an ability to learn from training data as well as being
able to handle noisy, incomplete data. Once trained, such mechanisms can respond to a new set
of data instantly. However, they can take a long time to train, especially since training is still
largely by trial and error and complicated by the fact that incomplete training data can cause the
network to provide incorrect results. Perhaps the most important disadvantage is that it is difficult
to explain the specific reasoning leading to the output, and so it can be criticised as a black-box
approach.
Statistical data analysis has been suggested as another way to choose fuzzy membership func-
tions and form fuzzy rules (Hanna et al., 2002). However, it has not been used widely in spatial
analysis. An example of its application to a spatially explicit problem is illustrated by the prob-
lem of estimating parameters to use in a regional ecohydrological simulation model. Mackay et al.
(2003) use a two-stage methodology where in the first stage, many simulations are run in which
parameters affecting stomatal conductance are assigned values using Monte Carlo sampling. Then
each simulation result is evaluated by regressing the simulated evaporative fraction modelled by
the Regional Hydro-Ecologic Simulation System (RHESSys; Tague and Band, 2004) and surface
temperature from thermal remote sensing data. For each regression, the coefficient of determination
( R 2 ) is calculated and used as a fuzzy measure of the goodness of fit for its respective simulation
result. Hence, the fuzzy set is composed of the set of R 2 measures for all simulations, to which an
information-theoretic tool based on ordered possibility distributions is applied to form a restricted
set in which only good simulations are retained. A restricted set is used as an ensemble solution in
the second stage of parameter estimation, and a separate solution is produced for each land facet
(Mackay et al., 2003).
Another geographically informed AT method is that used by Fisher et al. (2004) in their evalu-
ation of the location of Helvellyn, the second highest mountain in England, and their exploration