Applications of Fuzzy Rule-based Systems in Medical Image Understanding - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 102

If two or more classes take the maximum value then the classification of x is rejected (i.e.
x is left as an unclassifiable pattern), otherwise x is assigned to the class C.

6.3.2 Weighted fuzzy classification
In this section, we regard pattern classification as a cost minimisation problem and intro-
duce the concept of weighted training patterns. The idea is based on the assumption that
in certain cases misclassification of a particular input pattern introduces an extra costs.
For example in cancer diagnosis, false positives cases could be penalised more than false
negatives, i.e. diagnosing healthy individuals as cancer candidates.
To reformulate the pattern classification problem as a cost minimisation problem for each
training pattern we introduce a concept of a weight. The weight of an input pattern can be
viewed as the cost of its misclassification or rejection. Fuzzy If-Then rules are generated by
considering the weights as well as the compatibility of training patterns.
In order to incorporate the concept of the weight, Eq. 6.8 of the fuzzy rule generation is
modified to

β C (j) = X
x p ∈C

µ j (x p )·ω p (6.15)

where ω p is the weight associated with training pattern p.
We note that this fuzzy rule generation method can also be applied to the standard
pattern classification problem with no pattern weights. In this case, the class and the grade
of certainty are determined from training patterns by specifying a pattern weight as ω p = 1
for p = 1, . . . , m.
Under the assumption that a weight is assigned to each training pattern which can
be viewed as the relative importance of the patterns, we use the concept of classifica-
tion/rejection cost to construct a weighted fuzzy classification. We define a cost function
Cost (F S) of a fuzzy classification system F S as

X

m

Cost (F S) =

ω p ·z p (F S), (6.16)

p=1

where m is the number of training patterns, ω p is the weight of the training pattern x p , and
z p (F S) is a binary variable set according to the classification result of the training pattern
x p by F S: z p (F S) = 0 if x p is correctly classified by F S, and z p (F S) = 1 otherwise (i.e.
x p is misclassified or rejected). We use this cost function as well as the classification rate
as performance measures.
The number of generated fuzzy If-Then rules in a fuzzy classification system depends
on the partition of attributes and the dimensionality of the pattern classification problem.
Since there are three fuzzy sets for each attribute, the possible number of combinations of
antecedent fuzzy sets is N = 3 n where n is the number of attributes.
We use two weight assigment methods for determining the weights of patterns. In the
first method, we assume it is important to correctly classify a certain class. Thus the
weights of training patterns of this focussed class are specified as ω p = 1.0. On the other
hand the weights of the other training patterns are specified as ω p = 0.5. That is, the cost
of misclassifying/rejecting a training pattern from the focussed class is twice as large as
that from the other classes. It is clear that different values of ω p can also be used to put
more or less emphasis on certain classes. The second weight assigning method considers
the distribution of classes in a data set. The weight of a class specified by this method is
large if the proportion of the class is small. Thus it is assumed that classification of minor