Image Processing Reference
classes with a small number of patterns is more important than major classes with a large
number of training patterns, a situation that is often the case for medical datasets where
the number of malignant cases far exceeds those of benign ones. The weight of a training
pattern x p from class C is specified by the inverse of the proportion of the class over the
given training patterns as
Z · m
ω p = ω C =
p = 1, . . . , m,
C = 1, . . . , M,
where ω p is the weight of the training pattern x p that is from class C, ω C is the weight of
patterns from the class C, m is the number of given training patterns, N c is the number of
patterns from the class C, and Z is a normalisation factor that makes the maximum value
of weights from the class C a unit value (i.e. max
ω C = 1).
Learning Fuzzy If-Then rules for weighted training patterns is a strategy that adjusts the
grades of certainty CF j and can be employed to achieve improved classification performance.
It is based on an incremental learning approach where the adjustment occurs whenever clas-
sification of training patterns is performed. When a training pattern is correctly classified
we reinforce the grade of certainty of the fuzzy rule that is used for the classification. On the
other hand, we decrease the grade of certainty of rule if a training pattern is not successfully
Let us assume that we have generated fuzzy If-Then rules by the rule-generation procedure
detailed by Eq. 6.8 and Eq. 6.9. We also assume that a fuzzy If-Then rule R j is used for
the classification of a training pattern x p . That is, R j has the maximum product of the
compatibility and the grade of certainty (see Eq. 6.13).
The proposed learning method
adjusts the grades of certainty of R j as
= CF ol j −η·ω p ·CF old
if x p is misclassified
= CF old
+ η·ω p ·(1−CF old
if x p is correctly classified
where ω p is the weight of the training pattern x p , and η (the learning rate) is a positive
constant value in the interval [0; 1].
One epoch of the proposed learning method involves examining all given training patterns.
Thus, there will be m adjustments of fuzzy If-Then rules after all m training patterns are
examined. The learning process is summarised as follows:
1. Generate fuzzy If-Then rules from m given training patterns by the procedure
described by Eq. 6.4 and Eq. 6.5.
2. Set K as K = 1.
3. Set p as p = 1.
4. Classify x p by using the rules generated in Step 1.
5. After x p is classified, adjust the grades of certainty using Eq. 6.18 or eq. 6.19.
6. If p < m let p := p + 1 and go to Step 4. Otherwise go to Step 7.
7. If K reaches a pre-specified value, stop the learning procedure.
K := K + 1 and go to Step 3.
Note that K in the above learning procedure corresponds to the number of epochs.