Applications of Fuzzy Rule-based Systems in Medical Image Understanding - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 111

FIGURE 6.5: Illustration of definition of intervals r 1 , r 2 , . . . , r q for a single feature x

Step 2: Generate fuzzy rules from given training data set Produce a rule for
each input-output data pair included in the training data by assigning the given
inputs to the regions with maximum membership value, e.g. for a 2-dimensional
feature vector it is processed as:

x (1)
1

= 0.3, x (2)
1

= 0.8, C K = 1⇒

x (1)
1

(µ(x (1)
1

))(C k = 1) = 0.65 in r (1)

(max)),

2

x (1)
1

(µ(x (2)
1

))(C k = 1) = 0.75 in r (max))⇒

4

: IF x (1)
1

is R (1)
2

AND x (2)
1

is R (2)
4

Rule THEN C is Class = 1

1

Step 3: Minimisation of fuzzy rules As mentioned above, each data pair from the
training data set generates one rule. Usually there are a large number (several
thousand) of available data pairs, so it is very likely that some conflicting rules
are produced. These conflicting rules have the same IF part but different THEN
parts. One way to solve this problem is to assign a soundness degree SD, and
then to select from the subset of conflicting rules that rule with the maximal
soudness degree. Two strategies are proposed to assign a soundness degree to a
rule R j , j = 1, 2, . . . , N where N denotes the number of rules.
Strategy 1 The soudness degree SD(R j ) for j-th rule is determined by the ratio
of the number of training data pairs which supports the rule W R j

and the

total number of patterns which have the same IF part W IF j :

W R j
W IF j

SD(R j ) = (6.34)

This strategy works better when a large number of training patterns are
available. Using this strategy we incorporate the statistical information
into the fuzzy system resulting in more reliable decision.
Strategy 2 The soundness degree is determined by the membership grades of
inputs and outputs:

SD(R j ) = [µ(x (1) )(v = C)∗. . .∗µ(x (n)

j

)(v = C)]∗µ(x j )(v = C) (6.35)

j