Applications of Fuzzy Rule-based Systems in Medical Image Understanding - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 105

is chosen as a parent string. A pair of parent strings is chosen by iterating this procedure
twice.
From the selected pair of parent strings, two new strings are generated by a crossover
operation. We use a uniform crossover operator where the crossover points are randomly
chosen for each pair of parent strings. The crossover operator is applied to each pair of
parent strings with a pre-specified crossover probability p c . After new strings are generated,
each symbol of the generated strings is randomly replaced with a different symbol by a
mutation operator with a pre-specified mutation probability p m . Usually the same mutation
probability is assigned to every position of each string. Selection, crossover, and mutation
are iterated until a pre-specified number N replace of new strings are generated.
Finally, the N strings with the smallest fitness values in the current population

are removed, and the newly generated N replace strings added to form a new population.
Because the number of removed strings is the same as the number of added strings, every
population consists of the same number of strings. That is, every rule set has the same
number of rules. This generation update can be viewed as an elitist strategy where the
number of elite strings is (N rule −N replace ).
The above procedures are applied to the new popaulation again. The generation update
is iterated until a pre-specified stopping condition is satisfied. In our experiments we use
the total number of iterations (i.e., the total number of generation updates) as stopping
condition.
Algorithm summary
To summarise, our hybrid fuzzy rule-based classifier works as follows:

replace

Step 1: Parameter specification. Specify the number of rules N , the number of

rule

, the crossover probability p c , the mutation probability p m ,
and the stopping condition.
Step 2: Initialisation. Randomly generate N

replaced rules N

replace

rules (i.e., N strings of length n)

rule rule

as an initial population.
Step 3: Genetic operations. Calculate the fitness value of each rule in the current

rules using selection, crossover, and mutation from
existing rules in the current population.
Step 4: Generation update (elitist strategy). Remove the worst N replace rules from the
current population and add the newly generated N replace rules to the current
population.
Step 5: Termination test. If the stopping condition is not satisfied, return to Step 3.
Otherwise terminate the execution of the algorithm.

population. Generate N

replace

During the execution of the algorithm, we monitor the classification rate of the current
population on the given training patterns. The rule set (i.e. population) with the highest
classification rate is chosen as the final solution.

6.3.5 Fuzzy rule generation with clustering and learning by examples
This section presents how a fuzzy clustering approach can be used to generate fuzzy If-Then
rules. In the approaches described in the previous section the antecedent part of the rules
is initialised manually. Very often, heuristically chosen membership functions do not reflect
the actual data distribution in the input space. Moreover, in image understanding tasks the
amount of analysed data can be enormous and complex. According to the original tendency
in fuzzy classification systems, membership functions should be subjective, in contrary to
probabilities that are objective. However, a set of heuristically chosen membership functions