In GGA the encoding procedure and crossover and mutation operators of traditional GAs are modified to yield a compact algorithm, with improved performance in groupingbased problems.
The proposed GGA for fuzzy clustering is a variable-length genetic algorithm, with a novel encoding to deal with this specific problem.
The proposed GGA will be run with different objective (fitness) functions to lead the search.
The position of the individuals in the list is called rank of the individual and is denoted as [R.sub.i] (i = 1,..., [xi], with [xi] standing for the number of individuals in the population of the GGA).
Let us consider two different individuals [[xi].sub.1] and [[xi].sub.2] that have been randomly chosen among all individuals in a given GGA population so as to perform crossover on them.
In the proposed GGA, the population at a given generation j + 1 is obtained by replacement of the individuals in the population at generation j, through the application of the selection, crossover, and mutation operators described above.
In order to improve the performance of the proposed GGA, an island model is considered for its parallelization.
This section summarizes and discusses the experimental work we have carried out in order to assess the performance of our proposed GGA approach.
In this first experiment, we test the performance of the proposed GGA in a two-dimensional clustering problem, defined by 300 observations randomly generated using a Gaussian distribution from 8 equiprobable classes, with mean values [[mu].sub.1] = (-1,1), [[mu].sub.2] = (2,-2), [[mu].sub.3] = (1,0), [[mu].sub.4] = (3,-1), [[mu].sub.5] = (-1, -1), [[mu].sub.6] = (-1, -3), [[mu].sub.7] = (1,2), and [[mu].sub.8] = (3,1) and covariance matrices:
At this point it is important to emphasize that the proposed GGA is able to infer the number of clusters within the problem, whereas the FCM requires this parameter to be set before execution (namely, C in the above description of FCM).
Note that the proposed GGA with the three different objective functions obtains better results than the FCM algorithm.