## 3 Results and Discussion
### Performance of the Three Hybrid Classifiers
Figure 7 shows the evolution of the training set size \(S\) of the fittest individual of each generation (blue line with respect to the right \(y\)-axis of the plot) and the corresponding classification accuracy \(acc\) of that individual (red line with respect to the left \(y\)-axis of the plot) for the GAKNN hybrid algorithm. The population of the GA consisted of 100 individuals, the crossover probability for the uniform crossover operator was set to 0.8, the mutation probability for the uniform mutation operator was set to 0.05 and the GA evolved for 500 generations. The value of the balancing factor _alpha_ in the fitness function (Equation (2)) was set to 0.35, and the KNN algorithm was operated for \(k=3\) neighbors. The size of the training set gradually descended for the first 250 generations and then converged to an optimal set with size 106 (16.06% of the total 660 cases of the dataset). The classification accuracy on the test set reached its highest value (100%) in the early stage of evolution, even before the 65 th generation, and consequently was maintained there. Figure 5: Average values and error bars for KNN classification accuracy vs. the size \(S\) of the training set for 100,000 independent experiments. plot and the evolution of the classification accuracy _acc_ of that individual is shown by the red line with respect to the left \(y\)-axis of the plot. The population of the GA consisted of 100 individuals, the crossover probability for the uniform crossover operator was set to 0.8, the mutation probability for the uniform mutation operator was set to 0.05, and the GA evolved for 300 generations. The value of the balancing factor _alpha_ in the fitness function (Equation (2)) was set to 0.70. The GA converged to an optimal training set with size 222 (33.64% of the total 660 cases of the dataset), while the accuracy on the test set reached its highest value (100%) in the few last generations. In Figure 8, it is important to note the way that the training set size \(S\) evolved. It clearly followed a steep descent in the first 50 generations, reaching a minimum of approximately 28.82%, but after that, it gradually increased for the next 180 generations, up to generation 230. It then converged to its final value in the few last generations at almost the end of its evolution, where the size of the training set was 33.64%. It seems like the GA was fluctuating between the two objectives that were to be optimized and, eventually, the minimization of \(S\) was sacrificed, since \(S\) increased by 4.82%, for the maximized accuracy _acc_ of the optimal value (100%) to be achieved. Such a behavior is related to the selected value of the parameter _alpha_ of the fitness function (Equation (2)) and can be considered as _emergent behavior_ that implies intelligent characteristics [60; 61]. Regarding the third hybrid algorithm, representative results obtained by the GACLUST classification algorithm are shown in Figure 9, in the same manner as the ones of the two other hybrid algorithms presented above. The population of the GA consisted of 100 individuals, the crossover probability and the mutation probability were kept at 0.8 and 0.05, respectively, and the GA evolved for 4000 generations. The value of the balancing factor _alpha_ in the fitness function (Equation (2)) was set to 0.60. According to these results, the size \(S\) of the training set rapidly decreased for the first 500 generations, reaching a value close to 20%, while, at the same time, the classification accuracy climbed close to 100%. Afterward, \(S\) descended very slowly for the next 4500 generations, moving toward a set of  
Figure 8: Evolution of the training set size \(S\) and classification accuracy on the test set for the GAKMEANS hybrid algorithm. 108 cases (16.37% of the total cases of the dataset), while the classification accuracy on the test set finally reached its optimal value of 100%. Table 6 presents a summary four representative results for each of the three hybrid algorithms.