## 2 Method Description
### The Second Phase of the Proposed Method
Specifically, the p-values indicate strong statistical differences:

Figure 5: Comparison of error rates across the machine learning models utilized. Proposed vs. BFGS \(\left(p=10^{-8}\right)\), proposed vs. ADAM \(\left(p=6.2\times 10^{-8}\right)\), proposed vs. NEAT \(\left(p=2.9\times 10^{-9}\right)\), proposed vs. RBF-KMEANS \(\left(p=1.2\times 10^{-6}\right)\), and proposed vs. GENRBF \(\left(p=1.2\times 10^{-10}\right)\). These values suggest that the proposed model is significantly better than the others with high reliability. In Figure 7, which concerns the regression datasets, a similar pattern is observed, though the \(p\)-values are generally higher compared to the classification datasets. The proposed model demonstrates statistically significant superiority over the other models in all comparison pairs: Proposed vs. BFGS (\(p=0.00011\)), proposed vs. ADAM (\(p=0.015\)), proposed vs. NEAT (\(p=0.00016\)), proposed vs RBF-KMEANS (\(p=0.0016\)), and proposed vs. GENRBF (\(p=0.00049\)). Although the significance is not as strong as in the classification datasets, the proposed model's superiority remains clear. ### Experiments on the Perturbation Factor a

In order to determine the stability of the proposed technique, another experiment was performed in which the perturbation factor \(a\), presented in the second stage of the proposed technique, took a series of different values. Table 4 presents the error rates of the proposed machine learning model for three different values of the perturbation factor \(a\) (0.001, 0.005, 0.01) across various classification datasets. Each row represents a dataset, and the values indicate the model's error rate for each value of \(a\). The last row includes the average error rate for each value of \(a\). Analysis of the data shows that the smallest value of \(a\) (0.001) achieves the lowest average error rate (18.67%), while the largest value (0.01) results in the highest average (19.06%). This suggests that the model generally performs better with smaller values of \(a\), although the difference in averages is minimal. At the dataset level, there are cases where the model's performance is significantly affected by changes in the parameter. For instance, on the \"Lymography\" dataset, increasing \(a\) from 0.001 to 0.01 leads to a significant increase in the error rate, from 20.64% to 30.33%. A similar trend is observed on the \"ZOO\" dataset, where the error rate rises from 4.50% to 6.87% for \(a=0.005\), but decreases again to 4.60% for \(a=0.01\). On the other hand, on datasets like \"ZO_NF_S\", the error remains unchanged at 3.63%, regardless of changes in \(a\). Datasets such as \"Z_F_S\" and \"ZONF_S\" exhibit nonlinear behavior. On \"Z_F_S\", the error rate

Figure 6: Statistical comparison of the experimental results for the classification datasets. significantly decreases from 3.16% to 2.79% as \(a\) increases from 0.001 to 0.01, while on \"ZONF_S\", a similar decrease is observed from 1.79% to 1.74%. In conclusion, the analysis indicates that the perturbation factor \(a\) has a notable impact on the performance of the proposed model. Smaller values of \(a\) are generally associated with better performance; however, the optimal value may depend on the characteristics of each dataset. Instances where error rates increase or decrease nonlinearly with changes in \"a\" suggest the need for further investigation into the tuning of \"a\" for specific applications. Table 5 displays the absolute error values of the proposed machine learning model across various regression datasets for three different values of the perturbation factor \(a\) (0.001, 0.005, 0.01). Data analysis reveals that the parameter \(a=0.005\) yields the lowest average error (4.92), while the values \(a=0.001\) and \(a=0.01\) result in slightly higher averages (5.48 and 5.13, respectively). This difference indicates that 0.005 is generally the most suitable value for the model, ensuring better performance in most cases. At the dataset level, the impact of \(a\) varies.