## 2 Method Description
### The Second Phase of the Proposed Method
### Experimental Results

The code used in the experiments was implemented in the C++ programming language and a machine equipped with 128 GB RAM running Debian Linux was utilized in the conducted experiments. The code was written with the assistance of the freely available GlobalOptimus optimization environment, that can be downloaded from [[https://github.com/itsoulos/GlobalOptimus](https://github.com/itsoulos/GlobalOptimus)]([https://github.com/itsoulos/GlobalOptimus](https://github.com/itsoulos/GlobalOptimus)) (accessed on 12 April 2025). Each experiment was executed 30 times and the average classification error was measured for the classification datasets and the average regression error was measured for the regression datasets. The classification error was computed using the following equation:

\[E_{C}(R(x))=100\times\frac{\sum_{i=1}^{K}(\text{class}(R(x_{i}))-y_{i})}{K} \tag{8}\]

where the set \(T=\{x_{i},y_{i}\},\ i=1,\ldots,K\) stands for the test set of the current problem and \(R(x)\) is the RBF model. The regression error was calculated through the following equation:

\[E_{R}(R(x))=\frac{\sum_{i=1}^{K}(R(x_{i})-y_{i})^{2}}{K} \tag{9}\]

The values for the parameters of the proposed method are mentioned in Table 1. The selection of values for the experimental parameters was performed in such a way that there was a compromise between the speed and reliability of the proposed methodology. In the following tables, that describe the experimental results, the following notation is used:

1. The column Dataset represents the name of the objective problem. 2. The column BFGS denotes the application of the BFGS optimization method [80] in the training of a neural network [81; 82] with 10 processing nodes. 3. The column ADAM stands for the incorporation of the ADAM optimizer [83] to train an artificial neural network with 10 processing nodes. 4. The column NEAT represents the usage of the NEAT method (NeuroEvolution of Augmenting Topologies) [84]. 5. The column RBF-KMEANS stands for the usage of the original two-phase method to train an RBF network with 10 processing nodes. 6. The column GENRBF represents the incorporation of the method proposed in [85] to train an RBF network with 10 processing nodes. 7. The column Proposed denotes the usage of the proposed method to train an RBF network with 10 processing nodes. 8. The row Average represents the average classification or regression error. 9. The row W-average denotes the average classification error for all datasets and for each method. In this average, each individual classification error was multiplied by the number of patterns for the corresponding dataset. The results from the application of the previously mentioned machine learning methods to the classification datasets are depicted in Table 2, and for the regression datasets the results are presented in Table 3.

Table 2 presents the error rates of the various machine learning models (BFGS, ADAM, NEAT, RBF-KMEANS, GENRBF, proposed) on the different classification datasets. Each row corresponds to a dataset, while each column represents the error rate of a specific model. These values indicate the percentage of incorrect predictions, with lower values reflecting better performance. The last row of the table includes the average error rates for each model. Statistical analysis of the data reveals significant insights. The proposed model exhibits the lowest average error rate (18.67%) compared to the other models, establishing it as the optimal choice based on the table. Conversely, the other models demonstrate higher average error rates, with GENRBF showing the highest average error (34.64%). Additionally, significant variations in error rates across datasets are observed. For instance, on the \"Circular\" and \"ZONF_S\" datasets, the proposed model outperforms others, with very low error rates (4.19% and 1.79%, respectively). Conversely, on datasets like \"Cleveland\", the NEAT model shows a lower error rate (53.44%) compared to the proposed model (50.82%).