## 2 Method Description
### The Second Phase of the Proposed Method
Create the set \(T=\{r_{1},r_{2},\dots,r_{N_{s}}\}\) with \(N_{s}\) random samples in \([L,R]\). 2. Set \(E_{\text{min}}=\infty\), \(E_{\text{max}}=-\infty\)
3. For\(i=1,\dots,N_{s}\)do 1. **Create** the RBF network \(R_{i}\left(\overrightarrow{x}\right)\) using as parameter set the corresponding sample \(r_{i}\in T\). 2. **Calculate** the training error \(f_{R_{i}}=\sum_{j=1}^{M}\bigl{(}R_{i}\left(\overrightarrow{x_{j}}\right)-y_{ j}\bigr{)}^{2}\) for the training set of the objective problem. 3. **If \(f_{R_{i}}\leq E_{\text{min}}\) then\(E_{\text{min}}=f_{R_{i}}\)** 4. **If \(f_{R_{i}}\geq E_{\text{max}}\) then\(E_{\text{max}}=f_{R_{i}}\)** 5. **End For**
```

**Algorithm 5** The sampling function used in the Simulated Annealing algorithm. ### The Final Phase of the Proposed Method

In the last phase of the proposed procedure, a genetic algorithm is executed to train the RBF network. The training is performed within the optimal value interval identified in the previous phase of the proposed procedure. The main steps of the proposed genetic algorithm are shown in Algorithm 6. ```
ProcedurefinalPhase\(\big{(}N_{c},N_{g},p_{s},p_{m},[L_{b},R_{b}]\big{)}\) Inputs: Number of chromosomes \(N_{c}\),number of generations \(N_{g}\), selection rate \(p_{s}\), mutation rate \(p_{m}\), the bounds the bounds \([L_{b},R_{b}]\) of previous phase. 1. Initialize the chromosomes \(g_{i},\ i=1,\ldots,N_{c}\) as random vectors inside the bounds \([L_{b},R_{b}]\) of the second phase of the suggested algorithm. 2. Set\(k=0\)
3. For\(i=1,\ldots,N_{c}\)do 1. Create the RBF network \(R_{i}\big{(}\overrightarrow{x}\big{)}\) using as parameters the chromosome \(g_{i}\). 2. Calculate the corresponding fitness value \(f_{i}=\sum_{j=1}^{M}\big{(}R_{i}\big{(}\overrightarrow{x_{j}}\big{)}-y_{j} \big{)}^{2}\)
4. End For
5. Selection procedure. The \(p_{s}\times N_{c}\) best chromosomes with the lowest fitness values are copied intact to the next generation. The remaining chromosomes will be replaced by new chromosomes produced in crossover and mutation procedure. 6. Crossover procedure. During the application of crossover, pairs of chromosomes are chosen from the current population with tournament selection. For every pair \((z,w)\) of parents two new chromosomes \(\tilde{z}\) and \(\tilde{w}\) are will be produced using the following equations: \[\tilde{z_{i}} = a_{i}z_{i}+(1-a_{i})w_{i}\] \[\tilde{w_{i}} = a_{i}w_{i}+(1-a_{i})z_{i}\] (7) where \(i=1,\ldots,n\) and \(a_{i}\) are randomly selected values with \(a_{i}\in[-0.5,1.5]\)[37]. 7. Mutation procedure. For every element of each chromosome a random number \(r\in[0,1]\) is produced. This element will be altered randomly when \(r\leq p_{m}\). 8. Set\(k=k+1\)
9. If\(k<N_{g}\)then go to step 3. 10. Obtain the best chromosome \(g_{b}\) from the genetic population, with the lowest fitness value. 11. Create the corresponding RBF network \(R_{b}\big{(}\overrightarrow{x}\big{)}\)
12. Apply\(R_{b}\big{(}\overrightarrow{x}\big{)}\) to the corresponding test set and measure the associated test error and Terminate. End procedure
```

**Algorithm 6** The final phase of the proposed method. ## 3 Experiments

A series of classification and regression datasets from various websites was incorporated to test the proposed method and to measure its reliability. The datasets used in the conducted experiments were downloaded from the following online databases:

1. The UCI database: [[https://archive.ics.uci.edu/](https://archive.ics.uci.edu/)]([https://archive.ics.uci.edu/](https://archive.ics.uci.edu/)) (accessed on 26 March 2025) [38]
2. The Keel website: [[https://sci2s.ugr.es/keel/datasets.php](https://sci2s.ugr.es/keel/datasets.php)]([https://sci2s.ugr.es/keel/datasets.php](https://sci2s.ugr.es/keel/datasets.php)) (accessed on 26 March 2025) [39]. 3. The Statlib URL: [[https://lib.stat.cmu.edu/datasets/](https://lib.stat.cmu.edu/datasets/)]([https://lib.stat.cmu.edu/datasets/](https://lib.stat.cmu.edu/datasets/)) (accessed on 26 March 2025).