## 3 Experimental Results
### Experimental Results  
All the methods used here were coded in ANSI C++. The freely available optimization of Optimus, which 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 28 September 2024), was also utilized for the optimization process. Each experiment was conducted 30 times, and the average classification or regression value was measured. In each run, a different seed for the random number generator was used, and the drand48() function of the C programming language was incorporated. For the case of classification datasets, the displayed classification error for a model \(M(x)\) and the test dataset \(T\) is calculated as  
\[E_{C}(M(x))=100\times\frac{\sum_{i=1}^{N}(\text{class}(M(x_{i}))-y_{i})}{N} \tag{6}\]  
The test set \(T\) is defined as \(T=(x_{i},y_{i}),\ i=1,\ldots,N\). For the case of regression problems, the regression error \(E_{R}(M(x))\) is defined as \begin{table}
\begin{tabular}{c c} \hline
**DATASET** & **INPUTS** \\ \hline ABALONE & 8 \\ \hline AIRFOIL} & 5 \\ \hline BASEBALL & 16 \\ \hline BK & 4 \\ \hline BL & 7 \\ \hline CONCRETE & 8 \\ \hline DEE & 6 \\ \hline HO & 13 \\ \hline HOUSING & 13 \\ \hline FY & 4 \\ \hline LASER & 4 \\ \hline LW & 9 \\ \hline MORTGAGE & 15 \\ \hline MUNDIAL & 3 \\ \hline PL & 2 \\ \hline QUAKE & 3 \\ \hline REALSTATE & 5 \\ \hline SN & 11 \\ \hline TREASURY & 15 \\ \hline TZ & 60 \\ \hline VE & 7 \\ \hline \end{tabular}
\end{table} 
Table 2: Number of inputs for every regression dataset. \[E_{R}(M(x))=\frac{\sum_{i=1}^{N}(M(x_{i})-y_{i})^{2}}{N} \tag{7}\]  
The method of ten-fold cross-validation was incorporated to validate the experimental results. The experiments were carried out on an AMD Ryzen 5950X with 128 GB of RAM, running the Debian Linux operating system. The values for the parameters of the algorithms are shown in Table 3. In all tables, the bold font is used to mark the machine learning method that achieved the lowest classification or regression error. Table 4 contains the experimental results for the classification datasets and Table 5 contains the experimental results for the regression datasets. The neural network used in the experimental results has one processing unit with \(H=10\) processing nodes and the sigmoid function as an activation function. The following notation was used in these tables:  
1. The column BFGS represents the results obtained by the application of the BFGS method to train a neural network with \(H=10\) processing nodes. 2. The column GENETIC represents the results produced by the training of a neural network with \(H=10\) processing nodes using a genetic algorithm. The parameters of this algorithm are mentioned in Table 3.
3. The column NNC represents the results obtained by the neural construction technique. 4. The column INNC represents the results obtained by the proposed method. 5. The last row AVERAGE contains the average classification or regression error, as measured on all datasets. The methods BFGS and GENETIC train the neural network \(N\big{(}\overrightarrow{x},\overrightarrow{w}\big{)}\) by obtaining the global minimum of the error function defined in Equation (1). The original technique of constructing artificial neural networks has significantly lower classification or regression errors than the other two techniques, and the proposed method managed to improve the performance of this technique in the majority of the datasets. The percentage improvement in error is even greater in regression problems. In some cases, the improvement exceeds 50% in the test error. This effect is evident in the box plots for classification and regression errors outlined in Figures 4 and 5, respectively.