# Using Optimization Techniques in Grammatical Evolution
## 3 Results
### Experimental Results
Figure 4 is a scatter plot that provides a detailed comparative analysis of classification error percentages for six distinct machine learning and optimization algorithms: ADAM, NEAT, GENETIC, RBF, NNC, and NNC-S. Each point on the plot represents the outcome of an individual run of a model, thus showcasing the range of variation in the error rates associated with these classification methods. The vertical dispersion of points for each method reflects the spread of error rates, which is critical for evaluating the reliability of each algorithm. The medians of these error rates are indicated by horizontal lines intersecting the clusters of dots, offering a snapshot of each algorithm's central tendency in performance. The asterisk-based notation above the clusters denotes statistical significance levels: one asterisk (*) signifies a p-value less than 0.05, two asterisks (**) denote \(p<0.01\), three asterisks (***) represent \(p<0.001\), and four asterisks (***) indicate an extremely low p-value of less than 0.0001, suggesting strong evidence against the null hypothesis. Of note is the performance of the NNC-S method, which not only shows a significantly lower median error rate when compared to the NNC method but also displays a notably tighter cluster of data points. This implies that the NNC-S method not only tends to be more accurate on average but also provides greater consistency in its error rates across different runs, underscoring its robustness as a classification tool. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Dataset** & **ADAM** & **NEAT** & **GENETIC** & **RBF** & **NNC** & **NNC-S** \\ \hline ABALONE & 4.30 & 9.88 & 7.17 & 7.37 & 5.11 & **4.95** \\ \hline AIRFOIL} & 0.005 & 0.067 & 0.003 & 0.27 & 0.003 & **0.003** \\ \hline BASEBALL & 77.90 & 100.39 & 103.60 & 93.02 & 59.40 & **57.30** \\ \hline CONCRETE & 0.078 & 0.081 & 0.0099 & 0.011 & 0.008 & **0.006** \\ \hline DEE & 0.63 & 1.512 & 1.013 & 0.17 & 0.26 & **0.23** \\ \hline HO & 0.035 & 0.167 & 2.78 & 0.03 & 0.016 & **0.012** \\ \hline HOUSING & 80.20 & 56.49 & 43.26 & 57.68 & 25.56 & **18.82** \\ \hline LASER & 0.03 & 0.084 & 0.59 & 0.024 & 0.026 & **0.015** \\ \hline LW & 0.028 & 0.17 & 1.90 & 1.14 & 0.97 & **0.038** \\ \hline MORTGAGE & 9.24 & 14.11 & 2.41 & 1.45 & 0.29 & **0.12** \\ \hline PL & 0.117 & 0.097 & 0.28 & 0.083 & 0.046 & **0.033** \\ \hline SN & 0.026 & 0.174 & 2.95 & 0.90 & 0.026 & **0.024** \\ \hline TREASURY & 11.16 & 15.52 & 2.93 & 2.02 & 0.47 & **0.18** \\ \hline TZ & 0.07 & 0.097 & 5.38 & 4.10 & 0.06 & **0.028** \\ \hline
**AVERAGE** & **13.12** & **14.20** & **12.45** & **12.02** & **6.60** & **5.84** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental results as measured on the regression datasets. The numbers in cells denote the average regression errors for the associated machine learning method, as measured on the test set. The bold values it is and indication if the suggested method outperforms the original NNC method. Building on the previous analysis of classification methods, Figure 5 extends the evaluation to regression algorithms, providing a comparative study of regression error rates for the same six methods: ADAM, NEAT, GENETIC, RBF, NNC, and NNC-S. Each data point reflects the regression error from a specific trial, with the array of points for each method revealing the range of performance outcomes. The median error rates are again represented by horizontal lines across the clusters of dots, serving as a summary statistic that facilitates a direct comparison of the method's central performance trends. The notation of statistical significance is consistent with the previous figure, where asterisks convey the \(p\)-value levels, identifying statistically meaningful differences in performance between the methods. This plot reveals that the NNC-S method maintains its superior performance in the context of regression tasks, demonstrating lower median regression errors compared to the other methods.