# The Effect of Multi-Generational Selection in Geometric Semantic Genetic Programming
## 5 Results and Discussion
Moreover, it is still the case that U2, U10 and U20 slightly outperform GSGP, while U50, U100, G0.50 and G0.75 bring poor fitness values. Taking into account the %PPB dataset (Figures 3d and 4d), it is clear that all models are affected by overfitting, and a lower error in the training set entails a bigger error in the test set. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & **GSGP** & **G0.25** & **G0.50** & **G0.75** \\ \hline \multirow{2}{*}{air} & train & 34.43 & 37.48 & **32.97** & 40.76 \\ \cline{2-6}  & test & 34.44 & **32.93** & 40.63 & 43.51 \\ \hline \multirow{2}{*}{\%F} & train & 41.92 & **41.16** & 44.60 & 44.92 \\ \cline{2-6}  & test & 42.17 & **41.49** & 44.63 & 44.73 \\ \hline \multirow{2}{*}{conc} & train & 9.54 & **9.35** & 10.58 & 10.86 \\ \cline{2-6}  & test & 9.52 & **9.37** & 10.48 & 10.76 \\ \hline \multirow{2}{*}{\%PPB} & train & 36.62 & **32.06** & 57.37 & 58.73 \\ \cline{2-6}  & test & 255.51 & 300.03 & 119.20 & **106.33** \\ \hline \multirow{2}{*}{LD50} & train & 2183.65 & **2176.51** & 2234.56 & 2264.43 \\ \cline{2-6}  & test & 2262.15 & **2216.47** & 2258.10 & 2305.45 \\ \hline \multirow{2}{*}{yac} & train & 13.71 & **13.37** & 14.55 & 14.73 \\ \cline{2-6}  & test & 13.55 & **13.30** & 14.52 & 14.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fitness values obtained by selecting the ancestors with geometric multi-generational selection. The values in bold are the best results obtained. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **U2** & **U5** & **U10** & **U20** & **U50** & **U100** & **G0.25** & **G0.50** & **G0.75** \\ \hline airfoil & 0.158 & **0.000** & **0.000** & **0.000** & 0.280 & 0.000 & **0.000** & 0.000 & 0.000 \\ bioav & 0.741 & **0.000** & **0.001** & **0.000** & 0.000 & 0.000 & **0.007** & 0.000 & 0.000 \\ concrete & 0.763 & **0.001** & **0.042** & 0.445 & 0.001 & 0.000 & 0.557 & 0.000 & 0.000 \\ ppb & 0.000 & **0.000** & **0.000** & **0.000** & 0.000 & 0.000 & **0.000** & 0.000 & 0.000 \\ toxicity & 0.783 & **0.049** & 0.365 & 0.128 & 0.281 & 0.001 & 0.275 & 0.001 & 0.000 \\ yacht & 0.135 & **0.000** & **0.000** & **0.000** & 0.000 & 0.000 & **0.001** & 0.000 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: \(p\)-values returned by the Wilcoxon rank-sum test under the alternative hypothesis that the median errors on the test set obtained from classical GSGP are equal with respect to the errors obtained with the methods introduced in this paper. Highlighted in bold, the \(p\)-values below 0.05 where the direction of the difference shows an improvement with respect to standard GSGP. Applying our methods to the LD50 dataset (Figures 3e and 4e), we can once more recognize that U5 and G0.25 represent the best improvement in GSGP. Further, it is important to highlight that for this dataset, while U2, U10, U20, U50, U100, G0.50, and G0.75 are indeed less well performing than the standard GSGP, none of these methods result in actually poor fitness values; instead, each of them reaches performance very similar to that obtained by GSGP. Finally, the _yac_ dataset (Figures 3f and 4f) is the last confirmation of the behavior we have been observing so far. Once more, U5 and G0.25 are a significant improvement of the standard GSGP, U2, U10 and U20 outperform GSGP, whereas U50, U100, G0.50 and G0.75 result in poor fitness values. All in all, experiments performed revealed that U5 and G0.25 provide us with a meaningful improvement of the standard GSGP. Regarding results obtained with uniform multi-generational selection, a number \(k\) of previous generations equal to 10 and 20 (and 5, as aforementioned) leads to statistically significant improvement in the fitness. This confirms our intuition: selecting ancestors also from previous generations (that, anyway, are not too far away) led to better results as a more wide set of genotypes is considered for recombination, and good characteristics of an individual that may have been lost during generation can be retrieved, thus decreasing the likelihood of being stuck in local minima.