# The Effect of Multi-Generational Selection in Geometric Semantic Genetic Programming
## 5 Results and Discussion  
Figures 3 and 4 show, via box-plots, the distribution of the fitness, calculated over 100 independent runs, achieved by the different configurations, discussed in Section 4.2, compared to classical GSGP, which is denoted here as U1 whereas, as explained before, it considers only the previous generation from which to select parents, exactly as in GSGP. Table 3 stores the fitness values obtained by selecting the ancestors with uniform multi-generational selection for both the training and testing set among all the considered benchmark problems. Similarly, Table 4 stores the geometric multi-generational selection method's fitness values. Table 5 reports a statistical significance assessment of the achieved result, displaying the \(p\)-values obtained under the hypothesis that the median fitness resulting from the considered technique is equal to the one obtained with standard GSGP. This means that, if the resulting \(p\)-values are zero, the fitness obtained is statistically significantly better \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & **GSGP** & **U2** & **U5** & **U10** & **U20** & **U50** & **U100** \\ \hline \multirow{3}{*}{air} & train & 34.43 & 33.89 & **32.02** & 33.28 & 32.85 & 34.93 & 34.93 \\ \cline{2-8}  & test & 34.44 & 34.01 & **31.83** & 33.26 & 33.12 & 34.72 & 37.51 \\ \hline \multirow{3}{*}{\%F} & train & 41.92 & 41.78 & **40.37** & 41.13 & 40.72 & 43.00 & 43.53 \\ \cline{2-8}  & test & 42.17 & 42.54 & **41.27** & 42.10 & 41.36 & 43.56 & 43.94 \\ \hline \multirow{3}{*}{conc} & train & 9.54 & 9.41 & **9.15** & 9.27 & 9.36 & 9.82 & 10.08 \\ \cline{2-8}  & test & 9.52 & 9.49 & **9.21** & 9.31 & 9.35 & 9.69 & 10.07 \\ \hline \multirow{3}{*}{\%PPB} & train & 36.62 & 38.26 & **29.48** & 31.02 & 30.82 & 44.80 & 51.94 \\ \cline{2-8}  & test & 255.51 & 243.46 & 335.42 & 371.03 & 298.03 & 206.88 & **148.83** \\ \hline \multirow{3}{*}{LD50} & train & 2183.65 & 2183.17 & **2165.20** & 2171.09 & 2165.36 & 2199.38 & 2243.06 \\ \cline{2-8}  & test & 2262.15 & **2233.41** & 2250.19 & 2242.84 & 2240.93 & 2274.87 & 2280.51 \\ \hline \multirow{3}{*}{yac} & train & 13.71 & 13.77 & **13.04** & 13.24 & 13.18 & 14.19 & 14.44 \\ \cline{2-8}  & test & 13.55 & 13.69 & **12.99** & 13.12 & 13.00 & 14.11 & 14.27 \\ \hline \end{tabular}
\end{table} 
Table 3: Fitness values obtained by selecting the ancestors with uniform multi-generational selection. The values in bold are the best results obtained. Figure 4: Box-plots of the RMSE on the test set over 100 independent runs of the considered benchmark dataset for all the proposed methods. (**a**) Air; (**b**) %F; (**c**) conc; (**d**) %PPB; (**e**) LD50; (**f**) yac. Concerning the _air_ dataset (Figures 3a and 4a), it is possible to see that, if we consider a uniform distribution for the multi-generational selection, the best performance is obtained by selecting ancestors from the 5 previous generations (Table 3); on the other hand, with a geometric distribution, better results are achieved with \(p=0.25\) (Table 4). Both of these methods meaningfully outperform standard GSGP, as confirmed by the statistical results of Table 5. For what concerns the other variations of the proposed methods, combining information provided by fitness distribution shown in the box plots with the corresponding \(p\)-values, it is possible to conclude that: U2, U10 and U20 also outperform GSGP with statistical significance, U50 leads to results similar to GSGP, while U100, G0.50 and G0.75 results are substantially less performing with respect to GSGP. Considering the %F dataset (Figures 3b and 4b), a similar behavior as the one observed for the previous benchmark problem appears. Again, U5 and G0.25 represent the two best candidates and outperform GSGP together with U2, U10, and U20. Here, U50, U100, G0.50, and G0.75 lead to worse fitness values compared to classical GSGP. Moving to the _conc_ dataset, (Figures 3c and 4c provide us with more evidence that U5 and G0.25 are a significant improvement of the standard GSGP.