## 3 Results
### Results Acquired Using GPSR with RHS and 5-CV  
The GPSR algorithm with the 5-CV was executed multiple times. Before each execution, the hyperparameters were randomly selected from the predefined ranges shown in Table 3. From the obtained results, the three best cases were selected that achieved the highest estimation accuracy. The combination of the hyperparameters that were used to obtain the highest estimation accuracies is listed in Table 4.  
From Table 4, it can be seen that, for Cases 1 and 3, the population size was very large. In Case 1, the point mutation (0.43) was the dominating genetic operation, while in Cases 2 and 3, the crossover (0.41) and subtree mutation (0.469) dominated the genetic operations. However, in Case 3, both crossover and subtree mutation had higher probabilities than the other two types of mutations. The stopping criteria, as planned, were never reached by any population member in all three cases, and each execution stopped after the maximum possible number of generations was attained. The parsimony coefficient (Pcoef) value was much higher in Cases 2 and 3 than in Case 1. The results of these three cases are shown in Figure 6 and Table 5.  
As seen from Figure 6 and Table 5, in all three cases, the mean performance metric values obtained with the 5-CV on the training data were very high with low \(\sigma\) values. Table 6 shows the depth and length of each SE with the average depth and length of the SEs in each case. To clarify the terms length and depth in Table 6, the length is a representation of several elements (constants, functions, and possible variables) in the SEs, while the depth is measured when the SE is shown in tree form. The depth is measured from the required root node to the deepest leaf of the SE. Since the tree form of the SE is not important for this investigation, the length of the SE will be investigated. The highest SE lengths and average length were achieved in Case 1 followed by Case 3 and Case 2. From the length and the estimation accuracy in Table 5, the best case is Case 2, since the estimation accuracy is slightly lower than Case 1, while the average length of the SEs is the lowest. Figure 6: The mean values of the obtained performance metric during GPSR 5-CV. The \(\sigma\) values are presented as error bars. As seen from Figure 7 and Table 7, the mean values of the performance metric are almost the same as those obtained on the training dataset. The \(\sigma\) values are slightly higher when compared to those obtained on the training dataset. Although Case 2 showed better performance on the training dataset, the results of the testing dataset showed that Case 3 slightly outperformed Case 2. It can be noticed that the highest scores in Table 7 are marked with red color. The marked scores represent the chosen SE of three cases that will be used for a customized solution. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Case No.** & **Symbolic Expression No.** & **Length** & **Depth** & **Average Length** & **Average Depth** \\ \hline \multirow{4}{*}{3} & 1 & 145 & 31 & & & \\ \cline{2-7}  & 2 & 156 & 19 & & & \\ \cline{2-7}  & 3 & 127 & 27 & 161.6 & 24.8 \\ \cline{2-7}  & 4 & 230 & 17 & & & \\ \cline{2-7}  & 5 & 150 & 30 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: _Cont._