# Optimum Design of Cylindrical Walls Using Ensemble Learning Methods
## 3 Results and Discussion  
Table 3 and Figures 4 and 5 show the performance of four different ensemble learning models, which were trained with including the structural cost in the training set. In Figure 4, the horizontal axis shows the h  
Figure 4: Comparison of the optimized and predicted h values (4 design variables). **(a)** LightGBM, **(b)** Random Forest, **(c)** XGBoost, **(d)** CatBoost.  
Figure 5: Model accuracies with 4 design variables. **(a)** LightGBM, **(b)** Random Forest, **(c)** XGBoost, **(d)** CatBoost.  
Since, in practice, the optimum cost may not be readily available, predictive models for the optimum wall thickness were also generated using only the geometric variables H, r, and the material variable C/S ratio as the design variables. The comparison of the h values predicted using the ensemble learning algorithms with the h values obtained through the harmony search optimization can be seen in Figure 6 for the case of three design variables. The accuracy values corresponding to the design with three variables are listed in Table 3 and visually represented in Figure 7. The comparison between Tables 3 and 4 shows that the models exhibited relatively similar performance with the model trained without the cost variable performing slightly better. In the case of four input variables, the CatBoost model has the best performance on the test set with an R\({}^{2}\) value of 0.9999, while reducing the number of input variables increased the R\({}^{2}\) value of all models to the same level. Furthermore, the CatBoost model also has the lowest RMSE, MAE and MAPE values in the case of four input variables, whereas the Random Forest model is observed to have at least an order of magnitude smaller RMSE, MAE and MAPE values on the test set compared to the other models in the case of three input variables. The LightGBM, XGBoost, and CatBoost models had RMSE, MAE and MAPE values close to each other, with XGBoost having the largest RMSE, MAE and MAPE values in the case of three input variables. However, it should be noted that all of the models were able to achieve R\({}^{2}\) values close to 1, which indicates high model accuracy. The high accuracy of the models can also be seen in Figure 8 where the actual and predicted wall thickness values are plotted for the random forest model with three input variables.  
Figure 8: Comparison of the actual and predicted (Random Forest) wall thickness values.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\mathbf{R^{2}}\) & \(\mathbf{RMSE}\) & \(\mathbf{MAE}\) & \(\mathbf{MAPE}\) \\ \hline Train (LightGBM) & 0.9999 & \(1.86\times 10^{-4}\) & \(1.39\times 10^{-4}\) & \(3.86\times 10^{-4}\) \\ Test (LightGBM) & 0.9999 & \(1.98\times 10^{-4}\) & \(1.50\times 10^{-4}\) & \(4.09\times 10^{-4}\) \\ Train (RF) & 0.9999 & \(1.60\times 10^{-5}\) & \(6.40\times 10^{-7}\) & \(2.10\times 10^{-6}\) \\ Test (RF) & 0.9999 & \(2.90\times 10^{-5}\) & \(1.20\times 10^{-6}\) & \(4.10\times 10^{-6}\) \\ Train (XGBoost) & 0.9999 & \(4.00\times 10^{-4}\) & \(2.49\times 10^{-4}\) & \(7.76\times 10^{-4}\) \\ Test (XGBoost) & 0.9999 & \(4.50\times 10^{-4}\) & \(2.74\times 10^{-4}\) & \(8.51\times 10^{-4}\) \\ Train (CatBoost) & 0.9999 & \(1.10\times 10^{-4}\) & \(8.10\times 10^{-5}\) & \(2.35\times 10^{-4}\) \\ Test (CatBoost) & 0.9999 & \(1.40\times 10^{-4}\) & \(9.45\times 10^{-5}\) & \(2.75\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Performances of the ensemble learning algorithms trained without the cost variable.  
Figure 7: Model accuracies with 3 design variables. (a) LightGBM, (b) Random Forest, (c) XGBoost, (d) CatBoost.