## 6 Conclusions  
An investigation on the use of ML methods based on the Ensemble of Trees (ET) for the prediction of the torsional capacity of RC beams was conducted in this study. A database with 202 RC beam samples was created and randomly divided into testing and training sets. Using a five-fold cross-validation procedure paired with a grid search strategy, optimal hyperparameters for all the models were found based on the training dataset. The testing dataset was used to validate the performance of the built models. From the obtained results, the following main conclusions that can be drawn:  
* A simple Decision Tree (DT) regressor predicts the torsional capacity of RC beams with satisfactory accuracy. The model has a \(R^{2}\) value of 0.973 and a _cvRMSE_ value of 16.78% for the testing set;
* It was shown that ET reach higher accuracies than a simple DT. The Bagging Meta-Estimator (BME), the Forests of Randomized Trees (FRT), the AdaBoost (AB) and the Gradient Tree Boosting (GTB) reached values of \(R^{2}\) in the range between 0.982 and 0.990, and values of _cvRMSE_ in the range between 10.04% and 13.92%. These results indicate that the prediction capability of these four models can be trusted with high confidence, in particular for the GTB and FRT models; \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Models**} & \multirow{2}{*}{**INPUT VARIABLE**} & \multicolumn{8}{c}{**Testing Set**} \\ \cline{3-10}  & & **MAE** & **RMSE** & _cvRMSE_ (\%) & \(\mathbf{R^{2}}\) & _MAPE_ (\%) & _RMSE_ & _VAF_ \\ \hline \multirow{4}{*}{DT} & \(A_{c},f_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 10.713 & 17.749 & 16.78 & 0.973 & 12.18 & 0.163 & 0.975 \\  & \(f_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 27.620 & 46.919 & 44.36 & 0.813 & 35.85 & 0.432 & 0.815 \\  & \(A_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 13.694 & 22.280 & 21.06 & 0.958 & 14.94 & 0.205 & 0.960 \\  & \(A_{c}\), \(f_{c}\) & 19.959 & 31.359 & 29.65 & 0.917 & 23.09 & 0.289 & 0.926 \\  & \(A_{c}\), \(f_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 7.616 & 12.064 & 11.41 & 0.988 & 8.83 & 0.111 & 0.989 \\  & \(f_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 29.184 & 48.128 & 45.50 & 0.804 & 32.68 & 0.443 & 0.806 \\  & \(A_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 10.030 & 16.123 & 15.24 & 0.978 & 11.06 & 0.148 & 0.980 \\  & \(A_{c}\), \(f_{c}\) & 20.905 & 34.497 & 32.61 & 0.899 & 20.00 & 0.318 & 0.910 \\  & \(A_{c}\), \(f_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 7.434 & 11.461 & 10.83 & 0.989 & 8.72 & 0.106 & 0.990 \\  & \(f_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 28.391 & 44.576 & 42.14 & 0.832 & 35.61 & 0.410 & 0.833 \\  & \(A_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 9.974 & 16.468 & 15.57 & 0.977 & 11.01 & 0.152 & 0.979 \\  & \(A_{c}\), \(f_{c}\) & 20.586 & 31.897 & 30.16 & 0.914 & 20.20 & 0.294 & 0.925 \\  & \(A_{c}\), \(f_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 9.901 & 14.725 & 13.92 & 0.982 & 10.83 & 0.136 & 0.983 \\  & \(f_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 32.522 & 54.019 & 51.07 & 0.753 & 36.96 & 0.497 & 0.755 \\  & \(A_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 12.225 & 21.030 & 19.88 & 0.963 & 12.22 & 0.194 & 0.964 \\  & \(A_{c}\), \(f_{c}\) & 18.927 & 29.762 & 28.14 & 0.925 & 20.83 & 0.274 & 0.931 \\  & \(A_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 6.856 & 10.618 & 10.04 & 0.990 & 7.21 & 0.098 & 0.991 \\  & \(f_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 29.846 & 46.988 & 44.42 & 0.813 & 45.15 & 0.433 & 0.815 \\  & \(A_{c}\), \(A_{l}f_{yl}A_{t}f_{yt}/s\) & 13.664 & 23.794 & 22.49 & 0.952 & 14.52 & 0.219 & 0.952 \\  & \(A_{c}\), \(f_{c}\) & 24.378 & 41.416 & 39.155 & 0.855 & 21.97 & 0.381 & 0.896 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Effect of input variables on the performance of the ML models.