## 3 Results and Discussion
### Weighted Average Ensemble Models  
We developed a weighted average ensemble (WAE) method to predict the compressive strength of LWAC using the tree-based ensemble methods and the SVR method. The RF, SVR and XGBoost models were combined to give a more powerful model than each of these approaches alone. The triplet \((\alpha_{1},\alpha_{2},\alpha_{3})\) denotes the weights associated with each of these models, respectively. These parameters were varied from zero to one, such that \(\alpha_{1}+\alpha_{2}+\alpha_{3}=1\). The predictions from the WAE method were obtained following the flowchart shown in Figure 23 as a weighted sum of the results from the RF, SVR and XGBoost models, using the weights \(\alpha_{1}\), \(\alpha_{2}\) and \(\alpha_{3}\). For the cross-validation procedure, the dataset was split into training and test sets with a ratio of 75:25, using the same seeds as in the previous experiments. This randomised process was repeated 10 times. For each of these sets, the training process was carried out to tune the hyperparameters associated with these methods. The initialisation of the training procedure was conducted using good values for each individual model. To search for the optimal weights, the regularisation was controlled by means of the \(C\) parameter of the SVR method. In addition, the behaviour of this ensemble method was explored for varying proportions of the samples (\(p\)) used to train each base learner. The results showed good performance for \(p=0.75\) (see Figure 24). \begin{table}
\begin{tabular}{c c c c} \hline \(\mathbf{c}\) & **Measures** & **Mean of Test Sets** & **With Validation (Mean of 10 Models)** \\ \hline \multirow{3}{*}{0.3} & \(R^{2}\) & 0.8120 & 0.8237 \\  & MAE & 3.0000 & 2.8778 \\  & MAPE & 16.2627 & 15.5709 \\  & RMSE & 3.8586 & 3.7637 \\ \hline \multirow{3}{*}{0.4} & \(R^{2}\) & 0.8124 & 0.8272 \\  & MAE & 2.9944 & 2.852 \\  & MAPE & 16.1798 & 15.4092 \\  & RMSE & 3.8543 & 3.7252 \\ \hline \multirow{3}{*}{0.5} & \(R^{2}\) & 0.8115 & 0.8299 \\  & MAE & 3.0004 & 2.832 \\  & MAPE & 16.182 & 15.2920 \\  & RMSE & 3.8639 & 3.690 \\ \hline \multirow{3}{*}{0.6} & \(R^{2}\) & 0.8111 & 0.8134 \\  & MAE & 2.9986 & 2.8152 \\  & MAPE & 16.1617 & 15.1992 \\  & RMSE & 3.8679 & 3.6751 \\ \hline \multirow{3}{*}{0.7} & \(R^{2}\) & 0.8101 & 0.8333 \\  & MAE & 3.0035 & 2.8016 \\ \cline{1-1}  & MAPE & 16.1789 & 15.1239 \\ \cline{1-1}  & RMSE & 3.8783 & 3.6591 \\ \hline \end{tabular}
\end{table} 
Table 12: Statistical measures on the test data for SVR models; 10 models randomly generated with a 75:25 ratio by means of simple random sampling, RBF kernel, \(\gamma=\frac{1}{4\text{Var}(X)}\), \(\epsilon=0.04\). Figure 23: Flowchart of the WAE method. The WAE models were also rated based on the values of their goodness-of-fit measures, using both the discrete and continuous rankings described in Section 2.1.6 (see Table 13). Based on these rankings, Table 14 summarises several goodness-of-fit measures for the best configurations of hyperparameters found. For a tolerance error of 0.04, a good compromise between the accuracy of the models on the training sets and the accuracy on the test sets was obtained for \(0.6\leq C\leq 6\). For these configurations, suitable models were obtained with \(0\leq\alpha_{1}\leq 0.15\), \(0.65\leq\alpha_{3}\leq 1\), and \(\alpha_{2}=1-\alpha_{1}-\alpha_{3}\), with a mean RMSE of between 3.75 and 3.757, a mean MAE of between 2.872 and 2.877, and a mean MAPE of approximately 15.8% for the test datasets. These values were lower than those obtained from each individual estimator. We conclude that a good choice of weights for this ensemble method allows it to outperform the XGBoost models. For setting competitive models, we first identified the three WAE models with the lowest values of the overall continuous ranking; these were models (6), (4) and (5), with an overall rank of approximately 7.89 (see Table 13). We then analysed the corresponding training and test ratings to detect possible overfitting.