## 6 Results and Discussions
Figure 6: The development process of the classical XGBoost model for estimating HL. Figure 7: The development process of the SVM model for estimating HL  
For the development of the RF model, the number of the tree in the forest (\(n\)) and randomly selected predictor (\(\omega\)), were used to adjust the accuracy/performance of the RF model. According to the recommendation of [PERSON], [PERSON] [67], \(n\) should be set equal to 2000 to ensure the enrichment of the forest. Then, \(\omega\) was tested to check the accuracy of the RF model. Since the predictors used in this study was eight, \(\omega\) was set in the range of 1 to 8. Ultimately, the performance of the RF model based on the 2000 trees and randomly predictors was computed in Figure 8. As a result, the best RF model for predicting HL in this study was the RF3 (i.e., \(\omega=3\)). Note that, the same techniques, as well as the same training dataset, were used for the development of the RF model, like those used to develop the previous models (i.e., PSO-XGBoost, XGBoost, SVM). Like to the SVM model, kernel functions can also be applied to develop the GP model. RBF was also used to develop the GP model for estimating HL in this work with \(\sigma\) was the only parameter used to control the accuracy of the GP model. A grid search, as well as the same techniques, like those used for the previous models, was also applied to develop the GP model. As a result, an optimal GP model was developed in this study with an \(\sigma\) of 0.009, for estimating the HL of buildings (Figure 9). Figure 8: The development process of the random forest (RF) model for estimating HL. Figure 9: The development process of the Gaussian process (GP) model for estimating HL. Finally, an optimal CART model was also developed based on the same techniques and the training dataset, like those used for the previous models. Note that, only the complexity parameter (\(\psi\)) was used to develop the CART model with the grid search lies in the range of [0,0.1] (Figure 10). After the HL predictive models were developed, 165 observations of the testing dataset were used to evaluate the performance of the model through the statistical criteria, i.e., RMSE, R\({}^{2}\), VAF, MAE, and MAPE, as shown in Table 3. The intensity of color and ranking method were also applied to evaluate the models. The results in Table 3 confirmed the perfect predictability of the proposed AI techniques for the HL of buildings systems in the present study. Of those AI techniques, the proposed PSO-XGBoost model was defined as the most superior model in predicting the HL of buildings with an RMSE of 1.124, R\({}^{2}\) of 0.990, MAE of 0.615, VAF of 98.934, MAPE of 0.024, and total ranking of 29. Figure 11 illustrates the exact predictability of the proposed PSO-XGBoost model on the testing dataset. Figure 11: The results of HL prediction from the proposed PSO-XGBoost model. Figure 5:0 The development process of the classification and regression trees (CART) model for estimating HL. \begin{table}
\begin{tabular}{l|l|l|l|l|l|l|l|l|l|l|l|l} \hline \hline
**Technique** & **RMSE** & **R\({}^{2}\)** & **MAE** & **VAF** & **MAPE** & **Rank for** & **Rank for** & **Rank for** & **Rank for** & **Rank for** & **Total** \\  & & & & & & & & & & & & & \\ \hline PSO-XGBosest & 1.74 & 0.90 & 0.615 & 0.904 & 0.08 & 6 & 6 & 5 & 6 & 6 & 30 & \\ XGBost & 1.631 & 0.977 & 0.720 & 97.644 & 0.088 & 3 & 3 & 4 & 3 & 4 & 17 & \\ SVM & 1.767 & 0.973 & 0.910 & 9.315 & 0.072 & 2 & 1 & 1 & 2 & 1 & 7 \\ RF & 1.589 & 0.978 & 0.887 & 9.783 & 0.088 & 5 & 4 & 6 & 5 & 5 & 25 \\ GP & 1.632 & 0.978 & 0.798 & 97.22 & 0.033 & 4 & 4 & 2 & 4 & 2 & 16 \\ CART & 1.779 & 0.973 & 0.273 & 97.236 & 0.031 & 1 & 1 & 3 & 1 & 3 & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The results of the developed model on the testing dataset through several statistical indices. Compared with the classical XGBoost model (without being optimized by the PSO algorithm), the proposed PSO-XGBoost model provided superior performance in estimating the HL of buildings.