## 6 Results and Discussions  
In this stage, 80% of the whole dataset was randomly selected to develop the HL of buildings forecasting models; the remaining 20% of the dataset was used to test and re-evaluate the accuracy, as well as the performance of the developed models. The re-sampling method of 10-fold cross-validation was applied to reduce the error of the models. Note that, the same training/testing datasets, as well as  
Figure 4: Scheme of the development of the PSO and extreme gradient boosting (PSO-XGBoost) model for estimating the HL of building systems. the re-sampling techniques, were used for all models. It is of interest to consider the feasibility of the proposed PSO-XGBoost model in estimating the HL of buildings systems. Indeed, a combined process between the XGBoost model and the PSO algorithm was made based on the framework introduced in Figure 4. The parameters of the PSO algorithm has been optimally set before performing the optimization of the XGBoost model, as shown in Table 2. After the parameters of the PSO algorithm have been set, the process of searching and optimizing for the hyper-parameters of the XGBoost was performed. Its performance is shown in Figure 5.  
As shown in Figure 5, an optimal PSO-XGBoost model was found with the swarm size of 400 and stopped at the iteration of 209 with the lowest RMSE (i.e., RMSE = 1.776). For comparison and overall performance evaluation of the developed PSO-XGBoost model, the remaining models have also been developed as previously introduced, including XGBoost, SVM, RF, GP, and CART models. For the development of the classical XGBoost model, the same parameters were used to control the accuracy of the classical XGBoost model, like those used for the PSO-XGBoost model. However, a grid search technique was applied to determine the hyper-parameters of the classical XGBoost model. The parameters of the grid were set as follows: \(\eta\) = [0.1, 0.4]; \(\delta\) = 0.6 and 0.8; \(k\) = 50, 100, 150; \(\gamma\) = 0; \(d\) = [1,3]; \(\zeta\) = 0.5, 0.75, 1; and \(\mu\) = 1. Subsequently, 10-fold cross-validation was applied during the development of the classical XGBoost model. Finally, the optimal XGBoost model was found with \(k\) = 50; \(d\) = 3; \(\eta\) = 0.3, \(\gamma\) = 0, \(\delta\)= 0.6, \(\zeta\) = 1, and \(\mu\) = 1 (Figure 6). \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameters** & **Acronym** & **Value** \\ \hline The number population & \(p\) & 50,100,150,200,250,300,350,400,450,500 \\ Maximum particle’s velocity & \(V_{max}\) & 2.00 \\ Individual cognitive & \(\phi_{1}\) & 1.8 \\ Group cognitive & \(\phi_{2}\) & 1.8 \\ Inertia weight & \(w\) & 0.95 \\ Maximum number of iteration & \(m_{i}\) & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The optimal values of the PSO algorithm in this study. Figure 5: Performance of the proposed PSO-XGBoost technique in estimating the HL of building systems. For the development of the SVM model, the radial basis function (RBF) was used as a kernel function to estimate the HL in the present study. Two hyper-parameters of the SVM model with the RBF were selected and considered to construct the SVM model, including Sigma (\(\sigma\)) and cost (\(C\)). A grid search was established to find the optimal values for \(\sigma\) and \(C\) with \(\sigma\) lies in the range of [0,1]; C lies in the range of [0.25,5]. The stepwise for \(\sigma\) and \(C\) in the grid search is 0.05 and 0.25, respectively. The \"scale\" method was applied to reduce the skewness of the data in this study. The ten-fold cross-validation technique was also applied to increase the accuracy of the model. Ultimately, the optimal SVM model for estimating HL in this study was defined with the following hyper-parameters: \(\sigma\) = 0.05 and \(C\) = 1.75, as shown in Figure 7. Note that, the same training dataset (i.e., 672 experimental datasets) was used to develop the SVM model, like those used for the development of the PSO-XGBoost and XGBoost models.