# Plum Tree Algorithm and Weighted Aggregated Ensembles for Energy Efficiency Estimation
## 5 PTA and Machine Learning Ensembles for Energy Efficiency Estimation
### Energy Efficiency Estimation Results  
The energy efficiency estimation experiments were performed using the _sklearn_ package. The values of the specific configuration parameters of the PTA were the default ones presented in Table 5. Table 10 summarizes the adapted parameters of the PTA for the optimization of the ensemble weights.  
For each dataset used in the experiments, namely, the Cooling Dataset and the Heating Dataset, and for each configuration resulting from the cross-validation operation, 30 runs were performed. The total number of runs was equal to \(2\times 5\times 30=300\).  
The std values were either zero or a very small number for both the Cooling Dataset and the Heating dataset; therefore, the mean values of the RMSE, \(R^{2}\), MAE, and MAPE were also the best ones.  
The best mean results for the Cooling Dataset and the Heating Dataset were returned for fold 4 and fold 1, respectively. Another remark is that the mean RMSE results were much better for the Heating Dataset for all five folds.  
Figure 10 presents the detailed results summary that corresponds to the runs which resulted in the best RMSE values. The values for the \(R^{2}\), MAE, and MAPE were equal in each of the 30 runs for each fold; therefore, the results were the same when the best values of these indicators were used as selection criteria for the most representative runs.  
The detailed results for each fold show that the \(R^{2}\) prediction results were comparable for the two datasets for each fold, which means that the PTA  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline number of iterations (\(I\)) & 500 \\ number of dimensions (\(D\)) & 4 \\ number of plums (\(N\)) & 50 \\ maximum possible position value (\(X_{max}\)) & 2000 \\ minimum possible position value (\(X_{min}\)) & \(-2000\) \\ objective function (\(OF\)) & formula (10) \\ \hline \hline \end{tabular}
\end{table}
Table 10: PTA-Adapted Parameters for Ensemble Weight Optimization Summary.  
Figure 9: PTA Ensemble Mean RMSE Results Summary.  
models fit the data well for both datasets and for all folds. The best MAE and MAPE results were obtained for the Heating Dataset for all folds.  
Figure 11 presents a comparison of the running times of the runs which resulted in the best RMSE values.  
The running time results varied more for the Heating Dataset than for the Cooling Dataset. The differences are justified by the different durations required to train each regressor of the ensemble.  
Figure 12 presents the weight results for each algorithm of the ensemble for each fold.  
As can be seen in the figure, the PTA-optimized ensemble weights varied more in the case of the Heating Dataset than in the case of the Cooling Dataset. Although the GBR received the highest weight for each fold, the AdaBoost received a weight equal to zero for each fold of the Cooling Dataset. On the other hand, the ETR received the highest weight for three out of the five folds, whereas AdaBoost received non-zero weights for fold 4 and fold 5 of the Heating Dataset.