## 5 Simulation Study
### Simulation Results
Concerning the number of trees, although the total number of trees selected by ECAPRAF-EN is more than by ECAPRAF-Lasso, the values of WMSE, WRMSE, and WMAE are the lowest compared to the others. The simulation results of the accuracy measurements, i.e., MSE, RMSE, and MAE, are reported in Table 3 for each cluster. As can be seen, similar to the results of WRMSE in Table 2 and Figure 6, the RMSE value of ECAPRAF-EN in each cluster is the lowest value among the three algorithms. Elastic net also has the highest reduction compared to other shrinkage methods. Therefore, it can be concluded that both within clusters and for the sum of two clusters, the ECAPRAF-EN algorithm has the lowest error and the highest accuracy among the other proposed methods. As shown in Table 3, the number of RF trees used within some clusters may be the same, but the number of trees reduced by shrinkage methods can be different. This leads to different prediction accuracies. The reason for this can be the amount of correlation and the homogeneity of the data within the clusters. Generally, it can be said that the trees produced from the RF due to high variance and low bias can be associated with high error and low accuracy [36]. As a result, the usage of shrinkage methods can help to improve the performance of RF. This improvement varies in different methods. For example, in lasso regression, if the number of trees is bigger than the observations, then the number of selected trees will be less than the observations. There is no such defect in the elastic-net regression. Although a larger number of trees is selected, it has less error and greater accuracy than lasso. Concerning group lasso, although group selection removes a group of trees and can create a better model than other methods, it removes the trees that lead to the improvement of the model performance. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Algorithm** & **WMSE** & **WRMSE** & **WMAE** & **Selected Trees** \\ \hline \multirow{2}{*}{Ntree = 300} & Random Forest & 9.907040504 & 3.135776591 & 2.457571958 & 300 \\  & ECAPRAF–Lasso & 8.667368271 & 2.944039448 & 2.307557453 & 84 \\  & ECAPRAF–EN & **8.643681852** & **2.94001392** & **2.304445756** & **99** \\  & ECAPRAF–GL & 8.769322968 & 2.961304268 & 2.325545926 & 135 \\ \hline \multirow{2}{*}{Ntree = 500} & Random Forest & 9.690688581 & 3.102433591 & 2.435397318 & 500 \\  & ECAPRAF–Lasso & 8.565587148 & 2.926702436 & 2.292483911 & 83 \\  & ECAPRAF-EN & **8.545308173** & **2.923235908** & **2.288376565** & **106** \\  & ECAPRAF–GL & 8.616568 & 2.935399121 & 2.30589166 & 175 \\ \hline \multirow{2}{*}{Ntree = 800} & Random Forest & 9.855558666 & 3.127227327 & 2.44665839 & 800 \\  & ECAPRAF–Lasso & 8.621290954 & 2.936203493 & 2.293232464 & 90 \\  & ECAPRAF–EN & **8.581137457** & **2.929357857** & **2.288603167** & **113** \\  & ECAPRAF–GL & 8.628389783 & 2.937412089 & 2.302874334 & 160 \\ \hline \multirow{2}{*}{Ntree = 1000} & Random Forest & 9.694166557 & 3.104002527 & 2.428734774 & 1000 \\  & ECAPRAF–Lasso & 8.591221483 & 2.931078553 & 2.29209065 & 103 \\ \cline{1-1}  & ECAPRAF–EN & **8.561964742** & **2.926083516** & **2.287156371** & **118** \\ \cline{1-1}  & ECAPRAF–GL & 8.572411865 & 2.927868143 & 2.297112936 & 165 \\ \hline \end{tabular}
\end{table} 
Table 2: The results of the proposed method for the linear model.