## 5 Simulation Study
### Simulation Results  
In this subsection, the performance of our approaches, described in Section 5.1, is compared with RF. The results of the methods are calculated with 500 repetitions to evaluate the prediction accuracy using mean-squared error (MSE), root-mean-squared error (RMSE), mean absolute error (MAE), weighted mean-squared error (WMSE), weighted root-mean-squared error (WRMSE), and weighted mean absolute error (WMAE), which are used to assess the results as follows:  
\[MSE=\frac{\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2}}{N} \tag{16}\]  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2}}{N}} \tag{17}\]  
\[MAE=\frac{\sum_{i=1}^{N}|y_{i}-\hat{y}_{i}|}{N} \tag{18}\]  
\[WMSE=\frac{\sum_{i=1}^{N}w_{i}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{N}w_{i}} \tag{19}\]  
\[WRMSE=\sqrt{\frac{\sum_{i=1}^{N}w_{i}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{N}w _{i}}} \tag{20}\]  
\[WMAE=\frac{\sum_{i=1}^{N}w_{i}|y_{i}-\hat{y}_{i}|}{\sum_{i=1}^{N}w_{i}} \tag{21}\]  
where \(\hat{y}_{i}\), \(y_{i}\) and \(w_{i}\) are the predicted values, the true values of the \(i\)-th sample, and the number of samples in the \(k\)-th cluster, respectively. In Figure 5a,c, which show the relationship between MSE and log (\(\lambda\)), the red dots indicate the cross-validation error, and the top and bottom lines display the standard deviation for ECAPRAF-EN. The left and right vertical lines show the \(\lambda_{min}\) and \(\lambda_{1 se}\), respectively. The \(\lambda_{min}\) shows the minimum lambda with the lowest error and the \(\lambda_{1 se}\) represents the lambda value based on the standard error. The above characters show the number of non-zero coefficients in each cluster. Figure 4(b),d show the ECAPRAF-EN regression coefficients in each cluster. To the best of our knowledge, different variable coefficients have various effects on the response variable. The first variable entered into the model is the most effective one, and the subsequently entered variables have different effects. Finally, the coefficients of ineffective variables on the model are eliminated. This is true for lasso and group lasso, too. According to Figure 4(c), \(\log(\lambda)\), \(58\) trees out of 240 and 48 trees out of 260 are selected from the first and second clusters, respectively. In the total of two clusters, 106 trees out of 500 are selected in ECAPRAF-EN. The left vertical dashed line refers to \(\lambda_{min}=0.6891\) with the minimum \(MSE=8.4355\) and \(\lambda_{min}=6.6574\) with the minimum \(MSE=8.6256\) for the first and second cluster, respectively. Moreover, the bar plot of the predicted error for 500 trees is presented in Figure 5, which shows the lowest value of WRMSE for ECAPRAF-EN. According to Table 2, the results of WMSE, WRMSE, and WMAE are obtained from the simulation of a linear-regression model for 300, 500, 800, and 1000 trees, from which 99,  
Figure 5: **(a,b)** The simulation results of the algorithm in the first cluster. (**c,d**) The simulation results of the algorithm in the second cluster. (**a,c**) The left and right dashed lines show the \(\lambda_{min}\) and \(\lambda_{1\text{se}}\), respectively. The \(\lambda_{min}\) shows the minimum lambda with the lowest error and the \(\lambda_{1\text{se}}\) represents lambda value based on the one-standard error. The upper part of the plot shows the number of non-zero coefficients in each cluster for a given \(\log(\lambda)\). (**b,d**) The number of effective variables in the model. 106, 113, and 118 trees are selected by the ECAPRAF-EN algorithm, respectively. As can be seen, the weighted MSE for the proposed ECAPRAF-EN model is equal to 8.5453, in which 106 trees out of 500 are selected. In other words, it is less than RF and other shrinkage methods. Additionally, according to the weighted MAE and RMSE, the performance of the proposed ECAPRAF-EN is better than other methods. Generally, the proposed algorithm is better than the RF algorithm.