## 3 Results and Discussion
### Residual Modeling Using Regression Trees  
The RSM model provided a statistically significant fit and revealed the key influences of individual process parameters on surface roughness. However, deviations between predicted and experimental values remained, with a coefficient of determination (R\({}^{2}\)) of 0.8227 and a root mean square error (RMSE) of 0.2169, indicating that further improvement in predictive accuracy was possible. To address these residual discrepancies and capture nonlinearities not modeled by quadratic regression, a residual modeling approach using regression trees was implemented. Rather than predicting surface roughness directly, the algorithm was trained on residuals defined in Equation (6), representing the differences between experimental and RSM-predicted values. This residual-based modeling strategy was chosen because the RSM model had already captured the dominant linear and quadratic trends. By applying regression trees only to the residuals, the model focused on identifying the remaining nonlinear structures, thereby reducing the risk of overfitting and preserving model interpretability. The dataset consisted of 17 samples obtained from a Box-Behnken design, with cutting speed, laser power, and assist gas pressure as input variables. A grid search was conducted by varying the MaxNumSplits parameter from 1 to 10 to determine the optimal tree complexity. The best results were achieved with MaxNumSplits = 3, yielding a minimum RMSE of 0.1717. This indicates that a moderately complex regression tree was sufficient to capture the residual structure without overfitting a crucial consideration, given the limited number of samples. The structure of the regression tree model is shown in Figure 6. Each internal node represents a binary partition based on a threshold value of one of the input parameters  
Figure 5: Percentage contribution of each significant term in the reduced quadratic model to the variance in surface roughness (Ra), based on ANOVA results. cutting speed, laser power, or auxiliary gas pressure chosen to minimize the prediction error. These decision rules divide the input space into regions with similar residual behavior. The end nodes (leaves) indicate the predicted correction values that are added to the RSM output, along with the number of samples associated with each region. The overall depth and branching pattern of the tree reflect the ability of the model to isolate localized nonlinear effects within the residuals. This graphical representation provides a transparent and interpretable view of how the hybrid model adjusts the original predictions to bring them more in line with the experimental data. The resulting hybrid model, which combines RSM predictions with machine learning-based residual correction, achieved an improved R\({}^{2}\) of 0.8889. This demonstrates that the regression tree successfully modeled the nonlinear deviations unaccounted for by the RSM model. Figure 7 shows a comparison between the experimental surface roughness values, the predictions from the RSM model, and the corrected predictions from the hybrid model. The hybrid model matches the experimental data more closely across the entire range, especially in areas where the RSM model showed noticeable deviation. Figure 6: Graphical representation of the regression tree model trained on RSM residuals. Figure 7: Comparison of experimental surface roughness values, RSM model predictions, and hybrid model predictions. The performance gains of the hybrid model are further quantified in Figures 8 and 9, which compare the RMSE and R\({}^{2}\) values, respectively. As shown in Figure 8, the hybrid model reduced the RMSE from 0.2169 \(\upmu\)m to 0.1717 \(\upmu\)m, representing a 20.85% reduction in prediction error. Similarly, Figure 9 shows that the R\({}^{2}\) value increased from 0.8227 to 0.8889, corresponding to an 8.04% improvement in the explained variance.