## 3 Results and Discussion
### Validation of the Hybrid Model Using LOOCV  
Leave-one-out cross-validation (LOOCV) was employed to rigorously evaluate the generalizability of the developed hybrid model. In this approach, each of the 17 data points in the dataset was iteratively excluded from the training set and used as an independent test case. The regression tree model was retrained on the remaining samples, and the residual for the excluded point was predicted. This process was repeated for all data points, ensuring an unbiased assessment of model performance while mitigating the risk of overfitting, which is critical for small datasets.  
RMSE and R\({}^{2}\) in LOOCV are performance metrics that evaluate how well a model predicts and explains variance in the dependent variable during leave-one-out cross-validation. RMSE measures the average prediction error, while R-squared reflects how much variance is captured by the model. These metrics help assess generalization, especially for small datasets, but can be sensitive to outliers. Interpretation should consider model complexity and the potential for overfitting.  
In this research, the LOOCV results revealed an RMSE of 0.3241 \(\upmu\)m and an R\({}^{2}\) of 0.6039, indicating that the predicted surface roughness (Ra) deviates by approximately 0.32 \(\upmu\)m from experimental values. By applying LOOCV, the R\({}^{2}\) obtained is considered acceptable, depending on the application area, with values ranging from approximately 0.2 to 0.9 (i.e., 20% to 90%) [24]. This variation reflects the differing levels of model performance required across various domains. This accuracy is acceptable for industrial applications, considering typical Ra values in the laser cutting range from 2 to 3 \(\upmu\)m [25]. Although the performance declined compared to the training results (RMSE = 0.1717, R\({}^{2}\) = 0.8889), LOOCV offers a realistic estimate of generalization. These findings confirm the practical value of the proposed model and provide a solid basis for the concluding observations in the next section.  
As shown in Table 5, the R\({}^{2}\) value obtained for the proposed hybrid model (\(\approx\)0.89) is comparable to those reported in similar studies using traditional RSM approaches. For example, [PERSON] et al. [26] and [PERSON] [27] achieved R\({}^{2}\) values of \(\approx\)0.93 using Box-Behnken and Taguchi designs, respectively, with a similar number of experimental trials. [PERSON] et al. [28] reported an R\({}^{2}\) of \(\approx\)0.90 based on 32 experiments. Although our study involved only 17 experimental points, the predictive performance remains consistent with these references, highlighting the effectiveness of the proposed hybrid approach.  
These are often constrained by practical limitations such as machine settings, control accuracy, and technological stability during processing. Despite these constraints, the hybrid model successfully captured key process relationships, demonstrating its applicability, even with a limited dataset. It is also well recognized in the statistical literature that what is considered a good R\({}^{2}\) depends heavily on the field and context. According to published sources, R\({}^{2}\) values in the range of 0.60 to 0.90 are generally regarded as acceptable to strong for engineering and manufacturing processes, particularly when using empirical  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Study (Ref)** & **Method** & **Trials** & **R\({}^{2}\)** \\ \hline Obtained model & RSM + Tree (Hybrid) & 17 (Box–Benkken) & \(\approx\)0.89 \\ [PERSON] et al. [26] & RSM & 15 (Box–Benkken) & \(\approx\)0.93 \\ [PERSON] [27] & RSM & 27 (Taguchi) & \(\approx\)0.93 \\ [PERSON] et al. [28] & RSM & 32 (Taguchi) & \(\approx\)0.90 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of the obtained model with relevant literature in terms of R\({}^{2}\).  
models such as RSM or hybrid learning frameworks [29]. In this regard, the obtained R\({}^{2}\) of \(\approx\)0.89 supports the practical validity of the proposed model.