## 2 Materials and Methods
### Data Preparation and Processing
#### 2.4.2 Training and Optimization of the Model
Here, the blue markers represent the Sigma values recorded during Bayesian optimization and their corresponding RMSE results. The red markers denote the average RMSE values predicted by the model for Sigma, while the error bars indicate the estimated uncertainty range and the impact of natural noise in the dataset. Figure 6: Change in the objective function depending on iterations in the Bayesian optimization process. Bayesian optimization effectively identifies uncertain regions in the hyperparameter space and determines optimal solutions for these areas. The graph indicates that, when Sigma is approximately \(10^{-1}\), the objective function reaches its minimum value. At this point, further improvements in model performance were not observed, and the objective function remained stable. This confirms that Bayesian optimization conducted a sufficiently thorough exploration of the solution space. As a result, the Bayesian optimization process successfully determined the optimal value of Sigma and minimized the objective function, thereby enhancing model performance. Figure 8 illustrates the linear correlation between the laser intensity values predicted by the GPR model--optimized using the hyperparameters derived from the Bayesian optimization process--and the actual intensity values. Here, the blue circles represent the comparison between the predicted and actual values for each data point, while the linear trend of the slope clearly demonstrates the model's high predictive accuracy. The prediction accuracy of the optimized GPR model was evaluated using the obtained hyperparameters. Analysis of the model's performance metrics revealed that the average RMSE value decreased from 0.87451 to 0.83162, while the average R\({}^{2}\) value increased from 0.99245 to 0.99924. These results indicate that the model is capable of predicting actual intensity values with very high accuracy. The reduction in RMSE confirms that the model  
Figure 8: Relationship between actual laser intensity values and laser intensity values predicted by the optimized GPR model. Figure 7: Solution space and estimated objective function value for the Sigma hyperparameter. significantly minimizes prediction errors, while the increase in R\({}^{2}\) demonstrates that the model explains a larger proportion of the variance in the dataset. These enhancements clearly highlight the effectiveness of the Bayesian optimization process in improving the performance of the GPR model. In conclusion, the hyperparameter optimization process enhanced the laser intensity estimation performance of the GPR model. Bayesian optimization iteratively explored uncertain regions in the hyperparameter space, minimizing suboptimal or incorrect parameter selections. This process significantly improved the model's capacity. Selecting the optimal hyperparameters enhanced the model's prediction accuracy, mitigated the risk of overfitting, and strengthened its generalization ability. These findings underscore the effectiveness of the GPR model and emphasize the critical role of hyperparameter selection in Bayesian optimization.