## 2 Materials and Methods
### Data Preparation and Processing
#### 2.4.2 Training and Optimization of the Model
The literature also highlights that GPR provides stable predictions with low error rates in small datasets [28]. Specifically, in cases where other models struggle to learn complex relationships or face the risk of overfitting, the kernel-based structure of GPR and its ability to compute statistical uncertainty make it the most effective solution for laser intensity prediction. Furthermore, the ability to directly estimate uncertainty in the model's outputs positions GPR as a more reliable modeling approach, especially in applications such as LIDAR data analysis, where measurement accuracy is critical. To enhance the performance of the GPR model and increase prediction accuracy, a hyperparameter optimization process was conducted. In this study, the kernel function was fixed as a squared exponential, while the optimal values of other hyperparameters (e.g., sigma and noise level) were determined using Bayesian optimization. The squared \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Mean** & **Mean** & **Mean** \\  & **RMSE** & **R-Squared** & **Relative Error (\%)** \\ \hline GPR & 0.83162 & 0.99245 & 8.852 \\ \hline Polynomial Regression & 252.056 & 0.29727 & 947.620 \\ \hline SVM & 204.918 & 0.52893 & 620.980 \\ \hline Decision Tree & 26.274 & 0.99164 & 14.070 \\ \hline Random Forest & 112.436 & 0.86051 & 281.790 \\ \hline Gradient Boosting & 21.294 & 0.9937 & 39.615 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: K-Fold Cross-Validation performance values for the models examined. exponential kernel function offers significant advantages for modeling continuous and nonlinear data, such as laser intensity values. This function was specifically chosen because it facilitates the smooth and precise modeling of laser intensity variations across different distances and angles. Bayesian optimization is widely recognized as an effective method when evaluating an objective function with a high computational cost. In this process, Expected Improvement (EI) and Upper Confidence Bound (UCB) acquisition functions were employed. EI is designed to select hyperparameters that are most likely to enhance model performance, whereas UCB ensures an optimal trade-off between exploration and exploitation. In this study, a value of K = 2 was chosen for UCB and retained as the default setting in Matlab. This selection provides an optimal balance between reliability and performance. Figure 6 illustrates the minimization of the objective function over multiple iterations in the Bayesian optimization process. In this figure, the blue color denotes the lowest RMSE values obtained through direct observations, while the green color represents the lowest RMSE values estimated by the Bayesian model. The observed and predicted values show a significant decline over the iterations of the Bayesian optimization, demonstrating that the optimization process effectively enhances performance. The optimal value for the sigma hyperparameter was determined as 0.77256 using the Bayesian model prediction and 0.68476 through direct observations. The discrepancy between these two values stems from the Bayesian model's capability to account for uncertain regions within the hyperparameter space. This characteristic enhances the generalization capacity of the model without negatively affecting its performance. Additionally, both values exhibit a plateau particularly between iterations 8, 9, and 10, indicating that Bayesian optimization has reached convergence, where the objective function attains its minimum value and further improvements become challenging. This confirms that the optimization process has been carried out in a stable and successful manner. Figure 7 illustrates the solution space for the Sigma hyperparameter and its correlation with the objective function (RMSE). The figure depicts the predictions made for Sigma values during the Bayesian optimization process, along with the observed results of the objective function for these values.