## 4 Model Development and Evaluation Indices  
This study employed three tree-based ensemble models to predict the MFS distribution of rock: RF, ET, and GB. The accuracy of these models depends on the optimal combination of hyperparameters. To identify this optimal combination, tree-structured Parzen estimators (TPEs) were used for hyperparameter tuning. Consequently, three hybrid models were developed: TPE-RF, TPE-ET, and TPE-GB, as illustrated in Figure 4.  
The optimization process for all three models follows a similar and consistent procedure:  
1. Data preprocessing: The dataset is divided into training and test sets with 75% (2805 samples) being allocated for training and 25% (935 samples) being reserved for testing. This 75/25 data split was chosen for several reasons: (1) it provides enough samples for the models to learn patterns, which prevents underfitting; (2) smaller test sets tend to be biased while larger test sets offer a more representative sample; and (3) this split ratio generalizes better than a 70/30 split or 80/20 split. Figure 5 show the data distribution between the training and test datasets. It can be observed that the training and test sets are normally distributed; therefore, there is no need for further pre-processing. Additionally, the training and test sets are distributed the same;
2. Optimization process: Many influencing parameters can influence the performance of the algorithms, but only a limited number of hyperparameters were selected to balance performance and computational cost. Five hyperparameters were optimized for the RF and ET algorithms, while six were used for the GB algorithm, as shown in Table 4. These hyperparameters are critical for improving the accuracy of the models. The performance of the Bayesian optimization algorithm is influenced by the number of trials or iterations used to search for the optimal hyperparameter combinations. More trials result in longer training times and higher costs, whereas fewer trials may lead to underfitting. To maintain consistency across models, the number of trials was set to 150 for all models;  
Figure 4: The framework of the three proposed models. Model evaluation: In this study, four indices, including the root mean squared error (RMSE), mean absolute error (MAE), coefficient of determination (R\({}^{2}\)), and max error were employed to evaluate the performance of the three models. These metrics are described using Equations (15)-(18):  
\[\text{RMSE}=\sqrt{\sum_{i=1}^{n}\frac{\left(\hat{y_{i}}-y_{i}\right)^{2}}{n}} \tag{15}\]  
\[\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}\lvert\hat{y_{i}}-y_{i}\rvert \tag{16}\]  
\[\text{R}^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y_{i}})^{2}}{\sum_{i=1}^{n}(y_{ i}-\overline{y}_{i})^{2}} \tag{17}\]  
\[\text{Max Error}(y,\hat{y})=\text{max}(\lvert y_{i}-\hat{y_{i}}\rvert) \tag{18}\]  
where \(y_{i}\) represents the actual value, \(\overline{y}_{i}\) represents the mean value of the predicted target, and \(\hat{y_{i}}\) represents the predicted value. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Hyperparameters**} & \multicolumn{3}{c}{**Regression Algorithms Hyperparameter Search Spaces**} & \multirow{2}{*}{**Data Type**} & \multirow{2}{*}{**Description**} \\ \cline{2-2} \cline{5-6}  & **TPE-ET** & **TPE-RF** & & **TPE-GB** \\ \hline n estimators & [10, 3000] & [10, 3000] & [10, 3000] & Integer & Number of trees in the forest \\ max depth & [2; 40] & [2; 40] & [2; 40] & Integer & Maximum depth of \\  & & & & & each tree \\  & & & & & The minimum number of \\ min samples split & [2; 35] & [2; 35] & [2; 35] & Integer & samples required to split an \\  & & & & & internal code \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Hyperparameters and their search space.