## 5 Results and Discussion  
### Performance Comparison of Models for MFS Prediction  
The three optimized models were compared to determine the best model for predicting the MFS distribution of rock. As outlined in Section 4, aside from the TPE-GB model, the number of trials and hyperparameters for each model was kept consistent to ensure fairness in the results. Figure 6 presents a bar graph that shows the importance values of different hyperparameters during the optimization process. Figure 6 illustrates that all models prioritized the 'criterion' hyperparameter, which is rational, as it assesses potential splits at each node to optimize information gain. In the TPE-RF model, the 'criterion' hyperparameter significantly influenced the optimization process, whereas the other hyperparameters had a minimal impact. Figure 7 displays the performance of the three models during the optimization process involving 150 trials. The results indicate that the performance differences among the models were not substantial. However, the TPE-ET model exhibits better performance than the other two models. Additionally, it was observed that some hyperparameter combinations during the optimization process resulted in negative values of the objective function, as shown in the subfigure in Figure 7.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Hyperparameters**} & \multicolumn{2}{c}{**Regression Algorithms Hyperparameter Search Spaces**} & \multirow{2}{*}{**Data Type**} & \multirow{2}{*}{**Description**} \\ \cline{2-2} \cline{5-5}  & **TPE-ET** & **TPE-RF** & & **TPE-GB** \\ \hline \multirow{4}{*}{criterion} & ‘squared\_error’, & ‘squared\_error’, & \multirow{2}{*}{\ \begin{tabular}{} \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{} \end{tabular}  } \\  & ‘absolute\_error’, & ‘absolute\_error’, & & \\ \cline{1-1} \cline{3-5}  & ‘friedman\_mse’, & ‘friedman\_mse’, & ‘friedman\_mse’, & & \\ \cline{1-1} \cline{3-5}  & ‘poisson’ & ‘poisson’ & & & \\ \cline{1-1} \cline{3-5}  & & & & \\ \cline{  
According to Table 5, the TPE-GB model outperformed the other two models during the training phase, achieving R\({}^{2}\), RMSE, MAE, and max error values of 0.97, 0.03, 0.02, and 0.14, respectively. However, as shown in Table 6, the TPE-GB model underperformed compared to the other models during the testing phase. In contrast, the TPE-ET model demonstrated superior performance during the testing phase, with R\({}^{2}\), RMSE, MAE, and max error values of 0.93, 0.04, 0.03, and 0.25, respectively. However, the evaluation metrics across the models are closely matched, with minimal differences, which makes it challenging to definitively identify the best-performing model. The lower max error values suggest that TPE-GB and TPE-ET were the most reliable during the training and testing phases, respectively. To further analyze and compare the model performances, score rankings were visualized using stacked bar charts. As illustrated in Figure 8a,b, the TPE-GB model achieved the highest score during the training phase but ranked lower during the testing phase. Meanwhile, the TPE-ET and TPE-RF models exhibited comparable scores during the testing phase. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Models** & **R\({}^{2}\)** & **RMSE** & **MAE** & **Max Error** & **Scores** \\ \hline TPE-ET & 0.93 & 0.04 & 0.03 & 0.25 & \\ Rank & 1 & 1 & 1 & 3 & 6 \\ TPE-GB & 0.92 & 0.04 & 0.03 & 0.28 & \\ Rank & 2 & 1 & 1 & 1 & 5 \\ TPE-RF & 0.92 & 0.04 & 0.03 & 0.26 & \\ Rank & 2 & 1 & 1 & 2 & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Evaluation indices of models during the testing phase. Figure 7: Comparison of the models during the optimization process. To further understand the models' performance, regression plots are utilized to explore the relationship between the predicted X\({}_{50}\) values and actual X\({}_{50}\) values. Figures 9 and 10 depict scatter plots of the three models, where the predicted X\({}_{50}\) is on the y-axis and the actual X\({}_{50}\) is on the x-axis.