## 4 Results and Discussion
### Relationship Surveying through Machine Learning Algorithms  
The accuracy measures (MAE, MSE, and RMSE) obtained from three predictive machine learning (ML)-based models are presented in Table 4. This table compares the MAE, MSE, and RMSE values across different machine learning algorithms. The results indicate that for the left rail, the RMSE values are approximately 0.05 (linear regression), 0.07 (decision tree), and 0.06 (random forest). Similarly, for the right rail, the RMSE values are approximately 0.05 (linear regression), 0.06 (decision tree), and 0.06 (random forest). Notably, the linear regression model exhibits the lowest accuracy measures, followed by the random forest and decision tree models.  
Table 5 shows the model accuracy measures of the relationship between vertical ballast stiffness (\(\mathrm{k_{b}}\)) and longitudinal levels mined by ML algorithms.  
Based on Table 5, the accuracy measures for both the left and right rails exhibit remarkable similarity. These values suggest an appropriate relationship between vertical ballast stiffness and longitudinal levels. The root mean square error (RMSE) of the linear regression model is approximately 0.04, which stands as the lowest RMSE among the various machine learning algorithms investigated. Additionally, the RMSE values for the decision tree and random forest models fall within the range of 0.04 to 0.05. Consequently, the linear regression, random forest, and decision tree models yield the best accuracy results, respectively. The slight discrepancy between the decision tree and random forest can be attributed to the fact that the random forest model generalizes the decision tree approach. [PERSON] et al. (2019) highlighted a relation between rail deflection and track alignments using  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**ML Algorithms**} & \multicolumn{3}{c}{**y-LLL**} & \multicolumn{3}{c}{**y-LLR**} \\ \cline{2-7}  & **MAE** & **MSE** & **RMSE** & **MAE** & **MSE** & **RMSE** \\ \hline Linear regression & 0.038 & 0.003 & 0.054 & 0.038 & 0.003 & 0.054 \\ \hline Decision tree & 0.050 & 0.005 & 0.071 & 0.041 & 0.003 & 0.058 \\ \hline Random forest & 0.046 & 0.004 & 0.064 & 0.041 & 0.003 & 0.058 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Model accuracy measures of relationship between vertical rail deflections (y) and longitudinal levels (LLL, LLR) mined by machine learning algorithms.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **LLL-PSD** & **LLR-PSD** & **y-PSD** \\ \hline LLL-PSD & 1 & 0.902 & 0.672 \\ \hline LLR-PSD & 0.902 & 1 & 0.674 \\ \hline k\({}_{\mathrm{b}}\)-PSD & 0.672 & 0.674 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Pearson correlation matrix of vertical ballast stiffness (\(\mathrm{k_{b}}\)) and longitudinal levels (LLL, LLR) extracted by PSD analyses.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**ML Algorithms**} & \multicolumn{3}{c}{**\(\mathrm{k_{b}}\)-LLL**} & \multicolumn{3}{c}{**\(\mathrm{k_{b}}\)-LLR**} \\ \cline{2-7}  & **MAE** & **MSE** & **RMSE** & **MAE** & **MSE** & **RMSE** \\ \hline Linear regression & 0.027 & 0.001 & 0.038 & 0.027 & 0.001 & 0.038 \\ \hline Decision tree & 0.036 & 0.003 & 0.050 & 0.030 & 0.002 & 0.042 \\ \hline Random forest & 0.033 & 0.002 & 0.045 & 0.030 & 0.002 & 0.041 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Model accuracy measures of relationship between vertical ballast stiffness (\(\mathrm{k_{b}}\)) and longitudinal levels (LLL, LLR) mined by machine learning algorithms.  
the decision tree method. They clarified that rail deflections and track alignments in the range of 16.3 to 8.1 mm have a close relationship when utilizing this ML technique.