## 4 Case Study
### Analysis of the Prediction Results
#### 4.2.2 Prediction Error Analysis
 } & \multirow{2}{*}{ \begin{tabular}{c} **MAE (Days)** \\ \end{tabular}  } & \multicolumn{6}{c}{**Percentage of Absolute**} & \multicolumn{6}{c}{**Percentage of Absolute**} \\ \cline{5-16}  & & & & & \multicolumn{6}{c}{**Error \(\leq\) 30 Days**} & \multicolumn{6}{c}{**Error \(\leq\) 60 Days**} & \multicolumn{6}{c}{**Error \(\leq\) 90 Days**} \\ \cline{5-16}  & & & & & & & & & & & & & & & & & & \\ \hline
1 & 1002 & 19 & 20 & 28 & 25 & 79\% & 77\% & 72\% & 73\% & 9  
methods took about 2 h each. MLE methods are more suitable for the prediction needs of line-level and network-level tamping maintenance cycles, especially when dynamically updating model parameter estimates such as inspection data accumulate. Among the MLE methods, the MLE-MPA performs better than the MLE-GA and MLE-GD, confirming the general conclusions pointed out by [PERSON] et al. [43]. In addition to analyzing the prediction errors, we further conducted a comparative analysis of the uncertainty in the parameter estimation results by calculating standard deviations that reflect the confidence level of the estimators for the four methods. For the MLE framework, we utilized the gradient-based method to calculate the standard deviation of the parameter estimates for these three methods: MLE-MPA, MLE-GA, and MLE-GD. Specifically, we approximated the covariance of the parameter estimates by computing the Hessian matrix. Subsequently, we depicted the standard deviation bars of the parameters \(\beta_{k},\sigma_{k}\), and \(\theta_{k}\) for the 2171 track segments in Figures 11-13, respectively. In each figure, subplots (a), (b), and (c) correspond to the results of the MLE-MPA, MLE-GA, and MLE-GD methods, respectively. The results reveal inconsistencies in uncertainty across different methods, which underscores the importance of employing robust global optimization algorithms to avoid local optima traps. For the adaptive MCMC method, we randomly selected a track segment (\(k=1986\), starting mileage K945+200) to record the sampling results of the posterior distribution of the parameters \(\beta_{k},\sigma_{k}\), and \(\theta_{k}\), as shown in Figure 14. The uncertainty of the parameters for this track segment is high, possibly due to its location at a slope change point of the railway line, leading to changes in track dynamics forces. Additionally, inherent track irregularities and vehicle system dynamics contribute to increased uncertainty in track geometry degradation. Interestingly, we observed that the adaptive MCMC method yields smaller prediction errors for this specific track segment with high uncertainty. Future research could explore the relationship between parameter uncertainty and factors such as track environment and structure to propose a more robust and widely applicable parameter estimation method. Figure 11: Uncertainty analysis of the estimated parameter \(\beta_{k}\) obtained by the three parameter estimation methods under the MLE framework. Figure 12: Uncertainty analysis of the estimated parameter \(\sigma_{k}\) obtained by the three parameter estimation methods under the MLE framework. Figure 13: Uncertainty analysis of the estimated parameter \(\theta_{k}\) obtained by the three parameter estimation methods under the MLE framework. Model Validity Analysis  
Based on the preceding analysis, we employed the prediction results obtained using the MLE-MPA parameter estimation method to assess the validity of the model and determine whether it met the requirements for tamping maintenance planning. We calculated the coefficient of determination R\({}^{2}\) of the model to be 0.98. The 80 th, 85 th, 90 th, and 95 th percentiles of the absolute error were 32, 38, 45, and 57 days, respectively. Regarding the performance of the model, the overall prediction accuracy for the tamping maintenance cycle of the 2171 track sections was robust, as indicated by the average MAE and R\({}^{2}\) values.