## 4 Case Study
### Analysis of the Prediction Results
#### 4.2.2 Prediction Error Analysis
* Comparative Analysis of Four Parameter Estimation Methods  
Based on the mean absolute errors (MAE) presented in Table 1, it is observed that for various \(X_{lim}\) values, the maximum MAE value (21 days) of the prediction results obtained from the MLE-MPA method exhibits reductions of 4.5%, 33.3%, and 48.8% compared to those of the MLE-GA, MLE-GD, and AMCMC methods, respectively. Furthermore, the mean MAE value (19 days) achieved by the MLE-MPA method demonstrates decreases below 5%, 26.9%, and 44.1% when contrasted with those of the MLE-GA, MLE-GD, and AMCMC methods. Additionally, Figure 9 illustrates that for each given \(X_{lim}\) value, the MAE associated with predictions using the MLE method consistently outperforms that of the adaptive MCMC method. Figure 5: Absolute error distribution of prediction results obtained by MLE-MPA method for the time interval required for SDLL to deteriorate to the different given \(X_{lim}\) values. Figure 6: Absolute error distribution of prediction results obtained by MLE-GA method for the time interval required for SDLL to deteriorate to the different given \(X_{lim}\) values. Figure 8: Absolute error distribution of prediction results obtained by adaptive MCMC method for the time interval required for SDLL to deteriorate to the different given \(X_{lim}\) values. Figure 7: Absolute error distribution of prediction results obtained by MLE-GD method for the time interval required for SDLL to deteriorate to the different given \(X_{lim}\) values. Figure 10: Stacked bar chart of the distribution of different absolute error ranges for prediction results obtained by the four methods. Figure 9: Comparison of prediction errors of four methods for the time interval required for SDLL to deteriorate to the different given \(X_{lim}\) values. In addition to examining MAE values, an analysis of prediction stability under different scenarios can be conducted through root mean square error (RMSE). As depicted in Figures 5-8, both MLE-GD and AMCMC methods exhibit notably higher average RMSE values, which not only indicate their inferior performance in terms of prediction errors as previously demonstrated but also reveal their instability and sensitivity to varying scenarios. This behavior may be attributed to susceptibility of the gradient descent method towards local optima when dealing with non-convex and complex objective functions as well as heavy dependence on initial distributions exhibited by the AMCMC method leading to significant discrepancies in prediction outcomes across different dataset scenarios. Regarding the error distribution characteristics outlined in Table 1 and Figure 10, the prediction results generated by the MLE-MPA method demonstrate that all absolute errors fall within three months. A substantial proportion (79%) occurs within a span of just 30 days, while only a mere four percent exceed 60 days. The proportions of absolute errors within one and two months were both higher than those of the other three methods. Furthermore, from Figure 9, we observe that the absolute errors do not significantly increase alongside rising \(X_{lim}\) values, contrary to general expectations. This is mainly because a larger \(X_{lim}\) value during a tamping cycle spans a longer time and provides more inspection data, resulting in an abundance of effective data for constructing the likelihood function of each track segment. As a result, prediction accuracy improves as reflected in the trend of prediction errors with the \(X_{lim}\) value for the MLE-GA and MLE-MPA methods. It is evident that the estimation results of the MLE methods are superior to those of the adaptive MCMC method. Furthermore, in terms of computational resource requirements and solving speed, using adaptive MCMC to solve the model parameters for the 2171 segments in this case took nearly 26 h, while MLE-MPA, MLE-GA, and MLE-GD  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(X_{lim}\) (mm)} & \multirow{2}{*}{\ \begin{tabular}{c} **Sample** \\ **Size** \\ \end{tabular}