## 4 Case Study
### Analysis of the Prediction Results
#### 4.2.2 Prediction Error Analysis  
As the tamping maintenance strategy of combining \"periodic preventive maintenance\" with \"local corrective maintenance\" is still widely adopted in the Chinese railway system, most segments carried out preventive tamping maintenance operations before the SDLL degraded to the maintenance threshold (3-3.6 mm). As a result, the actual time interval of each segment during which the SDLL deteriorates from its initial value to the maintenance threshold cannot be extracted from the historical inspection data at the longitudinal level, resulting in the inability to support a prediction error analysis of the tamping maintenance cycle. The absence of benchmark data hinders prediction error analysis of tamping maintenance cycles. The tamping cycle prediction model proposed in this study essentially serves as a deterioration prediction model for the SDLL between two consecutive tamping operations. It can predict the time interval required for the SDLL to deteriorate to any specified threshold value \(X_{lim}\). Given the situation described above, we set SDLL threshold values (i.e., \(X_{lim}\)) between 1 and 3 mm at intervals of 0.1 mm and then predicted the time required for degradation to the different given \(X_{lim}\) values to indirectly verify the predictive accuracy of the model by comparing the predicted results with the actual time intervals recorded in the field SDLL historical inspection data. Figures 5-8 show box plots of the prediction errors corresponding to the parameter estimation results for MLE-MPA, MLE-GA, MLE-GD, and adaptive MCMC, respectively. The horizontal coordinate is the given \(X_{lim}\). For each \(X_{lim}\) value, a box plot was generated to display the distribution characteristics of the prediction error for the time interval required for the SDLL to deteriorate to that value. In these box plots, the box itself represents the interquartile range, with the central line indicating the median. The whiskers extend to the most extreme data points that fall within 1.5 times the interquartile range. Each figure also includes two additional reference lines: a yellow dashed line representing the Inner Limit, which is the maximum value of the upper whiskers across all \(X_{lim}\) values, and a green dashed line showing the Maximum Error across all \(X_{lim}\) values. At the top of each figure, the average Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) are displayed, providing overall performance metrics for each method across all scenarios. Figure 9 shows a comparison of the mean absolute error (MAE) corresponding to the three estimation methods. Table 1 shows the statistical values of the prediction errors corresponding to the four parameter estimation methods, including the MAE, the percentage of absolute error within one month (\(\leq\)30 days), the percentage of absolute error within two months (\(\leq\)60 days), and the percentage of absolute error within three months (\(\leq\)90 days). It should be noted that the percentages here represent cumulative frequencies, indicating that the count of samples with an absolute error of less than two months includes those with an absolute error of less than 30 days. Figure 10 visually illustrates the distribution of error ranges for different methods across various scenarios through stacked bars, providing a graphical representation of the data from Table 1.  
The errors in the prediction results obtained using the four parameter estimation methods were compared and analyzed. Based on the prediction results for the parameter estimation method with the least error, the validity of the model established in this study is illustrated in combination with the tamping maintenance planning demand.