## 3 Data and Analysis
### Analysis of Special-Cause Variation  
#### 3.3.1 Example of Control Chart  
Figures 5 and 6 show the individual control chart and moving range control chart for R-S-R-D13, where sixteen and twenty-two outliers are detected, respectively. For a stationary normal independent and identical distribution process, the individual control chart is expected to produce three outliers for each measuring point, and the moving range control chart is expected to produce eight outliers [24]. More detected outliers suggest the presence of special causes. Figure 7 exhibits the individual control chart for R-S-R-D13 with supplementary runs. Rule N2 detects an upward shift in the mean of the monitoring sequence during April 2017. Rule N3 detects downward drifts in the mean of the monitoring sequence during June and December 2015, and February 2018. Rule N7 detects increases in the variance of the monitoring sequence occurring in October 2017 and July 2018. Rule N8 detects a decrease in the variance of the monitoring sequence occurring in September 2015. Figure 4: R-S-R-D13: Autocorrelation functions of (**a**) the regression residual and (**b**) the difference equation residual. Figure 5: Individual control chart: R-S-R-D13. Black line is the centerline. Red lines are the control limits. Red circles indicate the values beyond the control limits. The adjustable parameter of EWMA control charts can be customized in the range \(0<\lambda\leq 1\). The smaller the \(\lambda\) is, the smoother the corresponding control chart. That is, more attention is paid to the sustained changes of the mean instead of outliers [24]. According to the characteristics of the required control chart and the discontinuous monitoring data of the track system, \(\lambda\) = 0.1 is selected. From the EWMA control chart with \(\lambda\) = 0.1 for R-S-R-D13 shown in Figure 8, it is noted that a small continuous shift from 19 September 2015 to 26 September 2015 is detected. clearly seen that 121, 215, 195, and 121 outliers are, respectively, detected by the individual control charts, supplementary runs rules, moving range control charts, and EWMA control charts. According to the discussion in Section 2.2, the individual control chart, moving range control chart, and supplementary runs rules are expected to produce 19 outliers (878 \(\times\) 8 \(\times\) 0.0027), 63 outliers (878 \(\times\) 8 \(\times\) 0.009), and 163 outliers (878 \(\times\) 8 \(\times\) 0.02321) [33]. The detected outliers exceed the expected value obviously, which indicates the existence of special events. These special events change the mean and variance of the monitoring data. The numbers of outliers detected by control charts for R-S-R-D15 and S-G-R-D17 are much larger than those detected by control charts for other measuring points. This suggests that the track system at these two measuring points is sensitive to special causes, such as cold waves, extreme high temperatures, structural deterioration, and damage, and should be paid more attention. Figure 14: Control charts: S-G-R-D18. **(a)** Individual control chart; **(b)** moving range control charts; **(c)** EWMA control chart. Black line is the centerline. Red lines are the control limits. Red circles indicate the values beyond the control limits. Figure 12: Control charts: S-G-R-D16. **(a)** Individual control chart; **(b)** moving range control charts; **(c)** EWMA control chart. Black line is the centerline. Red lines are the control limits. Red circles indicate the values beyond the control limits. Figure 13: Control charts: S-G-R-D17. **(a)** Individual control chart; **(b)** moving range control charts; **(c)** EWMA control chart. Black line is the centerline. Red lines are the control limits. Red circles indicate the values beyond the control limits. Figure 15: Control charts: G-D21. **(a)** Individual control chart; **(b)** moving range control charts; **(c)** EWMA control chart. Black line is the centerline. Red lines are the control limits. Red circles indicate the values beyond the control limits.