## 3 Data and Analysis
### Analysis of Common-Cause Variation
#### 3.2.2 Application of Time Series Difference Equation Model  
The monitoring data of the track system are discontinuous, so the autocorrelation, partial autocorrelation, Akaike's information criteria, and Bayesian information criteria are no longer suitable for selecting the order of difference equation. The coefficient of R\({}^{2}\) is adopted as the selection standard of model order and should satisfy the following two conditions: (1) the variation of R\({}^{2}\) with the model order is stable; (2) the \(t\)-stat of each coefficient is greater than 1.96 (with 95% significance level). When the R\({}^{2}\) of the model  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Measuring Point** & **Data Variation Caused by the Linear Trend (mm)** & **Data Variation Caused by the Temperature Effect (mm)** & **Proportion (\%)** \\ \hline S-G-R-D16 & 4.884 & 42.426 & 11.51\% \\ S-G-R-D17 & 1.099 & 11.146 & 9.86\% \\ S-G-R-D18 & 2.076 & 18.778 & 11.05\% \\ G-D21 & 18.803 & 93.975 & 20.01\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of total data variations caused by the linear trends and temperature effects.  
Figure 2: Structural decomposition of girder displacement 21 (G-D21).  
is greater than 0.7, the first order of R\({}^{2}\) > 0.7 is selected. Since the monitoring data of the track system were averaged, the remnant effects of early measurements are relatively small. The lowest order of the model should be selected once the above conditions are achieved. Figure 3 presents the relationship between R\({}^{2}\) and the order of the difference equation model. The model order of each measuring point is selected as 3, 1, 1, 3, 1, 1, 1, 1. The parameter estimation results of the time series difference equation models are shown in Table 3.  
Serial dependence in the regression residuals and difference equation residuals is evaluated by analyzing the corresponding autocorrelation functions (ACFs). The longest continuous segment (97 days) of the monitoring sequence is selected to verify the feasibility of the aforementioned method for capturing the serial dependence. Figure 4 shows the ACFs of the regression residuals and difference equation residuals for R-S-R-D13. In terms of a stationary and serially-independent time series, the values of ACF are small, with lag \(\geq\) 1, and vary irregularly with lags. By comparison, the time series difference equation model effectively eliminates the autocorrelation in the monitoring data. In addition, the ACF values of the difference equation residuals fall mostly in the blue shaded area in Figure 4b. The blue shaded area represents the 95% confidence interval. Therefore, the existence of serial dependence in the difference equation residual is rejected with 95% confidence.  
Figure 3: R\({}^{2}\) of the time series difference equation model with various orders.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Measuring Point** & \(\varphi_{1}\) & **f-stat** & \(\varphi_{2}\) & **f-stat** & \(\varphi_{3}\) & **f-stat** & **R\({}^{2}\)** \\ \hline R-S-R-D10 & 0.8562 & 23.943 & \(-\)0.2224 & \(-\)4.854 & 0.1348 & 3.813 & 0.596 \\ R-S-R-D13 & 0.9586 & 96.654 & - & - & - & - & 0.919 \\ R-S-R-D14 & 0.9233 & 67.120 & - & - & - & - & 0.845 \\ R-S-R-D15 & 0.8964 & 25.545 & \(-\)0.3436 & \(-\)7.484 & 0.2429 & 7.156 & 0.616 \\ S-G-R-D16 & 0.8565 & 48.274 & - & - & - & - & 0.738 \\ S-G-R-D17 & 0.7028 & 29.630 & - & - & - & - & 0.515 \\ S-G-R-D18 & 0.8159 & 42.707 & - & - & - & - & 0.688 \\ G-D21 & 0.7227 & 32.142 & - & - & - & - & 0.555 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Time series difference equation model: parameter estimates.