## 3 Data and Analysis
### Analysis of Common-Cause Variation  
#### 3.2.1 Application of Multilinear Regression Model  
The parameter estimation results of multilinear regression models are shown in Table 1. All absolute values of \(t\)-stat are larger than 1.96, indicating that corresponding variables are statistically significant at a 95% confidence level. In terms of fit goodness, the R\({}^{2}\) in the models of S-G-R-D16, S-G-R-D17, S-G-R-D18, and G-D21 are over 0.9, which means that the variations of monitoring data are mainly caused by temperature effects and linear trends. The R\({}^{2}\) of the models for the remaining measuring points are between 0.15 and 0.80, which may be related to the direct train loadings on rails.  
Taking the girder displacement 21 (G-D21) in Figure 2 as an example, a monitoring sequence can be decomposed into a linear trend, temperature effect, and residual, according to the multilinear regression model. During the whole period of analysis, the girder at G-D21 moves to one side at an average speed of 0.0154 mm/day. The displacement variations  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Measuring Point**} & \multirow{2}{*}{**Abbreviation**} & \multicolumn{2}{c}{**Intercept**} & \multicolumn{2}{c}{**Linear Trend**} & \multicolumn{2}{c}{**Air Temperature**} & \multirow{2}{*}{**R\({}^{2}\)**} \\ \cline{3-3} \cline{5-10}  & & \(\beta_{0}\) & \(t\)**-stat** & \(\beta_{1}\) & \(t\)**-stat** & \(\beta_{2}\) & \(t\)**-stat** & \\ \hline Rail–track slab relative & R-S-R-D10 & 6.3372 & 55.5 & 0.0025 & 18.4 & \(-\)0.2355 & \(-\)51.8 & 0.767 \\ Rail–track slab relative & R-S-R-D13 & \(-\)0.5130 & \(-\)10.2 & \(-\)0.0001 & \(-\)2.3 & 0.0394 & 19.7 & 0.308 \\ displacement 13 & R-S-R-D14 & \(-\)1.0387 & \(-\)24.4 & \(-\)0.0002 & \(-\)4.3 & \(-\)0.0228 & \(-\)13.5 & 0.196 \\ Rail–track slab relative & R-S-R-D15 & \(-\)0.6734 & \(-\)20.1 & \(-\)0.0011 & \(-\)26.8 & 0.0341 & 25.6 & 0.591 \\ displacement 15 & S-G-R-D16 & \(-\)23.0683 & \(-\)91.5 & \(-\)0.0040 & \(-\)13.5 & 0.9166 & 91.4 & 0.906 \\ Track slab–girder relative & S-G-R-D17 & \(-\)6.4153 & \(-\)123.9 & \(-\)0.0009 & \(-\)15.5 & 0.2408 & 117.0 & 0.940 \\ Track slab–girder relative & S-G-R-D18 & 10.2280 & 98.2 & 0.0017 & 13.7 & \(-\)0.4057 & \(-\)97.9 & 0.917 \\ displacement 18 & G-D21 & 57.5937 & 145.6 & 0.0154 & 33.2 & \(-\)2.0303 & \(-\)129.1 & 0.952 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Multilinear regression model: parameter estimates.  
Figure 1: Schematic of the monitoring system of a track system. The “number#” represents the serial number of the sensor.  
of the observation are consistent with the one caused by the temperature effect, which indicates that the temperature is a key factor for the displacement variation at G-D21.  
The total data variations caused by the linear trend and temperature effect are calculated as Equations (16) and (17). The results for four measuring points closely related to the temperature effects together with linear trends are listed in Table 2. The proportions of data variations caused by the linear trends to the ones caused by the temperature effects almost exceed 10%, suggesting that linear trends are also unneglectable components of the measurement sequences. The linear trends may be related to the changes of material properties [24], including concrete creep in the girder, the aging and deterioration of girder bearings, etc. Moreover, the proportion at G-D21 is the largest and demonstrates that the changes of material properties have more effects on the rail-track slab relative displacement than the track slab-girder relative displacement.  
\[D_{\text{LT}}=\beta_{1}\times T \tag{16}\]  
\[D_{\text{TT}}=\beta_{2}\times\Delta_{T} \tag{17}\]  
where \(D_{\text{LT}}\) and \(D_{\text{TT}}\) are the total data variations explained by the linear trend and temperature effect, respectively. \(T\) is the whole period of analysis and \(\Delta_{T}\) is the difference between maximum and minimum temperatures throughout the whole period of analysis.