## 3 Methodology and Details of Data Analysis
### Case Study-I: Analyzing Movements of the Tall Building  
In this section, we discuss the results from the inclination sensors and GPS data for clarifying the behavior of the building. First, we present the raw temperature and inclination data from the sensors. In Figure 6, the first chart shows the indoor temperatures recorded by the sensors, varying between 19 \({}^{\circ}\)C and 24 \({}^{\circ}\)C. The outdoor temperatures over the observation period ranged between 4 \({}^{\circ}\)C and 11 \({}^{\circ}\)C. The second chart shows the inclination data in milliradians (mrad) along the Y (east-west) and X (north-south) axes. In the legends, the data are labeled dX and dY, since these values are relative to an initial value of tilt, which was set to approximately zero at the beginning of the observations. In Table 2, the'magnitude' corresponds to the maximum displacement of the observed object [30]. \begin{table}
\begin{tabular}{c c c c c} \hline
**Sensor No\(\backslash\)Axis** & **Sensor-I\(\backslash\)Y** & **Sensor-II\(\backslash\)Y** & **Sensor-I\(\backslash\)X** & **Sensor-II\(\backslash\)X** \\ \hline
**maximum** & 0.107 & 0.144 & 0.036 & 0.031 \\
**minimum** & \(-\)0.059 & \(-\)0.001 & \(-\)0.141 & \(-\)0.221 \\
**magnitude** & 0.166 & 0.145 & 0.177 & 0.252 \\ \hline \end{tabular}
\end{table} 
Table 2: Statistics of _the tall building_ inclinations in mrad. Figure 6: Temperature changes and inclinations of _the tall building_ from 8 February to 15 March 2003. The LSSA of the inclination data was performed in two steps. In the first step, all observations shown in Figure 6 were taken into account and analyzed. In the second step, involving more rigorous investigation of the spectrum, the data was divided into weeks and LSSA was applied to weekly data. To expose higher frequencies in the data, the significant signal with a 24-hour period was suppressed. In the end, as expected, the LSSA of Sensor-I and Sensor-II data revealed identical results, and therefore only the results from Sensor-I data are shown here. The outputs were as follows: the periods, frequencies (cycle/hour), amplitudes with root mean square errors (RMSE), percentage variance levels (%var, which is a ratio showing how much of the signal \(\hat{g}\) is contained in the observed time series \(f\) from Equation (5)), and respective PSDs of the periodic signals. Significance tests were performed on the parameters. The variance levels, as derived in the first step, are plotted against frequency in Figure 7, where the significance of the signal with a 24-hour period (T = 24 h) is seen in both the Y and X lines. Nearly 40% of the signal with T = 24 h can be recognized in the observed data with LSSA. This outcome is supported by the evaluation results of the weekly observations (see Figure 8). Figure 8: LS spectrum of _weekly observations_. Figure 7: Least Squares (LS) spectrum of _the entire observations_. After the signal with T = 24 h, which was marked as significant in the previous steps, was suppressed, the significant peak in the X-axis data (inclinations on the north-south direction) with T = 12 h was recognized. In the observations on the Y-axis (east-west direction), an additional signal with T = 8 h was also identified. The significance of the higher frequencies in the Least Squares spectrum is clearly visible in Figure 9. The results of the evaluations are summarized in Table 3.  
After analyzing the building movements in the frequency domain based on inclination data, we evaluated them in the space domain using GPS data [30]. The data from three GPS campaigns conducted in September 2002, February 2003, and August 2003 were processed and hence three coordinate sets were calculated for the deformation points after the adjustment. The RMSE of the coordinates was around \(\pm\)3 millimeters, and three times the RMSE was adopted as the significance level with 99% confidence. In the assessments, the coordinate differences among the sets were also considered.