## 4 Methodological Approach
### Detection of Change-Points in Time Series of Individual Components  
The time series of the measured movements are not stationary but often express sudden changes. We assume that these changes are caused by external factors. Over time, several methods have been developed to detect such changes such as CUSUM (Cumulative Sum) and its modifications (firstly introduced by [61]). Limiting factors of a wide range of methods are very often the computational demands in combination with a large volume of data or the requirement for time-series homogeneity. The problem of geotechnical measurements (field measurements in general) is the incompleteness of the time series or its inhomogeneity--different time spacing between recorded measurements. In this work, the PARCS (Paired Adaptive Regressors for Cumulative Sum) method was used, a method for detecting a change in the mean value of the multidimensional time series [20]. The time series of measurements are burdened with random errors. The mean value is then close to the value of the monitored variable in a large number of measurements. The PARCS method looks for a predefined number of change-points in the time series. The input data to the algorithm are an array containing the time-series data and the scalar value indicating the number of searched change-points (N). The advantage over other methods is that the time series may not be completely consistent and may contain holes. The PARCS method was modified for data analysis so that the input parameter N was used only as an a priori (initial) value and the target mean error parameter was added. After the detection of change-points, the time series is divided into segments, which are approximated by linear regression. If the standard deviation on one of the sections exceeds the given limit value, the parameter N is increased (Figure 4). The cut-off target value of the standard deviation was derived from the assumed accuracy of the dilatometer measurement as twice the standard deviation of the measurement found during laboratory testing [62]. A relatively large number of change-points was detected. Taking a closer look at the trajectory of motion, it is clear that some of these change-points do not lead to a change in the trajectory of motion but are the result of chaotical behavior. Therefore, we decided to directly analyze the trajectory of motion in the plane (2D) and space (3D) to find the key points of its change and their time stamps (Figure 5). Figure 3: Example of the time-series decomposition: (**a**) close to a linear decrease of the velocity on HIE3 \(Y\)-axis; (**b**) significant seasonal periodicity visible on HIE2 \(X\)-axis. The trajectory of the movement is represented by an array of coordinates with a removed periodic component. At first, the trajectory was smoothed using a 30-day moving average of the position to remove random noise. Then, the key breaking points of the motion curve were found, the curve was  
Figure 4: Modified PARCS method applied to the time series of residual values (HIE3 \(z\)-axis): (**a**) time stamps of detected change-points marked in green, target standard deviation of 0.05 mm; (**b**) comparison of detected time stamps and seismic events. Figure 5: Example of the processing of the movement trajectory on a Z-Y plane on: (**a**) HIE2 device; (**b**) HIE3 device: raw trajectory (blue), smoothed trajectory using 30-day moving average (red), simplified trajectory (green). expressed by the minimum number of linear segments so that the accuracy criterion was preserved (Figure 5). For this purpose, the Ramer-Douglas-Peuckner (RDP) simplification algorithm was used [63]. The advantage of this approach is that all three components of motion can be analyzed together. Compared to other change-point detection methods applied to components, the RDP algorithm is simpler and enables very fast sampling of the curve and detection of change-points. Importance of the change-points was rated to highlight the most significant changes in the pathway of movement of the monitored SAL detachment plane.