# Automatic Detection of Maintenance Scenarios for Equipment and Control Systems in Industry
## 3 Results  
### The Study of the MSSA Method in Horizontal and Vertical Trajectory Matrix Compilation Method  
For the study, we chose six normalized time series of equal length as input data, i.e., \(s=6\) and \(N_{1}=N_{2}=N_{3}=N_{4}=N_{5}=N_{6}\). We made trajectory matrices vertically and horizontally (Figure 6) and performed singular value decomposition into components.  
Figure 6: Trajectory matrix with vertical (**top**) and horizontal (**bottom**) methods of composition.  
As one can see from Table 1, the horizontal method is more time-consuming than the vertical. The obtained time results largely depend on the equipment used for calculations as well as the length of time series. In our case, all six time series have the same length, equal to 5303 units.  
Moreover, the number of extracted components in the vertical method is naturally multiple times smaller. We also estimated the relative and total contributions of the components. According to the component contribution figures (Figure 7), it can be concluded that the horizontal trajectory matrix compilation obviously produces fewer components than the vertical one. However, the relative contribution of the components makes it clear that the contribution decreases more slowly when the trajectory matrix is compiled vertically. This enables us to identify periodic or transient components in addition to the zero-trend component. In the horizontal method, a sharp decrease in component contribution indicates that almost all the original time series data are contained in the trend (zero component \(X_{0}\)).  
In the horizontal method, the last components are noise ones, while in the vertical method, they carry some non-noise information (Figure 8). This is due to the fact that eigenvectors \(U_{i}\) reflect common features of time series, whereas in the vertical method, these vectors are multiple times larger and, accordingly, contain more information.  
Consequently, with the horizontal method, the Hankel matrices are stacked horizontally next to each other, resulting in a broader matrix. This approach is appropriate when variables or time series are either equally significant or when the focus is on analyzing the relationships between variables. In the vertical method, Hankel matrices are stacked vertically on top of each other, resulting in a taller matrix. This approach is useful when the focus is on analyzing the dynamics of the time series.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**The Number of Time Series** & & \\ **(N = 5303)** & **Horizontal Method** & **Vertical Method** \\ \hline s = 1 & 435 ms \(\pm\) 26.6 ms & 435 ms \(\pm\) 26.6 ms \\ s = 2 & 1.63 s \(\pm\) 55.7 ms & 568 ms \(\pm\) 21.8 ms \\ s = 3 & 4.25 s \(\pm\) 545 ms & 812 ms \(\pm\) 25.6 ms \\ s = 4 & 12.7 s \(\pm\) 977 ms & 893 ms \(\pm\) 10.6 ms \\ s = 5 & 40.8 s \(\pm\) 1.09 ms & 1.08 s \(\pm\) 28.2 ms \\ s = 6 & 1 min 7 s \(\pm\) 1.67 ms & 1.31 s \(\pm\) 83.6 ms \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of trajectory matrix decomposition time for horizontal and vertical methods of composition.  
Figure 7: Comparison of relative and cumulative component contributions for horizontal and vertical trajectory matrix composition.