## 3 Numerical Example  
The hypothetical data used in this paper include daily values of displacements of markers along the \(x\)- and \(y\)-axis, respectively. The period of monitoring is thirty days. This set of displacements is divided into two subsets, fitting and forecasting subsets. The period ranging from the first to the twenty-fifth day (25 values) was used as the fitting subset, and within this interval, we checked the model accuracy by comparing the fitted and monitored values of displacements. The second subset ranging from twenty-six to thirty days (5 values) was used as the validity subset, and in this period, we made a comparison between forecasted and monitored displacements. The positions of seven markers are defined around the internal rim of the steel arch support; see Figure 4.  
Figure 4: Steel arch support with positions of the markers. The coordinates of markers during the monitoring period are presented in Table 6, while the corresponding displacements are shown in Table 7.  
Time series related to displacements of marker M4 along the \(x\)-axis was used to show how the model works. Figure 5 presents displacements of marker M4 along the \(x\)-axis. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{10}{c}{**MONITORED COORDINATES**} \\ \hline \multirow{2}{*}{**Day**} & \multicolumn{2}{c}{**M1**} & \multicolumn{2}{c}{**M2**} & \multicolumn{2}{c}{**M3**} & \multicolumn{2}{c}{**M4**} & \multicolumn{2}{c}{**M5**} & \multicolumn{2}{c}{**M6**} & \multicolumn{2}{c}{**M7**} \\ \cline{2-13}  & \(x\) & \(y\) & \(x\) & \(y\) & \(x\) & \(y\) & \(x\) & \(y\) & \(x\) & \(y\) & \(x\) & \(y\) & \(x\) & \(y\) \\ \hline
0 & 0.0 & 981.0 & 122.0 & 2736.0 & 498.0 & 3302.0 & 1700.0 & 3800.0 & 2902.0 & 3302.0 & 3 \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline \(t\) & \(\boldsymbol{1}\) & \(\boldsymbol{2}\) & \(\boldsymbol{3}\) & \(\boldsymbol{4}\) & \(\boldsymbol{5}\) & \(\boldsymbol{6}\) & \(\boldsymbol{7}\) & \(\boldsymbol{8}\) & \(\boldsymbol{9}\) & \(\boldsymbol{10}\) & \(\boldsymbol{11}\) & \(\boldsymbol{12}\) & \(\boldsymbol{13}\) & \(\boldsymbol{14}\) & \(\boldsymbol{15}\) \\ \hline MD & \(-0.1\) & \(0.4\) & \(-0.4\) & \(0.3\) & \(-0.3\) & \(0.3\) & \(-0.2\) & \(0.1\) & \(-0.2\) & \(-0.1\) & \(0.4\) & \(0.3\) & \(-0.5\) & \(0.3\) \\ K & \(-1\) & \(1\) & \(-1\) & \(1\) & \(-1\) & \(1\) & \(-1\) & \(1\) & \(-1\) & \(-1\) & \(1\) & \(1\) & \(-1\) & \(1\) \\ NN & \(0.1\) & \(0.4\) & \(0.4\) & \(0.3\) & \(0.3\) & \(0.3\) & \(0.2\) & \(0.1\) & \(0.2\) & \(0.1\) & \(0.4\) & \(0.3\) & \(0.5\) & \(0.3\) \\ \hline \(t\) & \(\boldsymbol{16}\) & \(\boldsymbol{17}\) & \(\boldsymbol{18}\) & \(\boldsymbol{19}\) & \(\boldsymbol{20}\) & \(\boldsymbol{21}\) & \(\boldsymbol{22}\) & \(\boldsymbol{23}\) & \(\boldsymbol{24}\) & \(\boldsymbol{25}\) & \(\boldsymbol{26}\) & \(\boldsymbol{27}\) & \(\boldsymbol{28}\) & \(\boldsymbol{29}\) & \(\boldsymbol{30}\) \\ \hline MD & \(-0.2\) & \(-0.1\) & \(-0.2\) & \(-0.1\) & \(0.3\) & \(0.1\) & \(-0.3\) & \(0.2\) & \(-0.1\) & \(0.1\) & \(0.2\) & \(0.2\) & \(-0.4\) & \(-0.2\) & \(0.2\) \\ K & \(-1\) & \(-1\) & \(-1\) & \(-1\) & \(1\) & \(1\) & \(-1\) & \(1\) & \(1\) & \(1\) & \(-1\) & \(-1\) & \(1\) \\ NN & \(0.2\) & \(0.1\) & \(0.2\) & \(0.1\) & \(0.3\) & \(0.1\) & \(0.3\) & \(0.2\) & \(0.1\) & \(0.1\) & \(0.2\) & \(0.4\) & \(0.2\) & \(0.2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Transformation from _MD_ stationary to _NN_ state of series. Figure 8: Transformation of stationary _MD_ state of series. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(t\) & \(\boldsymbol{P(t)}\) & \(\boldsymbol{P^{(1)}(t)}\) & \(\boldsymbol{t}\) & \(\boldsymbol{P(t)}\) & \(\boldsymbol{P^{(1)}(t)}\) \\ \hline \(1\) & & & \(14\) & \(0.5\) & \(3.6\) \\ \(2\) & \(0.1\) & \(0.1\) & \(15\) & \(0.3\) & \(3.9\The following matrices are used to calculate parameters \(a\) and \(b\) of the grey modeling:  
\[