## 3 Numerical Example
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(t\)} & \multirow{2}{*}{\(\mathbf{\Delta X}(t)\)} & \multirow{2}{*}{\(\mathbf{\overset{\ast}{k}}(t)\)} & \multirow{2}{*}{\(\mathbf{\overset{\ast}{k}}(t)\)} & \multirow{2}{*}{\(\mathbf{\overset{\ast}{k}}(t)\)} & \multirow{2}{*}{\(\mathbf{\overset{\ast}{\Delta X}}(t)\)} & \multirow{2}{*}{\(\mathbf{\overset{\ast}{\Delta X}}(t)\)} & \multirow{2}{*}{\(\mathbf{\overset{\ast}{\Delta X}}(t)\)} & \multirow{2}{*}{**Monitored** Displacement} & \multirow{2}{*}{**APE(\%)**} \\ \cline{3-3} \cline{8-11}  & & & \(\mathbf{\overset{\ast}{k}}_{t}\in[-1,1]\) & & & & \(\mathbf{\overset{\ast}{k}}_{t}\in[-1,1]\) & & & \\ \hline
**AP** & \multicolumn{4}{c}{**Equations (33)**} & \multicolumn{4}{c}{**Equations (33)**} & \multicolumn{4}{c}{**Equations (33)**} & \multicolumn{4}{c}{**Equation (37)**} & \\ \cline{3-11}  & & \multicolumn{4}{c}{**and (34)**} & & \multicolumn{4}{c}{**(19)–(21) and (23)**} & & & & & & \\ \hline
26 & \(\overset{-}{\Delta\overset{-}{\Delta\overset{-}{\Delta\overset{-}{\Delta\overset{-}{ \Delta\overset{-}{\Delta\overset  
The comparison of steel arch support between the monitored, fitted, and forecasted shapes is presented in Figure 17.  
Calculation of model efficiency, which is based on the position error of a marker, is presented in a few successive steps, and we also used marker M4 as an illustrative marker for \(t=10\):  
Step 1: input data:  
- monitored position of marker M4, defined by coordinates, is \(M(\boldsymbol{x},\boldsymbol{y})=(1691.0,3753.4)\);  
- fitted position of marker M4, defined by coordinates, is \(M(x,y)=(1735.7,3754.0)\);  
- monitored position vector of marker M4 is \(\overset{\rightarrow}{\boldsymbol{R}}_{\boldsymbol{M}}=1691.0\,\overset{ \rightarrow}{l}+3753.4\,\overset{\rightarrow}{l}\);  
- fitted position vector of marker M4 is \(\overset{\rightarrow}{R}_{M}=1735.7\,\overset{\rightarrow}{l}+3754.0\, \overset{\rightarrow}{l}\). Step 2: the projection of fitted position vector on monitored position vector  
\[\overset{\rightarrow}{P}_{M}=proj_{\overset{\rightarrow}{R}_{M}} \overset{\rightarrow}{R}_{M} =\frac{1691.0\,\cdot 1735.7+3753.4\,\cdot 3754.0}{\sqrt{1691.0\,\cdot 2 3753.4}\,\cdot\,\sqrt{1691.0\,\cdot 2 3753.4}}. \tag{54}\] \[=1698.8\,\overset{\rightarrow}{l}+3770.6\,\overset{\rightarrow}{l}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{10}{c}{**Two State Series (1 and \(-\)1)**} \\ \cline{2-13} \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}  & \(\boldsymbol{\Delta x}\) & \(\boldsymbol{\Delta y}\) & \(\boldsymbol{\Delta x}\) & \(\boldsymbol{\Delta y}\) & \(\boldsymbol{\Delta x}\) & \(\boldsymbol{\Delta y}\) & \(\boldsymbol{\Delta x}\) & \(\boldsymbol{\Delta y}\) & \(\boldsymbol{\Delta x}\) & \(\boldsymbol{\Delta y}\) & \(\boldsymbol{\Delta x}\) & \(\boldsymbol{\Delta y}\) & \(\boldsymbol{\Delta x}\) & \(\boldsymbol{\Delta y}\) \\ \hline
26 & 90.5 & 885.3 & 226.3 & 2631.5 & 612.4 & 3187.7 & 1797.3 & 3678.0 & 2788.2 & 3189.7 & 3175.8 & 2628.7 & 3308.9 & 888.7 \\
27 & 94.0 & 881.4 & 229.8 & 2627.3 & 616.8 & 3183.2 & 1792.9 & 3673.4 & 2783.9 & 3185.6 & 3171.8 & 2624.5 & 3305.0 & 885.0 \\
28 & 97.3 & 877.6 & 233.0 & 2623.0 & 621.1 & 3178.8 & 1797.4 & 3669.0 & 2779.5 & 3181.7 & 3168.0 & 2620.5 & 3301.0 & 881.3 \\
29 & 100.8 & 874.0 & 236.5 & 2618.5 & 625.1 & 3174.3 & 1801.8 & 3664.7 & 2775.0 & 3177.6 & 3164.0 & 2616.7 & 3297.2 & 877.7 \\
30 & 104.5 & 870.3 & 240.2 & 2614.2 & 629.3 & 3170.0 & 1806.1 & 3660.3 & 2770.6 & 3173.7 & 3159.9 & 2613.1 & 3293.5 & 874.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 29: Forecasted coordinates of markers (forecasting-validity phase). Figure 17: Steel arch support shape for \(t=0\), 10, 25, 30 days (monitored, fitted, and forecasted data).