## 2 Materials and Methods
### The Model of Support Deformation Forecasting
#### 2.2.3 Model Accuracy Based on Marker Position Error  
Accuracy of the fitting model, which is expressed by MAPE, treats displacements along \(x\)- and \(y\)-axes as two separated time series. However, our problem concerns the fitted position of marker, and it means that we must estimate the closeness between fitted and monitored position of marker. Accordingly, this approach of model accuracy estimation deals with analysis of an error between monitored and fitted position of marker [29; 30]. Figure 3 shows the principle used to define the position error. The mechanism that we use to estimate fitting error concerns how much a vector is equal to another vector, in the same direction. Desired direction is direction of monitored  
Figure 3: Error between monitored and fitted position of the marker. position vector. Accordingly, we calculate the vector projection of \(\overset{\rightarrow}{R}_{M}\) on the vector \(\overset{\rightarrow}{R}_{M}\) as follows:  
\[\overset{\rightarrow}{P}_{M}=proj_{\overset{\rightarrow}{R}_{M}}\overset{ \rightarrow}{R}_{M}=\frac{\overset{\rightarrow}{R}_{M}\cdot\overset{ \rightarrow}{R}_{M}}{\left|\overset{\rightarrow}{R}_{M}\right|}\overset{ \rightarrow}{R}_{M}\overset{\rightarrow}{R}_{M}=P_{M_{x}}\cdot\overset{ \rightarrow}{l}+P_{M_{y}}\cdot\overset{\rightarrow}{J} \tag{39}\]  
where \(\overset{\rightarrow}{l}\) and \(\overset{\rightarrow}{J}\) are unit vectors along \(x\)- and \(y\)-axis, respectively. Magnitude of the vector \(\overset{\rightarrow}{P}_{M}\) equals  
\[\left|\overset{\rightarrow}{P}_{M}\right|=\sqrt{(P_{M_{x}})^{2}+\left(P_{M_{ y}}\right)^{2}} \tag{40}\]  
Similarly, magnitude of the vector \(\overset{\rightarrow}{P}_{E}\) is as follows:  
\[\left|\overset{\rightarrow}{P}_{E}\right|=\sqrt{(P_{E_{x}})^{2}+\left(P_{E_{ y}}\right)^{2}} \tag{41}\]  
Monitored position vector of marker \(\boldsymbol{M}(\boldsymbol{x},\boldsymbol{y})\) can be presented as a sum of vector \(\overset{\rightarrow}{R}_{M}\) and vector \(\overset{\rightarrow}{E}\), i.e., vector \(\overset{\rightarrow}{R}_{M}\) is a resultant of these two vectors:  
\[\overset{\rightarrow}{R}_{\boldsymbol{M}}=\overset{\rightarrow}{R}_{M}+ \overset{\rightarrow}{E} \tag{42}\]  
Vector \(\overset{\rightarrow}{R}_{\boldsymbol{M}}\) can also be presented as resultant of the following two vectors:  
\[\overset{\rightarrow}{R}_{\boldsymbol{M}}=\overset{\rightarrow}{P}_{M}+ \overset{\rightarrow}{P}_{E} \tag{43}\]  
Note that the following equality does not hold, according to the triangle inequality:  
\[\left|\overset{\rightarrow}{R}_{M}\right|+\left|\overset{\rightarrow}{E} \right|=\left|\overset{\rightarrow}{P}_{M}\right|+\left|\overset{ \rightarrow}{P}_{E}\right| \tag{44}\]  
In our case (see Figure 4), the triangle inequality is  
\[\left|\overset{\rightarrow}{R}_{\boldsymbol{M}}\right|<\left|\overset{ \rightarrow}{R}_{M}\right|+\left|\overset{\rightarrow}{E}\right| \tag{45}\]  
The projection of error vector on monitored position vector of marker can be calculated from Equation (43) as follows:  
\[\overset{\rightarrow}{P}_{E}\overset{\rightarrow}{R}_{\boldsymbol{M}}- \overset{\rightarrow}{P}_{M} \tag{46}\]  
Vectors \(\overset{\rightarrow}{R}_{\boldsymbol{M}}\), \(\overset{\rightarrow}{P}_{M}\), \(\overset{\rightarrow}{P}_{E}\) are collinear, since they lie on the same line, so we can transform Equation (46) from vector to a scalar form in the following way:  
\[\left|\overset{\rightarrow}{P}_{E}\right|=\left|\overset{\rightarrow}{R}_{ \boldsymbol{M}}\right|-\left|\overset{\rightarrow}{P}_{M}\right| \tag{47}\]  
Obviously, Equation (47) presents the absolute error of marker position prediction.