## 2 Materials and Methods
### The Model of Support Deformation Forecasting
#### 2.2.2 Forecasting of Displacements  
Displacement forecasting is based on the simulation of Equation (19) from \(T\) to \(T+h\), where \(h\) presents the number of points in time ahead. The prediction phase is over at point \(T\), and forecasting phase starts. Having in mind that Equation (19) concerns simulation of AGO series, we obtain future values of AGO series beyond \(T\) as follows:  
\[p_{t}^{(1)}=p_{t-1}^{(1)}+\left(b-ap_{t-1}^{(1)}\right)\Delta t+N\bigg{(}0, \frac{\sigma^{2}}{\omega}\bigg{)},t=T+1,T+2,\ldots,T+h \tag{29}\]  
Simulation of Equation (29) produces expected values for specific points in the future of the AGO stochastic process:  
\[ \begin{array}{l}\mathrm{E}_{t}^{AGO}=\frac{1}{S}\sum\limits_{s=1}^{S}p_{t,s }^{(1)},t=T+1,T+2,\ldots,T+h;p_{t,s}^{(1)}\\ =p_{t-1,s}^{(1)}+\left(b-ap_{t-1,s}^{(1)}\right)\Delta t+N\bigg{(}0,\frac{ \sigma^{2}}{\omega}\bigg{)},t\\ =T+1,T+2,\ldots,T+h;s=1,2,\ldots,S\end{array}  \tag{30}\]  
Applying the same IAGO approach as we did in the prediction phase, the following forecasted series is obtained:  
\[\overset{=}{\overline{Q}}(t)=\left\{\overset{=}{\overline{q}}_{t}\right\},t =T+1,T+2,\ldots,T+h \tag{31}\]  
Forecasted \(\overset{=}{\overline{q}}_{t}\) series is also in the \(NN\) state. Transformation from \(NN\) state to \(MD\) state of series is performed in the following way:  
\[MD\overset{=}{\overline{Q}}(t)=\overset{=q}{k}_{t}\overset{=q}{q}_{t},t=T+1, T+2,\ldots,T+h,\overset{=}{\overline{q}}_{t}>0 \tag{32}\]  
Analyzing Equation (32), we can conclude that transformation coefficients series \(K^{q}(t)\) should also be forecasted. For that purpose, we apply autoregressive process (AR) over original \(K^{q}(t)=\left\{k_{t}^{q}\right\}_{T\times 1},t=1,2,\ldots,T\) series, and obtain reconstructed series as follows:  
\[\hat{K}^{q}(t)=\left\{\hat{k}_{t}^{q}\right\}=\beta_{0}^{q}+\beta_{1}^{q}k_{t -1}^{q}+\beta_{2}^{q}k_{t-2}^{q}+\ldots+\beta_{\rho}^{q}k_{t-\rho}^{q} \tag{33}\]  
where \(\rho\) is the order of the autoregressive process, and coefficients \(\beta_{0}^{q},\beta_{1}^{q},\beta_{2}^{q},\ldots,\beta_{\rho}^{q}\) of the linear combination are the parameters of the AR process. If we take into consideration that \(\hat{k}_{t}^{q}\) can take only \(1\) or \(-1\) value, then following conditions are used to define this series:  
\[\hat{K}^{q}(t)=\left\{\hat{k}_{t}^{q}\right\}=\left\{ \begin{array}{c}1,if\hat {k}_{t}^{q}>0\\ -1,if\hat{k}_{t}^{q}<0\end{array} ,t=\rho+1,\rho+2,\ldots,T \tag{34}\]  
Accuracy of the AR(\(\rho\)) model is calculated according to the following equation:  
\[AC(\%)=100\times\frac{\Sigma_{t-\rho}^{T}n_{t}^{TRUE}}{T-\rho} \tag{35}\] \begin{table}
\begin{tabular}{c c} \hline
**Linguistic Description** & **MAPE (\%)** \\ \hline High accuracy & \textless{}10 \\ Good accuracy & 10–20 \\ Reasonable accuracy & 20–50 \\ Inaccurate & \textgreater{}50 \\ \hline \end{tabular}
\end{table} 
Table 1: Linguistic description of model accuracy [23; 24]. where \(n_{t}^{TRUE}\) is assigned value one if \(\hat{k}_{t}^{q}=k_{t}^{q},t\in[\rho,T]\), otherwise, zero. To define the order of the autoregressive process, we propose the modified window length selection for singular spectrum analysis (SSA). We only propose a way of selecting the order (\(\rho\)) without further discussion in theory. Hence, we validate our approach via the following test. The range of window length \(L\) in SSA is in \(2\leq L\leq T/2\)[25; 26; 27; 28]. Modified approach defines order of AR process as \(\rho\cong T/2-w\), where \(w\) presents the number of states of variable. Since we have only 1 and -1 values, i.e., only two states exist, we set \(w=2\). So, the order of regression is \(\rho\cong T/2-2\). Suppose there is the series that is presented in Table 2.