## 2 Materials and Methods
### The Model of Support Deformation Forecasting  
#### 2.2.1 Fitting the Monitored Displacements  
The proposed model of support deformation forecasting will be explained with respect to displacement of one marker \(M\) along \(x\)-axes. Let us consider a monitored time series of marker displacement along \(x\)-axes over time \(t=1,2,\ldots,T\). Denote monitored time series as \(\Delta X(t)=(\Delta x_{1},\Delta x_{2},\ldots,\Delta x_{T})\). Next, we define three different states of displacement series, namely non-negative, negative, and mixed. The series is in the non-negative state if all values of \(\Delta x_{t}\) are greater than zero, while the series is in the negative state if all values of \(\Delta x_{t}\) are  
Equation (11) presents ordinary differential equation of the form:  
\[dp_{t}^{(1)}=f\Big{(}t,p_{t}^{(1)}\Big{)}dt \tag{14}\]  
with initial condition \(p_{t=1}^{(1)}=p_{1}\). Displacement is subject to random perturbation caused by random variation of the loading forces over time. It is hard to specify it in the model description, working within the system. Such perturbations create noise in the monitored displacements and AGO series as well, and cannot be captured by Equation (14). When we take Equation (14) and assume that \(f\Big{(}t,p_{t}^{(1)}\Big{)}\) is not deterministic function but stochastic, and add the noisy part to it, we obtain stochastic differential equation. So, we can create stochastic differential equation with additive noise part as follows:  
\[dp_{t}^{(1)}=f\Big{(}t,p_{t}^{(1)}\Big{)}dt+dW_{t},t=1,2,\ldots,T \tag{15}\]  
where:  
\(dW_{t}\)--Brownian motion, \(dW_{t}\sim N(0,Var)\)  
The grey system model is now transformed into stochastic system [21]  
\[dp_{t}^{(1)}=\Big{(}b-ap_{t}^{(1)}\Big{)}dt+dW_{t},p_{t=1}^{(1)}=p_{1} \tag{16}\]  
Numerical approximation of \(p_{t}^{(1)}\) is obtained from the explicit Euler-Maruyama discretization [22]:  
\[p_{t}^{(1)}=p_{t-1}^{(1)}+\Big{(}b-ap_{t-1}^{(1)}\Big{)}\Delta t+\Delta W_{t}, t=1,2,\ldots,T \tag{17}\]  
using a uniform timestep of \(\Delta t\) with Brownian increments \(\Delta W_{t}\), and there is a fixed initial value \(p_{t=1}^{(1)}=p_{1}\). Behavior of \(p_{t}^{(1)}\), in the form of Equation (17), is only discrete in the time variable but not as random variable. Set \(\Delta t=\frac{T}{\varphi}\), \(\varphi\leq T\) and \(\Delta W_{t}=W_{t}-W_{t-1}\sim N\Big{(}0,\frac{\varphi^{2}}{\omega}\Big{)}\), where \(\varphi\) is the number of timesteps in time \(T\) and \(N\Big{(}0,\frac{\varphi^{2}}{\omega}\Big{)}\) is the normal distribution with expected value of 0 and variance \(\frac{\varphi^{2}}{\omega}\). Variance in AGO series is \(\varphi^{2}\) and \(\omega\) is the time resolution coefficient, which is defined as follows:  
\[\omega=\left\{ \begin{array}{l}1,annual\text{ }time\text{ }resolution\\ 12,\text{ }monthly\text{ }time\text{ }resolution\\ 365,\text{ }daily\text{ }time\text{ }resolution\end{array} \right. \tag{18}\]  
Substituting \(\Delta W_{t}\sim N\Big{(}0,\frac{\varphi^{2}}{\omega}\Big{)}\) in Equation (17), we obtain solution of the stochastic AGO process defined by Equation (16):  
\[p_{t}^{(1)}=p_{t-1}^{(1)}+\Big{(}b-ap_{t-1}^{(1)}\Big{)}\Delta t+N\Big{(}0, \frac{\varphi^{2}}{\omega}\Big{)},t=1,2,\ldots,T \tag{19}\]  
Assume we have a function, \(f\): \(\mathbb{R}{\rightarrow}\mathbb{R}\), which depends on the solution, \(p_{t}^{(1)}\), of the stochastic differential Equation (16) on the time interval [1,7]. Equation (19) presents the solution, and simulation of Equation (19) produces expected values for specific point in time of the AGO stochastic process:  
\[\text{E}_{t}^{AGO}=\frac{1}{S}\sum_{s=1}^{S}p_{t,s}^{(1)},t=2,3,\ldots,T,\text {E}_{1}^{AGO}=p_{1} \tag{20}\]  
where  
\(S\)--the total number of simulations. Obviously, at every point in time, there is an adequate probability density function of AGO. Next, the Inverse Accumulated Generation Operation (IAGO) is applied to reconstruct \(NN\) state of \(P(t)\) series as  
\[\left\{