## 3 Numerical Example
\begin{bmatrix}-0.30&1\\ -0.70&1\\ -1.05&1\\ -1.35&1\\ -1.65&1\\ -1.90&1\\ -2.05&1\\ -2.20&1\\ -2.35&1\\ -2.60&1\\ -2.95&1\\ -3.35&1\\ -3.35&1\\ -3.75&1\\ -4.00&1\\ -4.15&1\\ -4.30&1\\ -4.45&1\\ -4.65&1\\ -4.85&1\\ -5.05&1\\ -5.30&1\\ -5.45&1\\ -5.55&1\\ \end{bmatrix} ,\quad \begin{bmatrix}0.4\\ 0.4\\ 0.3\\ 0.3\\ 0.2\\ 0.1\\ 0.2\\ 0.3\\ 0.4\\ 0.3\\ 0.2\\ 0.3\\ 0.2\\ 0.1\\ 0.3\\ 0.3\\ 0.2\\ 0.1\\ 0.3\\ 0.3\\ 0.2\\ 0.1\\ 0.3\\ 0.2\\ 0.1\\ 0.1\\ \end{bmatrix} ,\quad \begin{bmatrix}0.4\\ 0.4\\ 0.3\\ 0.3\\ 0.2\\ 0.1\\ 0.2\\ 0.1\\ 0.2\\ 0.3\\ 0.2\\ 0.3\\ 0.2\\ 0.1\\ 0.3\\ 0.2\\ 0.3\\ 0.3\\ 0.2\\ 0.1\\ 0.3\\ 0.2\\ 0.1\\ 0.1\\ \end{bmatrix} ,\quad \begin{bmatrix}0.4\\ 0.4\\ 0.3\\ 0.3\\ 0.2\\ 0.1\\ 0.2\\ 0.1\\ 0.2\\ 0.3\\ 0.2\\ 0.3\\ 0.2\\ 0.1\\ 0.3\\ 0.2\\ 0.3\\ 0.3\\ 0.2\\ 0.1\\ 0.1\\ \end{bmatrix} ,\quad \begin{bmatrix}0.4\\ 0.4\\ 0.3\\ 0.3\\ 0.3\\ 0.2\\ 0.1\\ 0.2\\ 0.3\\ 0.2\\ 0.1\\ 0.3\\ 0.2\\ 0.3\\ 0.3\\ 0.2\\ 0.1\\ 0.3\\ 0.2\\ 0.1\\ 0.1\\ \end{bmatrix} . \tag{52}\]  
According to Equation (12), we obtain \(a=0.03454\) and \(b=0.35021\). The parameters of the grey stochastic process are presented in Table 12. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Variance of AGO series \((\sigma^{2})\) & 2.88 \\ Time \((T)\) & 30 days \\ Number of steps \((\varphi)\) & 30 \\ Timestep of \(\Delta t\) & 1 day \\ Daily time resolution—coefficient \((\omega)\) & 365 \\ Brownian increments—\(\Delta W_{t}=W_{t}-W_{t-1}\)\(\sim N\Big{(}0,\frac{\sigma^{2}}{\omega}\Big{)}\) & \(\sim N\Big{(}0,\frac{2.88}{365}\Big{)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Parameters of stochastic AGO series of transformed stationary series. Figure 9: Accumulated Generation Operation for transformed stationary series (fitting period). To obtain the reconstructed (fitted) \(MD\) state of the series of displacements, we applied Equation (27), and the results with the corresponding absolute percentage errors are shown in Table 15 and Figure 12.  
The mean absolute percentage error (MAPE) is 1.88%, and according to Table 1, the fitting model of the displacement of marker M4 along the \(x\)-axis belongs to a highly accurate \begin{table}
\begin{tabular}{c c c c} \hline \hline \(t\) & **Original Displacement** & **Fitted Displacement** & **APE (\%)** \\ \hline
1 & \(-4.4\) & & \\
2 & \(-4.3\) & \(-4.30\) & \\  
\hline \hline \end{tabular}
\end{table} 
Table 15: Reconstructed displacements and absolute percentage errors. Figure 12: Original and fitted displacements of marker M4 along \(x\)-axis. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \(t\) & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** & **14** & **15** \\ \hline E(IAGO) & \(0.10\) & \(0.34\) & \(0.33\) & \(0.32\) & \(0.31\) & \(0.30\) & \(0.29\) & \(0.28\) & \(0.26\) & \(0.26\) & \(0.25\) & \(0.24\) & \(0.23\) & \(0.23\) \\ K & \(-1\) & \(1\) & \(-1\) & \(1\) & \(-1\) & \(1\) & \(-1\) & \(1\) & \(-1\) & \(1\) & \(1\) & \(-1 class. Fitting errors of displacement for all markers along the \(x\)- and \(y\)-axis are presented in Table 16.  
The MAPE of the model is \(1.98\%\approx 2\%\), which points out that the model is capable of forecasting future values of displacement. AGO series beyond \(T=25\) are forecasted by simulation of Equation (52), for \(t=26\), \(27\), \(28\), \(29\), \(30\); and outcomes of expected AGO and corresponding IAGO series are shown in Table 17.  
Application of Equation (5) for the forecasting period requires the transformation coefficient series \(K^{q}(t)\) to also be forecasted. The two-state time series of the first difference monitored data is presented in Table 18.  
The order of the AR process equals \(\rho\cong\frac{24}{2}-2=10\).