## 4 Application Examples
### Landslide Prediction Performance  
In this subsection, we will first describe the sources of the data of three cases as well as the split of the training and verification sets. Furthermore, the data features of the three sets of data are examined. Then, the effectiveness of the proposed model will be demonstrated by these cases, and the whole information of the data set is displayed in the picture. Finally, we will compare the errors produced by various approaches. Each sensing unit has a list of data including the displacement and tilt angle, since from the fifth unit to the tenth are below the sliding surface, which has no obvious displacements changes throughout the experimental process (the maximum displacement does not exceed 15 mm), we will focus on units from 1 to 4. Figure 13 is the original data of the experiment. The horizontal position data in Figure 13a shows that these units have a similar variation trend of horizontal displacement, although the horizontal displacement of each element is quite different, however, in Figure 13b, not only the vertical displacement but also its variation trend has a distinct difference. A total of 1035 sets of data were collected in the experiment, but considering that the grey model is a small sample prediction model, the 950 th to 1000 th set of data was selected as the training set, and the 1001 st to 1035 th set of data was selected as the test set. When predicting the data of a sensing unit, the same data of the other 3 units will be brought in as dependent variables. Figures 14 and 15 are the prediction results of sensing units 1 and 2 through the FOBGM (1, N) model, respectively. Figures 14 and 15 are cases 1 and 2, respectively. As we can see from \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**N = 3**} & \multicolumn{3}{c}{**N = 6**} & \multicolumn{3}{c}{**N = 9**} \\ \cline{2-10} \(\mathbf{x_{1}}^{(0)}\) & \(\mathbf{\kappa_{1}}^{(0)}\) (\(\mathbf{k}\)) & \(\mathbf{\alpha(k)}\) & \(\mathbf{\alpha_{1}^{(0)}}\) (\(\mathbf{k}\)) & \(\mathbf{\alpha(k)}\) & \(\mathbf{\Delta(k)}\) & \(\mathbf{\alpha_{1}^{(0)}}\) (\(\mathbf{k}\)) & \(\mathbf{\varepsilon(k)}\) & \(\mathbf{\Delta(k)}\) \\ \hline
13.042 & 13.042 & 0.000 & 0.000\% & 13.042 & 0.000 & 0.000\% & 13.042 & 0.000 & 0.000\% \\
12.307 & 11.465 & \(-\)0.842 & 6.842\% & 12.164 & \(-\)0.143 & 1.162\% & 12.306 & 0.001 & 0.008\% \\
17.518 & 16.649 & \(-\)0.870 & 4.966\% & 17.376 & \(-\)0.143 & 0.816\% & 17.498 & \(-\)0.020 & 0.114\% \\
18.599 & 17.387 & \(-\)1.213 & 6.522\% & 18.413 & \(-\)0.187 & 1.005\% & 18.676 & 0.077 & 0.414\% \\
19.456 & 18.371 & \(-\)1.085 & 5.577\% & 19.267 & \(-\)0.189 & 0.971\% & 19.403 & \(-\)0.053 & 0.272\% \\
19.760 & 18.463 & \(-\)1.298 & 6.569\% & 19.502 & \(-\)0.259 & 1.311\% & 19.726 & \(-\)0.034 & 0.172\% \\
19.903 & 18.550 & \(-\)1.353 & 6.798\% & 19.727 & \(-\)0.176 & 0.884\% & 19.916 & 0.013 & 0.065\% \\
20.010 & 18.667 & \(-\)1.343 & 6.711\% & 19.767 & \(-\)0.243 & 1.214\% & 20.040 & 0.030 & 0.150\% \\
20.135 & 18.788 & \(-\)1.348 & 6.695\% & 19.926 & \(-\)0.210 & 1.043\% & 20.130 & 0.005 & 0.025\% \\ \hline \(\overline{\Delta}\) & \multicolumn{3}{c}{6.335\%} & \multicolumn{3}{c}{1.051\%} & \multicolumn{3}{c}{0.136\%} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fitting effect of FOBGM with a different number of explanatory variables (\(\xi=0.5\)). Figure 13: The curve of original displacement data. (**a**) Graph of original horizontal displacement versus time. (**b**) Graph of original vertical displacement versus time.