## 4 Application Examples
### Performance of Model Parameters

Table 1: Fitting effect of FOBGM with different background values. Figure 12: Curve of accumulative deformation. (**a**) Graph of cumulative horizontal displacement versus time (**b**) Graph of cumulative vertical displacement versus time. By using three indicators, the residual error \(\varepsilon\), absolute percentage error \(\Delta\), and mean of absolute percentage error \(\overline{\Delta}\), defined as follows, to appraise the effect of different background values on the model. \[ \begin{array}{l}\varepsilon(k)=\hat{x}_{1}^{(0)}(k)-x_{1}^{(0)}(k)\\ \Delta(k)=\frac{1\varepsilon(k)!}{x_{1}^{(0)}(k)}\times 100\%\\ \overline{\Delta}=\frac{1}{n}\overline{\Sigma}_{k=1}^{n}\Delta(k)\end{array}  \tag{32}\]  
From the above table, it is known that the fitting error of the FOBGM model varies with the \(\xi\). In this case, since the whole data set is randomly selected, there is no obvious regularity in the changing trend between the data, resulting in a large fitting error. It can be seen that when \(\xi\) is in the range of 0.2 to 0.3, its MAPE is the smallest, and the best \(\xi\) calculated by WOA is 0.267. It can also be seen that a small change in \(\xi\) also leads to a large change in MAPE, for example, when \(\xi\) is 0.6 and 0.7, the MAPE between them has a huge difference. Therefore, the background value in the traditional grey model is 0.5 which is a simplification and is unreasonable. The background value coefficient should be optimized according to the specific data series. In the actual process, the variation of the data is relatively smooth, so this model has a better fitting effect than this case. Nine sequential groups of horizontal displacement data from the original experimental data have been selected as the data that need to be fitted. Table 2 shows the fitting effect of the FOBGM model with 3, 6, and 9 variables in the same background value. The system characteristic sequence \(X_{1}^{(0)}\) and the explanatory variable sequences, from \(X_{2}^{(0)}\) to \(X_{9}^{(0)}\), are defined as follows:  
\[ \begin{array}{l}X_{1}^{(0)}=(x_{1}^{(0)}(1),x_{1}^{(0)}(2),\cdots,x_{1}^{(0) }(9))=(13.042,12.307,17.518,18.599,19.456,19.760,19.903,20.010,20.135)\\ X_{2}^{(0)}=(x_{2}^{(0)}(1),x_{2}^{(0)}(2),\cdots,x_{2}^{(0)}(9))=(14.060,13.324,17.613,18.541,19.260,19.519,19.643,19.742,19.851)\\ X_{3}^{(0)}=(x_{3}^{(0)}(1),x_{3}^{(0)}(2),\cdots,x_{3}^{(0)}(9))=(13.839,13.054,16.636,17.486,18.028,18.243,18.362,18.455,18.549)\\ X_{4}^{(0)}=(x_{4}^{(0)}(1),x_{4}^{(0)}(2),\cdots,x_{4}^{(0)}(9))=(11.347,10.5 42,13.640,14.348,14.763,14.947,15.047,15.131,15.205)\\ X_{5}^{(0)}=(x_{5}^{(0)}(1),x_{5}^{(0)}(2),\cdots,x_{5}^{(0)}(9))=(7.988,7.17 8,9.922,10.528,10.850,10.995,11.095,11.163,11.226)\\ X_{6}^{(0)}=(x_{6}^{(0)}(1),x_{6}^{(0)}(2),\cdots,x_{6}^{(0)}(9))=(6.462,5.685,8.291,8.771,9.013,9.135,9.221,9.285,9.342)\\ X_{7}^{(0)}=(x_{7}^{(0)}(1),x_{7}^{(0)}(2),\cdots,x_{7}^{(0)}(9))=(4.357,3.573,5.847,6.223,6.408,6.500,6.573,6.610,6.660)\\ X_{8}^{(0)}=(x_{9}^{(0)}(1),x_{8}^{(0)}(2),\cdots,x_{8}^{(0)}(9))=(2.369,1.480,3.166,3.434,3.573,3.625,3.677,3.710,3.751)\\ X_{9}^{(0)}=(x_{9}^{(0)}(1),x_{9}^{(0)}(2),\cdots,x_{9}^{(0)}(9))=(0.423,0.607,1.592,1.754,1.833,1.874,1.907,1.937,1.961)\end{array} \]From the above table, we know that the fitting error of the FOBGM model decreases with the increasing number of correlated factors in the data series of correlated factors. This is mainly due to the strong correlation between the explanatory variable sequences involved in the calculation and the system characteristic sequence.