## 4 Application Examples
### Performance of Model Parameters  
In order to verify whether the different background values and different independent variables will affect the fitting results, randomly selected 9 sets of horizontal displacement data from one experiment for fitting. The system characteristic sequence \(X_{1}\)(0) and the explanatory variable sequences, from \(X_{2}\)(0) to \(X_{9}\)(0), are defined as follows:  
\[ \begin{array}{l}X_{1}^{(0)}=[x_{1}^{(0)}(1),x_{1}^{(0)}(2),\cdots,x_{1}^{(0)}( 9)]=[12.30,10.91,5.39,2.32,3.62,14.84,14.76,40.37,526.06]\\ X_{9}^{(0)}=[x_{2}^{(0)}(1),x_{2}^{(0)}(2),\cdots,x_{2}^{(0)}(9)]=[15.75,13.23,7.92,10.60,0.04,12.76,12.81,26.76,380.04]\\ X_{3}^{(0)}=[x_{3}^{(0)}(1),x_{3}^{(0)}(2),\cdots,x_{3}^{(0)}(9)]=[18.29,14.52,9.62,3.83,1.18,11.43,11.58,13.29,219.50]\\ X_{4}^{(0)}=[x_{4}^{(0)}(1),x_{4}^{(0)}(2),\cdots,x_{4}^{(0)}(9)]=[16.36,12.7 1,8.22,3.47,1.15,9.57,9.71,0.77,45.27]\\ X_{5}^{(0)}=[x_{5}^{(0)}(1),x_{5}^{(0)}(2),\cdots,x_{5}^{(0)}(9)]=[12.44,9.01,4.94,1.26,0.77,9.67,9.78,8.26,0.34]\\ X_{6}^{(0)}=[x_{6}^{(0)}(1),x_{6}^{(0)}(2),\cdots,x_{6}^{(0)}(9)]=[9.85,6.82,3. 35,0.67,2.35,9.57,9.67,10.86,8.61]\\ X_{7}^{(0)}=[x_{7}^{(0)}(1),x_{7}^{(0)}(2),\cdots,x_{7}^{(0)}(9)]=[7.30,4.96,2. 26,0.80,2.11,10.18,10.25,11.24,10.36]\\ X_{8}^{(0)}=[x_{8}^{(0)}(1),x_{8}^{(0)}(2),\cdots,x_{8}^{(0)}(9)]=[4.25,2.60, 0.52,2.01,3.06,9.13,9.16,9.70,9.31]\\ X_{9}^{(0)}=[x_{9}^{(0)}(1),x_{9}^{(0)}(2),\cdots,x_{9}^{(0)}(9)]=[1.49,0.50,0. 79,2.46,3.13,6.81,6.82,6.91,6.67]\end{array} \]  
Five of the above eight explanatory variables were selected and brought into the FOBGM model to testify to the background values effect. Table 1 shows the fitting effect of the FOBGM model under different background values. In order to maximize the fitting effect of different \(\zeta\), use the FOBGM(1, 6) model mentioned above as the test model. The \(\zeta\) will be chosen from 0.1 to 0.7 and the best background value selected by WOA to prove the importance of the background value. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\xi\)} & \multirow{2}{*}{\(\mathbf{x_{1}}\)(0)} & \multirow{2}{*}{**12.297**} & \multirow{2}{*}{**10.911**} & \multirow{2}{*}{**5.391**} & \multirow{2}{*}{**2.324**} & \multirow{2}{*}{**3.619**} & \multirow{2}{*}{**14.838**} & \multirow{2}{*}{**14.757**} & \multirow{2}{*}{**40.369**} & \multirow{2}{*}{**526.060**} & \multirow{2}{*}{\(\overline{\Delta}\)} \\ \cline{3-3} \cline{8-11}  & & & & & & & & & & & & & \\ \hline \multirow{4}{*}{0.1} & \(\hat{x}_{1}^{(0)}(k)\) & 12.297 & 12.248 & 8.014 & 1.367 & 0.151 & 15.435 & 17.529 & 42.799 & 553.553 & \\  & \(\varepsilon\)(k) & 0.000 & 1.337 & 2.623 & 0.957 & 3.468 & 0.597 & 2.772 & 2.430 & 27.493 & 25.772\% \\  & \(\Delta\)(k) & 0.000\% & 12.255\% & 48.643\% & 41.174\% & 95.828\% & 4.020\% & 18.785\% & 6.020\% & 5.226\% & \\ \hline \multirow{4}{*}{0.2} & \(\hat{x}_{1}^{(0)}(k)\) & 12.297 & 12.221 & 8.214 & 1.707 & 0.344 & 15.659 & 18.149 & 44.577 & 571.029 & \\  & \(\varepsilon\)(k) & 0.000 & 1.310 & 2.823 & 0.617 & 3.275 & 0.821 & 3.392 & 4.208 & 44.969 & 25.432\% \\  & \(\Delta\)(k) & 0.000\% & 12.007\% & 52.352\% & 26.543\% & 90.495\% & 5.530\% & 22.986\% & 10.425\% & 8.548\% & \\ \hline \multirow{4}{*}{0.3} & \(\hat{x}_{1}^{(0)}(k)\) & 12.297 & 12.187 & 8.458 & 2.145 & 0.652 & 16.096 & 19.199 & 47.456 & 599.211 & \\  & \(\varepsilon\)(k) & 0.000 & 1.276 & 3.067 & 0.179 & 2.967 & 1.258 & 4.442 & 7.087 & 73.151 & 25.366\% \\ \cline{1-1}  & \(\Delta\)(k) & 0.000\% & 11.696\% & 56.878\% & 7.694\% & 81.984\% & 8.475\% & 30.101\% & 17.557\% & 13.905\% & \\ \hline \multirow{4}{*}{0.4} & \(\hat{x}_{1}^{(0)}(k)\) & 12.297 & 12.145 & 8.759 & 2.728 & 1.152 & 16.922 & 21.062 & 52.422 & 647.347 & \\ \cline{1-1}  & \(\varepsilon\)(k) & 0.000 & 1.234 & 3.368 & 0.404 & 2.467 & 2.084 & 6.305 & 12.053 & 121.287 & 29.891\% \\ \cline{1-1}  & \(\Delta\)(k) & 0.000\% & 11.311\% & 62.461\% & 17.394\% & 68.168\% & 14.041\% & 42.726\% & 29.858\% & 23.056\% & \\ \hline \hline \end{tabular}
\end{table}