## 4 Engineering Examples and Results of Statistical Analyses
### Results and Analyses  
There are 177 observation points available in working face 110801, including 96 observation points on the strike observation line and 81 observation points on the inclination observation line. The subsidence values of the 177 observation points are obtained as the initial data, 5% of the observation points are randomly selected to join the 100 mm gross error, and the probability integral parameter inversion is performed by using the BFGS  
Figure 7: Schematic diagram of 110801 working face. algorithm and the IGGIII-BFGS algorithm, respectively. The two algorithms were inverted 15 times, respectively, and the mean value was taken as the final inversion parameter. The subsidence value was predicted according to the obtained parameter, and the predicted curve was plotted. The results are shown in Figure 8, and the results of the parameter inversion and the root mean square error are shown in Table 3.  
According to Figure 8, it can be seen that the trends of the subsidence curves obtained by the two methods are basically the same as the measured values, but there is an obvious difference in the predicted curves in the area of maximum subsidence of the subsiding basin. The expected curves of the IGGIII-BFGS method in the area of maximum subsidence of the subsiding basin are more closely matched with the measured values, while the expected curves of the BFGS method are shifted upward significantly, and the expected curves of both methods at the inflection point are shifted to a smaller extent from the measured values. The predicted curves of the two methods at the inflection point also have a small degree of deviation from the measured values. Comparing the absolute value of the errors between the predicted values and observed values of the two methods, it can be seen that the absolute value of the errors of the BFGS method are between 0 and 300 mm, and the absolute value of the errors of the IGGIII-BFGS method are between 0 and 200 mm, the errors of the BFGS method are larger than that of the IGGIII-BFGS method in most cases, and the errors of both methods are larger near the maximum subsidence value and the inflection point. According to Table 3, it can be seen that the parameter inversion results of the two methods are still different, and the difference is bigger compared with the simulation results, in which the difference between the inversion results of q, tan \(\beta\), and \(\theta_{0}\) is relatively small; the difference between the inversion results of the four parameters of s\({}_{1}\), s\({}_{2}\), s\({}_{3}\), and s\({}_{4}\) is bigger; and the root mean square error of the BFGS method is 109.482 mm and that of the IGGIII-BFGS method is 81.278 mm. However, the root mean square error of the IGGIII-BFGS method is still smaller than that of the BFGS method, and it can be seen that the predicted results of the IGGIII-BFGS are more accurate. Figure 8: Surface subsidence predicted curve. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Method** & **q** & **tan \(\beta\)** & **\(\theta_{0}\)** & **s\({}_{1}\)** & **s\({}_{2}\)** & **s\({}_{3}\)** & **s\({}_{4}\)** & **RMSE/mm** \\ \hline BFGS & 1.530 & 1.665 & 93.825 & 5.023 & 0.235 & 62.479 & 59.481 & 109.482 \\ IGGIII-BFGS & 1.513 & 1.808 & 93.825 & 15.809 & 4.605 & 71.095 & 72.479 & 81.278 \\ \hline \end{tabular}
\end{table} 
Table 3: Comparison of the predicted results of the parameters of the two methods. Overall, due to the complexity of the measured data in the actual engineering, which leads to its incomplete conformity to the probability integral prediction curve, compared with the simulation experiments, the accuracy of the prediction of the inversion parameters by both methods is reduced. However, the IGGIII-BFGS algorithm demonstrates superior practical performance. The RMSE decreases from 109.482 mm (BFGS) to 81.278 mm (IGGIII-BFGS), representing a 25.8% improvement. It is proved that the prediction results obtained by using the IGGIII-BFGS method have good validity and reliability.