## 4 Experimental Results and Analysis  
To validate the correctness of the variable-weight combination model proposed in this paper, real measurement data from an inclined observation line at a coal mine were taken as an example. The study area and the layout of the mining face and the designed observation station positions are illustrated in Figure 12. The average mining depth was 145 m, with a coal seam thickness of approximately 4.5 m, and a coal seam dip ranging from 0 to 3 degrees. The observation line C was located within the secondary mining area, consisting of 22 observation points spaced at 15 m intervals, labeled as C01, C02, and so on up to C22. For the research, we selected point C22, situated within the subsidence basin, as the focal point. The TRF algorithm mentioned above was applied to fit real measurement data using four different time function models: Usher, Weibull, Richards, and MMF. The fitting curves for each model alongside the actual subsidence curve can be seen in Figure 13. The resulting equations for the four time function models are as follows:Usher time function model:  
\[W(t)=\frac{-3563.18}{(1+11.45\times e^{-0.58t})^{3.82}} \tag{27}\]  
Weibull time function model:  
\[W(t)=-3532.77\times(1-e^{-6.37\times 10^{-4}t^{3.56}}) \tag{28}\]  
[PERSON] time function model:  
\[W(t)=-3573.99\times\left(1-3.15\times 10^{-3}e^{-0.52t}\right)^{\frac{1}{1.2 \times 10^{-4}}} \tag{29}\]  
MMF time function model:  
\[W(t)=-3587.56\times\frac{t^{5.26}}{2.82\times 10^{4}+t^{5.26}} \tag{30}\]  
To verify the accuracy of the TRF algorithm in the parameter optimization process, this study compared it to three other algorithms: genetic algorithm (GA) [60], differential evolution algorithm (DE) [61], and particle swarm optimization algorithm (PSO) [62]. These comparisons were carried out 50 times for each of the [PERSON], [PERSON], Richards, and MMF time function models. The accuracy of the algorithms was evaluated by comparing the mean, maximum, and minimum values of the root mean square error (RMSE) between the predicted values and the actual measurements obtained from 50 runs of each algorithm. The results of this evaluation are presented in Figure 14.  
Figure 12: **(a)** Location of Yulin City in China; (b) location of Shenmu City in Yulin City; (c) location of the working face and the observation line layout. Figure 14: (**a**) Comparison of RMSE minimum values; (**b**) comparison of RMSE maximum values; (**c**) comparison of RMSE mean values. Figure 13: Comparison of predicted and measured subsidence curves of four prediction models: (**a**) comparison of predicted and measured subsidence curves of [PERSON] time function model; (**b**) comparison of predicted and measured subsidence curves of Weibull time function model; (**c**) comparison of predicted and measured subsidence curves of [PERSON] time function model; (**d**) comparison of predicted and measured subsidence curves of MMF time function model. From Figure 14, it can be observed that among the four algorithms, the average, maximum, and minimum values of the root mean square error (RMSE) obtained via the TRF algorithm were superior to the other three algorithms. This indicated that using the TRF algorithm for parameter estimation resulted in higher accuracy and greater stability. Compared to the TRF algorithm, the maximum RMSE values obtained via the PSO algorithm were consistently greater than those obtained via the TRF algorithm. This indicated that the PSO algorithm exhibited lower stability in contrast to the TRF algorithm. For the [PERSON] time function model with poor fitting accuracy, both the DE and GA algorithms exhibited significant fluctuations in the calculation of RMSE. This was primarily due to their inherent vulnerability to local optima and slower convergence. In particular, the DE algorithm yielded a maximum RMSE value of 348.65, while the TRF algorithm demonstrated higher reliability.