# Hybrid Improved PSO Algorithm for Soil Property Parameter Estimation
## 4 Simulation Results and Discussion  
### Simulation Results  
Based on the established soil property parameter estimation model, simulation experiments were conducted using MATLAB R2020a version. The simulation data were obtained from [24], and the true values of the corresponding vehicle parameters and soil properties are listed in Table 6. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Iteration** & **F1** & **F2** & **F3** & **F4** & **F5** & **F6** \\ \hline
1 & 6.39 \(\times\) 10\({}^{8}\) & 4.58 \(\times\) 10\({}^{51}\) & 2.93 \(\times\) 10\({}^{6}\) & 20.67 & 443.87 & 610.03 \\ \hline
2 & 1.18 \(\times\) 10\({}^{8}\) & 1.74 \(\times\) 10\({}^{47}\) & 3.31 \(\times\) 10\({}^{5}\) & 7.84 & 123.95 & 305.93 \\ \hline
3 & 1.72 \(\times\) 10\({}^{7}\) & 2.09 \(\times\) 10\({}^{45}\) & 9.03 \(\times\) 10\({}^{4}\) & 4.24 & 26.41 & 128.03 \\ \hline
4 & 8471.73 & 3.02 \(\times\) 10\({}^{39}\) & 1007.56 & 2.01 & 1.65 & 115.90 \\ \hline
5 & 362.42 & 9.04 \(\times\) 10\({}^{30}\) & 32.21 & 1.32 & 0.02 & 75.45 \\ \hline
6 & 13.47 & 3.49 \(\times\) 10\({}^{27}\) & 23.72 & 0.15 & 0.00 & 46.28 \\ \hline
7 & 0.03 & 2.23 \(\times\) 10\({}^{20}\) & 24.16 & 0.00 & 0.00 & 40.38 \\ \hline
8 & 0.00 & 7.92 \(\times\) 10\({}^{8}\) & 23.86 & 0.00 & 0.00 & 45.85 \\ \hline
9 & 0.00 & 0.00 & 24.01 & 0.00 & 0.00 & 38.42 \\ \hline
10 & 0.00 & 0.00 & 24.00 & 0.00 & 0.00 & 0.23 \\ \hline
11 & 0.00 & 0.00 & 23.73 & 0.00 & 0.00 & 0.00 \\ \hline
12 & 0.00 & 0.00 & 23.76 & 0.00 & 0.00 & 0.00 \\ \hline
13 & 0.00 & 0.00 & 23.57 & 0.00 & 0.00 & 0.00 \\ \hline
14 & 0.00 & 0.00 & 23.71 & 0.00 & 0.00 & 0.18 \\ \hline
15 & 0.00 & 0.00 & 23.87 & 0.00 & 0.00 & 0.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The mean F1–F6 fitness evolution curves by using PSO-EDO in 30 independent runs. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Comparison Groups** & **Test Statistic** & \(p\)**-Value** & **Significance Conclusion** \\ \hline PSO-EDO vs. PSO & 0.0 & \textless{}0.001 (\(\approx\)3.33 \(\times\) 10\({}^{-165}\)) & Significant difference \\ \hline PSO-EDO vs. SPSO & 271.0 & \textless{}0.001 (\(\approx\)7.50 \(\times\) 10\({}^{-165}\)) & Significant difference \\ \hline PSO-EDO vs. PSO-GA & 226,060.0 & \textless{}0.01 (\(\approx\)0.0081) & Significant difference \\ \hline PSO-EDO vs. PSO-WOA & 15,510.0 & \textless{}0.001(\(\approx\)1.32 \(\times\) 10\({}^{-145}\)) & Significant difference \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Wilcoxon signed-rank test results for fitness comparison. Among the four types of PSO algorithms, the PSO-WOA algorithm achieved better optimization results, and the performance of the EDO algorithm was similar to that of the improved algorithm. Therefore, PSO-WOA, EDO, and PSO-EDO algorithms were selected for simulation comparison. Table 7 presents the simulation results of the improved algorithm, with a maximum of 1000 iterations and 30 independent runs. Global Best refers to the set of values with the lowest fitness among the 30 optimization results, Mean Best represents the mean of the optimal values (0) across the 30 runs, and Mean Error denotes the error rate between the mean value and the true value. As shown in the table, after 30 simulations, the parameter estimation results for the three algorithms are identical, with no significant differences. The only difference lies in the average computation time: PSO-EDO has the shortest computation time, followed by the EDO algorithm, with the PSO-WOA algorithm taking the longest. Figure 5 shows the average fitness over the first 30 iterations for the simulation results of the three algorithms. As observed in Figure 5, both the PSO-WOA and PSO-EDO algorithms converge within the first 10 iterations, while the EDO algorithm takes until the 15 th iteration to complete convergence. This indicates that the EDO algorithm has a slower convergence speed during the simulation compared to the other two algorithms.