# Hybrid Improved PSO Algorithm for Soil Property Parameter Estimation
## 3 Improvement of the PSO Algorithm
### Comparative Test Experiment
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{
\begin{tabular}{c} **Performance** \\ **Metrics** \\ \end{tabular} } & \multirow{2}{*}{**PSO**} & \multirow{2}{*}{**SPSO**} & \multirow{2}{*}{**PSO-SO-GA**} & \multirow{2}{*}{**PSO-WOA**} & \multirow{2}{*}{**EDO**} & \multirow{2}{*}{**PSO-EDO**} \\ \hline \multirow{4}{*}{F1} & Mean & 46.6667 & 18.4343 & 1.9908 \(\times\) 10\({}^{8}\) & 779.8416 & 0 & 0 \\  & Best & 2.5343 \(\times\) 10\({}^{-7}\) & 0.0585 & 6.9623 \(\times\) 10\({}^{7}\) & 0.0537 & 0 & 0 \\  & Std & 50.7416 & 38.0227 & 4.8996 \(\times\) 10\({}^{7}\) & 869.0725 & 0 & 0 \\  & Time (s) & 0.1930 & 0.1904 & 0.2640 & 0.1445 & 0.3424 & 0.2043 \\ \hline \multirow{4}{*}{F2} & Mean & 6.7014 \(\times\) 10\({}^{34}\) & 3.3667 \(\times\) 10\({}^{32}\) & 6.9615 \(\times\) 10\({}^{42}\) & 5.92 \(\times\) 10\({}^{4}\) & 0 & 0 \\  & Best & 1.0001 \(\times\) 10\({}^{14}\) & 1887.8 & 1.0061 \(\times\) 10\({}^{34}\) & 6.46 \(\times\) 10\({}^{-10}\) & 0 & 0 \\  & Std & 2.5362 \(\times\) 10\({}^{35}\) & 1.8252 \(\times\) 10\({}^{33}\) & 2.5332 \(\times\) 10\({}^{43}\) & 3.18 \(\times\) 10\({}^{5}\) & 0 & 0 \\  & Time (s) & 0.3217 & 0.3005 & 0.3639 & 0.2461 & 0.4775 & 0.3178 \\ \hline \multirow{4}{*}{F3} & Mean & 726.5842 & 105.8248 & 2.6968 \(\times\) 10\({}^{5}\) & 1.0566 & 29 & 23.4577 \\  & Best & 16.6368 & 17.8986 & 8.9816 \(\times\) 10\({}^{4}\) & 2.03 \(\times\) 10\({}^{-4}\) & 29 & 0.1445 \\  & Std & 2.5265 \(\times\) 10\({}^{3}\) & 195.028 & 1.2390 \(\times\) 10\({}^{5}\) & 1.2549 & 0 & 5.8286 \\  & Time (s) & 1.5197 & 1.483 & 1.5197 & 1.3997 & 1.5758 & 1.4501 \\ \hline \multirow{4}{*}{F4} & Mean & 2.1070 & 0.094 & 18.3392 & 0.0744 & 8.8818 \(\times\) 10\({}^{-16}\) & 8.8818 \(\times\) 10\({}^{-16}\) \\  & Best & 3.2920 \(\times\) 10\({}^{-11}\) & 0.0013 & 16.0595 & 2.59 \(\times\) 10\({}^{-4}\) & 8.8818 \(\times\) 10\({}^{-16}\) & 8.8818 \(\times\) 10\({}^{-16}\) \\  & Std & 1.6360 & 0.3153 & 0.9442 & 0.058 & 0 & 0 \\  & Time (s) & 0.2049 & 0.2164 & 0.2867 & 0.1869 & 0.4092 & 0.2431 \\ \hline \multirow{4}{*}{F5} & Mean & 106.7421 & 45.9901 & 300.5527 & 0.6695 & 0 & 0 \\  & Best & 53.7277 & 21.8909 & 245.1119 & 7.44 \(\times\) 10\({}^{-5}\) & 0 & 0 \\  & Std & 33.7884 & 14.8105 & 27.1314 & 2.3247 & 0 & 0 \\  & Time (s) & 0.2174 & 0.2243 & 0.2872 & 0.1638 & 0.4060 & 0.2425 \\ \hline \multirow{4}{*}{F6} & Mean & 0.1170 & 0.012 & 198.0842 & 0.3537 & 0 & 0 \\  & Best & 0 & 6.83 \(\times\) 10\({}^{-5}\) & 136.4232 & 0.0105 & 0 & 0 \\  & Std & 0.1870 & 0.0121 & 44.3387 & 0.2968 & 0 & 0 \\  & Time (s) & 1.5253 & 1.413 & 1.5829 & 1.4101 & 1.6735 & 1.4597 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The optimization results of the comparative testing experiment. In the tests on unimodal functions F1 and F2, the computational time difference between the PSO and PSO-EDO was not significant. However, PSO-EDO demonstrated superior stability, with a standard deviation of 0, indicating that the algorithm consistently converges precisely to the global optimal solution in every run. For multimodal functions F4 and F5, the computational efficiency of the standard PSO algorithm was within 15.5%, faster than that of PSO-EDO. In contrast, for functions F3 and F6, the computational time of PSO-EDO was reduced by up to 4.6% compared to PSO. From a performance metric perspective, PSO-EDO significantly outperforms the standard PSO algorithm in terms of mean error, best value, and standard deviation, with its optimization results nearly attaining the theoretical optimal values. Notably, in optimization problems with highly volatile solution spaces (e.g., function F2), the performance of the standard PSO algorithm degraded substantially, exhibiting a standard deviation as high as \(2.5\times 10^{35}\). In stark contrast, PSO-EDO maintained a stable standard deviation of 0, unequivocally validating the superiority and robustness of the improved algorithm in solving complex optimization problems.