## 3 Numerical Experiments and Analysis
### Nonparametric Tests  
Conducting a thorough analysis of the algorithm data is also a crucial part of the conclusion [20]. Nonparametric tests can provide significant results when disparities exist among the algorithms. In this paper, the [PERSON] test [21; 22] is used to rank the algorithms. The top performance is assigned a rank of 1, the second-best one is given a rank of 2, and this ranking method continues accordingly [23]. The results of [PERSON]'s test calculations for the four algorithms are presented in Table 4. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{2}{c}{**WaOA**} & \multicolumn{2}{c}{**SCSO**} & \multicolumn{2}{c}{**ALO**} & \multicolumn{2}{c}{**SOA**} \\  & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** \\ \hline F1 & \(5.50\times 10^{-283}\) & 0.00 & \(1.49\times 10^{-113}\) & \(4.61\times 10^{-113}\) & \(1.08\times 10^{-3}\) & \(7.18\times 10^{-4}\) & \(1.70\times 10^{-11}\) & \(3.42\times 10^{-11}\) \\ F2 & \(3.89\times 10^{-144}\) & \(1.36\times 10^{-143}\) & \(8.07\times 10^{-61}\) & \(2.86\times 10^{-61}\) & \(5.64\times 10^{-1}\) & \(4.81\times 10^{1}\) & \(2.38\times 10^{-8}\) & \(3.19\times 10^{-8}\) \\ F3 & \(7.22\times 10^{-215}\) & 0.00 & \(1.84\times 10^{-100}\) & \(4.47\times 10^{-100}\) & \(4.93\times 10^{3}\) & \(2.13\times 10^{3}\) & \(2.00\times 10^{-5}\) & \(2.85\times 10^{-5}\) \\ F4 & \(3.40\times 10^{-135}\) & \(1.10\times 10^{-134}\) & \(7.31\times 10^{-52}\) & \(2.62\times 10^{-51}\) & \(1.71\times 10^{1}\) & \(5.10\times 10^{0}\) & \(5.37\times 10^{-3}\) & \(1.55\times 10^{-2}\) \\ F5 & 0.00 & 0.00 & 28.2 & 0.70 & \(4.62\times 10^{2}\) & \(4.83\times 10^{2}\) & \(2.84\times 10^{1}\) & \(4.2\times 10^{-1}\) \\ F6 & 0.00 & 0.00 & 20.4 & 0.56 & \(2.26\times 10^{-3}\) & \(1.93\times 10^{-3}\) & \(3.18\times 10^{0}\) & \(3.66\times 10^{-5}\) \\ F7 & \(5.04\times 10^{-5}\) & \(3.97\times 10^{-5}\) & \(2.34\times 10^{-4}\) & \(2.25\times 10^{-4}\) & \(2.47\times 10^{-1}\) & \(8.50\times 10^{-2}\) & \(3.04\times 10^{-3}\) & \(1.50\times 10^{-3}\) \\ F8 & \(-8.57\times 10^{3}\) & \(6.04\times 10^{2}\) & \(-6.31\times 10^{3}\) & \(1.08\times 10^{3}\) & \(-5.46\times 10^{3}\) & \(5.92\times 10^{1}\) & \(-5.02\times 10^{3}\) & \(3.50\times 10^{2}\) \\ F9 & 0.00 & 0.00 & 0.00 & 0.00 & \(8.15\times 10^{1}\) & \(2.17\times 10^{1}\) & \(2.94\times 10^{1}\) & \(4.58\times 10^{0}\) \\ F10 & \(2.22\times 10^{-15}\) & \(1.78\times 10^{-15}\) & \(4.44\times 10^{-16}\) & \(1.97\times 10^{-31}\) & \(6.03\times 10^{0}\) & \(2.72\times 10^{0}\) & \(2.00\times 10^{1}\) & \(1.50\times 10^{-3}\) \\ F11 & 0.00 & 0.00 & 0.00 & 0.00 & \(5.24\times 10^{-2}\) & \(3.02\times 10^{-2}\) & \(2.33\times 10^{-2}\) & \(3.80\times 10^{-2}\) \\ F12 & \(1.57\times 10^{-32}\) & \(5.47\times 10^{-48}\) & \(9.83\times 10^{-2}\) & \(4.61\times 10^{-2}\) & \(1.49\times 10^{1}\) & \(6.96\times 10^{0}\) & \(3.53\times 10^{-1}\) & \(1.79\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance comparison of four algorithms. It can be seen from Table 4 that the WaOA ranks first among the four algorithms, and the Freidman rank is 1.16, which suggests that it has excellent comprehensive performance. When analyzing the performance of the algorithm, we apply the Wilcoxon rank sum test method and set the significance level \(p\) at 0.05. Under this setting, when the \(p\)-value is below 0.05, it shows that the WaOA differs significantly from other algorithms. On the contrary, it is not significant [24]. Table 5 lists the \(p\)-values of the WaOA in relation to the other three algorithms. As shown in Table 5, the \(p\)-value of the majority of the algorithms is below 0.05 compared to the WaOA, regardless of unimodal or multimodal functions. It indicates that the WaOA has more obvious advantages. In functions F9 and F11, the \(p\)-values of the WaOA relative to the SCSO are both 1.