## 4 Simulation Experiment Analysis
### Wilcoxon Rank-Sum Test  
From the analysis of each experiment above, it can be concluded that the CG-TSA performed well in the benchmark test function search. To reflect the algorithm's performance more comprehensively, the CG-TSA was compared with the basic TSA and the other algorithms in terms of the optimisation-seeking effect. Wilcoxon nonparametric statistical tests were performed at the 0.05 significance level. The results of the CG-TSA run for the 10 benchmark test functions in Table 2 were selected to perform the Wilcoxon rank-sum tests and calculate the \(p\) values with the TSA after adding the Halton initialisation (HTSA), TSA after grouping improvements (FTSA), TSA after adding the cosine variation (CTSA), and the results of the GWO algorithm and SCA runs. When \(p<5\%\), it was considered a strong validation of the rejection of the null hypothesis [36]. The experimental results NaN indicated that there were no data for comparison and \(+\), \(=\), and \(-\) indicated that the CG-TSA outperformed, equalled, and underperformed the compared algorithms, respectively. The results of the Wilcoxon rank-sum tests are shown in Table 4.  
According to the results of the Wilcoxon rank-sum statistics in Table 4, except for the absence of data comparison, the Wilcoxon rank-sum test \(p\) values of the CG-TSA were less than 5%, and the performance of the CG-TSA was better than the various algorithms compared. This shows that the optimal performance advantage of the CG-TSA for basic functions was obvious. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Fun No.** & **TSA (\(p_{1}\))** & **HTSA (\(p_{2}\))** & **FTSA (\(p_{3}\))** & **CTSA (\(p_{4}\))** & **GWO (\(p_{5}\))** & **SCA (\(p_{6}\))** \\ \hline F\({}_{1}\) & \(1.34\times 10^{-16}\) & \(1.34\times 10^{-16}\) & \(3.31\times 10^{-20}\) & \(3.31\times 10^{-20}\) & \(3.31\times 10^{-20}\) & \(9.91\times 10^{-16}\) & \(7.06\times 10^{-18}\) \\ F\({}_{2}\) & \(3.31\times 10^{-20}\) & \(3.31\times 10^{-20}\) & \(3.31\times 10^{-20}\) & NaN & \(3.31\times 10^{-20}\) & \(3.31\times 10^{-20}\) \\ F\({}_{3}\) & \(4.55\times 10^{-4}\) & \(3.46\times 10^{-14}\) & \(3.06\times 10^{-10}\) & \(3.31\times 10^{-20}\) & \(6.85\times 10^{-4}\) & \(7.06\times 10^{-18}\) \\ F\({}_{4}\) & \(2.94\times 10^{-13}\) & \(7.67\times 10^{-15}\) & \(7.06\times 10^{-18}\) & \(3.31\times 10^{-20}\) & \(7.77\times 10^{-4}\) & \(7.06\times 10^{-18}\) \\ F\({}_{5}\) & \(7.55\times 10^{-6}\) & \(5.37\times 10^{-10}\) & \(7.06\times 10^{-18}\) & \(1.09\times 10^{-1}\) & \(7.50\times 10^{-18}\) & \(7.50\times 10^{-18}\) \\ F\({}_{6}\) & \(7.06\times 10^{-18}\) & \(7.06\times 10^{-18}\) & \(3.83\times 10^{-1}\) & \(2.26\times 10^{-2}\) & \(7.06\times 10^{-18}\) & \(7.06\times 10^{-18}\) \\ F\({}_{7}\) & \(3.31\times 10^{-20}\) & \(3.31\times 10^{-20}\) & \(2.80\times 10^{-11}\) & NaN & \(3.17\times 10^{-20}\) & \(3.31\times 10^{-20}\) \\ F\({}_{8}\) & \(3.31\times 10^{-20}\) & \(2.62\times 10^{-23}\) & \(3.27\times 10^{-1}\) & \(9.46\times 10^{-9}\) & \(2.97\times 10^{-20}\) & \(3.31\times 10^{-20}\) \\ F\({}_{9}\) & \(1.51\times 10^{-14}\) & \(4.07\times 10^{-15}\) & \(3.31\times 10^{-20}\) & \(3.31\times 10^{-20}\) & \(2.33\times 10^{-17}\) & \(7.96\times 10^{-18}\) \\ F\({}_{10}\) & \(7.06\times 10^{-18}\) & \(1.34\times 10^{-16}\) & \(7.06\times 10^{-18}\) & \(1.63\times 10^{-17}\) & \(8.00\times 10^{-17}\) & \(7.96\times 10^{-18}\) \\ +/=/\(-\) & 10/0/0 & 10/0/0 & 8/0/2 & 7/2/1 & 10/0/0/0 & 10/0/0 \\ \hline \end{tabular}
\end{table} 
Table 4: Wilcoxon rank-sum test results. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \multirow{2}{*}{**Fun No.