# Search Patterns Based on Trajectories Extracted from the Response of Second-Order Systems
## 7 Experimental Results
### Multimodal Functions
To statistically validate the conclusions from Table 1, a non-parametric study is considered. In this test, the Wilcoxon rank-sum analysis [35] is adopted with the objective to validate the performance results. This statistical test evaluates if exists a significant difference when two methods are compared. For this reason, the analysis is performed considering a pairwise comparison such as SOA versus ABC, SOA versus CMAES, SOA versus CSA, SOA versus DE, SOA versus MFO and SOA versus PSO. In the Wilcoxon analysis, a null hypothesis (\(H_{0}\)) was adopted that showed that there is no significant difference in the results. On the other hand, it is assumed as an alternative hypothesis (\(H_{1}\)) that the result has a similar structure. For the Wilcoxon analysis, it is assumed a significance value of 0.05 considering 30 independent execution for each test function. Table 2 shows the \(p\)-values assuming the results of Table 2 (where \(n=30\)) produced by the Wilcoxon study. For faster visualization, in the Table, we use the following symbols \(\blacktriangle\)\(\blacktriangledown\), and \(\blacktriangleright\). The symbol \(\blacktriangle\) refers that the SOA algorithm produces significantly better solutions than its competitor. \(\blacktriangledown\) symbolizes that SOA obtains worse results than its counterpart. Finally, the symbol \(\blacktriangleright\) denotes that both compared methods produce similar solutions. A close inspection of Table 2 demonstrates that for functions \(f_{1}\), \(f_{4}\), \(f_{5}\), \(f_{6}\), \(f_{8}\), \(f_{9}\), \(f_{10}\), \(f_{11}\) and \(f_{12}\) the proposed SOA scheme obtain better solutions than the other methods. On the other hand, for functions \(f_{2}\), \(f_{2}\) and \(f_{7}\), it is clear that the groups SOA versus ABC, SOA versus CMAES, SOA versus DE and SOA versus MFO and EA-HC versus SCA present similar solutions. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & **ABC** & **DE** & **CMAES** & **CSA** & **PSO** & **MFO** & **SOA** \\ \hline \multirow{3}{*}{\(f_{10}(\mathbf{x})\)} & **AB** & \(110,282.54\) & \(665,278.86\) & \(-4930\) & \(1,170,939.0\) & \(45,556.260\) & \(222,833.73\) & \(-501.79356\) \\  & **MD** & \(96,461.061\) & \(673,449.49\) & \(-4930\) & \(1,126,234.2\) & \(5076.8152\) & \(71,582.051\) & \(-332.82466\) \\  & **SD** & \(44,933.417\) & \(129,147.27\) & \(3,7318\times 10^{-9}\) & \(159,175.64\) & \(75,990.880\) & \(305,159.37\) & \(663.92006\) \\ \hline \multirow{3}{*}{\(f_{11}(\mathbf{x})\)} & **AB** & \(-18.26109\) & \(-26.056561\) & \(-29.6576\) & \(-16.504756\) & \(-28.367666\) & \(-28.863589\) & \(-30\) \\  & **MD** & \(-18.131984\) & \(-26.092349\) & \(-29.9286\) & \(-16.183447\) & \(-28.14029\) & \(-29.070145\) & \(-30\) \\  & **SD** & \(1.6366873\) & \(0.5428906\) & \(0.466\) & \(1.16917142\) & \(1.5408536\) & \(1.2837415\) & \(0\) \\ \hline \multirow{3}{*}{\(f_{12}(\mathbf{x})\)} & **AB** & \(1502.3129\) & \(369.60375\) & \(786.36819\) & \(519.17242\) & \(196.95838\) & \(261.52332\) & \(11.905761\) \\  & **MD** & \(1457.6865\) & \(368.82916\) & \(778.72369\) & \(465.93238\) & \(213.00730\) & \(252.76747\) & \(0.3841574\) \\ \cline{1-1}  & **SD** & \(420.70611\) & \(35.389136\) & \(215.93893\) & \(228.69860\) & \(86.542862\) & \(106.52353\) & \(29.544101\) \\ \hline \hline \end{tabular}
\end{table}