## 3 Results
### Results of Computational Experiments
Table 9 contains the results of pairwise Mann-Whitney statistical tests applied to mQSO with two population sizes and mQSDE. The statistical tests were performed with a significance level \(p=0.01\), with normal approximation and tie-breaking. The values in Table 3 are the test result (Z-score), where the test result is either -1 (worse), 0 (equal), or 1 (better). For example, if the absolute value of Z was smaller than 2.58, the test result was 0. With the same population size, the mQSDE algorithms were able to significantly outperform the mQSO in the third setting, i.e., when the number of components is increased to 25. Comparing with mQSO\({}_{29}\), the mQSDE\({}_{21,0.1,p}\) always performs better, except for the fourth setting, although these improvements are not significant according to the Mann-Whitney test. Figure 2 shows the current error graphs of mQSO\({}_{29}\) and mQSDDE\({}_{21,0.1,p}\) algorithms for four considered settings, and Figure 3 shows the offline error graphs. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**Setting 1**} & \multicolumn{2}{c}{**Setting 2**} \\  & \(E_{\text{BBC}}\) & \(E_{O}\) & \(E_{\text{BBC}}\) & \(E_{O}\) \\ \hline mQSDDE\({}_{21,0.1,p}^{0.3,0.1}\) & \(8.70\pm 1.70\) & \(12.38\pm 1.79\) & \(12.39\pm 1.52\) & \(17.95\pm 1.66\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.2}\) & \(8.35\pm 1.26\) & \(12.00\pm 1.36\) & \(12.41\pm 1.38\) & \(18.05\pm 1.69\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.3}\) & \(8.29\pm 1.40\) & \(11.94\pm 1.44\) & \(12.38\pm 1.24\) & \(18.06\pm 1.47\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.4}\) & \(8.63\pm 1.77\) & \(12.26\pm 1.89\) & \(12.37\pm 1.42\) & \(18.02\pm 1.48\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.5}\) & \(8.49\pm 1.21\) & \(12.14\pm 1.25\) & \(12.17\pm 1.69\) & \(17.71\pm 1.76\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.6}\) & \(8.25\pm 1.10\) & \(11.85\pm 1.11\) & \(12.06\pm 1.18\) & \(17.69\pm 1.35\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.7}\) & \(8.56\pm 1.34\) & \(12.24\pm 1.37\) & \(\mathbf{12.01\pm 1.24}\) & \(\mathbf{17.52\pm 1.31}\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.8}\) & \(8.46\pm 1.51\) & \(12.19\pm 1.56\) & \(12.59\pm 1.84\) & \(18.25\pm 2.01\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.9}\) & \(8.50\pm 1.11\) & \(12.08\pm 1.31\) & \(12.59\pm 1.74\) & \(18.31\pm 1.87\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,1.0}\) & \(\mathbf{8.10\pm 0.96}\) & \(\mathbf{11.80\pm 1.02}\) & \(12.19\pm 1.24\) & \(17.87\pm 1.25\) \\ \hline \multirow{2}{*}{Algorithm} & \multicolumn{2}{c}{Setting 3} & \multicolumn{2}{c}{Setting 4} \\  & \(E_{\text{BBC}}\) & \(E_{O}\) & \(E_{\text{BBC}}\) & \(E_{O}\) \\ \hline mQSDDE\({}_{21,0.1,p}^{0.3,0.1}\) & \(9.11\pm 0.85\) & \(12.46\pm 0.83\) & \(\mathbf{13.16\pm 1.33}\) & \(17.66\pm 2.01\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.2}\) & \(9.30\pm 1.08\) & \(12.64\pm 1.11\) & \(13.17\pm 1.75\) & \(\mathbf{17.25\pm 1.98}\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.5}\) & \(9.18\pm 0.87\) & \(12.51\pm 0.89\) & \(13.84\pm 1.53\) & \(18.32\pm 2.03\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.4}\) & \(9.28\pm 1.40\) & \(12.62\pm 1.49\) & \(13.24\pm 1.56\) & \(17.67\pm 1.86\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.5}\) & \(\mathbf{8.85\pm 1.14}\) & \(\mathbf{12.13\pm 1.18}\) & \(13.55\pm 1.81\) & \(17.82\pm 1.87\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.6}\) & \(9.00\pm 1.08\) & \(12.43\pm 1.22\) & \(13.53\pm 1.36\) & \(17.95\pm 1.48\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.7}\) & \(9.50\pm 1.42\) & \(12.93\pm 1.54\) & \(13.51\pm 1.51\) & \(17.69\pm 1.85\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.8}\) & \(9.21\pm 0.95\) & \(12.59\pm 1.04\) & \(13.57\pm 1.58\) & \(18.32\pm 2.14\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,0.9}\) & \(9.30\pm 1.11\) & \(12.65\pm 1.18\) & \(13.83\pm 1.31\) & \(18.15\pm 1.81\) \\ mQSDDE\({}_{21,0.1,p}^{0.3,1.0}\) & \(9.12\pm 1.04\) & \(12.54\pm 1.12\) & \(13.70\pm 1.61\) & \(18.21\pm 1.79\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of mQSDE with different crossover rates.