# A Novel Binary QUasi-Affine TRansformation Evolutionary (QUATRE) Algorithm
## 4 Experimental Results and Analysis
### Benchmark Function
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**BDE**} & \multicolumn{2}{c}{**ABGWO-V3a**} & \multicolumn{2}{c}{**BQUATE1-S2**} & \multicolumn{2}{c}{**BQUATE1-V2**} & \multicolumn{2}{c}{**BQUATE2-S2i**} & \multicolumn{2}{c}{**BQUATE2-V2i**} & \multicolumn{2}{c}{**BPSO**} \\ \cline{2-13}
**Function** & \multicolumn{2}{c}{**AVG**} & \multicolumn{2}{c}{**STD**} & \multicolumn{2}{c}{**AVG**} & \multicolumn{2}{c}{**STD**} & \multicolumn{2}{c}{**AVG**} & \multicolumn{2}{c}{**STD**} & \multicolumn{2}{c}{**AVG**} & \multicolumn{2}{c}{**STD**} & \multicolumn{2}{c}{**AVG**} & \multicolumn{2}{c}{**STD**} & \multicolumn{2}{c}{**AVG**} & \multicolumn{2}{c}{**STD**} \\ \hline \(f1\) & 3.7333 & 1.8742 & 3.4303 & 1.3047 & 3.8333 & 0.5307 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 7.8333 & 1.176  
Figure 8: The optimal values of BDE, ABGWO-V3a, BQUATE1-S2, BQUATE1-V2, BQUATE2-S2i, BQUATE2-V2i, and BPSO on f14-f23. Figure 6: The optimal values of BDE, ABGWO-V3a, BQUATE1-S2, BQUATE1-V2, BQUATE2-S2i, BQUATE2-V2i, and BPSO on f1-f7. Figure 7: The optimal values of BDE, ABGWO-V3a, BQUATE1-S2, BQUATE1-V2, BQUATE2-S2i, BQUATE2-V2i, and BPSO on f8-f13. Moreover, the U-shaped and Z-shaped families of transfer functions are compared to BDE, ABGWO-V3a, and BPSO, that is the first approach of BQUATRE algorithm with \(U_{2}\) transfer function (record as BQUATRE1-U2), the first approach of BQUATRE algorithm with \(Z_{2}\) transfer function (record as BQUATRE1-Z2), the second approach of BQUATRE algorithm with the improved transfer function \(U_{2i}\) (record as BQUATRE2-U2i) and the second approach of BQUATRE algorithm with the improved transfer function \(Z_{2i}\) (record as BQUATRE2-Z2i). The results are shown in Table 14.  
From Table 14, it can be seen that the seven algorithms get the same solutions on six benchmark functions. However, BQUATE1-U2 gets the best results on other 12 functions. Furthermore, BQUATRE1-Z2, BQUATRE2-U2i and BQUATRE2-Z2i obtained 16, 15 and 7 red results, respectively. However, BDE, ABGWO-V3a, and BPSO only obtained 6, 3 and 1 red results, respectively. On the whole, BQUATRE1-U2, BQUATRE1-Z2, BQUATRE2-U2i and BQUATRE2-Z2i are superior to BDE, ABGWO-V3a, and BPSO. From Tables 13 and 14, we can conclude that V-shaped, U-shaped and Z-shaped functions in BQUATRE1 and BQUATRE2 are superior to BDE, ABGWO-V3a, and BPSO. The BQUATRE2 with S-shaped transfer function can also perform well. The proposed second approach of BQUATRE can perform well on the unimodal functions and common multimodal functions. However, at the same time, we can see the improved algorithm does not improve much on multimodal functions in low dimensions. Figure 9-11 are the visualization results of Table 14.  
To further compare the results, t-test is used for a significance test. The t-test is used to compare the mean value of two groups of data. It is suitable for a normal distribution with a small sample size and unknown population standard deviation. In the experiments, a two tailed t-test is used with significant level of 0.01, which means very significant. Table 15 is shows the results between ABGWO-3 Va and QUATRE1-S2, QUATRE1-V2, QUATRE1-U2, QUATRE1-Z2, QUATRE2-S2i, QUATRE2-V2i, QUATRE2-U2i, QUATRE2-Z2i. \"+\" appears that the compared algorithm is superior to ABGWO-3 Va, and \"-\" indicates that the algorithm is inferior to ABGWO-3 Va. \"=\" implies that the performance of the algorithms is consistent.