# A Modified Artificial Bee Colony Algorithm Based on the Self-Learning Mechanism
## 4 Experiments and Results
### Comparison Regarding the t-Test  
This section mainly analyzes the experimental data and study whether the experimental results are statistically significantly different between SLABC and other algorithm. The F-test was used to analyze the homogeneity of variances and two-tailed t-test was used to determine if two sets of data are significantly different from each other. The sample size and number of degrees of freedom were set as 20 and 38, respectively. The confidence level was set to 95%. In the F-test, \(F_{0.05}(19,19)=2.17\) and if the F-value is larger than 2.17, the corresponding experimental results are heterogeneity of variance. If the p-value received in the t-test is less than 5%, it illustrates that the corresponding experimental results are significantly different. Table 5 lists the results of t-tests between SLABC and the best results of the other algorithms regarding the indexes \"Mean\" for different benchmark functions (listed are the F-value, p-value, and the significance of the results). \"YES\" indicates that the experimental results are significantly different between SLABC and the best one in other algorithms. \"NO\" suggests that no significant difference between the results of SLABC and of other algorithms.  
From Tables 2-5, although other algorithms (distABC) have higher solution accuracy on function \(f4\) with \(D=30,60,100\), there exist no significant difference between SLABC and distABC. The same as function \(f6\) with \(D=100\) and function \(f8\) with \(D=60\), there exist no significant difference between SLABC and other algorithm. Moreover, SLABC and other algorithm obtain the same optimal solution on functions \(f5,f6,f8\) with \(D=30\).  
Based on the results of statistical tests in Table 5, almost most of the results have obvious differences and it is clear that the results of SLABC algorithm are significantly better than the results of other algorithms.  
Figure 7: The success ratio \(Srat_{k}\) of each SSE. The horizontal ordinate denotes success ratio of each solution search equation (operator) at the midpoint of the experiments (\(Max.FE/2\)) and the end of the experiments (\(Max.FE\)) with 100 dimensions. (\(f_{1}\)) Sphere Function; (\(f_{2}\)) Quartic Function; (\(f_{3}\)) Schwefel’s Problem 2.22; (\(f_{4}\)) Rosenbrock Function; (\(f_{5}\)) Rastrigin Function; (\(f_{6}\)) Griewank Function; (\(f_{7}\)) Ackley Function; (\(f_{8}\)) Schwefel’s Problem 2.26; (\(f_{9}\)) Penalized Function.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Functions** & **Dim.** & **F-Value** & \(p\)**-Value** & **Significance** & **Two-Tailed P** \\ \hline \multirow{3}{*}{\(f_{1}\)} & D = 30 & 39.930 & \(1.10\times 10^{-4}\) & YES & 0.05 \\  & D = 60 & 29.663 & \(2.80\times 10^{-5}\) & YES & 0.05 \\  & D = 100 & 60.363 & \(1.74\times 10^{-8}\) & YES & 0.05 \\ \hline \multirow{3}{*}{\(f_{2}\)} & D = 30 & 14.894 & 0.022 & YES & 0.05 \\  & D = 60 & 20.124 & \(1.42\times 10^{-3}\) & YES & 0.05 \\ \cline{1-1}  & D = 100 & 15.007 & \(4.00\times 10^{-5}\) & YES & 0.05 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Results of \(t\)-tests between SLABC and the best results of the other algorithms regarding the indexes “Mean” for different benchmark functions.