## 5 Statistical Comparison for ICA and FICA with \(\beta\) Increasing  
The statistical test was performed between the ICA algorithm and the FICA algorithm with dynamic adjustment of the \(\beta\) parameter with the range from 1 to 2, and the experiments correspond to the results obtained in tests for the ICA and FICA algorithms. Each algorithm was applied separately to each of the mathematical functions, 6 functions were used for the algorithms and 30 experiments were realized. Table 7 contains the average and standard deviation obtained after 30 executions for algorithm with 1000 decades. The values of parameters utilized in the fuzzy imperialist competitive algorithm and the ICA Algorithm [24; 31] are the following:  
* No. of Countries: 200. * No. of Imperialists: 10. * Revolution Rate: 0.2. * \(\beta\): 1.4. * \(\xi\): 0.02. * Dimensions: 30. * Simulations: 30. * Decades: 1000. Table 7 shows the performance of the FICA algorithm with dynamic adjustment of the \(\beta\) parameter and the ICA algorithm for 30 times we find the mean and standard deviation results. The statistical test used for the comparison is the Wilcoxon test [32] that is commonly utilized to analyze data, where its parameters are shown in Table 8.  
The null hypothesis tells us that the average of the results obtained by the fuzzy imperialist competitive algorithm is equal to the average of the ICA algorithm, and the alternative hypothesis states that the average of the results obtained by the fuzzy imperialist competitive algorithm is different to the average performance of the ICA algorithm. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{2}{c}{**ICA**} & \multicolumn{2}{c}{**FICA**} \\ \hline \multirow{2}{*}{**Sphere**} & **Mean** & \(2.51\times 10^{-21}\) & \(2.27\times 10^{-25}\) \\  & **S.D.** & \(1.37\times 10^{-20}\) & \(4.17\times 10^{-25}\) \\ \hline \multirow{2}{*}{**Quartic**} & **Mean** & \(9.75\times 10^{-41}\) & \(2.96\times 10^{-39}\) \\  & **S.D.** & \(5.34\times 10^{-40}\) & \(1.58\times 10^{-38}\) \\ \hline \multirow{2}{*}{**Griewank**} & **Mean** & \(0.3591025\) & \(0.5033387\) \\  & **S.D.** & \(0.3903724\) & \(0.3845768\) \\ \hline \multirow{2}{*}{**Rosenbrock**} & **Mean** & \(18.32843\) & \(17.302077\) \\  & **S.D.** & \(5.6853778\) & \(5.9223444\) \\ \hline \multirow{2}{*}{**Rastrigin**} & **Mean** & \(131.0165\) & \(95.81005\) \\  & **S.D.** & \(22.167742\) & \(29.2925518\) \\ \hline \multirow{2}{*}{**Ackley**} & **Mean** & \(5.0099715\) & \(4.6910324\) \\  & **S.D.** & \(2.2992776\) & \(1.8764895\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison with ICA and FICA with increasing \(\beta\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{**No.**} & **F1** & **F2** & **Abs** & **Signed** & **Signed** \\  & & **ICA** & **FICA(\(\beta\))** & **Difference** & **(Difference)** & **Rank** & **Rank (\(-\))** & **Rank (\(+\))** \\ \hline \multirow{2}{*}{**Spherical**} & 1 & \(2.51\times 10^{-21}\) & \(2.27\times 10^{-25}\) & \(2.51\times 10^{-21}\) & \(2.51\times 10^{-21}\) & \(2\) & \(2\) \\  & 2 & \(9.75\times 10^{-41}\) & \(2.96\times 10^{-39}\) & \(-2.86\times 10^{-39}\) & \(2.86\times 10^{-39}\) & \(1\) & \(1\) & \\  & 3 & \(0.3591025\) & \(0.50333873\) & \(-0.14423619\) & \(0.14423619\) & \(3\) & \(3\) & \\  & 4 & \(18.3284301\) & \(17.3020772\) & \(1.02635282\) & \(1.02635282\) & & \(5\) \\  & 5 & \(131.016497\) & \(95.8100497\) & \(35.2064469\) & \(35.2064469\) & \(6\) & \(6\) \\  & 6 & \(5.00997147\) & \(4.6910324\) & \(0.31893908\) & \(0.31893908\) & \(4\) & \(4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Parameters for the statistical test. To test the hypothesis, the absolute values \(\left|Z_{i}\right|\dots\left|Z_{n}\right|\) are ordered and assigned its range on Rank, the order of these values are from lowest to highest, Sign (+) column indicates the values are positive and Sign (-) column indicates the values are negative.