## 5 Statistical Comparison for ICA and FICA with \(\beta\) Increasing
The results obtained from the statistical test are as follows: for the FICA combination (\(\beta\)-\(\xi\)) algorithm, it fails to reject \(\mathrm{H}_{0}\) the null hypothesis and the alternative hypothesis is rejected, which means that the average performance of the FICA algorithm is not significantly different to the performance of the FCS (pa) algorithm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{2}{c}{\(\beta\)**Inc**} & \(\xi\)**Inc** & \(\beta\)**Inc-\(\xi\)**Dec** & **FCS** \\ \cline{2-5}  & **1–2** & **0–1** & **1–2 and 0–1** & **Pa** \\ \hline
**Spherical** & \(5.16\times 10^{-130}\) & \(1.06\times 10^{-164}\) & \(1.93\times 10^{-128}\) & \(2.23\times 10^{-40}\) \\
**Griewank** & \(9.02\times 10^{-2}\) & \(2.90\times 10^{-1}\) & \(1.68\times 10^{-1}\) & \(6.06\times 10^{-12}\) \\
**Rosenbrock** & \(1.68\times 10^{-1}\) & \(1.67\times 10^{-1}\) & \(2.89\times 10^{-2}\) & \(2.39\times 10^{-5}\) \\
**[PERSON]** & \(7.2952\) & \(23.8532\) & \(7.3277\) & \(11.6309\) \\
**[PERSON]** & \(4.5930\) & \(4.2932\) & \(3.5919\) & \(7.47\times 10^{-14}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparison with FICA and Fuzzy Cuckoo Search (FCS) (pa). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{**No.**} & \multicolumn{2}{c}{**F1**} & \multicolumn{2}{c}{**F2**} & \multirow{2}{*}{**Abs**} & \multirow{2}{*}{**Signal**} & \multirow{2}{*}{**Rank (-)**} & \multirow{2}{*}{**Signal**} \\ \cline{3-3} \cline{6-7}  & & **FCS (pa)** [33] & **FICA(\(\beta\)-L)** & & & & \\ \hline
**Spherical** & \(1\) & \(2.23\times 10^{-40}\) & \(1.93\times 10^{-128}\) & \(2.23\times 10^{-40}\) & \(2.23\times 10^{-40}\) & \(1\) & & \\
**Griewank** & \(3\) & \(6.06\times 10^{-12}\) & \(1.68\times 10^{-1}\) & \(-0.16876\) & \(0.1687\) & \(3\) & \(-3\) & \\
**Rosenbrock** & \(4\) & \(2.39\times 10^{-5}\) & \(2.89\times 10^{-2}\) & \(-0.0289451\) & \(0.0289451\) & \(2\) & \(-2\) & \\
**Rastrigin** & \(5\) & \(11.6309\) & \(7.3277\) & \(4.3032\) & \(4.3032\) & \(5\) & & \(5\) \\
**Ackley** & \(6\) & \(7.47\times 10^{-14}\) & \(3.5919\) & \(-3.5919\) & \(3.5919\) & \(4\) & & \(4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Parameters for the statistical test for FICA (\(\beta\)) and FCS (pa). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{**No.**} & \multicolumn{2}{c}{**F1**} & \multicolumn{2}{c}{**F2**} & \multirow{2}{*}{**Difference**} & \multirow{2}{*}{**Abs**} & \multirow{2}{*}{**(Difference)**} & \multirow{2}{*}{**Rank (-)**} & \multirow{2}{*}{**Signal**} \\ \cline{3-3} \cline{6-7}  & & **FCS (pa)** [33] & **FICA(\(\beta\)-L)** & & & & \\ \hline
**Spherical** & \(1\) & \(2.23\times 10^{-40}\) & \(1.93\times 10^{-128}\) & \(2.23\times 10^{-40}\) & \(2.23\times 10^{-40}\) & \(1\) & & \\
**Griewank** & \(3\) & \(6.06\times 10^{-12}\) & \(1.68\times 10^{-1}\) & \(-0.1687\) & \(0.1687\) & \(3\) & \(-3\) & \\
**Rosenbrock** & \(4\) & \(2.39\times 10^{-5}\) & \(2.89\times 10^{-2}\) & \(-0.0289451\) & \(0.0289451\) & \(2\) & \(-2\) & \\
**Rastrigin** & \(5\) & \(11.6309\) & \(7.3277\) & \(4.3032\) & \(4.3032\) & \(5\) & & \(5\) \\
**Ackley** & \(6\) & \(7.47\times 10^{-14}\) & \(3.5919\) & \(-3.5919\) & \(3.5919\) & \(4\) & & \(4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Parameters for the statistical test for FICA combination (\(\beta\)-\(\xi\)) and FCS (pa). rejected, which means that the average performance of the FICA combination (\(\beta\)-\(\xi\)) algorithm is not significantly different to the average performance of the FCS (pa) algorithm.