# An Improved Squirrel Search Algorithm for Global Function Optimization
## 3 The improved Squirrel Search Algorithm
### The Jumping Search Method
#### 3.2.3 Characters of the Jumping Search Method
A Holm test was carried out and the results are shown in Table 10:

It can be seen from Table 10 that \(P_{1}<\alpha(k-1)\), \(P_{2}<\alpha(k-2)\), \(P_{3}<\alpha(k-3)\), \(P_{4}>\alpha(k-4)\), \(P_{5}>\alpha(k-5)\); the original hypothesis is rejected at the 5% significance level. Therefore, compared with AND-RSN-PSO, distABC and IGSA/PSO, ISSA has significantly better performance. ISSA has a smaller average rank, though it does not outperform PSO-GWO and MDE. In summary, compared with five other algorithms, the proposed algorithm ISSA has better performance on the low dimensional functions. Table 11 shows the convergence results on the high dimensional functions. It can be seen that MDE can converge to a certain precision on F11, F12, F13, F16, F18 and F20, but the obtained precision has obvious gaps compared with ISSA; distABC almost has no efficient convergence performance on the high dimensional functions; IGSA/PSO can converge to a certain precision on F11, F12 and F13, but the results are not good enough; ADN-RSN-PSO can converge to a certain precision on all the high dimensional functions except on F18, but the obtained precision is far from ISSA; PSO-GWO can converge to a better precision except on F18, whose convergence result is not good enough and on F21 which cannot converge efficiently, with the obtained convergence precision still being worse than that of ISSA. For ISSA, all the functions can converge to the optimal except F15, F16 and F20. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(i\)} & \multirow{2}{*}{**Algorithm**} & \multirow{2}{*}{\(z=(R_{i}-R_{6})/\sqrt{\frac{k(k+1)}{6\alpha}}=(R_{i}-R_{6})/0.8367\)} & \multirow{2}{*}{\(P_{i}\)} & \multirow{2}{*}{\(a/(k-i)\)} \\  & & & \(R_{6}\)\(\sqrt{\frac{6\times(6+1)}{6\times 10}}=(R_{i}-R_{6})/0.8367\) & \\ \hline
1 & AND-RSN-PSO & \((5.3-1.7)/0.8367=4.3026\) & 2e\(-05\) & 0.01 \\
2 & distABC & \((5.15-1.7)/0.8367=4.1233\) & 4e\(-05\) & 0.0125 \\
3 & IGSA/PSO & \((3.9-1.7)/0.8367=2.6294\) & 0.0087 & 0.0167 \\
4 & PSO-GWO & \((2.9-1.7)/0.8367=1.4342\) & 0.1513 & 0.025 \\
5 & MDE & \((2.05-1.7)/0.8367=0.4183\) & 0.6781 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The results of the Holm test. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{5}{c}{**Rank**} \\ \cline{2-7}  & **MDE** & **distABC** & **IGSA/PSO** & **AND-RSN-PSOPSO-GWO** & **ISSA** \\ \hline F1 & 1 & 5 & 4 & 6 & 3 & 2 \\ F2 & 3 & 6 & 4 & 5 & 2 & 1 \\ F3 & 2 & 4 & 5 & 6 & 2 & 2 \\ F4 & 2.5 & 2.5 & 5 & 6 & 2.5 & 2.5 \\ F5 & 2 & 5 & 4 & 6 & 2 & 2 \\ F6 & 1 & 6 & 2 & 4 & 5 & 3 \\ F7 & 2 & 6 & 5 & 4 & 3 & 1 \\ F8 & 3 & 5 & 4 & 6 & 1.5 & 1.5 \\ F9 & 2 & 6 & 3 & 5 & 4 & 1 \\ F10 & 2 & 6 & 3 & 5 & 4 & 1 \\ \hline
**Total rank** & 20.5 & 51.5 & 39 & 53 & 29 & 17 \\
**Average rank** & 2.05 & 5.15 & 3.9 & 5.3 & 2.9 & 1.7 \\
**Sort** & 5 & 2 & 3 & 1 & 4 & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The results of the Friedman test. Aside from this, the best fitness value, the worst fitness value, the mean fitness value and the standard deviation of F11, F12, F13, F14, F17, F19 and F21 are all equal to zero and keep unchanged even if the dimensions become higher. The best fitness value of F18 is also equal to zero no matter whether the dimension is 30, 50 or 100, and the deviation is much smaller than the other five algorithms for F15, F16 and F20, which cannot converge to the optimal. The best fitness value, the worst fitness value, the mean fitness value and the standard deviation are still much better than the other five algorithms. In order to compare the differences of each method, a Friedman test is taken to check the data in Table 11 when the dimension is 30. The specific process is shown below: