## 6 Experimental Results
### Analyzing the Influence of Using Islands  
We start studying the effect on the quality of solutins of using _islands_. Note that splitting particles into sets which independently evolve and update the global optimum only from time to time has two effects. On the one hand, enabling such an independent evolution of particles helps to preserve the diversity of solutions. On the other hand, out-of-date global optima may prevent particles from searching for solutions in more interesting areas for a while, until the global optimum is updated again. In order to obtain results that are statistically relevant, for each function and number of islands, we perform \(50\) executions and we compute the corresponding averages. In Table 2 we depict the results of our PSO skeleton for 1, 2, 3, and 4 islands (note that using \(1\) island is the same as not considering islands at all). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **Dimensions (n)** & \([min,max]^{n}\) & \(f(x)_{min}\) & **Asym. Init. Range** \\ \hline \(f_{1}(x)\) & 30 & \([-100,100]^{n}\) & 0 & \([-100,-33.3]^{n}\) \\ \(f_{2}(x)\) & 30 & \([-10,10]^{n}\) & 0 & \([-10,-3.3]^{n}\) \\ \(f_{3}(x)\) & 30 & \([-100,100]^{n}\) & 0 & \([-100,-33.3]^{n}\) \\ \(f_{4}(x)\) & 30 & \([-100,100]^{n}\) & 0 & \([-100,-33.3]^{n}\) \\ \(f_{5}(x)\) & 30 & \([-30,30]^{n}\) & 0 & \([-30,-10]^{n}\) \\ \(f_{6}(x)\) & 30 & \([-100,100]^{n}\) & 0 & \([-100,-33.3]^{n}\) \\ \(f_{8}(x)\) & 30 & \([-500,500]^{n}\) & \(-12569.5\) & \([-500,500]^{n}\) \\ \(f_{9}(x)\) & 30 & \([-5.12,5.12]^{n}\) & 0 & \([-5.12,-1.7]^{n}\) \\ \(f_{10}(x)\) & 30 & \([-32,32]^{n}\) & 0 & \([-32,-10.7]^{n}\) \\ \(f_{11}(x)\) & 30 & \([-600,600]^{n}\) & 0 & \([-600,-200]^{n}\) \\ \(f_{12}(x)\) & 30 & \([-50,50]^{n}\) & 0 & \([-50,-16.7]^{n}\) \\ \(f_{13}(x)\) & 30 & \([-50,50]^{n}\) & 0 & \([-50,-16.7]^{n}\) \\ \(f_{14}(x)\) & 2 & \([-65.536,65.536]^{n}\) & 1 & \([-65.536,-21.84]^{n}\) \\ \(f_{15}(x)\) & 4 & \([-5,5]^{n}\) & 0.0003075 & \([-5,-1.67]^{n}\) \\ \(f_{16}(x)\) & 2 & \([-5,5]^{n}\) & \(-1.0316285\) & \([-5,-1.67]^{n}\) \\ \(f_{17}(x)\) & 2 & \([-5,10]\times[0,15]\) & 0.398 & \([-5,-1.67]\times[0,5]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Benchmark functions. In order to statistically analyze the influence of using islands on the optimality of the solutions, we use [41] to perform a statistical test of the mean results given in Table 2. In general, a Friedman test [42, 43] can be used to check whether the hypothesis that all methods behave similarly (the null hypothesis) holds or not. However, since the number of variables (methods) under consideration is low, using a _Friedman aligned ranks_ test [44] or a _Quade_ test [45] is more recommended in this case (see e.g., [46]). These tests do not rank methods for each problem separately (as Friedman test does), but construct a global ranking where values of all methods and problems are ranked together. In Friedman aligned ranks test, for each problem the difference of each method with respect to the average value for all methods is considered, and next all values of all problems are ranked together. In Quade test, problems where the difference between the best result and the worst result is high are given more weight in the comparison. Let us consider \(\alpha=0.05\), a standard significance level. From results given in Table 2, we calculate that the _p_-value for aligned Friedman is 0.0049, which allows to reject the null hypothesis with a high level of significance (the _p_-value is much lower than 0.05). So, the test concludes that methods using 1, 2, 3 and 4 islands are not considered similar. That is, algorithms using different numbers of islands are not considered equivalent. Since the null hypothesis was rejected, we can apply a post-hoc analysis to compare the control method with the other three. Ranks assigned by this test to these methods are 33.1563, 33.8750, 28.3438 and 34.6250, respectively. Thus, the control method consists in using 3 islands.