## 4 Results and Discussion
### Sum of Distances  
Table 2 as well as Figures 3 and 4 show the quantification of diversity using the sum of distance measure (SD) and its proposed variants, i.e., modified SD (mSD) and niche SD (nSD).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & **Mean** & **Stdev** & **Mean** & **Stdev** & **Mean** & **Stdev** \\  & **(SD)** & **(SD)** & **(mSD)** & **(mSD)** & **(nSD)** & **(nSD)** \\ \hline \(f_{1}\) & \(3.02\times 10^{2}\) & \(9.12\times 10^{0}\) & \(2.76\times 10^{1}\) & \(1.01\times 10^{1}\) & \(1.41\times 10^{1}\) & \(4.40\times 10^{0}\) \\ \hline \(f_{2}\) & \(3.62\times 10^{3}\) & \(4.87\times 10^{2}\) & \(7.93\times 10^{0}\) & \(8.11\times 10^{0}\) & \(1.31\times 10^{3}\) & \(4.47\times 10^{2}\) \\ \hline \(f_{3}\) & \(7.39\times 10^{1}\) & \(2.30\times 10^{0}\) & \(1.13\times 10^{0}\) & \(2.67\times 10^{-1}\) & \(2.15\times 10^{1}\) & \(2.57\times 10^{0}\) \\ \hline \(f_{4}\) & \(2.26\times 10^{2}\) & \(5.43\times 10^{0}\) & \(1.38\times 10^{1}\) & \(2.74\times 10^{0}\) & \(1.31\times 10^{1}\) & \(2.17\times 10^{0}\) \\ \hline \(f_{5}\) & \(3.32\times 10^{1}\) & \(2.50\times 10^{1}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) & \(3.32\times 10^{1}\) & \(2.50\times 10^{1}\) \\ \hline \(f_{6}\) & \(9.22\times 10^{1}\) & \(1.82\times 10^{1}\) & \(2.14\times 10^{-1}\) & \(3.37\times 10^{-1}\) & \(1.94\times 10^{1}\) & \(1.14\times 10^{1}\) \\ \hline \(f_{7}\) & \(4.30\times 10^{2}\) & \(5.75\times 10^{1}\) & \(4.28\times 10^{1}\) & \(2.19\times 10^{1}\) & \(1.70\times 10^{1}\) & \(8.44\times 10^{0}\) \\ \hline \(f_{8}\) & \(3.35\times 10^{3}\) & \(4.11\times 10^{2}\) & \(2.03\times 10^{-1}\) & \(2.03\times 10^{-1}\) & \(3.32\times 10^{3}\) & \(4.15\times 10^{2}\) \\ \hline \(f_{9}\) & \(1.41\times 10^{2}\) & \(1.15\times 10^{1}\) & \(8.32\times 10^{0}\) & \(3.47\times 10^{0}\) & \(1.09\times 10^{1}\) & \(3.45\times 10^{0}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mean (\(\mu\)) and standard deviation (\(\sigma\)) for SD measure at the last iteration.  
The results for all the considered multimodal problems show that the diversity quantified using the standard SD is high. For swarm diversity, the expectation is for diversity to decrease monotonically as the particles converge towards the locations of optima. However, for the standard SD, since the pairwise distance of each particle to all other particles in the swarm is considered, the diversity predicted will be high. These results show that the standard SD is not suitable for niching algorithms.  
For the mSD, the expectation is that, as the particles converge towards the location of optima, the measured diversity should monotonically decrease. It is also expected that, if each particle forms a niche, then the diversity predicted should be low. The results obtained correlates to the expectations, that is, the obtained mSD predicts low diversity towards convergence.  
Figure 3: Quantification of diversity using the sum of distance measure.  
Figure 4: Quantification of diversity using the sum of distances measure (_Cont._).  
The nSD calculates diversity as standard SD does, but only using the unique solutions. As such, achieved diversity should be high. In addition, even in the case where the unique solutions are close to each other, it is expected that the diversity will never converge. The obtained results agree with this expectation.  
In conclusion, the standard SD is not a good measure to quantify diversity in niching algorithms, as is expected. If the diversity of all candidate solutions is required, the mSD should be used. Finally, if only the diversity of found solutions is required, then nSD is a good measure.