## 4 Results and Discussion
### Average Distance around the Swarm Centre  
Figures 7 and 8 and Table 4 show the quantification of diversity using the average distance around the swarm centre measure (ADSC) and its proposed variants, i.e., modified ADSC (mADSC) and niche ADSC (nADSC). The intial posit in the article was that standard ADSC and the nASDC will show high diversity because the distances will be computed from a swarm \"best\" in both cases. The results do not seem to correlate with this expectation. Analysis of the ADSC led to the following observations. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & **Mean** & **Stdev** & **Mean** & **Stdev** & **Mean** & **Stdev** \\  & **(SDNN)** & **(SDNN)** & **(mSDNN)** & **(mSDNN)** & **(nSDNN)** & **(nSDNN)** \\ \hline \(f_{1}\) & \(4.28\times 10^{1}\) & \(1.29\times 10^{1}\) & \(9.77\times 10^{0}\) & \(4.09\times 10^{  
Figure 7: Quantification of diversity using the average distance around the swarm centre measure. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & **Mean** & **Stdev** & **Mean** & **Stdev** & **Mean** & **Stdev** \\  & **(ADSC)** & **(ADSC)** & **(mADSC)** & **(mADSC)** & **(nADSC)** & **(nADSC)** \\ \hline \(f_{1}\) & \(4.55\times 10^{-1}\) & \(1.62\times 10^{-1}\) & \(3.42\times 10^{-1}\) & \(1.54\times 10^{-1}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) \\ \hline \(f_{2}\) & \(1.23\times 10^{1}\) & \(9.74\times 10^{0}\) & \(4.06\times 10^{0}\) & \(3.54\times 10^{0}\) & \(1.65\times 10^{0}\) & \(2.59\times 10^{0}\) \\ \hline \(f_{3}\) & \(6.54\times 10^{-2}\) & \(1.23\times 10^{-2}\) & \(3.52\times 10^{-2}\) & \(7.87\times 10^{-3}\) & \(5.95\times 10^{-3}\) & \(6.99\times 10^{-3}\) \\ \hline \(f_{4}\) & \(2.69\times 10^{-1}\) & \(6.93\times 10^{-2}\) & \(2.44\times 10^{-1}\) & \(1.32\times 10^{-1}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) \\ \hline \(f_{5}\) & \(8.45\times 10^{-10}\) & \(3.74\times 10^{-9}\) & \(8.45\times 10^{-10}\) & \(3.74\times 10^{-9}\) & \(8.45\times 10^{-10}\) & \(3.74\times 10^{-9}\) \\ \hline \(f_{6}\) & \(5.27\times 10^{-3}\) & \(9.36\times 10^{-3}\) & \(7.67\times 10^{-3}\) & \(1.21\times 10^{-2}\) & \(1.42\times 10^{-3}\) & \(5.35\times 10^{-3}\) \\ \hline \(f_{7}\) & \(6.81\times 10^{-3}\) & \(1.10\times 10^{-2}\) & \(1.03\times 10^{-3}\) & \(1.83\times 10^{-3}\) & \(1.41\times 10^{-6}\) & \(7.58\times 10^{-6}\) \\ \hline \(f_{8}\) & \(4.76\times 10^{0}\) & \(3.44\times 10^{0}\) & \(4.15\times 10^{0}\) & \(2.71\times 10^{0}\) & \(3.98\times 10^{0}\) & \(2.71\times 10^{0}\) \\ \hline \(f_{9}\) & \(1.13\times 10^{-1}\) & \(4.00\times 10^{-2}\) & \(6.54\times 10^{-2}\) & \(4.15\times 10^{-2}\) & \(6.34\times 10^{-3}\) & \(3.42\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mean (\(\mu\)) and standard deviation (\(\sigma\)) for ADSC measure and its variants at the last iteration. Figure 8: Quantification of diversity using the average distance around the swarm centre measure (_Cont._). The standard ADSC sums the distances from each particle to the best particle in the swarm and then computes the average. For niching algorithms, there is a high likelihood of multiple global bests, based on the modality of the problem being investigated. As such, each particle's distance from the global best is dependent on which optimum the particle was optimising. This behaviour explains why the diversity as quantified using the ADSC decreased monotonically as the search progressed. For the mADSC, it was expected that the diversity will decrease monotonically. This is because the computation is done using the best particle in the niche. Similar to the standard ADSC, the nADSC measure was computed using the nbests. As such, although the final particle positions (as shown in Appendices A-C) suggest that, in the presence of outliers, the use of nbests in the computation meant that there were multiple \"centres\" being used. This led to the low diversity. These results suggest that a better strategy needs to be devised in working with both ADSC and nADSC.