## 4 Results and Discussion
### Solow-Polasky Diversity
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & \begin{tabular}{c} **Mean** \\ **(SR)** \\ \end{tabular} & \begin{tabular}{c} **Stdev** \\ **(SR)** \\ \end{tabular} & \begin{tabular}{c} **Mean** \\ **(mSR)** \\ \end{tabular} & \begin{tabular}{c} **Stdev** \\ **(mSR)** \\ \end{tabular} & \
\begin{tabular}{c} **Mean** \\ **(nSR)** \\ \end{tabular} &
\begin{tabular}{c} **Stdev** \\ **(nSR)** \\ \end{tabular} \\ \hline \(f_{1}\) & \(1.67\times 10^{1}\) & \(1.44\times 10^{0}\) & \(3.81\times 10^{0}\) & \(1.63\times 10^{0}\) & \(1.54\times 10^{1}\) & \(1.40\times 10^{0}\) \\ \hline \(f_{2}\) & \(1.25\times 10^{3}\) & \(4.04\times 10^{1}\) & \(2.62\times 10^{1}\) & \(2.99\times 10^{1}\) & \(1.24\times 10^{3}\) & \(3.19\times 10^{1}\) \\ \hline \(f_{3}\) & \(9.95\times 10^{-1}\) & \(1.41\times 10^{-1}\) & \(1.50\times 10^{-1}\) & \(3.32\times 10^{-2}\) & \(9.84\times 10^{-1}\) & \(1.31\times 10^{-1}\) \\ \hline \(f_{4}\) & \(8.96\times 10^{0}\) & \(7.36\times 10^{-1}\) & \(2.10\times 10^{0}\) & \(5.99\times 10^{-1}\) & \(8.46\times 10^{0}\) & \(2.15\times 10^{-1}\) \\ \hline \(f_{5}\) & \(8.13\times 10^{-1}\) & \(6.34\times 10^{-1}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) & \(8.13\times 10^{-1}\) & \(6.34\times 10^{-1}\) \\ \hline \(f_{6}\) & \(1.44\times 10^{0}\) & \(2.01\times 10^{-1}\) & \(2.27\times 10^{-2}\) & \(4.81\times 10^{-2}\) & \(1.39\times 10^{0}\) & \(1.08\times 10^{-1}\) \\ \hline \(f_{7}\) & \(7.62\times 10^{0}\) & \(6.12\times 10^{-1}\) & \(1.47\times 10^{0}\) & \(7.20\times 10^{-1}\) & \(4.40\times 10^{0}\) & \(1.07\times 10^{0}\) \\ \hline \(f_{8}\) & \(1.17\times 10^{3}\) & \(6.37\times 10^{1}\) & \(3.30\times 10^{0}\) & \(3.64\times 10^{0}\) & \(1.17\times 10^{3}\) & \(6.37\times 10^{1}\) \\ \hline \(f_{9}\) & \(3.00\times 10^{0}\) & \(5.31\times 10^{-1}\) & \(6.07\times 10^{-1}\) & \(3.04\times 10^{-1}\) & \(2.33\times 10^{0}\) & \(7.54\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Mean (\(\mu\)) and standard deviation (\(\sigma\)) for SR measure and its variants at the last iteration. Figure 16: Quantification of diversity using the Swarm Radius (_Cont._). For the SR measure, both the standard SR and the nSR were computed using the \"swarm best\", i.e., the solution with the best objective function value. Since the goal of niching algorithms is to obtain multiple optima, the SR measure obtained the largest distance between any of the \"swarm best\" and a candidate solution. Since each candidate solution is optimising an optimum, the expectation is that the diversity quantified using SR will monotonically decrease as candidate solutions converge. This is because the diversity is not measured on how far apart the solutions are from each other but from the \"swarm best\" which are multiple. Following this, where SR reports high diversity, it is likely to be as a result of the presence of outliers. This is the case with the obtained results. In the case of mSR, the distances are computed with respect to each niche, and therefore the neighbourhood best is used. Diversity is thus expected to be low and tending towards zero as particles converge towards an optimum. As such, the achieved results are as expected. For the nSR, unique solutions are used. As such, the obtained results will be somewhat similar to those of the standard SR. This is because, for both cases, the particle with the best objective function value is used. Where two candidate solutions have the same objective function value, the computed distance is zero. Where the nSR value is smaller than SR, the likely explanation is that the furthest particle among the set of unique solutions was closer to a solution than in the general swarm. This too is expected. These results mirror those of the ADSC. For both cases, the distances are calculated from a candidate solution with the best objective fitness. For niching algorithms, the presence of multiple optima for which the distances can be calculated from means that lower diversity will be expected.