## 4 Results and Discussion
### Solow-Polasky Diversity  
Figures 13 and 14 and Table 7 show the diversity results as computed using the Solow-Polasky diversity measure, i.e., standard SPD, modified SPD (mSPD), and the niche diversity (nSPD). As shown in Equation (9), the SPD is computed based on an \(n\times n\) matrix, \(M\), with entries \(m_{i,j}=\exp(-\theta\|\mathbf{x}_{i}-\mathbf{x}_{j}\|_{2})\). For the standard SPD, the entries are therefore the exponent of the pairwise distance of each particle to another particle, normalised using a user defined parameter, \(\theta\). In this work, \(\theta=1\). Following this, the expectation is that the computed diversity will gradually decrease but stagnate at high values when particles converge to the location of the optima. The obtained results are therefore as expected. For the mSPD, the entries \(m_{i,j}=\exp(-\theta\|\mathbf{x}_{i}-\mathbf{x}_{j}\|_{2})\) are computed from the particles in each niche and then an average is calculated. Since it is expected for the intra-niche distances to be small as particles converge towards an optimum, the obtained diversity should be low and tend towards zero. The results shown are therefore as expected. The nSPD is computed similarly to the standard SPD, but with the exception that only the unique solutions are used. As such, whereas the distances for each of the unique solutions are the same as in SPD, the matrix is smaller because the number of niches is less. As such, the nSPD values will be smaller than those of the standard SPD, but the computed diversity will still be high. Results obtained from the experiment are therefore as expected. The results obtained here concurs with [4] on the suitability of using the SPD measure to compute diversity of solutions and candidate solutions of niching algorithms. Figure 13: Quantification of diversity using the Solow–Polasky Diversity. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & **Mean** & **Stdev** & **Mean** & **Stdev** & **Mean** & **Stdev** \\  & **(SPD)** & **(SPD)** & **(mSPD)** & **(mSPD)** & **(nSPD)** & **(nSPD)** \\ \hline \(f_{1}\) & \(1.11\times 10^{0}\) & \(6.63\times 10^{-3}\) & \(1.03\times 10^{0}\) & \(1.05\times 10^{-2}\) & \(1.08\times 10^{0}\) & \(1.30\times 10^{-4}\) \\ \hline \(f_{2}\) & \(1.03\times 10^{1}\) & \(1.35\times 10^{0}\) & \(1.04\times 10^{0}\) & \(3.99\times 10^{-2}\) & \(9.52\times 10^{0}\) & \(1.27\times 10^{0}\) \\ \hline \(f_{3}\) & \(1.01\times 10^{0}\) & \(3.34\times 10^{-4}\) & \(1.00\times 10^{0}\) & \(1.88\times 10^{-4}\) & \(1.01\times 10^{0}\) & \(1.91\times 10^{-4}\) \\ \hline \(f_{4}\) & \(1.06\times 10^{0}\) & \(4.19\times 10^{-3}\) & \(1.01\times 10^{0}\) & \(3.49\times 10^{-3}\) & \(1.05\times 10^{0}\) & \(5.76\times 10^{-5}\) \\ \hline \(f_{5}\) & \(1.00\times 10^{0}\) & \(3.49\times 10^{-3}\) & \(1.00\times 10^{0}\) & \(0.00\times 10^{0}\) & \(1.00\times 10^{0}\) & \(3.49\times 10^{-3}\) \\ \hline \(f_{6}\) & \(1.01\times 10^{0}\) & \(1.87\times 10^{-3}\) & \(1.00\times 10^{0}\) & \(2.41\times 10^{-4}\) & \(1.01\times 10^{0}\) & \(1.47\times 10^{-3}\) \\ \hline \(f_{7}\) & \(1.09\times 10^{0}\) & \(6.42\times 10^{-3}\) & \(1.01\times 10^{0}\) & \(6.84\times 10^{-3}\) & \(1.03\times 10^{0}\) & \(6.30\times 10^{-3}\) \\ \hline \(f_{8}\) & \(1.21\times 10^{1}\) & \(1.31\times 10^{0}\) & \(1.01\times 10^{0}\) & \(5.51\times 10^{-3}\) & \(1.19\times 10^{1}\) & \(1.28\times 10^{0}\) \\ \hline \(f_{9}\) & \(1.03\times 10^{0}\) & \(4.96\times 10^{-3}\) & \(1.00\times 10^{0}\) & \(1.96\times 10^{-3}\) & \(1.02\times 10^{0}\) & \(6.19\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Mean (\(\mu\)) and standard deviation (\(\sigma\)) for the SPD measure and its variants at the last iteration. Figure 14: Quantification of diversity using the Solow–Polasky Diversity (\(Cont.\)).