# Improved Particle Swarm Optimization for Sea Surface Temperature Prediction
## 4 Results and Discussion
### Analysis of Experimental Results

Table 2: RMSE obtained by predicting SST with three similarity measures. Figure 3: Pareto optimal solutions obtained by LSPSO for BNH. Table 3 shows the GD and SP of MODE, NSGA-II, OMOPSO, and LSPSO in solving BNH, SRN, and TNK, respectively. Overall, our LSPSO is better than the other three methods according to the mean and std values of GD and SP. When dealing with the BNH function, our LSPSO method outperforms the other three methods in the uniformity of the solution set, achieving a statistical significance of SP (p \(<\) 0.001). This can be observed from Figure 3, where the solution set of LSPSO is uniformly distributed on the Pareto front for BNH. The convergence of LSPSO is much better than that of MODE in term of GD (p \(<\) 0.001), but it is not significantly better than NSGA-II and OMOPSO (p \(=\) 0.37 and p \(=\) 0.15, respectively). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{2}{c}{**MODE**} & \multicolumn{2}{c}{**NSGA-II**} & \multicolumn{2}{c}{**OMOPSO**} & \multicolumn{2}{c}{**LSPSO**} \\ \cline{2-10}  & \multicolumn{2}{c}{**GD**} & \multicolumn{2}{c}{**SP**} & \multicolumn{2}{c}{**GD**} & \multicolumn{2}{c}{**SP**} & \multicolumn{2}{c}{**GD**} & \multicolumn{2}{c}{**SP**} & \multicolumn{2}{c}{**GD**} & \multicolumn{2}{c}{**SP**} \\ \hline \multirow{2}{*}{BNH} & Mean & 0.5498 & 1.9309 & 0.1386 & 1.1174 & 0.1405 & 0.9745 & 0.1350 & 0.7127 \\  & Std & 8.77 \(\times 10^{-2}\) & 0.5502 & 1.56 \(\times 10^{-2}\) & 6.50 \(\times 10^{-2}\) & 1.46 \(\times 10^{-2}\) & 7.94 \(\times 10^{-2}\) & 1.36 \(\times 10^{-2}\) & 6.95 \(\times 10^{-2}\) \\ \hline \multirow{2}{*}{SRN} & Mean & 28408 & 4.1774 & 0.6819 & 1.8991 & 0.5876 & 1.3269 & 0.4644 & 1.5033 \\  & Std & 0.4715 & 0.5601 & 9.02\(\times 10^{-2}\) & 0.1593 & 6.58\(\times 10^{-2}\) & 0.1020 & 3.77\(\times 10^{-2}\) & 0.1149 \\ \hline \multirow{2}{*}{TNK} & Mean & 4.96 \(\times 10^{-3}\) & 4.34 \(\times 10^{-2}\) & 2.45 \(\times 10^{-3}\) & 3.73 \(\times 10^{-2}\) & 2.68 \(\times 10^{-3}\) & 3.39 \(\times 10^{-2}\) & 2.41 \(\times 10^{-3}\) & 3.39 \(\times 10^{-2}\) \\  & Std & 6.91 \(\times 10^{-4}\) & 2.37 \(\times 10^{-3}\) & 2.24 \(\times 10^{-4}\) & 7.84 \(\times 10^{-4}\) & 2.44 \(\times 10^{-4}\) & 7.25 \(\times 10^{-4}\) & 1.46 \(\times 10^{-4}\) & 6.90 \(\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The GD and SP obtained by MODE, NSGA-II, OMOPSO and LSPSO in solving BNH, SRN and TNK. Figure 4: Pareto optimal solutions obtained by LSPSO for SRN. Figure 5: Pareto optimal solutions obtained by LSPSO for TNK. For the SRN function, the GD of LSPSO is significantly better than those on the other three methods (p < 0.001), indicating a good convergence. Regarding the uniformity of the solution set distribution (SP), although LSPSO is significantly better than MODE and NSGA-II (p < 0.001), it is worse than OMOPSO (p < 0.001). This maybe because the true Pareto frontier of the SRN function shows linearly distributed and global search has advantages when dealing with such problems, but when the true Pareto frontier of the test function is non-linear, our method can achieve much better results. For the TNK function, the average and std of the GD and SP obtained by LSPSO are better than those on the other methods, indicating that LSPSO handles TNK well. According to the significance test, our LSPSO has similar performance with NSGA-II in terms of GD (p = 0.48), and with OMOPSO in terms of SP (p = 0.95). In summary, LSPSO can better handle the bi-objective test functions, and provide effective support for the parameter optimization of DS. MODE, NSGA-II and OMPSO have strong global search capabilities and insufficient local search capabilities, so the solution sets obtained are not uniform and easily fall into a local optimum. The global and local search capabilities of PSO are mutually constrained and tend to fall into local optimum in the later stages of search.