# Improved Particle Swarm Optimization for Sea Surface Temperature Prediction
## 4 Results and Discussion
### Analysis of Experimental Results  
In this work we designed three sets of experiments: (1) to compare the advantages and disadvantages of the similarity measures in SSTP and choose the best method; (2) to verify the effectiveness of LSPSO by comparing it with the MODE, NSGA-II, and OMOPSO algorithms; (3) to compare the performance of DSL with other SSTP methods. **Experiment 1: Comparison of similarity measures. **_Applying AC to SSTP requires choosing a suitable similarity measure method to measure the similarity of SST sequences. Therefore, we first computed the Euclidean distance, the cosine distance, and the DTW distance to measure the similarity of SSTs. Then, we chose the optimal similarity measure based on this principle: the better the similarity is measured, the smaller the SSTP error is. The error of SSTP was measured by Root Mean Square Error (RMSE)._  
Table 2 shows the achieved RMSE in relation to SST predictions based, respectively, on Euclidean distance, Cosine distance, and DTW distance similarity measures. The first nine rows provide RMSE values for nine different SST sequences, with the average values given in the last row. When using the Euclidean distance to predict SST, the average RMSE is 0.4949, which is slightly higher than (at times comparable to) DTW, but much better than Cosine. The reason for this is that the number of days in which the SST changes regularly is not fixed, and the Euclidean distance does not support scaling. However, the DTW distance can better overcome this deficiency, so DTW can better reflect the similarity of SST changes. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Number of Variables** & **Number of Objectives** & **Analytical Pareto Frontier** \\ \hline BNH & 2 & 2 & Connected/Convex \\ SRN & 2 & 2 & Disconnected/Convex \\ TNK & 2 & 2 & Disconnected/Nonconvex \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Bi-objective optimization problems. The average RMSE when using the cosine distance to predict SST is 1.0655. This is much higher than the other two indexes, leading to a much worse prediction ability. This happens because the cosine distance uses the cosine of the angle of the SST vector to measure its similarity, which only reflects the trend of SST changes, and is not sensitive to the value of the SST itself. The DTW and Euclidean distances are based on the SST values. In summary, DTW has the highest prediction accuracy among the three. Therefore, we use only DTW in the remaining experiments, below. **Experiment 2: Verification of LSPSO. **_An LSPSO algorithm was proposed in this paper. To verify its effectiveness, three classical bi-objective test functions (BNH, SRN and TNK) were selected. GD and SP were used as evaluation indicators. The number of populations, N was set to 100, and the number of iterations G = 250 was compared with MODE, NSGA-II and OMOPSO, respectively. Each algorithm ran independently 30 times for each test function, computing the mean and variance of GD and SP values. Analysis of Variance (ANOVA) [42] was used to test the significant difference of the GD and SP indicators between LSPSO and the other models (MODE, NSGA-II, and OMOPSO). In general, P values smaller than 0.05 indicate that there is a significant difference._  
Figures 3-5 show the solution obtained by LSPSO for BNH, SRN and TNK functions and the true Pareto frontier. The red circle is the optimal solution obtained by LSPSO, and the black line is the true Pareto front. The optimal solution sets of LSPSO are convergent and evenly distributed on the true Pareto front. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Station** & **Euclidean** & **Cosine** & **DTW** \\ \hline
1 & 0.5289 & 1.3223 & 0.4866 \\
2 & 0.5525 & 1.2907 & 0.5136 \\
3 & 0.4813 & 0.8982 & 0.4606 \\
4 & 0.4641 & 0.6669 & 0.4637 \\
5 & 0.4788 & 1.4734 & 0.4544 \\
6 & 0.5247 & 0.6666 & 0.4630 \\
7 & 0.5364 & 0.5549 & 0.5204 \\
8 & 0.4437 & 1.3070 & 0.4429 \\
9 & 0.4439 & 1.4096 & 0.4161 \\ Avg & 0.4949 & 1.0655 & 0.4690 \\ \hline \hline \end{tabular}
\end{table}