# Optimal Prediction of Individual Vessel Trajectories Based on Sparse Gaussian Processes
## 5 Experiments
### Comparison of SGP to Other Baselines
Figure 6a,b show the prediction results for the inbound trajectories, with the SGPR-based model achieving the highest accuracy for both ADE and FED values. Furthermore, the prediction accuracy of all models was poor in the curved region of the upper right of the trajectory plot. Figure 6c,d show the predicted outbound trajectories. According to their findings, the FGPR-based model was the most accurate for ADE values, while the SGPR-based model was the best for FED values. Furthermore, all models' prediction accuracy was poor in the upper-right corner and middle sections of the trajectory plot. It is necessary to compare the accuracy of long-term predictions as prediction time increases among the developed models. Figure 7 depicts the average ADE for the test dataset for each model as the prediction time was increased by 1 min at a time. The closed-loop prediction is performed at 1 min intervals during the prediction time, and when the prediction time arrives, the actual observed data are fed into the prediction model. Figure 7 shows that the SGPR-based model has a higher prediction accuracy over time than the FGPR- and SVR-based models. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **Metric** & \(x_{IN}\) & \(y_{IN}\) & \(x_{OUT}\) & \(y_{OUT}\) \\ \hline \multirow{6}{*}{FGPR} & RMSE & 1.4908 & 2.0752 & 1.7833 & 2.7188 \\  & MAE & 0.9930 & 1.5122 & 1.3179 & 1.8442 \\  & Avg. ADE (NM) & & 0.0742 & & 0.0844 \\  & Avg. FDE (NM) & & 0.0714 & & 0.0862 \\  & Elapsed time (s) & 80.8644 & 53.2851 & 54.3169 & 77.2834 \\ \hline \multirow{6}{*}{SVR} & RMSE & 1.5108 & 1.9099 & 1.7960 & 2.3734 \\  & MAE & 0.9894 & 1.3922 & 1.3504 & 1.6395 \\  & Avg. ADE (NM) & & 0.0710 & & 0.0796 \\  & Avg. FDE (NM) & & 0.0728 & & 0.0867 \\  & Elapsed time (s) & 0.3612 & 0.3729 & 0.4120 & 0.3065 \\ \hline \multirow{6}{*}{SGPR} & RMSE & 1.4394 & 1.8872 & 1.7921 & 2.3233 \\  & MAE & 0.9654 & 1.3868 & 1.3246 & 1.6527 \\ \cline{1-1}  & Avg. ADE (NM) & & 0.0695 & & 0.0792 \\ \cline{1-1}  & Avg. FDE (NM) & & 0.0711 & & 0.0787 \\ \cline{1-1}  & Elapsed time (s) & 2.7413 & 0.6776 & 15.8502 & 1.1940 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of performance for model validations. Figure 5: Box chart of displacement error for each method. (FGPR) Median value 0.0556, quartiles [0.0313 0.1039], whisker [0.0004 0.2115]; (SGPR) median value 0.0529, quartiles [0.03 0.10], whisker [0.0005 0.205]; (SVR) median value 0.0537, quartiles [0.03 0.10], whisker [0.001 0.2068]. Figure 6: _Cont._  
Figure 6: Predicted trajectories using closed-loop method for each model. (**a**) Accuracy of each prediction model for inbound trajectory #1: (FGPR) ADE = 1.0338 NM, FDE = 1.2314 NM; (SGPR) ADE = 0.4390 NM, FDE = 0.6075; (SVR) ADE = 0.7020 NM, FDE = 0.8345. (**b**) Accuracy of each prediction for model inbound trajectory #2: (FGPR) ADE = 2.2383 NM, FDE = 0.8421 NM; (SGPR) ADE = 2.0965 NM, FDE = 0.5223 NM; (SVR) ADE = 2.3920 NM, FDE = 1.3517 NM. (**c**) Accuracy of each prediction model for outbound trajectory #1: (FGPR) ADE = 0.6365 NM; FDE = 1.0465 NM, (SGPR) ADE = 0.8448 NM; FDE = 0.6115 NM, (SVR) ADE = 1.4901 NM; FDE = 1.0491 NM. (**d**) Accuracy of each prediction model for outbound trajectory #2: (FGPR) ADE = 0.9787 NM; FDE = 0.8728 NM, (SGPR) ADE = 1.3794 NM; FDE = 0.3637 NM, (SVR) ADE = 2.1228 NM; FDE = 1.2397 NM.