# Optimal Prediction of Individual Vessel Trajectories Based on Sparse Gaussian Processes
## 5 Experiments
### Comparison of SGP to Other Baselines  
To verify the SGPR-based prediction model, a performance comparison was conducted using the FGPR and SVR models. Hyperparameter optimizations were conducted for the FGPR and SVR models using the same GA as in the SGPR model. For the SVR kernel function, the most commonly used SE kernel function was applied. The hyperparameters and optimization results of the models to be compared are listed in Table 7. In Table 7, \(\theta\) is the kernel parameter, \(C\) is the trade-off parameter of error and margin, and \(\epsilon\) is the insensitivity parameter. Table 8 presents the results of comparing the prediction accuracy and training time between the models with the optimized hyperparameters applied. The prediction accuracy is the result of calculating the error of the predicted trajectory by inputting the test dataset into the trained model, and the training time is the time excluding the hyperparameter optimization time by the GA. As demonstrated in Table 8, the SGPR-based model outperformed other models in both ADE and FDE, two metrics that reflect comprehensive prediction accuracy. Furthermore, the model achieved significant reductions in both training time and prediction error compared to the FGPR-based model. Specifically, the average elapsed time was reduced by 61.3 s, and the average ADE for inbound and outbound trajectories decreased by 8.7 m and 9.6 m, respectively. The training time of the SVR-based model was found to be the shortest among all models evaluated. Figure 5 depicts the results of the statistical analysis of displacement error by the methods for each trajectory in the test datasets. In Figure 5, the line inside each box represents the median value, while the top and bottom edges denote the upper and lower quartiles, respectively. Whiskers refer to the highest and lowest data points that have not been identified as outliers. The outlier values exceeded 1.5 times the distance between the top and bottom edges. Each outlier is represented by a circle marker symbol. Figure 5 shows that the displacement errors of the FGPR model were larger and distributed over a wide range, with outliers being the most deviant. The median value of the displacement error with the SGPR model was the lowest and this model was found to be robust to outliers. Furthermore, given that the overall median values were slightly lower than the boxes, there were many displacement errors below the average values. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **Metric** & \(x_{IN}\) & \(y_{IN}\) & \(x_{OUT}\) & \(y_{OUT}\) \\ \hline \multirow{4}{*}{GA} & RMSE & 1.5527 & 1.9915 & 1.3011 & 1.5198 \\  & MAE & 1.0829 & 1.5958 & 0.9478 & 1.2170 \\  & Avg. ADE(NM) & & 0.0739 & & 0.0604 \\  & Avg. FDE(NM) & & 0.0741 & & 0.0731 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Candidate**} & \multirow{2}{*}{**Type**} & \multicolumn{4}{c}{**Best Hyperparameter**} \\ \cline{5-8}  & & & & \(x_{IN}\) & \(y_{IN}\) & \(x_{OUT}\) & \(y_{OUT}\) \\ \hline \multirow{4}{*}{FGPR} & \(\theta_{I}\) & \([5\times 10^{-3},\,3]\) & Continuous & 1.6449 & 0.6850 & 0.8640 & 0.9886 \\  & \(\sigma_{I}\) & \([10^{-4},\,10]\) & Continuous & 8.5656 & 0.5871 & 0.7288 & 1.0181 \\  & \(k(\cdot)\) & \(\{\)SE, MT32, MT52\(\}\) & Categorical & MT32 & SE & MT32 & SE \\ \hline \multirow{4}{*}{SVR} & \(\theta\) & \([5\times 10^{-3},\,3]\) & Continuous & 0.2560 & 0.3701 & 0.1003 & 0.1434 \\  & \(C\) & \([0,\,2]\) & Continuous & 1.9795 & 0.8568 & 1.9238 & 0.8538 \\ \cline{1-1}  & \(\epsilon\) & \([2^{-5},\,2^{-1}]\) & Continuous & 0.3931 & 0.4872 & 0.4909 & 0.4981 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Best Hyperparameters for other frameworks obtained by Genetic Algorithm. Figure 6 depicts the long-term prediction results of each model for the trajectories in the test dataset.