# Trajectory Prediction of Marine Moving Target Using Deep Neural Networks with Trajectory Data
## 4 Experiments
### Comparisons with Other Algorithms  
To thoroughly test and analyze our proposed method, we run a series of experiments to compare our method with other algorithms, including the interpolation extrapolation method, fitting extrapolation method, grey prediction method, autoregressive prediction method, BP, and LSTM neural network model. The interpolation extrapolation method adopts cubic interpolation, which obtained the best results compared to nearest neighbor interpolation and linear interpolation. The fitting extrapolation method adopts polynomial fitting based on the least square method. The parametric experiment shows the best prediction results were obtained when the order of the polynomial is two. Grey prediction and autoregressive prediction methods are applied to data with consistent sampling intervals. Thus, polynomial interpolation and uniform sampling are first performed to obtain equal \begin{table}
\begin{tabular}{c c c} \hline \hline
**Training Steps** & **Loss of Training Set** & **Loss of Test Set** \\ \hline
1 & 0.239351 & 0.217599 \\
2000 & 4.55702 \(\times\) 10\({}^{-4}\) & 2.79363 \(\times\) 10\({}^{-4}\) \\
4000 & 3.19982 \(\times\) 10\({}^{-5}\) & 3.81833 \(\times\) 10\({}^{-5}\) \\
6000 & 2.06495 \(\times\) 10\({}^{-5}\) & 3.98102 \(\times\) 10\({}^{-5}\) \\
8000 & 1.58536 \(\times\) 10\({}^{-5}\) & 3.26941 \(\times\) 10\({}^{-5}\) \\
10,000 & 1.25557 \(\times\) 10\({}^{-5}\) & 2.85061 \(\times\) 10\({}^{-5}\) \\
20,000 & 5.28681 \(\times\) 10\({}^{-6}\) & 2.37062 \(\times\) 10\({}^{-5}\) \\
30,000 & 3.27563 \(\times\) 10\({}^{-6}\) & 4.25452 \(\times\) 10\({}^{-6}\) \\
40,000 & 3.11066 \(\times\) 10\({}^{-6}\) & 3.63867 \(\times\) 10\({}^{-6}\) \\
50,000 & 3.06778 \(\times\) 10\({}^{-6}\) & 4.76017 \(\times\) 10\({}^{-6}\) \\
60,000 & 3.14196 \(\times\) 10\({}^{-6}\) & 3.81424 \(\times\) 10\({}^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance metrics of the training set and test set in the training process. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number** & \begin{tabular}{c} **Prediction** \\ **Duration (h)** \\ \end{tabular}  & \begin{tabular}{c} **Longitude** \\ **Error (Degree)** \\ \end{tabular}  & \ \begin{tabular}{c} **Latitude** \\ **Error (Degree)** \\ \end{tabular}  & \begin{tabular}{c} **Distance** \\ **Error (km)** \\ \end{tabular}  \\ \hline
1 & 0.305 & \(-\)0.08594 & \(-\)0.00915 & 9.06339 \\
2 & 0.677 & \(-\)0.193 & \(-\)0.269 & 36.207 \\
3 & 0.854 & \(-\)0.071 & \(-\)0.118 & 15.112 \\
4 & 1.202 & \(-\)0.  
interval data, and predictions of future fixed time intervals are then carried out. As the original trajectory data are non-stationary sequences, the autoregressive prediction method uses the Autoregressive Integrated Moving Average Model (ARIMA), which converts the non-stationary sequences to stationary sequences using the difference method, and the parameters are fine-tuned with the number of autoregressive terms p = 0, the order of differencing d = 1, and the number of moving average terms q = 2. The BP method is based on the multi-dimensional trajectory features presented in this paper. It adopts a three-layer network structure and is well-fitting after 10,000 training sessions. The LSTM method uses time as input and position as output; it is trained until the loss function no longer drops. All of these methods are parametrically optimized based on the dataset to ensure optimal results for the dataset studied in this paper. The RTIME expressed as Equation (21) is computed to measure the prediction performance. The experimental results of RTIME are shown in Table 3.  
It can be seen from Table 3 that the prediction accuracy of this method is better than other methods. From the perspective of model accuracy and adaptability, the interpolation extrapolation method has inferior prediction accuracy. It can be concluded that the interpolation algorithm can only be used to complete the historical data under the constraint of both endpoints instead of predicting the extrapolation in the future. By contrast, fitting