## 5 De-Noising Filter Method and Experimental Results
### Experimental Results  
In this section we aim at comparing the position estimation performance of the two methods : the robust-EKF and the rEKF-LSTM.  
A total of 20 hidden layers were set up for the LSTM in this paper. The LSTM set up to H = 20 hidden layers, N = 3 inputs features (positions), S = 1440 number of timesteps and M = 3 outputs features (errors positions). The data used are still from ABMF station recorded on the 1 January 2019 (as in Section 4.2). Figure 13 represents the position errors with respect to time for the two methods: robust-EKF and rEKF-LSTM. The period of data is from 12 h to 24 h, since data from 0 h to 12 h is used to train the model.  
In Figure 13, the green curve represents the position errors for the robust-EKF method, whereas the purple curve represents the errors produced by the rEKF-LSTM method. It can be seen that the position accuracy of rEKF-LSTM method largely improves and remains stable. To have more insights, Table 3 summarizes the RMS position errors of the two methods.  
As shown in Table 3, the position accuracy improves by about 74.0 % using rEKF-LSTM compared to the robust EKF. To have a more general view, the five scenarios in Section 4.2 and two scenarios in this section are combined. As a result, six scenarios are considered since the scenario pertaining to robust-EKF is in common. The underlying results are summarized in Table 4.  
Figure 11: rEKF-LSTM Validation architecture.  
Figure 12: rEKF-LSTM Prediction architecture.  
As shown in Table 4, the position precision is improved by about 95% using the rEKF-LSTM compared to the non-robust data combination. This result proves that applying LSTM method can greatly improve the position accuracy. In order to assess more thoroughly the rEKF-LSTM method, other base station data are tested and the results are summarized in Table 5.  
Table 5 gives the RMS errors in position for three other base stations: AJAC, GRAC and LMMF. The position precision is enhanced by about 87%, 77% and 93% using the rEKF-LSTM compared to non-robust data combination from the three base stations AJAC, GRAC and LMMF in France  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**RMS Error (m)** & **GPS** & **GAL** & **GLO** & **GPS/GAL/GLO** & **Robust-EKF GPS/GAL/GLO** & **rEKF-LSTM** \\ \hline
**RMS-E** & 1.43 & 1.21 & 30.71 & 7.08 & 0.82 & 0.39 \\
**RMS-N** & 1.58 & 1.19 & 6.18 & 2.27 & 0.83 & 0.32 \\
**RMS-U** & 3.32 & 3.07 & 21.10 & 8.18 & 2.04 & 0.36 \\
**3D-RMS** & 3.95 & 3.50 & 37.76 & 11.05 & 2.35 & 0.62 \\ \hline \hline \end{tabular}
\end{table}
Table 4: RMS Errors in Position for the six scenarios at station ABMF in late 12 h.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**RMS Error (m)** & **Robust-EKF** & **rEKF-LSTM** \\ \hline
**RMS-E** & 0.82 & 0.39 \\ \hline
**RMS-N** & 0.83 & 0.32 \\ \hline
**RMS-U** & 2.04 & 0.36 \\ \hline
**3D-RMS** & 2.35 & 0.62 \\ \hline \hline \end{tabular}
\end{table}
Table 3: RMS Errors in position for the two methods using data from ABMF station in late 12h.  
Figure 13: Comparison of position errors between robust-EKF and rEKF-LSTM.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**RMS Error (m)** & **GPS** & **GAL** & **GLO** & **GPS/GAL/GLO** & **Robust-EKF GPS/GAL/GLO** & **rEKF-LSTM** \\ \hline
**RMS-E** & 1.43 & 1.21 & 30.71 & 7.08 & 0.82 & 0.39 \\
**RMS-N** & 1.58 & 1.19 & 6.18 & 2.27 & 0.83 & 0.32 \\
**RMS-U** & 3.32 & 3.07 & 21.10 & 8.18 & 2.04 & 0.36 \\
**3D-RMS** & 3.95 & 3.50 & 37.76 & 11.05 & 2.35 & 0.62 \\ \hline \hline \end{tabular}
\end{table}
Table 4: RMS Errors in Position for the six scenarios at station ABMF in late 12 h.  
on 1 January 2019, respectively. This corroborates the conclusion that the rEKF-LSTM method can significantly improve the position precision.