# Vehicle Trajectory Reconstruction Using Lagrange-Interpolation-Based Framework
## 5 Discussion  
This paper uses the 1942 nd trajectory in the NGSIM dataset to repair outliers and denoise, compared with the baseline method proposed by [PERSON] [37]. And it uses three indexes (root mean square error (RMSE), signal-to-noise ratio (SNR) and Pearson correlation coefficient (P)) to demonstrate that the proposed method has good reconstruction results. The three measurements are defined by Equations (17)-(19). \[RMSE=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}\left(S_{i}-\hat{S_{i}}\right)^{2}} \tag{17}\]  
\[SNR=10\log\left(\frac{\sum\limits_{i=1}^{N}{{S_{i}}^{2}}}{\sum\limits_{i=1}^{ N}\left(S_{i}-\hat{S_{i}}\right)^{2}}\right) \tag{18}\]  
\[P=\frac{\sum\limits_{i=1}^{N}\left(S_{i}-\overline{S}_{i}\right)\left(\hat{S}_ {i}-\overline{S}_{i}\right)}{\sum\limits_{i=1}^{N}\left(S_{i}-\overline{S}_{ i}\right)^{2}\sum\limits_{i=1}^{N}\left(\hat{S}_{i}-\overline{S}_{i}\right)} \tag{19}\]  
where \(S_{i}\) is the empirical vehicle trajectory data [41]; \(\hat{S}_{i}\) is the reconstructed trajectory data; \(\overline{S}_{i}\) and \(\overline{S}_{i}\) denote the average of the empirical vehicle trajectory data and the reconstructed trajectory data, respectively; and \(N\) is the length of trajectory data. This paper calculates the three measurements (RMSE, SNR and P) between the baseline method and the proposed method. The results are shown in Table 2.  
Table 2 shows that the RMSE of the proposed method is smaller than that of the baseline method. The SNR is greater than that of the baseline method. It shows that this proposed method retains more trajectory information and denoises better. By calculating the average increase percentage of the measurements, it finds that the RMSE of this proposed method is reduced by 45.76%, the SNR is increased by 26.43%, and the P is increased by 25.58% compared with the baseline method. It is observed that the results of the proposed method are better than those of the baseline framework in Figure 8. The paper uses the acceleration distribution of trajectory data to analyze the acceleration and its proportion. The acceleration distributions among the proposed method, the baseline method and the original trajectory are shown in Figure 9.  
Figure 9 shows that acceleration of more than 10% in original data is greater than \(\pm\)9 m/s\({}^{2}\). It exceeds the human body bearing limit. The acceleration of the baseline method is more than 20% outside the range of \(\pm\)5 m/s\({}^{2}\), and that of the proposed method is in the range of [-3 m/s\({}^{2}\), 3 m/s\({}^{2}\)]. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Index** & \multicolumn{2}{c}{**Baseline Method**} & **Proposed Method** \\ \hline \multirow{3}{*}{RMSE} & Position & 0.05 & 0.035 \\  & Speed & 0.08 & 0.05 \\  & Acceleration & 0.43 & 0.13 \\ \hline \multirow{3}{*}{SNR} & Position & 75.13 & 79.22 \\  & Speed & 36.8 & 44.7 \\  & Acceleration & 8.4 & 12.8 \\ \hline \multirow{3}{*}{P} & Position & 1.00 & 1.00 \\  & Speed & 1.00 & 1.00 \\ \cline{1-1}  & Acceleration & 0.77 & 0.967 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the measurements obtained by the two frameworks. The jerk value is the derivative value of acceleration. It indicates the change degree of acceleration in unit time. It is used to analyze whether trajectory reconstruction results are within the limitation of vehicle dynamic and driver driving behavior. The main indexes include the percentage of jerks that are greater than \(\pm\)15 m/s\({}^{3}\), the maximum jerk value, the minimum jerk value, and the percentage that the jerk symbol changes more than once in one second (N). The results are shown in Table 3.  
Table 3 shows that the percentage of jerks that are greater than \(\pm\)15 m/s\({}^{3}\) is 22.93% in the original track; the reconstruction results of the baseline method and the proposed method are 0.04% and 0.03%, respectively. The extreme jerks in the proposed paper are [\(-\)13.26 m/s\({}^{3}\), 16.83 m/s\({}^{3}\)].