## 4 The Results of Discriminative Parameter Training of the PAUKF  
The simulation experiment results and discussion are given in this section. To evaluate the performance of the manually tuned PAUKF and variance-trained PAUKF, the root mean square error (RMSE) is used as the assessment value. The smaller the value of RMSE, the better the algorithm is. The calculation of the RMSE is shown as Equation (5). The \(\text{RMSE}_{\text{est}}\) means the RMSE is calculated based on the filter's estimation and the \(\text{RMSE}_{\text{noise}}\) is calculated based on the noisy position without \begin{table}
\begin{tabular}{c c} \hline \hline
**Order** & **Process** \\ \hline
1 & Initialization of hyper-parameters’, \(a,\ b,\ c,\ \sigma_{x,y,\theta},\text{EC}\), training time \\
2 & Start training \(\sigma_{x,y,\theta}\) one by one \\
3 & Calculate the sigma in this iteration \\
4 & Run the PAUKF in the whole track with the calculated sigma \\
5 & Save the data from 1:T of the PAUKF and high-performance sensor \\
6 & Calculate the evaluation criteria based on data collected \\
7 & Compare the performance change \\
8 & Change the training parameters a and b based on the performance \\
9 & Start new training iteration based on the changed training parameters a and b \\
10 & End the iteration if the training process meets training times \\
11 & Save the \(\sigma_{x,y,\theta}\) as the final result \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The training process of the PAUKF based on the coordinate descent algorithm. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Value** \\ \hline Random seed & 50 \\ The noise of the high-accuracy sensor & \(\sim\)N(0, 0.01) \\ Initial value of \(\sigma_{x,y,\theta}\) & 10 \\ Initial RPE (m) & 9999 \\ Training times & 15 \\ a & 0.1 \\ b & 0.3 \\ Initial c & 0.2 \\ Velocity of vehicle (km/h) & 60, 70, 80, 90, 100, 110, 120 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters of the training process. filtering. The N represents the number of accumulated position data after the vehicle runs on the whole track. \[\left[ \begin{array}{c}\mathrm{RMSE_{est}}\\ \mathrm{RMSE_{noise}}\end{array} \right]=\left[ \begin{array}{c}\sqrt{\left[ \sum_{i=1}^{N}\left(\mathrm{Position_{esti}}-\mathrm{Position_{mean_{esti}}} \right)^{2}\right]/N}\\ \sqrt{\left[\sum_{i=1}^{N}\left(\mathrm{Position_{noise}}-\mathrm{Position_{ mean_{noise}}}\right)^{2}\right]/N}\end{array} \right], \tag{5}\]  
Figure 2 is the filtering result of the manually tuned PAUKF and the trained PAUKF at 120 km/h. Figure 2a is the trajectory of the noisy vehicle position, estimated by the PF, and the manually tuned PAUKF. Figure 2b is the trajectory of the noisy vehicle position estimated by the PF, and the PAUKF with the trained measurement variance. The blue line with the blue circles is the ground truth position of the vehicle, the red dashed line with the red triangles is the noisy GPS sensor measured position, the yellow dash line with the yellow squares is the estimation result of the PF and the black dash line with the black asterisks is the final estimation result of the PAUKF. The estimation results of the manually tuned PAUKF and the variance-trained PAUKF have many differences. From the figure, the algorithm with trained variance has better performance in the whole track. The most significant differences are marked with red circle A and red circle B for comparison. Red circle A shows the estimation results of the vehicle at the start. The vehicle localizes itself quickly in both of the figures. However, the error of estimation in (a) becomes larger than the error in (b). Since the parameters and random seed are fixed, the performance of the PF and UKF does not change, meaning that the only reason for the divergent performances is the weight of the measurement which is decided on the measurement variances in the PAUKF. Compared to the manually tuned PAUKF, the estimation error of the variance-trained PAUKF is smaller and smoother.