## 4 The Results of Discriminative Parameter Training of the PAUKF
The performance of the PAUKF becomes more distinct at circle B. The manually tuned PAUKF tends to put more weight on the prediction model. However, not only does the prediction model not describe the physical movement, but the values used in the prediction contain a lot of noise. Due to the non-optimized measurement variances, the algorithm cannot balance the information well. As a result, the PAUKF cannot fully use the information from the PF which provides an accurate measurement based on map matching. Due to the non-optimized variance value, the position error of the manually tuned PAUKF is larger than the error of the variance-trained PAUKF at circle B.  
Figure 2: The position estimation results of the PAUKF. (**a**) The position estimation result of the PAUKF with manually tuned variance; (**b**) the position estimation result of the PAUKF with trained variance. The results of the RMSE of the filter at different velocities are shown in Table 3. The title \"position of vehicle\" means the RMSE of the noisy position of the vehicle. The title \"manual\" means the data are generated by the manually tuned PAUKF. The title \"trained\" means the data are generated with variance-trained PAUKF. The \"Mean\" means the average of the RMSE of different velocity conditions and the \"RMSE change\" means the difference in the RMSE of the manually tuned PAUKF and the variance-trained PAUKF. As illustrated in Section 3, the parameters used in the trained variance PAUKF are the same as manually tuned PAUKF, meaning the only variable making the result change is the value of the measurement variance. The RMSE which is calculated based on the second row of Equation (5) shows that both the manually tuned PAUKF and variance-trained PAUKF are tested under the same noisy environment, and the RMSE of the manually tuned PF and trained PF shows the same value. This means the training process does not affect the performance of the PF. As a result, the parameter \"RMSE change\" in the title \"position of vehicle\" is equal to 0%. The change in the RMSE happens at the PAUKF column. The RMSE of the PF and PAUKF is calculated based on the first row of Equation (5). Compared to the manually tuned PAUKF, the RMSE of the variance-trained PAUKF decreases about 15.7% on average only because of the well-trained measurement covariance values. The RMSE results show that the trained variance PAUKF improves the performance without adding any computational burden. The numerical value of measurement variances affects the estimation performance, as seen from the results. The variance is trained by the coordinate descent algorithm in 15 iterations of each parameter. In the training algorithm, the variance of x, y, and \(\theta\) is calculated by the standard deviation. Therefore, Figures 3-5 show the record of the standard deviation of x, y, and \(\theta\). Since the initial measurement variance is equal to 10, the PAUKF tends to give greater weight to the prediction model than to the measurement. From Figure 3, the standard deviation of x converged at iteration 15-16. The final standard deviation converges to different values at different velocities. The reason for this phenomenon is the noise in the prediction model decreasing the precision of the estimation. Therefore, the error of the prediction model of the PAUKF becomes larger. Conversely, the estimation of the PF is not affected by the velocity of the vehicle, meaning the PAUKF should give more weight to the PF estimation. The smaller the standard deviation of the measurement, the more credence the PAUKF gives to the measurement value. So, the weight of the PF is represented as the standard deviation and the standard deviation decreases as the velocity increases, as shown in Figure 3. The standard deviation of y shows the same property of the standard deviation of x, as shown in Figure 4.