## 5 Simulation and Discussion
### Scenario 1
Figure 1: (**a**) The trajectory estimates of AGM-VB-TPHD until time 40. (**b**) The trajectory estimates of AGM-VB-TPHD until time 75. (**c**) The trajectory estimates of AGM-tag-VB-PHD until time 40. (**d**) The trajectory estimates of AGM-tag-VB-PHD until time 75. The blue line represents the true trajectories of the targets in the period, the circle represents the starting point of the trajectory, the triangle represents the end point of the trajectory, different trajectories are marked with different color styles and the black asterisk represents the measurement at the current time. Figure 2 shows the estimation process of AGM-VB-TPHD filter for the unknown measurement noise standard deviation. In the period with new targets' appearance, larger standard deviation estimation may occur. However, as time goes by, the two independent components of standard deviation approach the unknown true standard deviation adaptively and gradually tend to be stable. To highlight the performance of AGM-VB-TPHD filters, we conducted the simulations of the adaptive newborn intensity Gaussian mixture tag-VB-PHD (AGM-tag-VB-PHD) filter [20; 34], AGM-TPHD filter with larger estimated covariance (\(R_{1}\) = diag(40, 40)) and smaller estimated covariance (\(R_{2}\) = diag(0.25, 0.25)) for comparation under the same parameters. The effective number of tracking targets at each time can be seen from the cardinality distribution. The closer the cardinality to the true value is, the lower the target losing probability is, and the more stable the tracking performance is. Figure 3 shows that AGM-VB-TPHD filter and AGM-tag-VB-PHD filter have little difference in the estimation of target number and are more accurate than that whose covariance estimation is wrong. When the estimated covariance is \(R_{1}\), the position parameters of the covariance of the new intensity are much smaller than the estimated covariance parameters, resulting in the estimation mismatch, so that the filter cannot detect the target immediately while a new target is generated. When the estimated covariance is \(R_{2}\), it can be seen from Figure 3 and the GOSPA missed target cost in the Figure 4 that the degree of target missed detection is serious. This is because the covariance estimation is too small to cover the range of targets that should be detected, resulting in a large loss of targets. Figure 3: The average cardinalities of different filters. Figure 2: The average standard deviation of estimated measurement noise of AGM-VB-TPHD. From the square GOSPA error on the upper left of Figure 4, we can see that the error of AGM-VB-TPHD filter is the smallest and the tracking effect is the best. The performance of the AGM-VB-TPHD filter is obviously better than that of the AGM-tag-VB-PHD filter under the three indexes of localization, missed detections and false targets. The AGM-tag-VB-PHD filter is based on labeling each PHD component. It does not improve the accuracy of the PHD itself, and if multiple components extracted have the same label, it will cause missed target and false detection. A larger estimated covariance will lead to a larger range of target detection. As part of the measurement is redundant clutter, there will be the generation of wrong targets. A large covariance can lead to a small estimated error of target localization, but it is still inferior to the AGM-VB-TPHD filter under the overall GOSPA metric. If the estimated covariance is too small, the target will be missed. After the adaptive covariance is stabilized, the AGM-VB-TPHD filter shows the best performance in the three indicators clearly. Based on overall evaluation, the AGM-VB-TPHD filter has the smallest GOSPA error and the most accurate number of estimated trajectories.