## 5 Simulation and Discussion  
### Scenario 1  
To prove the tracking performance of Gaussian mixture VB-TPHD, a two-dimensional extended trajectory space scenario with unknown and constant noise covariance is designed in Scenario 1. Suppose there are four targets can form trajectories in the scenario and each target obeys the linear Gaussian dynamic and measurement model such as Equations (28) and (29), that is, it moves at a constant velocity. Each model is expressed as follows:  
\[F_{k}= \begin{bmatrix}I_{2}&\Delta I_{2}\\ 0_{2}&I_{2}\end{bmatrix} ,\;Q_{k}=\begin{array}{ll}\sigma_{v}^{2} \begin{bmatrix} \frac{\Delta^{4}}{I_{2}}&\frac{\Delta^{3}}{3}I_{2}\\ \Delta^{3}&I_{2}&\Delta^{2}I_{2}\end{bmatrix} , \tag{59}\]  
\[H_{k}= \begin{bmatrix}I_{2}&0_{2}\end{bmatrix} ,\;R_{k}=\text{diag}(\sigma_{v_{1 }}^{2},\;\sigma_{v_{2}}^{2}),\]  
where \(I_{2}\) and \(0_{2}\) represent \(2\times 2\) identity matrix and zero matrix, respectively, and \(\Delta\) represents the scanning period. In scenario 1, \(\Delta=1\) s, \(\sigma_{v}=\;1.8\) m/s represents the standard deviation of the process noise. Similarly, \(\sigma_{v_{1}}\) and \(\sigma_{v_{2}}\) represent the unknown quantity of the standard deviation of the measurement noise. The state of the target in the extended space can be written as follows: the first and second dimensions are the \(x\) and \(y\) coordinates of the target, respectively, and the third and fourth dimensions are the fractional velocities of the target in the \(x\) and \(y\) direction respectively. The first and second dimensions of measurement obtained are the \(x\) and \(y\) coordinates of the measurement, respectively. Next are some settings for the scenario parameters. The size of scenario is set to \([0,1000]\) m \(\times\)\([0,1000]\) m and the total tracing duration is set to \(100\) s. The trajectory survival probability is set as \(p_{s}=0.99\) and the detection probability is set as \(p_{D}=0.98\). The clutter generated in each scan obeys the uniform distribution and its number obeys the Poisson distribution with the parameter \(\lambda=5\). The true unknown measurement noise covariance \(R_{k}\) is set to be \(\text{diag}(4,\;4)\). The initial state, birth time and death time of the trajectory in the specific extended state space are shown in Table 1. The threshold of pruning is set as \(\varepsilon_{p}=\;10^{-4}\), the threshold of absorption is set as \(\varepsilon_{a}=\;4\) and the maximum Gaussian component number is set as \(J_{\text{max}}=\;70\). The simulation in scenario 1 adopts L-scan approximation whose \(L=5\). The covariance matrix for measurement noise obeys the inverse Gamma distribution. The initial parameters are \(\alpha_{1}=\;\beta_{1}=\;1,\;\alpha_{2}=\;\beta_{2}=\;1,\) and the given fading factor is \(\rho=0.95\). In scenario 1, we conduct 1000 Monte Carlo runs with MATLAB 2021a on 2.50 GHz Intel I5 laptop. GOSPA metric [39] is used to evaluate the performance of extended trajectory estimation. Simulation parameters are set as \(p=2\), \(c=10\), \(\alpha~{}=~{}2\). The GOSPA error of the target is decomposed into GOSPA localization cost, GOSPA missed target cost and GOSPA false target cost. From these aspects, the root mean square error (RMS) of Monte Carlo runs is obtained to analyze the filter. Figure 1 reflects the true tracking of the four targets and it can be seen that a good track is formed in the extended state space. Figures 2-4 respectively show the average standard deviation of the measurement noise, the average cardinality estimated by AGM-VB-TPHD and the comparison with the other filters under the GOSPA metric. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Initial Targets States** & **Birth Time/s** & **Death Time/s** \\ \hline Target 1 & \{100; 400; 12.14; 4.86\} & 2 & 45 \\ Target 2 & \{450; 300; \(-\)3.43; 4.29\} & 2 & 65 \\ Target 3 & \{100; 150; 6.15; 0.77\} & 15 & 85 \\ Target 4 & \{100; 400; 3.75; \(-\)3.75\} & 20 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The information of true targets.