# Particle Probability Hypothesis Density Filter Based on Pairwise Markov Chains
## 4 Experimental Simulation
### Performance Analysis  
Let us now analyze quantitatively the tracking performance of the PF-PMC-PHD filter and PF-HMC-PHD filter in a nonlinear system based on the PMC model. We compute at each time step the OSPA (Optimal Sub-Pattern Assignment) metric and the target number estimation. The OSPA metric can simultaneously evaluate the target number estimation error and position error of the multi-target tracking. Let \(X=\{\mathbf{x}_{1},\cdots,\mathbf{x}_{m}\}\) and \(\hat{X}=\{\hat{\mathbf{x}}_{1},\cdots,\hat{\mathbf{x}}_{n}\}\) be two finite sets. Here, \(\hat{X}\) represents the estimated finite set of the targets and \(X\) represents the true finite set of targets. For \(1\leq p<+\infty\) and \(c>0\), let \(d^{(\mathrm{c})}(\mathbf{x},\hat{\mathbf{x}})=\min(\mathrm{c},\|\mathbf{x}-\hat{\mathbf{x}}\|)\) (\(\|\cdot\|\) is the Euclidean norm), \(\Pi_{n}\) be the set of permutations on \(\{1,2,\cdots,n\}\), and let \(\pi(i)\) be the \(i\)th component of a given permutation \(\pi\). The OSPA metric is defined by [13]  
\[\overline{d}_{p}^{\mathrm{c}}(X,\hat{X})\triangleq\left(\frac{1}{n}\Bigg{(} \min_{\pi\in\Pi_{n}=1}^{m}d^{(\mathrm{c})}(\mathbf{x}_{i},\hat{\mathbf{x}}_{\pi(i)})^{ p}+c^{p}(n-m)\Bigg{)}\right)^{\frac{1}{p}} \tag{31}\]  
if \(m\leq n\) and by \(\overline{d}_{p}^{\mathrm{c}}(X,\hat{X})\triangleq\overline{d}_{p}^{\mathrm{ c}}(\hat{X},X)\) if \(m>n\). In our simulations, we set \(p=100\) and \(c=1\). Set  
\[F_{k}=\left[ \begin{array}{ccc}1&\frac{\sin\Omega T}{\Omega}&0&-\frac{1-\cos \Omega T}{\Omega}\\ 0&\cos\Omega T&0&-\sin\Omega T\\ 0&\frac{1-\cos\Omega T}{\Omega}&1&\frac{\sin\Omega T}{\Omega}\\ 0&\sin\Omega T&0&\cos\Omega T\end{array} \right],F_{k}^{2}=\left[ \begin{array}[] {ccc}0.7&0\\ 0&0\\ 0&0.7\\ 0&0\end{array} \right] \tag{32}\]  
\[H_{k}=\left[ \begin{array}{ccc}1&0&0&0\\ 0&0&1&0\end{array} \right],H_{k}^{2}=\left[ \begin{array}{ccc}0.1&0\\ 0&0.1\end{array} \right] \tag{33}\]  
\[Q_{k}=\left[ \begin{array}{ccc}25&1&0&0\\ 1&5&0&0\\ 0&0&25&1\\ 0&0&1&5\end{array} \right],\ R_{k}=\left[ \begin{array}{ccc}4&0\\ 0&4\end{array} \right] \tag{34}\]  
We generate uniformly a mean of 10 clutter measurements on the region \(V=[-2000,2000]\times[-2000,2000]\) with a sampling period of \(T=1s\) and a simulation experiment step of \(N=50\). There are 4 targets: targets 1 and 2 appear at time \(k=1\) and targets 3 and 4 appear at time \(k=20\). We track the position and velocity of the targets in Cartesian coordinates, \(\mathbf{x}_{k}=\left[p_{\mathbf{x}_{k}},\dot{p}_{\mathbf{x}_{k}},p_{\mathbf{y}_{k}},\dot{p}_{ \mathbf{y}_{k}}\right]^{T}\). We set \(p_{5,k}=0.98\) and \(p_{D,k}=0.9\) and select \(L=1000\) for the particle number of one target. The target trajectories and measurements of such a scenario is displayed in Figure 1.  
Figure 1: The target trajectories and measurements: Trajectories of four targets are represented by lines and the measurements taken from the sensor are represented by “Union Jack”. Figure 2 shows the tracking result of PF-PMC-PHD. The tracking result shows that the proposed PF-PMC-PHD filter can effectively achieve the target tracking. Figures 3 and 4 respectively show target number estimations and the OSPA metrics of the PF-PMC-PHD filter and PF-HMC-PHD filter. The simulation result shows that the tracking performance of the PF-PMC-PHD filter is better than that of PF-HMC-PHD filter although the two filters share the same \(p(\mathbf{x}_{k}|\mathbf{x}_{k-1})\) and \(p(\mathbf{y}_{k}|\mathbf{x}_{k})\). This is because the HMC model does not take into account the information given by the observation \(\mathbf{y}_{k-1}\) in the case of the given \(\mathbf{x}_{k-1}\) and \(\mathbf{y}_{k-1}\) and because the uncertainty of the state \(\mathbf{x}_{k}\) increases. The target number estimation error of the PF-HMC-PHD filter is large, and the situation of missing target is serious.