# Particle Probability Hypothesis Density Filter Based on Pairwise Markov Chains
## 3 PF-PMC-PHD Filter  
There are two main methods to implement a PHD filter, one is a particle implementation and the other is a GM implementation. The GM implementation of the PMC-PHD filter proposed by [PERSON] and [PERSON] is only suitable for a linear Gaussian multi-target tracking system. The particle PMC-PHD (PF-PMC-PHD) is given for the nonlinear multi-target tracking system based on the PMC model in this paper. A set of weighted random samples \(\left\{w_{k}^{(i)},(\mathbf{x}_{k}^{(i)},\mathbf{y}_{k}^{(i)})\right\}_{i=1}^{L}\) are used to approximate the posterior probability density function of the pair \((\mathbf{x},\mathbf{y})\) as follows:  
\[v_{k}(\mathbf{x},\mathbf{y})\approx\sum_{i=1}^{L}w_{k}^{(i)}\delta((\mathbf{x},\mathbf{y})-( \mathbf{x}_{k}^{(i)},\mathbf{y}_{k}^{(i)})) \tag{8}\]  
where \(w_{k}^{(i)}\) represents the expected value of the pair with the state \((\mathbf{x}_{k}^{(i)},\mathbf{y}_{k}^{(i)})\). PF-PMC-PHD filter can be summarized as follow. **Step 1:** Initialization of particles. At time \(k=0\), use \(L_{0}\) particles \(\left\{w_{0}^{(i)},(\mathbf{x}_{0}^{(i)},\mathbf{y}_{0}^{(i)})\right\}_{i=1}^{L_{0}}\) to represent the prior probability density \(v_{0}(\cdot)\) of pair \((\mathbf{x},\mathbf{y})\), and the particles number is proportional to the number of targets, that is, if there are \(\hat{N}_{0}\) targets, \(w_{0}^{(i)}=\hat{N}_{0}/L_{0}\), the joint PHD function \(v_{0}(\mathbf{x},\mathbf{y})\) of pair \((\mathbf{x},\mathbf{y})\) writes as \[v_{0}(\mathbf{x},\mathbf{y})=\sum\limits_{i=1}^{L_{0}}w_{0}^{(i)}\delta((\mathbf{x},\mathbf{y})-( \mathbf{x}_{0}^{(i)},\mathbf{y}_{0}^{(i)})) \tag{9}\]  
**Step 2:** Particles prediction. At time \(k\geq 1\), the particles \((\widetilde{\mathbf{x}}_{k}^{(i)},\widetilde{\mathbf{y}}_{k}^{(i)})\) generating for the surviving targets are sampled from the proposed importance probability density function \(q_{k}(\cdot\left|(\widetilde{\mathbf{x}}_{k-1}^{(i)},\widetilde{\mathbf{y}}_{k-1}^{( i)}),Z_{k}\right. )\), \(i=1,\cdots L_{k-1}\), and the particles \((\widetilde{\mathbf{x}}_{k}^{(i)},\widetilde{\mathbf{y}}_{k}^{(i)})\), \(i=L_{k-1}+1,\cdots L_{k-1}+J_{k}\) generating for the new birth targets are sampled from another suggested density function \(p_{k}(\cdot|Z_{k})\). The corresponding predicted weights are calculated as  
\[\widetilde{w}_{k|k-1}^{(i)}=\left\{ \begin{array}{ll}w_{k-1}^{(i)}\cdot\frac{ \phi_{kk-1}\left((\widetilde{\mathbf{x}}_{k}^{(i)},\widetilde{\mathbf{y}}_{k}^{(i)}), (\mathbf{x}_{k-1}^{(i)},\mathbf{y}_{k-1}^{(i)})\right)}{q_{k}(\widetilde{\mathbf{x}}_{k}^ {(i)},\widetilde{\mathbf{y}}_{k}^{(i)})|(\mathbf{x}_{k-1}^{(i)},\mathbf{y}_{k-1}^{(i)}),Z_ {k})}&i=1,\cdots L_{k-1}\\ \frac{1}{I_{k}}\cdot\frac{b_{k}(\widetilde{\mathbf{x}}_{k}^{(i)},\widetilde{\mathbf{y} }_{k}^{(i)})}{p_{k}((\widetilde{\mathbf{x}}_{k}^{(i)},\widetilde{\mathbf{y}}_{k}^{(i) })|Z_{k})}&i=L_{k-1}+1,\cdots L_{k-1}+J_{k}\end{array} \right. \tag{10}\]  
Assuming that there is no spawning,  
\[\phi_{k|k-1}\Big{(}(\widetilde{\mathbf{x}}_{k}^{(i)},\widetilde{\mathbf{y}}_{k}^{(i)} ),(\mathbf{x}_{k-1}^{(i)},\mathbf{y}_{k-1}^{(i)})\Big{)}=p_{S,k}(\mathbf{x}_{k-1}^{(i)})f_{ k|k-1}(\widetilde{\mathbf{x}}_{k}^{(i)},\widetilde{\mathbf{y}}_{k}^{(i)}|\mathbf{x}_{k-1}^{(i)},\mathbf{y}_{k-1}^{(i)}) \tag{11}\]  
The predicted joint PHD function \(v_{k|k-1}(\mathbf{x},\mathbf{y})\) of pair \((\mathbf{x},\mathbf{y})\) writes as  
\[v_{k|k-1}(\mathbf{x},\mathbf{y})=\sum\limits_{i=1}^{L_{k-1}+I_{k}}\widetilde{w}_{k|k-1 }^{(i)}\cdot\delta((\mathbf{x},\mathbf{y})-(\widetilde{\mathbf{x}}_{k}^{(i)},\widetilde{ \mathbf{y}}_{k}^{(i)})) \tag{12}\]  
**Step 3:** Particles update.