## 4 Analytical Implementation of VB-TPHD
### The Implementation of Gaussian Mixture VB-TPHD Filter
#### 4.1.2 The Step of Prediction  
Given the expansion of the \(\mathbf{k}-1\) time trajectory in space trajectory PHD, \(\sum_{j=1}^{j_{k-1}}w_{k-1}^{j}(\mathbf{N}(t,x^{1:k_{1-1}};t_{k-1}^{j},\hat{m}_{k -1}^{j},\hat{p}_{k-1}^{j})\prod_{l=1}^{d}\mathbf{IG}((\sigma_{k-1}^{j,j})^{2}; \alpha_{k-1}^{i,j},\beta_{k-1}^{i,j}))\), in which the mean \(\hat{m}_{k-1}^{j}\in\mathbb{R}^{i_{k-1}n_{x}}\), covariance \(\hat{P}_{k-1}^{j}\in\mathbb{R}^{i_{k-1}n_{x}\times i_{k-1}n_{x}}\), \(t_{k-1}^{j}\) represents the start time of the trajectory, \(i\,_{k-1}^{j}\) represents the duration of the trajectory and \(j\) is the serial number of the trajectory.  
**Assumption 8**.: _Survival probability and detection probability are set as constant values in trajectory filter:_  
\[p_{s,k}(\overline{X})=p_{s^{\prime}},p_{D,k}(\overline{X})=p_{D} \tag{35}\]  
**Proposition 3**.: _Based on Assumptions 1-3 and Assumptions 7 and 8, there is the Gaussian mixture form of VB-TPHD prediction:_  
\[D_{\omega_{k}}(\overline{X})=D_{\gamma_{k}}(\overline{X})+p_{s}\sum_{j=1}^{J_{ \xi_{k}}}w_{\xi_{k}}^{j}\Bigg{(}\mathbf{N}\Big{(}X;t_{\xi_{k}}^{j},\hat{m}_{ \xi_{k}}^{j},\hat{p}_{\xi_{k}}^{j}\Big{)}\prod_{l=1}^{d}\mathbf{IG}\Big{(} \Big{(}\sigma_{\xi_{k}}^{j,j}\Big{)}^{2};\alpha_{\xi_{k}}^{l,j},\beta_{\xi_{k} }^{l,j}\Big{)}\Bigg{)}, \tag{36}\]  
_where_  
\[w_{\xi_{k}}^{j}=p_{s}w_{k-1}^{j}, \tag{37}\]  
\[\hat{m}_{\xi_{k}}^{j}=\Bigg{[}(\hat{m}_{k-1}^{j})^{T},(\hat{F}\cdot\hat{m}_{k- 1}^{j})^{T}\,\Bigg{]}^{T}, \tag{38}\]  
\[\hat{P}_{\xi_{k}}^{j}=\Bigg{[}\begin{matrix}\hat{P}_{k-1}^{j}&\hat{P}_{k-1}^{ j}\hat{F}^{T}\\ \hat{F}\hat{P}_{k-1}^{j}&\hat{F}\hat{P}_{k-1}^{j}\hat{F}^{T}+\mathcal{Q}\end{matrix} \Bigg{]}, \tag{39}\]  
\[\dot{F}\,=\,\Big{[}0_{1,t_{k-1}^{j}-1},1\Big{]}\otimes F, \tag{40}\]  
\[\alpha_{\xi_{k}}^{l,j}=\rho\cdot\alpha_{k-1}^{l,j},\ \beta_{\xi_{k}}^{l,j}=\rho \cdot\beta_{k-1}^{l,j}, \tag{41}\]  
_where \(t_{\xi_{k}}^{l}\) represents the starting time of the extended trajectory state and \(\rho\) is the attenuation factor which is considered to be suitable for 0.9-0.95 according to [29]._  
The above expression can be mainly predicted based on the surviving extended trajectory state \(\overline{X}_{\xi_{k}}\) bearing at \(k-1\). The noise covariance parameters of the surviving extended tracking state at \(k-1\) should be multiplied by the fading factor \(\rho\). The proof of Proposition 3 is arranged in Appendix C. The surviving extended tracking component and the newly formed extended trajectory component should be combined as the integral Gaussian components after the step of prediction, \(\hat{m}_{\omega_{k}}^{1:J_{\xi_{k}}+J_{\eta_{k}}}=\Bigg{\{}\hat{m}_{\xi_{k}}^{1: [PERSON]_{\xi_{k}}},m_{\gamma_{k}}^{1:[PERSON]_{\eta_{k}}}\Bigg{\}},\hat{P}_{\omega_{k}}^{1:[PERSON]_{ \xi_{k}}+J_{\eta_{k}}}=\Bigg{\{}\hat{P}_{\xi_{k}}^{1:[PERSON]_{\xi_{k}}},P_{\gamma_{k} }^{1:[PERSON]_{\eta_{k}}}\Bigg{\}}\), \(w_{\omega_{k}}^{1:[PERSON]_{\xi_{k}}+J_{\gamma_{k}}}=\Bigg{\{}w_{\xi_{k}}^{1:[PERSON]_{\xi_{k}}}, w_{\gamma_{k}}^{1:[PERSON]_{\gamma_{k}}}\Bigg{\}}\), \(\alpha_{\omega_{k}}^{1:[PERSON]_{\xi_{k}}+[PERSON]_{\gamma_{k}}}=\)\(\Big{\{}\hat{\alpha}_{\xi_{k}}^{1:[PERSON]_{\xi_{k}}},\alpha_{\eta_{k}}^{1:[PERSON]_{\gamma_{k}}} \Bigg{\}}\), \(\beta_{\omega_{k}}^{1:[PERSON]_{\xi_{k}}+[PERSON]_{\gamma_{k}}}=\)\(\Big{\{}\hat{\beta}_{\xi_{k}}^{1:[PERSON]_{\xi_{k}}},\beta_{\gamma_{k}}^{1:[PERSON]_{\gamma_{k}}} \Bigg{\}}\).