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伯努利滤波器（七）——伯努利粒子滤波器

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强度模型下的伯努利粒子滤波：

输入： 存在概率 $\varphi$, ${\omega^i_{k-1},\mathbf{m}^i_{k-1}}^{N+N_B}_{i=1}$, $\mathbf{z}_k$

  1. 预测存在概率 $q_{k|k-1} = p_b(1-q_{k-1})+p_sq_{k-1}$。
  2. 取样$\mathbf{m}^i_{k|k-1} \sim \varrho _k(\mathbf{m}k|\mathbf{m}^i{k-1},\mathbf{z}_k)$ for $i = 1,…,N+ N_B$
  3. 预测概率 $\omega^i_{k|k-1}$，
    $\omega^i_{k|k-1} = \left { \begin{matrix} \frac{p_sq_{k-1}}{q_{k|k-1}}\frac{\pi_{k|k-1}(\mathbf{m}^i_{k|k-1}|\mathbf{m}^i_{k-1})}{\varrho_k(\mathbf{m}^i_{k|k-1}|\mathbf{m}^i_{k-1},\mathbf{Z}k)}\omega^i{k-1} & i = 1,…,N \ \frac{p_b(1-q_{k-1})}{q_{k|k-1}}\frac{\pi_{k|k-1}(\mathbf{m}^i_{k|k-1}|\mathbf{m}^i_{k-1})}{\varrho_k(\mathbf{m}^i_{k|k-1}|\mathbf{m}^i_{k-1},\mathbf{Z}_k)}\frac{1}{B} & i = N+1,…,N+B \end{matrix} \right .$
  4. 计算likelihood $h_k(\mathbf{z}k|\emptyset) = \Pi^n{s=1} g^r_0(z^r_k)$
  5. for $i = 1,…,N+N_B$ do
    1. 计算likelihood $h_k(\mathbf{z}k|\mathbf{m}^i{k|k-1}) =\Pi^n_{s=1} g^r_1(z^r_k |\mathbf{m}^i_{k|k-1})$,
    2. 计算likelihood 比率 $l_k(\mathbf{z}k|\mathbf{m})= \frac{h_k(\mathbf{z}_k|{\mathbf{m}{}})}{h_k(\mathbf{z}_k|\emptyset)}=\Pi^n{r=1} \frac{g^r_1(\mathbf{z}^r_k|\mathbf{m})}{g^r_0(\mathbf{z}^r_k)}$
  6. end for
  7. 近似积分$I_k \approx\sum^{N+N_{B}}{i=1}l_k(\mathbf{z}|\mathbf{m}^i_{k|k-1})\omega^i{k|k-1}$
  8. 更新存在概率 $q_{k} = \frac{\mathbf{I}kq{k|k-1}}{1-q_{k|k-1}+q_{k|k-1}I_k}$
  9. for $i = 1,…,N+N_B$ do
    1. $\hat {\omega}^i_k = l_k(\mathbf{z}k|\mathbf{m}^i{k|k-1})\omega^i_{k|k-1}$
  10. end for
  11. 归一化$\hat {\omega}^i_k-> {\omega}^i_k$
  12. 对 $i = 1,.., N$ Resample, $\mathbf{m}^i_{k|k-1}->\mathbf{m}^i_{k}$
  13. 粒子正规化(MCMC move)
  14. 对 $i = 1,.., N$ $\omega^i_k = 1/N$
  15. 产生birth particles, $\mathbf{m}^i_k \sim b_k(\mathbf{m};\mathbf{z}_k), i =N+1,…,N+B$
  16. $\mathbf{\omega}^i_k \sim 1/N_B, i =N+1,…,N+B$
  17. 输出：$q_{k}$, ${\omega^i_{k},\mathbf{m}^i_{k}}^{N+N_B}_{i=1}$.

检测模型下的伯努利粒子滤波：

输入： 存在概率 $\varphi$, ${\omega^i_{k-1},\mathbf{m}^i_{k-1}}^{N+N_B}_{i=1}$, $\mathbf{Z}_k$

  1. 预测存在概率 $q_{k|k-1} = p_b(1-q_{k-1})+p_sq_{k-1}$。
  2. 取样$\mathbf{m}^i_{k|k-1} \sim \varrho _k(\mathbf{m}k|\mathbf{m}^i{k-1},\mathbf{z}_k)$ for $i = 1,…,N+ N_B$
  3. 预测概率 $\omega^i_{k|k-1}$，
    $\omega^i_{k|k-1} = \left { \begin{matrix} \frac{p_sq_{k-1}}{q_{k|k-1}}\frac{\pi_{k|k-1}(\mathbf{m}^i_{k|k-1}|\mathbf{m}^i_{k-1})}{\varrho_k(\mathbf{m}^i_{k|k-1}|\mathbf{m}^i_{k-1},\mathbf{Z}k)}\omega^i{k-1} & i = 1,…,N \ \frac{p_b(1-q_{k-1})}{q_{k|k-1}}\frac{\pi_{k|k-1}(\mathbf{m}^i_{k|k-1}|\mathbf{m}^i_{k-1})}{\varrho_k(\mathbf{m}^i_{k|k-1}|\mathbf{m}^i_{k-1},\mathbf{Z}_k)}\frac{1}{B} & i = N+1,…,N+B \end{matrix} \right .$
  4. 近似积分$I_1 \approx \sum^{N+N_b}{i=1}p_d(\mathbf{m}^i_{k})\omega^i{k|k-1}$
  5. 对于$\mathbf{z}\in\mathbf{Z}k$计算近似积分$I{2}(\mathbf{z})\approx\sum^{N+N_B}{i=1}p_d(\mathbf{m}^i_{k|k-1})h_k(\mathbf{z}|\mathbf{m}^i_{k|k-1})\omega^i{k|k-1}$
  6. $\triangle_k \approx I_1-\sum_{\mathbf{z}\in\mathbf{Z}_k}\frac{I_2(\mathbf{z})}{\lambda c(\mathbf{z})}$
  7. 更新存在概率$q_k = \frac{1- \triangle_k}{1-\triangle_k q_{k|k-1}q_{k|k-1}}$
  8. 更新权重$\hat{\omega}^i_k = [1-p_d(\mathbf{m}^i_{k|k-1})+p_d(\mathbf{m}^i_{k|k-1})\sum_{\mathbf{z}\in\mathbf{Z}k}\frac{h_k(\mathbf{z}|\mathbf{m}^i_{k|k-1})}{\lambda c(\mathbf{z})}]\omega^i{k|k-1}$
  9. 归一化$\hat {\omega}^i_k-> {\omega}^i_k$
  10. 对 $i = 1,.., N$ Resample, $\mathbf{m}^i_{k|k-1}->\mathbf{m}^i_{k}$
  11. 粒子正规化(MCMC move)
  12. 对 $i = 1,.., N$ $\omega^i_k = 1/N$
  13. 产生birth particles, $\mathbf{m}^i_k \sim b_k(\mathbf{m};\mathbf{z}_k), i =N+1,…,N+B$
  14. $\mathbf{\omega}^i_k \sim 1/N_B, i =N+1,…,N+B$
  15. 输出：$q_{k}$, ${\omega^i_{k},\mathbf{m}^i_{k}}^{N+N_B}_{i=1}$.