伯努利滤波器(七)——伯努利粒子滤波器

强度模型下的伯努利粒子滤波:

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

  1. 预测存在概率 $q{k|k-1} = p_b(1-q{k-1})+psq{k-1}$。
  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 $\omega^i{k|k-1}$,
    $\omega^i
    {k|k-1} = \left { \begin{matrix} \frac{psq{k-1}}{q{k|k-1}}\frac{\pi{k|k-1}(\mathbf{m}^i{k|k-1}|\mathbf{m}^i{k-1})}{\varrhok(\mathbf{m}^i{k|k-1}|\mathbf{m}^i{k-1},\mathbf{Z}_k)}\omega^i{k-1} & i = 1,…,N \ \frac{pb(1-q{k-1})}{q{k|k-1}}\frac{\pi{k|k-1}(\mathbf{m}^i{k|k-1}|\mathbf{m}^i{k-1})}{\varrhok(\mathbf{m}^i{k|k-1}|\mathbf{m}^i_{k-1},\mathbf{Z}_k)}\frac{1}{B} & i = N+1,…,N+B \end{matrix} \right .$
  4. 计算likelihood $hk(\mathbf{z}_k|\emptyset) = \Pi^n{s=1} g^r_0(z^r_k)$
  5. for $i = 1,…,N+N_B$ do
    1. 计算likelihood $hk(\mathbf{z}_k|\mathbf{m}^i{k|k-1}) =\Pi^n{s=1} g^r_1(z^r_k |\mathbf{m}^i{k|k-1})$,
    2. 计算likelihood 比率 $lk(\mathbf{z}_k|\mathbf{m})= \frac{h_k(\mathbf{z}_k|{\mathbf{m}{}})}{h_k(\mathbf{z}_k|\emptyset)}=\Pi^n{r=1} \frac{g^r_1(\mathbf{z}^r_k|\mathbf{m})}{g^r_0(\mathbf{z}^r_k)}$
  6. end for
  7. 近似积分$Ik \approx\sum^{N+N{B}}{i=1}l_k(\mathbf{z}|\mathbf{m}^i{k|k-1})\omega^i_{k|k-1}$
  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. $\hat {\omega}^ik = l_k(\mathbf{z}_k|\mathbf{m}^i{k|k-1})\omega^i_{k|k-1}$
  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 $\varphi$, ${\omega^i{k-1},\mathbf{m}^i{k-1}}^{N+NB}{i=1}$, $\mathbf{Z}_k$

  1. 预测存在概率 $q{k|k-1} = p_b(1-q{k-1})+psq{k-1}$。
  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 $\omega^i{k|k-1}$,
    $\omega^i
    {k|k-1} = \left { \begin{matrix} \frac{psq{k-1}}{q{k|k-1}}\frac{\pi{k|k-1}(\mathbf{m}^i{k|k-1}|\mathbf{m}^i{k-1})}{\varrhok(\mathbf{m}^i{k|k-1}|\mathbf{m}^i{k-1},\mathbf{Z}_k)}\omega^i{k-1} & i = 1,…,N \ \frac{pb(1-q{k-1})}{q{k|k-1}}\frac{\pi{k|k-1}(\mathbf{m}^i{k|k-1}|\mathbf{m}^i{k-1})}{\varrhok(\mathbf{m}^i{k|k-1}|\mathbf{m}^i_{k-1},\mathbf{Z}_k)}\frac{1}{B} & i = N+1,…,N+B \end{matrix} \right .$
  4. 近似积分$I1 \approx \sum^{N+N_b}{i=1}pd(\mathbf{m}^i{k})\omega^i_{k|k-1}$
  5. 对于$\mathbf{z}\in\mathbf{Z}k$计算近似积分$I{2}(\mathbf{z})\approx\sum^{N+NB}{i=1}pd(\mathbf{m}^i{k|k-1})hk(\mathbf{z}|\mathbf{m}^i{k|k-1})\omega^i_{k|k-1}$
  6. $\trianglek \approx I_1-\sum{\mathbf{z}\in\mathbf{Z}_k}\frac{I_2(\mathbf{z})}{\lambda c(\mathbf{z})}$
  7. 更新存在概率$qk = \frac{1- \triangle_k}{1-\triangle_k q{k|k-1}q_{k|k-1}}$
  8. 更新权重$\hat{\omega}^ik = [1-p_d(\mathbf{m}^i{k|k-1})+pd(\mathbf{m}^i{k|k-1})\sum{\mathbf{z}\in\mathbf{Z}_k}\frac{h_k(\mathbf{z}|\mathbf{m}^i{k|k-1})}{\lambda c(\mathbf{z})}]\omega^i_{k|k-1}$
  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:$q{k}$, ${\omega^i{k},\mathbf{m}^i{k}}^{N+N_B}{i=1}$.