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2025-11-05 08:36:42 +08:00
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@@ -17,29 +17,29 @@ $$
### k <-> k+1
1. Birth move
$$
\log R=\log \frac{P\left( x_{k+1} \right) P\left( y|x_{k+1} \right)}{P\left( x_k \right) P\left( y|x_k \right)}+\log \frac{P\left( dead \right)}{P\left( birth \right)}+\log \frac{\small{\frac{1}{k+1}}}{q\left( u \right)}+\log \left| J_1 \right|
$$
辅助变量的先验：$u~U\left( -1,1 \right) $
提议分布:$q\left( \lambda _k,\lambda _{k+1} \right) =q\left( u \right) \cdot P\left( birth \right) $
接受率:$\alpha \left( \lambda _k,\lambda _{k+1} \right) =\min \left\{ 1,\frac{P\left( \lambda _{k+1} \right) P\left( y|\lambda _{k+1} \right)}{P\left( \lambda _k \right) P\left( y|\lambda _k \right)}\cdot \frac{P\left( dead \right)}{P\left( birth \right)}\cdot \frac{\small{\frac{1}{k+1}}}{q\left( u \right)}\cdot \left| J_1 \right| \right\} $
缩放因子:$\left| J_1 \right|=1$
2. Death move
$$
\log R=\log \frac{P\left( x_k \right) P\left( y|x_k \right)}{P\left( x_{k+1} \right) P\left( y|x_{k+1} \right)}+\log \frac{P\left( birth \right)}{P\left( dead \right)}+\log \frac{\small{q\left( u \right)}}{\frac{1}{k+1}}-\log \left| J_1 \right|
$$
辅助变量的先验：$\frac{1}{k+1} $
提议分布:$q\left( \lambda _{k+1},\lambda _k \right) =\frac{1}{k+1}\cdot P\left( dead \right) $
接受率:$\alpha \left( \lambda _{k+1},\lambda _k \right) =\min \left\{ 1,\frac{P\left( \lambda _k \right) P\left( y|\lambda _k \right)}{P\left( \lambda _{k+1} \right) P\left( y|\lambda _{k+1} \right)}\cdot \frac{P\left( birth \right)}{P\left( dead \right)}\cdot \frac{q\left( u \right)}{\frac{1}{k+1}}\cdot \frac{1}{\left| J_1 \right|} \right\} $
缩放因子:$\left| J_1 \right|=1$
### k <-> k+2
1. Birth move
$$
\log R=\log \frac{P\left( x_{k+2} \right) P\left( y|x_{k+2} \right)}{P\left( x_k \right) P\left( y|x_k \right)}+\log \frac{P\left( dead \right)}{P\left( birth \right)}+\log \frac{\small{\frac{1}{m+1}}}{q\left( \theta \right) q\left( r \right)}+\log \left| J_2 \right|
$$
映射关系: $\lambda _1,\lambda _2,\cdots \lambda _k,\theta ,\rho \longrightarrow \lambda _1,\lambda _2,\cdots \lambda _k,\lambda _{k+1},\lambda _{k+2}$ 其中: $\lambda _{k+1}=\rho \left( \cos \theta +i\sin \theta \right) $, $\lambda _{k+2}=\rho \left( \cos \theta -i\sin \theta \right) $
辅助变量的先验: $\theta ~U\left( 0,\pi \right) ;\rho ~U\left( 0,1 \right) $
提议分布:$ q\left( \lambda _k,\lambda _{k+1} \right) =\frac{\small{1}}{q\left( \theta \right) q\left( \rho \right)}\cdot P\left( birth \right) $
接受率:$\alpha \left( \lambda _k,\lambda _{k+1} \right) =\min \left\{ 1,\frac{P\left( \lambda _{k+2} \right) P\left( y|\lambda _{k+2} \right)}{P\left( \lambda _k \right) P\left( y|\lambda _k \right)}\cdot \frac{P\left( dead \right)}{P\left( birth \right)}\cdot \frac{\small{\frac{1}{k+1}}}{q\left( \theta \right) q\left( \rho \right)}\cdot \left| J_2 \right| \right\} $
缩放因子: $$ \left| J_2 \right|=\left| \det \left( \frac{\partial \left( \lambda _1,\lambda _2,\cdots \lambda _k,\lambda _{k+1},\lambda _{k+2} \right)}{\partial \left( \lambda _1,\lambda _2,\cdots \lambda _k,\theta ,\rho \right)} \right) \right|=\left| \det \left( \begin{matrix}\cos \theta +i\sin \theta&-\rho \sin \theta +i\rho \cos \theta\\\cos \theta -i\sin \theta&-\rho \sin \theta -i\rho \cos \theta\\\end{matrix} \right) \right|=2\rho $$
2. Death move
$$
\log R=\log \frac{P\left( x_k \right) P\left( y|x_k \right)}{P\left( x_{k+2} \right) P\left( y|x_{k+2} \right)}+\log \frac{P\left( birth \right)}{P\left( dead \right)}+\log \frac{\small{q\left( \theta \right) q\left( r \right)}}{\frac{1}{m+1}}-\log \left| J_2 \right|
$$
$其中 m$ 是当前系统中复共轭极点对的数量
$J_1$ 和 $J_2$ 是雅可比矩阵行列式
$\left| J_1 \right|=\left| \prod_{\boldsymbol{i}=1}^{\boldsymbol{k}}{\left( \lambda _i-u_i \right)} \right|$
$\left| J_2 \right|=\left| \prod_{\boldsymbol{i}=1}^{\boldsymbol{k}}{\left( \lambda _{k+1}-\lambda _i \right) \left( \lambda _{k+2}-\lambda _i \right)} \right|\left| \lambda _{k+1}-\lambda _{k+1} \right|$
辅助变量的先验: $\frac{1}{k+1}$
提议分布:$ q\left( \lambda _{k+1},\lambda _k \right) =\frac{1}{k+1}\cdot P\left( dead \right) $
接受率:$\log R=\log \frac{P\left( x_k \right) P\left( y|x_k \right)}{P\left( x_{k+1} \right) P\left( y|x_{k+1} \right)}+\log \frac{P\left( birth \right)}{P\left( dead \right)}+\log \frac{q\left( \theta \right) q\left( \rho \right)}{\frac{1}{k+1}}-\log \left| J_1 \right| $
缩放因子: $$ \left| J_2 \right|=\left| \det \left( \frac{\partial \left( \lambda _1,\lambda _2,\cdots \lambda _k,\lambda _{k+1},\lambda _{k+2} \right)}{\partial \left( \lambda _1,\lambda _2,\cdots \lambda _k,\theta ,\rho \right)} \right) \right|=\left| \det \left( \begin{matrix}\cos \theta +i\sin \theta&-\rho \sin \theta +i\rho \cos \theta\\\cos \theta -i\sin \theta&-\rho \sin \theta -i\rho \cos \theta\\\end{matrix} \right) \right|=2\rho $$
## 同维度游动
@@ -53,8 +53,8 @@ a=\frac{r_1+r_2}{2}\text{、}b=\frac{r_1-r_2}{2}
r_1=a+b\text{、}r_2=a-b
$$
再确定雅可比矩阵行列式
$$
J_{C\rightarrow R}=\left| \det \left( \begin{matrix}
\frac{\partial a}{\partial r_1}& \frac{\partial a}{\partial r_2}\\