100 lines
5.7 KiB
Markdown
100 lines
5.7 KiB
Markdown
# MOTIVATION
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论文中的辨识是认为系统的阶数已知的辨识，而且 MCMC 的抽样的变量是特征多项式矩阵，我想使用特征值来抽样，并同时考虑系统阶数未知的情况。
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采用RJMMC的方法
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## 细致平稳条件
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$$
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\alpha \left( x,y \right) =\min \left\{ 1,R \right\}
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\\
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\pi \left( x \right) p\left( x,y \right) =\pi \left( y \right) p\left( y,x \right)
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$$
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## 跨维度游动
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### k <-> k+1
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1. Birth move
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辅助变量的先验：$u~U\left( -1,1 \right) $
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提议分布:$q\left( \lambda _k,\lambda _{k+1} \right) =q\left( u \right) \cdot P\left( birth \right) $
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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\} $
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缩放因子:$\left| J_1 \right|=1$
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2. Death move
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辅助变量的先验：$\frac{1}{k+1} $
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提议分布:$q\left( \lambda _{k+1},\lambda _k \right) =\frac{1}{k+1}\cdot P\left( dead \right) $
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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\} $
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缩放因子:$\left| J_1 \right|=1$
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### k <-> k+2
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1. Birth move
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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) $
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辅助变量的先验: $\theta ~U\left( 0,\pi \right) ;\rho ~U\left( 0,1 \right) $
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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) $
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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\} $
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缩放因子: $$ \left| J_2 \right|=\left| \det \left( \frac{\partial \left( \lambda _1,\lambda _2,\cdots \lambda _k,\lambda _{k+1},\lambda _{k+2} \right)}{\partial \left( \lambda _1,\lambda _2,\cdots \lambda _k,\theta ,\rho \right)} \right) \right|=\left| \det \left( \begin{matrix}\cos \theta +i\sin \theta&-\rho \sin \theta +i\rho \cos \theta\\\cos \theta -i\sin \theta&-\rho \sin \theta -i\rho \cos \theta\\\end{matrix} \right) \right|=2\rho $$
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2. Death move
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辅助变量的先验: $\frac{1}{k+1}$
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提议分布:$ q\left( \lambda _{k+1},\lambda _k \right) =\frac{1}{k+1}\cdot P\left( dead \right) $
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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| $
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缩放因子: $$ \left| J_2 \right|=\left| \det \left( \frac{\partial \left( \lambda _1,\lambda _2,\cdots \lambda _k,\lambda _{k+1},\lambda _{k+2} \right)}{\partial \left( \lambda _1,\lambda _2,\cdots \lambda _k,\theta ,\rho \right)} \right) \right|=\left| \det \left( \begin{matrix}\cos \theta +i\sin \theta&-\rho \sin \theta +i\rho \cos \theta\\\cos \theta -i\sin \theta&-\rho \sin \theta -i\rho \cos \theta\\\end{matrix} \right) \right|=2\rho $$
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## 同维度游动
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### 根的类型转换
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先确定映射关系，令
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$$
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a=\frac{r_1+r_2}{2}\text{、}b=\frac{r_1-r_2}{2}
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\\
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r_1=a+b\text{、}r_2=a-b
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$$
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再确定雅可比矩阵行列式
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$$
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J_{C\rightarrow R}=\left| \det \left( \begin{matrix}
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\frac{\partial a}{\partial r_1}& \frac{\partial a}{\partial r_2}\\
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\frac{\partial b}{\partial r_1}& \frac{\partial b}{\partial r_2}\\
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\end{matrix} \right) \right|=\frac{1}{2}
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\\
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J_{R\rightarrow C}={J_{C\rightarrow R}}^{-1}=2
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$$
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1. 实根 <-> 复共轭根对
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在$n_r$个实根中任选两个实根$r_1$和$r_2$，转换为复共轭根对$a\pm jb$，其中$a=\frac{r_1+r_2}{2}$，$b=\frac{r_1-r_2}{2}$。
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$$
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q\left( x|x' \right) =p_m\times \frac{1}{C\left( n_r,2 \right)}
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$$
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此时的接受率写作:$Merge\text{：}\alpha \left( x',x \right) =\min \left\{ 1,\frac{\pi \left( x \right)}{\pi \left( x' \right)}\cdot \frac{1}{q\left( x|x' \right)}\cdot \frac{1}{2} \right\} $
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2. 复共轭根对 <-> 实根
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在$n_c$个复共轭根对中任选一个复共轭根对$a\pm jb$，转换为两个实根$r_1=a+b$和$r_2=a-b$。
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$$
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q\left( x'|x \right) =p_c\times \frac{1}{n_c}
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$$
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此时的接受率写作:$Split\text{：}\alpha \left( x,x' \right) =\min \left\{ 1,\frac{\pi \left( x' \right)}{\pi \left( x \right)}\cdot \frac{1}{q\left( x'|x \right)}\cdot 2 \right\} $
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### 同类型根的调整
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1. 实根调整
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选择一个实根$r$，通过添加噪声$\epsilon \sim N\left( 0,\sigma ^2 \right)$调整该实根的位置为$r' = r + \epsilon$。
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$$
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\alpha \left( x,x' \right) =\min \left\{ 1,\frac{\pi \left( x' \right)}{\pi \left( x \right)} \right\}
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$$
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2. 复共轭根对调整
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选择一个复共轭根对$a\pm jb$，通过添加噪声$\epsilon _a \sim N\left( 0,\sigma _a^2 \right)$和$\epsilon _b \sim N\left( 0,\sigma _b^2 \right)$调整该复共轭根对的位置为$a' = a + \epsilon _a$和$b' = b + \epsilon _b$。
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$$
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\alpha \left( x,x' \right) =\min \left\{ 1,\frac{\pi \left( x' \right)}{\pi \left( x \right)} \right\}
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$$
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