MOTIVATION

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。

采用RJMMC的方法

细致平稳条件


\alpha \left( x,y \right) =\min \left\{ 1,R \right\} 
\\
\pi \left( x \right) p\left( x,y \right) =\pi \left( y \right) p\left( y,x \right)

跨维度游动

k <-> k+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|

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|

k <-> k+2

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|

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|

同维度游动

根的类型转换

先确定映射关系，令


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}\\
	\frac{\partial b}{\partial r_1}&		\frac{\partial b}{\partial r_2}\\
\end{matrix} \right) \right|=\frac{1}{2}
\\
J_{R\rightarrow C}={J_{C\rightarrow R}}^{-1}=2

实根 <-> 复共轭根对在$n_r$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$a=\frac{r_1+r_2}{2}$，$b=\frac{r_1-r_2}{2}$。


q\left( x|x' \right) =p_m\times \frac{1}{C\left( n_r,2 \right)}

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} $

复共轭根对 <-> 实根在$n_c$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$r_2=a-b$。


q\left( x'|x \right) =p_c\times \frac{1}{n_c}

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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