I have four random variables${\left( {x,y,z,\varphi} \right)^T}$. And I know its covariance matrix $Cov=\left[ {\begin{array}{*{20}{c}} {\sigma _{xx}^2}&{\sigma _{yx}^2}&{\sigma _{zx}^2}&{\sigma _{\varphi x}^2}\\ {\sigma _{xy}^2}&{\sigma _{yy}^2}&{\sigma _{zy}^2}&{\sigma _{\varphi y}^2}\\ {\sigma _{xz}^2}&{\sigma _{yz}^2}&{\sigma _{zz}^2}&{\sigma _{\varphi z}^2}\\ {\sigma _{x\varphi }^2}&{\sigma _{y\varphi }^2}&{\sigma _{z\varphi }^2}&{\sigma _{\varphi \varphi }^2} \end{array}} \right]$

I want to derive the information matrix of $x,y,\varphi$. I have two ways in mind:

  1. Directly inverse the matrix$\left[ {\begin{array}{*{20}{c}} {\sigma _{xx}^2}&{\sigma _{yx}^2}&{\sigma _{\varphi x}^2}\\ {\sigma _{xy}^2}&{\sigma _{yy}^2}&{\sigma _{\varphi y}^2}\\ {\sigma _{x\varphi }^2}&{\sigma _{y\varphi }^2}&{\sigma _{\varphi \varphi }^2} \end{array}} \right]$ whose elements are selected from matrix $Cov$. Thus get the information matrix of $x,y,\varphi$.
  2. Firstly inverse the matrix $Cov$ $(Cov)^{-1}={\left[ {\begin{array}{*{20}{c}} {\sigma _{xx}^2}&{\sigma _{yx}^2}&{\sigma _{zx}^2}&{\sigma _{\varphi x}^2}\\ {\sigma _{xy}^2}&{\sigma _{yy}^2}&{\sigma _{zy}^2}&{\sigma _{\varphi y}^2}\\ {\sigma _{xz}^2}&{\sigma _{yz}^2}&{\sigma _{zz}^2}&{\sigma _{\varphi z}^2}\\ {\sigma _{x\varphi }^2}&{\sigma _{y\varphi }^2}&{\sigma _{z\varphi }^2}&{\sigma _{\varphi \varphi }^2} \end{array}} \right]^{ - 1}}=\left[ {\begin{array}{*{20}{c}} {\lambda _{xx}^{}}&{\lambda _{yx}^{}}&{\lambda _{zx}^{}}&{\lambda _{\varphi x}^{}}\\ {\lambda _{xy}^{}}&{\lambda _{yy}^{}}&{\lambda _{zy}^{}}&{\lambda _{\varphi y}^{}}\\ {\lambda _{xz}^{}}&{\lambda _{yz}^{}}&{\lambda _{zz}^{}}&{\lambda _{\varphi z}^{}}\\ {\lambda _{x\varphi }^{}}&{\lambda _{y\varphi }^{}}&{\lambda _{z\varphi }^{}}&{\lambda _{\varphi \varphi }^{}} \end{array}} \right]$. Afterwards, select the elements from the results and form the information matrix of variable $x,y,\varphi$ $\left[ {\begin{array}{*{20}{c}} {\lambda _{xx}^{}}&{\lambda _{yx}^{}}&{\lambda _{\varphi x}^{}}\\ {\lambda _{xy}^{}}&{\lambda _{yy}^{}}&{\lambda _{\varphi y}^{}}\\ {\lambda _{x\varphi }^{}}&{\lambda _{y\varphi }^{}}&{\lambda _{\varphi \varphi }^{}} \end{array}} \right]$

Could any one tell me which way is the correct way? Thank you very much.


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