非线性变换公式
设两个随机向量满足如下非线性关系:
\[\pmb{y} = \pmb{f}(\pmb{x})\]记随机向量$\pmb{x}$的均值与协方差分别为
\[\begin{align} &E(\pmb{x}) = \pmb{\mu_x} \\ &E((\pmb{x} - \pmb{\mu_x})(\pmb{x} - \pmb{\mu_x})^T) = \pmb{\Sigma_{xx}} \end{align}\]则随机向量$\pmb{y}$的均值与协方差分别为
\[\begin{align} \pmb{\mu_y} &= \pmb{f}(\pmb{\mu_x}) \\ \pmb{\Sigma_{yy}} &= \pmb{J}\pmb{\Sigma_{xx}}\pmb{J}^T \end{align}\]其中向量$\pmb{J}$为
\[\pmb{J} = \frac{d\pmb{f}}{d\pmb{x}}\]应用实例
考察如下变换:
\[\theta = \frac{z_r - z_f}{L}\]写成矩阵形式为:
\[\theta=\left[\frac{1}{L} \quad-\frac{1}{L}\right]\left[\begin{array}{l} z_{r} \\ z_{f} \end{array}\right]\]则雅可比矩阵为
\[\pmb{J} = \left[\begin{array}{ll} \frac{\partial \theta}{\partial z_{f}} & \frac{\partial \theta}{\partial z_{r}} \end{array}\right] = \left[\begin{array}{ll} \frac{1}{L} & -\frac{1}{L} \end{array}\right]\]