非线性函数的协方差传播

最新推荐文章于 2022-03-14 14:42:36 发布

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最新推荐文章于 2022-03-14 14:42:36 发布 · 置顶 · 2.1k 阅读

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设有观测值 $Xn1\mathop{X}\limits_{n1}$ 的非线性函数
$\tag{1}$

或写成 $,Xn)Z=f\left(X_{1}, X_{2}, \cdots, X_{n}\right)$

已知 $X$ 的协方差矩阵 $D_{xx}$ ,欲求Z的协方差 $D_{zz}$
假定观测值 $X$ 有近似值 $X0n1\mathop{X^0}\limits_{n1}$
$X0=[X10X20⋯Xn0]T\begin{aligned} &X^{0}=\left[\begin{array}{llll} X_{1}^{0} & X_{2}^{0} & \cdots & X_{n}^{0} \end{array}\right]^{T}\\ \end{aligned}$

可将函数式按泰勒级数在点 $X10,X20,⋯Xn0X_{1}^{0},X_{2}^{0}, \cdots X_{n}^{0}$ 处展开为
$Z=f(X10X20⋯Xn0)+(∂f∂X1)0(X1−X10)+(∂f∂X2)(X2−X20)+⋯+(∂f∂Xn)(Xn−Xn0)+(二次以上项).(2)\begin{aligned} &Z=f\left(X_{1}^{0} \quad X_{2}^{0} \quad \cdots \quad X_{n}^{0}\right)+\left(\frac{\partial f}{\partial X_{1}}\right)_{0}\left(X_{1}-X_{1}^{0}\right)\\ &+\left(\frac{\partial f}{\partial X_{2}}\right)\left(X_{2}-X_{2}^{0}\right)+\cdots+\left(\frac{\partial f}{\partial X_{n}}\right)\left(X_{n}-X_{n}^{0}\right)+(二次以上项). \end{aligned}\tag{2}$

式中， $(∂f∂Xi)0(\frac{\partial f}{\partial X_{i}})_0$ 是函数对各个变量所取的偏导数，并以近似值 $X^0$ 带入所算得的数值，它们都是常数。当 $X^0$ 与X非常接近时，上式中二次以上各项很微小，故可以略去。因此，可将上式写为
$,Xn)−∑i=1n(∂f∂Xi)0Xi(3)Z=\left(\frac{\partial f}{\partial X_{1}}\right)_{0} X_{1}+\left(\frac{\partial f}{\partial X_{2}}\right)_{0} X_{2}+\cdots+\left(\frac{\partial f}{\partial X_{2}}\right) _{0} X_{0}+f\left(X_{1}^{0}, X_{2}^{0}, \cdots, X_{n}\right)-\sum_{i=1}^{n}\left(\frac{\partial f}{\partial X_{i}}\right) _{0} X_i\tag{3}$