误差传播

在统计学上，由于变量含有误差，而使函数受其影响也含有误差，称之为误差传播。阐述这种关系的定律称为误差传播定律。

误差传播定律

设有一般函数（线性函数和非线性函数）

Z=$f(x_1,x_2,\dots,x_n)$

式中$x_1,x_2,\dots,x_n$ 为可直接观测的相互独立的未知量，z为不便于直接观测的未知量。已知$x_1,x_2,\dots,x_n$ 的标准差分别为$m_1,m_2,\dots,m_n$ ,现在要求z的标准差$m_z$ 。已知函数z的中误差关系式为$m_z^2$ =$k_1^2 m_1^2+k_2^2 m_2^2+\dots+ k_n^2 m_n^2$（其中$k_1,k_2,\dots,k_n$为任意常数）。由数学分析可知，变量的误差与函数的误差之间的关系，可以近似的用函数的全微分来表达，为此对上式求全微分，并以真误差的符号“Δ”替代微分的符号“d”得 $\Delta z=\frac{\partial f}{\partial x_1} \cdot \Delta x_1+\frac{\partial f}{\partial x_2}\cdot \Delta x_2+\cdots+\frac{\partial f}{\partial x_n}\cdot \Delta x_n$

式中$\frac{\partial f}{\partial x_i}$ （i=1，2，，…，n）是函数对各个变量变量所取得偏导数，对上式以标准差平方代替真误差,由函数z的中误差关系式可得

$m_z^2$ =$\left(\frac{\partial f}{\partial x_1} \right)^2 m_1^2+\left(\frac{\partial f}{\partial x_2} \right)^2 m_2^2+\dots+ \left(\frac{\partial f}{\partial x_n} \right)^2 m_n^2$

将上式取平方根可得误差传播定律的一般形式

$m_z$=±$\sqrt{\left(\frac{\partial f}{\partial x_1} \right)^2 m_1^2+\left(\frac{\partial f}{\partial x_2} \right)^2 m_2^2+\dots+ \left(\frac{\partial f}{\partial x_n} \right)^2 m_n^2}$

外部链接

• Uncertainties and Error Propagation, Appendix V from the Mechanics Lab Manual, Case Western Reserve University.
• Mathieu Rouaud, 2013: Probability, Statistics and Estimation Propagation of Uncertainties in Experimental Measurement.
• A detailed discussion of measurements and the propagation of uncertainty explaining the benefits of using error propagation formulas and monte carlo simulations instead of simple significance arithmetic.
• Uncertainties and Error Propagation, Vern Lindberg's Guide to Uncertainties and Error Propagation.