查看“︁误差传播”︁的源代码

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很多情况下，被测量Z 不能直接测得，而是由N 个其他量<math>x_1,x_2,\dots,x_n</math>通过函数关系来确定，在统计学上，由于变量含有误差，而使由函数计算出的被测量Z 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=<math>f(x_1,x_2,\dots,x_n)</math> 

式中<math>x_1,x_2,\dots,x_n</math> 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<math>x_1,x_2,\dots,x_n</math> 的[[標準差]]分别为<math>m_1,m_2,\dots,m_n</math> ,现在要求z的[[標準差]]<math>m_z</math> 。已知函数z的中误差关系式为<math>m_z^2</math> =<math>k_1^2 m_1^2+k_2^2 m_2^2+\dots+ k_n^2 m_n^2</math>（其中<math>k_1,k_2,\dots,k_n</math>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
<math>\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</math> 

式中<math>\frac{\partial f}{\partial x_i}</math> （i=1，2，，…，n）是函数对各个变量变量所取得[[偏导数]]，对上式以標準差平方代替真误差,由函数z的中误差关系式可得

<math>m_z^2</math> =<math> \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</math>

将上式取平方根可得误差传播定律的一般形式

<math>m_z</math>=±<math>\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}</math>

==参见==
* [[测量精度]]
* [[自動微分]]
* {{tsl|en|Delta method|Delta方法}}
* [[精度衰减因子]]
* [[误差]]
* {{tsl|en|Experimental uncertainty analysis|实验不确定度分析}}
* [[测量不确定度]]
* [[数值稳定性]]
* {{tsl|en|Probability bounds analysis|概率界限分析}}
* {{tsl|en|Significance arithmetic|有效位运算}}
* {{tsl|en|Uncertainty quantification|不确定性量化}}
* {{tsl|en|Random-fuzzy variable|随机模糊变量}}

==外部链接==
* [https://web.archive.org/web/20080516202656/http://physicslabs.phys.cwru.edu/MECH/Manual/Appendix_V_Error%20Prop.pdf Uncertainties and Error Propagation], Appendix V from the [https://web.archive.org/web/20090917235018/http://physicslabs.phys.cwru.edu/MECH/Manual/manual.htm Mechanics Lab Manual], Case Western Reserve University.
* [http://www.incertitudes.fr/book.pdf Mathieu Rouaud, 2013: Probability, Statistics and Estimation]{{Wayback|url=http://www.incertitudes.fr/book.pdf |date=20170403013924 }} Propagation of Uncertainties in Experimental Measurement.
* [http://www.av8n.com/physics/uncertainty.htm A detailed discussion of measurements and the propagation of uncertainty]{{Wayback|url=http://www.av8n.com/physics/uncertainty.htm |date=20090415140213 }} explaining the benefits of using error propagation formulas and monte carlo simulations instead of simple [[significance arithmetic]].
* [https://web.archive.org/web/20090307022925/http://www.rit.edu/cos/uphysics/uncertainties/Uncertainties.html Uncertainties and Error Propagation], Vern Lindberg's Guide to Uncertainties and Error Propagation.

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[[category:误差理论]]
[[category:测绘学]]
[[category:数学术语]]