SciLearn: error propagation

Thursday, September 18, 2008

error propagation

Recently, I've been working on my physics lab reports. The main theoretical concerns are about error analysis. This means there's a lot of equations to mess around: specifically the propagation of errors.

The first time I learned about error propagation (under a different name) was at pre-college level high school (i.e. AS-Level). It sounded simply, and the basic equations were as follows (in summarized form):
$\newcommand{\error}{\operator{\boldsymbol{\sigma}}} \begin{align*} \error_{a \pm b} &= \error_a + \error_b \\ \error_{Ba} &= B \error_a \\ {\error_{ab} \over \left|ab\right|} = {\error_{a \over b} \over \left|a \over b\right|} &= {\error_a \over \left|a\right|} + {\error_b \over \left|b\right|} \\ {\error_{a^B} \over \left|a^B\right| } &= \left|B\right| {\error_a \over \left|a\right|} \end{align*}$
Notice that:
$\newcommand{\error}{\operatorname{\boldsymbol{\sigma}}} \error_u \over \left|u\right|$
is simply the fractional error of u, whatever u might be.

However, the above equations can be replaced with quadratures if the uncertainty is a standard error and the measurements are independent:
$\newcommand{\error}{\operatorname{\boldsymbol{\sigma}}} \begin{align*} \error_{a \pm b}^2 &= \error_a^2 + \error_b^2 \\ \left({\error_{ab} \over ab}\right)^2 = \left({\error_{a \over b} \over {a \over b}}\right)^2 &= \left(\error_a \over a\right)^2 + \left(\error_b \over b\right)^2 \end{align*}$

Here's the general formula that applies to any multivariable function f of a, b, c, etc:
$\newcommand{\error}{\operator{\boldsymbol{\sigma}}} \begin{align*} \error_{f(a, b, c, \,\cdots)} = \left| \partial f \over \partial a \right| \error_a + \left| \partial f \over \partial b \right| \error_b + \left| \partial f \over \partial c \right| \error_c + \cdots \end{align*}$

Again, if quadratures are to be used (if uncertainties are standard errors and the variables are independent)
$\newcommand{\error}{\operator{\boldsymbol{\sigma}}} \begin{align*} \error_{f(a, b, c, \,\cdots)}^2 = \left( {\partial f \over \partial a} \error_a \right)^2 + \left( {\partial f \over \partial b} \error_b \right)^2 + \left( {\partial f \over \partial c} \error_c \right)^2 + \cdots \end{align*}$

(Note: most of the above formulas are approximations)