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Error Analysis

Introduction

In an experiment we want to increase the knowledge on $\mu $ given some outcome $x$. Before an experiment we don't know $\mu$ or $x$ and we would quantify this knowledge $f(x,\mu)$ which is in general unknown. After an experiment we get $x%$ and we want to know how to get %$f(\mu|x)$. This can be computed by:

  $$ f(\mu|x)= \frac{f(x,\mu)}{f(x)}= \frac{f(x|\mu)f(\mu)}{\int f(x,\mu)d\mu }= \frac{f(x|\mu)f(\mu)}{\int f(x|\mu) f(\mu) d\mu } .$$ (1)

Usually $f(\mu)$ is a subjective a priori. $f(x|\mu)$ is called likelihood.

If $\mu_3 =g(\mu_1, \mu_2) $ then

  $$ f(\mu_3)=\int f(\mu_1, \mu_2) \delta(\mu_3 - g(\mu_1,\mu_2) ) d\mu_2 d\mu_3 $$ (2)

Useful formulas

We assume that the limiting distribution of a set of $ N $ measurements values $ x_i $ is a gaussian

  $$ P ( x \le x_i < x + {\rm d}x) = \frac{1}{\sigma \sqrt{2\pi}} \exp \left( \frac{(x - \mu)^2}{\sigma^2} \right) {\rm d}x $$. (3)

The notation $a \cong b $ means the maximum likelihood estimator for a is b.

If $ x_i $ are a set of $ N $ measurements:

  • the average, $\mu \cong \frac1N \sum x_i = \bar x$.
  • the standard deviation, $\sigma \cong \sqrt{\frac1{N-1} \sum (x_i - \bar x)^2} = \sigma_x$ or $\sigma \cong \sqrt{\frac1{N} \sum (x_i - \mu )^2}=\sigma_x$, which means that if you repeat a measurement there is 68% chance to obtain a value in $ \bar x \pm \sigma_x $.
  • the error on $\bar x$ is $\sigma_{\bar x} = \sigma_x/\sqrt{N}$, which means that if you repeat $N$ measurements there is 68% chance to obtain an average in $\bar x \pm \sigma_{\bar x} $;
  • the error on $\sigma_{\bar x}$ is $ {\sigma_{\bar x} / \sqrt{2(N-1)}$

If $q=q(x,y)$ then

  $$ \sigma_q = \sqrt{\left( \frac{dq}{dx} \sigma_x \right)^2 +\left( \frac{dq}{dy} \sigma_y\right) ^2 + 2 \frac{dq}{dx}\frac{dq}{dy}\sigma_{xy} }$$ (4)

If $x$ is measured by several independent measurements, the maximum likelihood estimator for $ x $ is the weighted average with $1/\sigma^2$ as weight.

If $y=A x + B$ and there are N (x,y) measurements then

  • $A \cong \frac{\sum x^2 \sum y - \sum xy}{\Delta} $
  • $B \cong \frac{N \sum xy - \sum x\sum y}{\Delta} $
  • $ \Delta = N\sum x^2 - (\sum x)^2 $
  • $ \sigma_y \cong \sqrt{\frac1{(N-2)}\sum (y-Ax-B)^2} $
  • $ \sigma_A \cong \sigma_y \sqrt{\sum x^2/\Delta} $
  • $ \sigma_B \cong \sigma_y \sqrt{N/\Delta} $

Controversy

Lectures

References

(?? REFLATEX{} not defined in eqn list ??)

-- RiccardoDeMaria - 05 Jan 2009

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Topic revision: r5 - 2009-01-15 - RiccardoDeMaria
 
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