mirror of
https://github.com/slsdetectorgroup/slsDetectorPackage.git
synced 2025-04-28 17:10:03 +02:00
319 lines
9.2 KiB
HTML
319 lines
9.2 KiB
HTML
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
|
|
|
<!--Converted with LaTeX2HTML 2012 (1.2)
|
|
original version by: Nikos Drakos, CBLU, University of Leeds
|
|
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
|
|
* with significant contributions from:
|
|
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
|
<HTML>
|
|
<HEAD>
|
|
<TITLE>Poisson and normal statistics for diffraction</TITLE>
|
|
<META NAME="description" CONTENT="Poisson and normal statistics for diffraction">
|
|
<META NAME="keywords" CONTENT="slsDetectors-FAQ">
|
|
<META NAME="resource-type" CONTENT="document">
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2012">
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
<LINK REL="STYLESHEET" HREF="slsDetectors-FAQ.css">
|
|
|
|
<LINK REL="next" HREF="node54.html">
|
|
<LINK REL="previous" HREF="node52.html">
|
|
<LINK REL="up" HREF="node46.html">
|
|
<LINK REL="next" HREF="node54.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html841"
|
|
HREF="node54.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="file:/usr/share/latex2html/icons/next.png"></A>
|
|
<A NAME="tex2html837"
|
|
HREF="node46.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="file:/usr/share/latex2html/icons/up.png"></A>
|
|
<A NAME="tex2html831"
|
|
HREF="node52.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="file:/usr/share/latex2html/icons/prev.png"></A>
|
|
<A NAME="tex2html839"
|
|
HREF="node1.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="file:/usr/share/latex2html/icons/contents.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html842"
|
|
HREF="node54.html">Average vs. weighted average</A>
|
|
<B> Up:</B> <A NAME="tex2html838"
|
|
HREF="node46.html">How are different positions</A>
|
|
<B> Previous:</B> <A NAME="tex2html832"
|
|
HREF="node52.html">Advanced binning</A>
|
|
<B> <A NAME="tex2html840"
|
|
HREF="node1.html">Contents</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A NAME="SECTION00624000000000000000">
|
|
Poisson and normal statistics for diffraction</A>
|
|
</H2>
|
|
|
|
<P>
|
|
The normal situation for diffraction data
|
|
is that the observed signal is a photon count.
|
|
Therefore it follows a Poisson distribution.
|
|
If we have a count value <IMG
|
|
WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img128.png"
|
|
ALT="$ C_0$">
|
|
that follows a Poisson distribution,
|
|
we can assume immediately that the average is equal to <IMG
|
|
WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img128.png"
|
|
ALT="$ C_0$">
|
|
and the s.d. is <!-- MATH
|
|
$\sqrt{C_0}$
|
|
-->
|
|
<IMG
|
|
WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img129.png"
|
|
ALT="$ \sqrt{C_0}$">
|
|
.
|
|
I.e., repeated experiments would give values <IMG
|
|
WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img130.png"
|
|
ALT="$ n$">
|
|
|
|
distributed according to the normalized distribution
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
P(n)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{C_0^n{\ensuremath{\mathrm{e}}}^{-C_0}
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
n!}}}}}}}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="118" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img131.png"
|
|
ALT="$\displaystyle P(n)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{C_0^n{\ensuremath{\mathrm{e}}}^{-C_0}
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
n!}}}}}}}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
This obeys
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\mathop{\sum}_{n=0}^{+\infty}
|
|
P(n)=1\ ;
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="104" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img132.png"
|
|
ALT="$\displaystyle \mathop{\sum}_{n=0}^{+\infty}
|
|
P(n)=1\ ;
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\langle n\rangle=\mathop{\sum}_{n=0}^{+\infty}
|
|
nP(n)=C_0\ ;
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="168" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img133.png"
|
|
ALT="$\displaystyle \langle n\rangle=\mathop{\sum}_{n=0}^{+\infty}
|
|
nP(n)=C_0\ ;
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\langle n^2\rangle=\mathop{\sum}_{n=0}^{+\infty}
|
|
n^2 P(n)=C_0^2+C_0\ ;
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="221" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img134.png"
|
|
ALT="$\displaystyle \langle n^2\rangle=\mathop{\sum}_{n=0}^{+\infty}
|
|
n^2 P(n)=C_0^2+C_0\ ;
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
The standard deviation comes then to
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\sigma_{C_0}=\sqrt{\langle n^2\rangle-\langle n\rangle^2}=\sqrt{C_0}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="200" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img135.png"
|
|
ALT="$\displaystyle \sigma_{C_0}=\sqrt{\langle n^2\rangle-\langle n\rangle^2}=\sqrt{C_0}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
|
|
<P>
|
|
When the data have to be analyzed, one must compare observations with a model
|
|
which gives calculated values of the observations in dependence of a certain set of
|
|
parameters. The best values of the parameters (the target of investigation)
|
|
are the one that maximize the likelihood function [4,5]. The likelihood function for
|
|
Poisson variates is pretty difficult to use; furthermore, even simple data manipulations
|
|
are not straightforward with Poisson variates (see Sec. <A HREF="node63.html#sec:3">5.2.6</A>). The common choice is to approximate
|
|
Poisson variates with normal variates, and then use the much easier formalism
|
|
of normal distribution to a) do basic data manipulations and b) fit data with model.
|
|
To the latter task, in fact, the likelihood function is maximized simply by minimizing
|
|
the usual weighted-<IMG
|
|
WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1.png"
|
|
ALT="$ \chi ^2$">
|
|
[4] :
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({F_j-O_j}\right)}}^2
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\sigma_j^2
|
|
}}}}}}}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="151" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img136.png"
|
|
ALT="$\displaystyle \chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremath{\dis...
|
|
...\left({F_j-O_j}\right)}}^2
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\sigma_j^2
|
|
}}}}}}}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
where <IMG
|
|
WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img137.png"
|
|
ALT="$ O_j$">
|
|
are the experimentally observed values, <IMG
|
|
WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img138.png"
|
|
ALT="$ F_j$">
|
|
the calculated model values,
|
|
<IMG
|
|
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img139.png"
|
|
ALT="$ \sigma_j$">
|
|
the s.d.s of the observations.
|
|
|
|
<P>
|
|
Substituting directly the counts (and derived s.d.s) for the observations in the former :
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\chi_{(0)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({F_j-C_j}\right)}}^2
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
C_j
|
|
}}}}}}}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="160" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img140.png"
|
|
ALT="$\displaystyle \chi_{(0)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremat...
|
|
...remath{\left({F_j-C_j}\right)}}^2
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
C_j
|
|
}}}}}}}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
is the most common way. It is <I>slightly</I> wrong to do so, however [6],
|
|
the error being large only when the counts are low.
|
|
There is also a divergence for zero counts.
|
|
In fact, a slightly modified form [6] exists, reading
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\chi_{(1)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({F_j-{\ensuremath{\left({C_j+\min{\ensuremath{\left({1,C_j}\right)}}}\right)}}}\right)}}^2
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
C_j+1
|
|
}}}}}}}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="266" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img141.png"
|
|
ALT="$\displaystyle \chi_{(1)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremat...
|
|
...\right)}}}\right)}}}\right)}}^2
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
C_j+1
|
|
}}}}}}}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
Minimizing this form of <IMG
|
|
WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1.png"
|
|
ALT="$ \chi ^2$">
|
|
is equivalent - to an exceptionally good approximation [6]-
|
|
to maximizing the proper Poisson-likelihood.
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html841"
|
|
HREF="node54.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="file:/usr/share/latex2html/icons/next.png"></A>
|
|
<A NAME="tex2html837"
|
|
HREF="node46.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="file:/usr/share/latex2html/icons/up.png"></A>
|
|
<A NAME="tex2html831"
|
|
HREF="node52.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="file:/usr/share/latex2html/icons/prev.png"></A>
|
|
<A NAME="tex2html839"
|
|
HREF="node1.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="file:/usr/share/latex2html/icons/contents.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html842"
|
|
HREF="node54.html">Average vs. weighted average</A>
|
|
<B> Up:</B> <A NAME="tex2html838"
|
|
HREF="node46.html">How are different positions</A>
|
|
<B> Previous:</B> <A NAME="tex2html832"
|
|
HREF="node52.html">Advanced binning</A>
|
|
<B> <A NAME="tex2html840"
|
|
HREF="node1.html">Contents</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Thattil Dhanya
|
|
2018-09-28
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|