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manual/docs/html/slsDetectors-FAQ/node53.html
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manual/docs/html/slsDetectors-FAQ/node53.html
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<!--Converted with LaTeX2HTML 2012 (1.2)
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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* with significant contributions from:
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<TITLE>Poisson and normal statistics for diffraction</TITLE>
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<B> <A NAME="tex2html840"
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HREF="node1.html">Contents</A></B>
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<BR>
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<!--End of Navigation Panel-->
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<H2><A NAME="SECTION00624000000000000000">
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Poisson and normal statistics for diffraction</A>
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</H2>
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<P>
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The normal situation for diffraction data
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is that the observed signal is a photon count.
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Therefore it follows a Poisson distribution.
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If we have a count value <IMG
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WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img128.png"
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ALT="$ C_0$">
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that follows a Poisson distribution,
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we can assume immediately that the average is equal to <IMG
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WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img128.png"
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ALT="$ C_0$">
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and the s.d. is <!-- MATH
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$\sqrt{C_0}$
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-->
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<IMG
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WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img129.png"
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ALT="$ \sqrt{C_0}$">
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.
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I.e., repeated experiments would give values <IMG
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WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img130.png"
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ALT="$ n$">
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distributed according to the normalized distribution
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<P><!-- MATH
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\begin{displaymath}
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P(n)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{C_0^n{\ensuremath{\mathrm{e}}}^{-C_0}
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}}}}{{\ensuremath{\displaystyle{
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n!}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="118" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
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SRC="img131.png"
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ALT="$\displaystyle P(n)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{C_0^n{\ensuremath{\mathrm{e}}}^{-C_0}
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}}}}{{\ensuremath{\displaystyle{
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n!}}}}}}}
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$">
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</DIV><P>
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</P>
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This obeys
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<P><!-- MATH
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\begin{displaymath}
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\mathop{\sum}_{n=0}^{+\infty}
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P(n)=1\ ;
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="104" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
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SRC="img132.png"
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ALT="$\displaystyle \mathop{\sum}_{n=0}^{+\infty}
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P(n)=1\ ;
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$">
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</DIV><P>
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</P>
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<P><!-- MATH
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\begin{displaymath}
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\langle n\rangle=\mathop{\sum}_{n=0}^{+\infty}
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nP(n)=C_0\ ;
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="168" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
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SRC="img133.png"
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ALT="$\displaystyle \langle n\rangle=\mathop{\sum}_{n=0}^{+\infty}
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nP(n)=C_0\ ;
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$">
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</DIV><P>
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</P>
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<P><!-- MATH
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\begin{displaymath}
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\langle n^2\rangle=\mathop{\sum}_{n=0}^{+\infty}
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n^2 P(n)=C_0^2+C_0\ ;
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="221" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
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SRC="img134.png"
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ALT="$\displaystyle \langle n^2\rangle=\mathop{\sum}_{n=0}^{+\infty}
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n^2 P(n)=C_0^2+C_0\ ;
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$">
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</DIV><P>
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</P>
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The standard deviation comes then to
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<P><!-- MATH
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\begin{displaymath}
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\sigma_{C_0}=\sqrt{\langle n^2\rangle-\langle n\rangle^2}=\sqrt{C_0}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="200" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img135.png"
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ALT="$\displaystyle \sigma_{C_0}=\sqrt{\langle n^2\rangle-\langle n\rangle^2}=\sqrt{C_0}
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$">
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</DIV><P>
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</P>
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<P>
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When the data have to be analyzed, one must compare observations with a model
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which gives calculated values of the observations in dependence of a certain set of
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parameters. The best values of the parameters (the target of investigation)
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are the one that maximize the likelihood function [4,5]. The likelihood function for
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Poisson variates is pretty difficult to use; furthermore, even simple data manipulations
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are not straightforward with Poisson variates (see Sec. <A HREF="node63.html#sec:3">5.2.6</A>). The common choice is to approximate
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Poisson variates with normal variates, and then use the much easier formalism
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of normal distribution to a) do basic data manipulations and b) fit data with model.
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To the latter task, in fact, the likelihood function is maximized simply by minimizing
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the usual weighted-<IMG
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WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img1.png"
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ALT="$ \chi ^2$">
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[4] :
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<P><!-- MATH
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\begin{displaymath}
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\chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({F_j-O_j}\right)}}^2
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}}}}{{\ensuremath{\displaystyle{
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\sigma_j^2
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}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="151" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
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SRC="img136.png"
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ALT="$\displaystyle \chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\dis...
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...\left({F_j-O_j}\right)}}^2
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}}}}{{\ensuremath{\displaystyle{
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\sigma_j^2
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}}}}}}}
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$">
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</DIV><P>
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</P>
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where <IMG
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WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img137.png"
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ALT="$ O_j$">
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are the experimentally observed values, <IMG
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WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img138.png"
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ALT="$ F_j$">
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the calculated model values,
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<IMG
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WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img139.png"
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ALT="$ \sigma_j$">
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the s.d.s of the observations.
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<P>
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Substituting directly the counts (and derived s.d.s) for the observations in the former :
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<P><!-- MATH
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\begin{displaymath}
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\chi_{(0)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({F_j-C_j}\right)}}^2
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}}}}{{\ensuremath{\displaystyle{
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C_j
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}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="160" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
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SRC="img140.png"
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ALT="$\displaystyle \chi_{(0)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremat...
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...remath{\left({F_j-C_j}\right)}}^2
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}}}}{{\ensuremath{\displaystyle{
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C_j
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}}}}}}}
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$">
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</DIV><P>
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</P>
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is the most common way. It is <I>slightly</I> wrong to do so, however [6],
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the error being large only when the counts are low.
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There is also a divergence for zero counts.
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In fact, a slightly modified form [6] exists, reading
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<P><!-- MATH
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\begin{displaymath}
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\chi_{(1)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({F_j-{\ensuremath{\left({C_j+\min{\ensuremath{\left({1,C_j}\right)}}}\right)}}}\right)}}^2
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}}}}{{\ensuremath{\displaystyle{
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C_j+1
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}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="266" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
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SRC="img141.png"
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ALT="$\displaystyle \chi_{(1)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremat...
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...\right)}}}\right)}}}\right)}}^2
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}}}}{{\ensuremath{\displaystyle{
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C_j+1
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}}}}}}}
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$">
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</DIV><P>
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</P>
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Minimizing this form of <IMG
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WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img1.png"
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ALT="$ \chi ^2$">
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is equivalent - to an exceptionally good approximation [6]-
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to maximizing the proper Poisson-likelihood.
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HREF="node54.html">Average vs. weighted average</A>
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HREF="node46.html">How are different positions</A>
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<ADDRESS>
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Thattil Dhanya
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2018-08-23
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