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manual/docs/html/slsDetectors-FAQ/node51.html
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!--Converted with LaTeX2HTML 2008 (1.71)
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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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<HTML>
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<HEAD>
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<TITLE>Special nasty cases</TITLE>
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<META NAME="description" CONTENT="Special nasty cases">
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<META NAME="keywords" CONTENT="slsDetectors-FAQ">
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HREF="node52.html">Advanced binning</A>
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<B> Up:</B> <A NAME="tex2html814"
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HREF="node50.html">Basic binning</A>
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<B> Previous:</B> <A NAME="tex2html810"
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HREF="node50.html">Basic binning</A>
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<B> <A NAME="tex2html816"
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HREF="node1.html">Contents</A></B>
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<BR>
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<BR>
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<!--End of Navigation Panel-->
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<H3><A NAME="SECTION00622100000000000000">
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Special nasty cases</A>
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</H3>
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<P>
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Here we explore some special cases to see the robustness
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of the method.
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<P>
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1) If no experimental observation contributes to bin <IMG
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WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img98.png"
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ALT="$ B_\ell$">
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according to one of the criteria
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above, then we shall find <IMG
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WIDTH="53" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img118.png"
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ALT="$ X_\ell=0$">
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and especially <IMG
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WIDTH="49" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img119.png"
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ALT="$ Y_\ell=0$">
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. The latter condition is
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valid as an exclusion condition
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(meaning that we discard that point and we do not perform further operations on it,
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neither do we output it).
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<P>
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2) if only one experimental observation - call it interval <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img120.png"
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ALT="$ b$">
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, dropping indices - contributes
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to bin <IMG
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WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img98.png"
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ALT="$ B_\ell$">
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,
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then we have
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<P><!-- MATH
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\begin{displaymath}
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X_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}\
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e(C+1)}}}}{{\ensuremath{\displaystyle{m|b|}}}}}}}\
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{\ensuremath{\left({
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{|b|m}}}}{{\ensuremath{\displaystyle{e\sqrt{C+1}}}}}}}}
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}\right)}}^{2}
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={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{|b|m}}}}{{\ensuremath{\displaystyle{e}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="380" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
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SRC="img121.png"
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ALT="$\displaystyle X_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyl...
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...nsuremath{\displaystyle{\vert b\vert m}}}}{{\ensuremath{\displaystyle{e}}}}}}}
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$">
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</DIV><P>
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</P>
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<P><!-- MATH
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\begin{displaymath}
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Y_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}\
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{\ensuremath{\left({
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{|b|m}}}}{{\ensuremath{\displaystyle{e\sqrt{C+1}}}}}}}}
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}\right)}}^{2}
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={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{|b|^2m^2}}}}{{\ensuremath{\displaystyle{e^2(C+1)}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="344" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
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SRC="img122.png"
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ALT="$\displaystyle Y_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyl...
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...\displaystyle{\vert b\vert^2m^2}}}}{{\ensuremath{\displaystyle{e^2(C+1)}}}}}}}
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$">
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</DIV><P>
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</P>
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and so
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<P><!-- MATH
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\begin{displaymath}
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R_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{X_\ell}}}}{{\ensuremath{\displaystyle{Y_\ell}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e(C+1)}}}}{{\ensuremath{\displaystyle{m|b|}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="152" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
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SRC="img123.png"
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ALT="$\displaystyle R_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyl...
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...emath{\displaystyle{e(C+1)}}}}{{\ensuremath{\displaystyle{m\vert b\vert}}}}}}}
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$">
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</DIV><P>
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</P>
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that is the experimental rate as in pixel <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img120.png"
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ALT="$ b$">
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;
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<P><!-- MATH
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\begin{displaymath}
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\sigma_{R_\ell}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{\sqrt{Y_\ell}}}}}}}}=
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\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{B}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}}}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e\sqrt{(C+1)}}}}}{{\ensuremath{\displaystyle{|b|m}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="258" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
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SRC="img124.png"
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ALT="$\displaystyle \sigma_{R_\ell}={\ensuremath{\displaystyle{\frac{{\ensuremath{\di...
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...isplaystyle{e\sqrt{(C+1)}}}}}{{\ensuremath{\displaystyle{\vert b\vert m}}}}}}}
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$">
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</DIV><P>
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</P>
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that is the same s.d. that can be calculated directly for <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img120.png"
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ALT="$ b$">
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, augmented by factor
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<P><!-- MATH
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\begin{displaymath}
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\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{B}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="76" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
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SRC="img125.png"
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ALT="$\displaystyle \sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle...
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...ath{\displaystyle{{\ensuremath{\left\vert{ b\cap B_\ell }\right\vert}}}}}}}}}}
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$">
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</DIV><P>
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</P>
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that takes into account the extrapolation error.
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<P>
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<BR><HR>
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<ADDRESS>
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Thattil Dhanya
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2018-03-12
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</ADDRESS>
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</BODY>
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</HTML>
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