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206 lines
5.8 KiB
HTML
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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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<TITLE>Mighell-Poisson weighted average</TITLE>
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<B> Next:</B> <A NAME="tex2html922"
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HREF="node60.html">Comparison</A>
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<B> Up:</B> <A NAME="tex2html918"
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HREF="node54.html">Average vs. weighted average</A>
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HREF="node58.html">Straight Poisson (zero-skipping) weighted</A>
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HREF="node1.html">Contents</A></B>
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<H3><A NAME="SECTION00625500000000000000">
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Mighell-Poisson weighted average</A>
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</H3>
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<P>
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When <!-- MATH
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$O_j=C_j+\min{\ensuremath{\left({1,C_j}\right)}}$
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-->
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<IMG
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WIDTH="157" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img165.png"
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ALT="$ O_j=C_j+\min{\ensuremath{\left({1,C_j}\right)}}$">
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and <!-- MATH
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$\sigma_j^2=C_j+1$
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-->
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<IMG
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WIDTH="88" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img166.png"
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ALT="$ \sigma_j^2=C_j+1$">
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<P><!-- MATH
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\begin{displaymath}
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\langle x \rangle_{\!\mathrm{w(2)}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
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{N_{\mathrm{obs}}^*}
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}}}}{{\ensuremath{\displaystyle{
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\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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C_j+1
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}}}}}}}
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}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="180" HEIGHT="117" ALIGN="MIDDLE" BORDER="0"
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SRC="img167.png"
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ALT="$\displaystyle \langle x \rangle_{\!\mathrm{w(2)}}={\ensuremath{\displaystyle{\f...
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...uremath{\displaystyle{1
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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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$">
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</DIV><P>
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</P>
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<P><!-- MATH
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\begin{displaymath}
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\sigma_{\langle x \rangle_{\!\mathrm{w(2)}}} = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
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1
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}}}}{{\ensuremath{\displaystyle{\sqrt{
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\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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C_j+1
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}}}}}}}
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}}}}}}}}=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x \rangle_{\!\mathrm{w(2)}}}}}}{{\ensuremath{\displaystyle{
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N_{\mathrm{obs}}^*
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}}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="286" HEIGHT="141" ALIGN="MIDDLE" BORDER="0"
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SRC="img168.png"
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ALT="$\displaystyle \sigma_{\langle x \rangle_{\!\mathrm{w(2)}}} = {\ensuremath{\disp...
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..._{\!\mathrm{w(2)}}}}}}{{\ensuremath{\displaystyle{
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N_{\mathrm{obs}}^*
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}}}}}}}}
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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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\mathsf{GoF}_{(2)}=
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\sqrt{
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
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\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
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\!\!\!\!C_j+N_{\mathrm{obs}}^*
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-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
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{\ensuremath{\left[{
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N_{\mathrm{obs}}^*
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}\right]}}^2
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}}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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C_j+1
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}}}}}}} }}}}}}}
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}}}}{{\ensuremath{\displaystyle{
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N_{\mathrm{obs}}^*-1
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}}}}}}}
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}
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=\sqrt{
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*-1}}}}}}}
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{\ensuremath{\left({
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\langle x\rangle^*-\langle x \rangle_{\!\mathrm{w(2)}}+1
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}\right)}}
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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="568" HEIGHT="189" ALIGN="MIDDLE" BORDER="0"
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SRC="img169.png"
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ALT="$\displaystyle \mathsf{GoF}_{(2)}=
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\sqrt{
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{\ensuremath{\displaystyle{\frac{{\ens...
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...{\left({
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\langle x\rangle^*-\langle x \rangle_{\!\mathrm{w(2)}}+1
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}\right)}}
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}
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$">
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</DIV><P>
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</P>
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where <!-- MATH
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$\langle x\rangle^*$
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-->
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<IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img163.png"
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ALT="$ \langle x\rangle^*$">
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is the simple average of the non-zero data points; and of course
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<P><!-- MATH
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\begin{displaymath}
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{\sigma}_{\langle x \rangle_{\!\mathrm{w(2)}}}^{\mathrm{corrected}} = \mathsf{GoF}_{(2)}\ \sigma_{\langle x \rangle_{\!\mathrm{w(2)}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="185" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img170.png"
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ALT="$\displaystyle {\sigma}_{\langle x \rangle_{\!\mathrm{w(2)}}}^{\mathrm{corrected}} = \mathsf{GoF}_{(2)}\ \sigma_{\langle x \rangle_{\!\mathrm{w(2)}}}
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$">
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</DIV><P>
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</P>
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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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2018-09-28
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</ADDRESS>
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</BODY>
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</HTML>
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