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243 lines
6.7 KiB
HTML
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<TITLE>Straight Poisson (zero-skipping) weighted average</TITLE>
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HREF="node59.html">Mighell-Poisson weighted average</A>
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HREF="node54.html">Average vs. weighted average</A>
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HREF="node57.html">Weighted average: definition and</A>
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<H3><A NAME="SECTION00625400000000000000">
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Straight Poisson (zero-skipping) weighted average</A>
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</H3>
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<P>
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When <IMG
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WIDTH="63" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img158.png"
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ALT="$ O_j=C_j$">
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and <!-- MATH
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$\sigma_j^2=C_j$
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-->
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<IMG
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WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img159.png"
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ALT="$ \sigma_j^2=C_j$">
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<P><!-- MATH
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\begin{displaymath}
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\langle x \rangle_{\!\mathrm{w(1)}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
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{N_{\mathrm{obs}}}
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}}}}{{\ensuremath{\displaystyle{
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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C_j
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}}}}}}}
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}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="126" HEIGHT="110" ALIGN="MIDDLE" BORDER="0"
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SRC="img160.png"
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ALT="$\displaystyle \langle x \rangle_{\!\mathrm{w(1)}}={\ensuremath{\displaystyle{\f...
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...nsuremath{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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C_j
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}}}}}}}
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}}}}}}}
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$">
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</DIV><P>
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</P>
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Here we need to eliminate the singularity when <IMG
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WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img148.png"
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ALT="$ C_j=0$">
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. In order to do so, we skip data points which are zero.
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Then if <!-- MATH
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$N_{\mathrm{obs}}^*$
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-->
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<IMG
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WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img149.png"
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ALT="$ N_{\mathrm{obs}}^*$">
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is the number of non-zero data points,
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<P><!-- MATH
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\begin{displaymath}
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\langle x \rangle_{\!\mathrm{w(1)}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
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{N_{\mathrm{obs}}^*}
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}}}}{{\ensuremath{\displaystyle{
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\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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C_j
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}}}}}}}
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}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="126" HEIGHT="110" ALIGN="MIDDLE" BORDER="0"
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SRC="img160.png"
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ALT="$\displaystyle \langle x \rangle_{\!\mathrm{w(1)}}={\ensuremath{\displaystyle{\f...
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...nsuremath{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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C_j
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}}}}}}}
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}}}}}}}
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$">
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</DIV><P>
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</P>
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<P><!-- MATH
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\begin{displaymath}
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\sigma_{\langle x \rangle_{\!\mathrm{w(1)}}} = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
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1
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}}}}{{\ensuremath{\displaystyle{\sqrt{
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\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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C_j
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}}}}}}}
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}}}}}}}}=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x \rangle_{\!\mathrm{w(1)}}}}}}{{\ensuremath{\displaystyle{
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N_{\mathrm{obs}}^*
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}}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="258" HEIGHT="141" ALIGN="MIDDLE" BORDER="0"
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SRC="img161.png"
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ALT="$\displaystyle \sigma_{\langle x \rangle_{\!\mathrm{w(1)}}} = {\ensuremath{\disp...
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..._{\!\mathrm{w(1)}}}}}}{{\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}_{(1)}=
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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
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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
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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(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="463" HEIGHT="189" ALIGN="MIDDLE" BORDER="0"
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SRC="img162.png"
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ALT="$\displaystyle \mathsf{GoF}_{(1)}=
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\sqrt{
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{\ensuremath{\displaystyle{\frac{{\ens...
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...th{\left({
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\langle x\rangle^*-\langle x \rangle_{\!\mathrm{w(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^*$">
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is the simple average of the non-zero data points; and of course
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<P><!-- MATH
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\begin{displaymath}
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{\sigma}_{\langle x \rangle_{\!\mathrm{w(1)}}}^{\mathrm{corrected}} = \mathsf{GoF}_{(1)}\ \sigma_{\langle x \rangle_{\!\mathrm{w(1)}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="185" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img164.png"
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ALT="$\displaystyle {\sigma}_{\langle x \rangle_{\!\mathrm{w(1)}}}^{\mathrm{corrected}} = \mathsf{GoF}_{(1)}\ \sigma_{\langle x \rangle_{\!\mathrm{w(1)}}}
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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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</HTML>
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