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<TITLE>Robust Averaging</TITLE>
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<H1><A NAME="SECTION001100000000000000000"></A><A NAME="ap:robust"></A>
<BR>
Robust Averaging
</H1>
<P>
A robust averaging algorithm is implemented by <SMALL>GCX </SMALL>and used in
several places, most notably for zeropoint fitting by the aperture
photometry and multiframe reduction routines.
The algorithm calculates the robust average of a number of values
(for the zeropoint routines, these are the differences between the
standard and instrumental magnitudes of standard stars).
<P>
The data used consists of the values we want to calculate, and
the estimated error of each value. For fitting frame zeropoints
they are:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{eqnarray}
y_k = S_k - I_k\\
\epsilon^2_k = \epsilon_{ik}^2 + \epsilon_{sk}^2
\end{eqnarray}
-->
<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG
WIDTH="100" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img174.png"
ALT="$\displaystyle y_k = S_k - I_k$"></TD>
<TD> </TD>
<TD> </TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(B.<SPAN CLASS="arabic">1</SPAN>)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG
WIDTH="105" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img175.png"
ALT="$\displaystyle \epsilon^2_k = \epsilon_{ik}^2 + \epsilon_{sk}^2$"></TD>
<TD> </TD>
<TD> </TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(B.<SPAN CLASS="arabic">2</SPAN>)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
where <SPAN CLASS="MATH"><IMG
WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img135.png"
ALT="$S$"></SPAN> is the standard magnitude, <SPAN CLASS="MATH"><IMG
WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img13.png"
ALT="$I$"></SPAN> is the instrumental magnitude,
<SPAN CLASS="MATH"><IMG
WIDTH="17" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img176.png"
ALT="$\epsilon_i$"></SPAN> is the estimated error of the instrumental magnitude,
<SPAN CLASS="MATH"><IMG
WIDTH="18" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img177.png"
ALT="$\epsilon_s$"></SPAN> is the error of the standard magnitude of each star.
Each star is assigned a <EM>natural weight</EM>, calculated as
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
<!-- MATH
\begin{equation}
W_k = \frac{1}{\epsilon_k^2}
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
WIDTH="64" HEIGHT="47" BORDER="0"
SRC="img178.png"
ALT="\begin{displaymath}
W_k = \frac{1}{\epsilon_k^2}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(B.<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
We start with a very robust estimate of the average:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
<!-- MATH
\begin{equation}
\widetilde{Z}={\rm median}(y_k)
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
WIDTH="119" HEIGHT="31" BORDER="0"
SRC="img179.png"
ALT="\begin{displaymath}
\widetilde{Z}={\rm median}(y_k)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(B.<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
and calculate the <EM>residuals</EM> of each value:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
<!-- MATH
\begin{equation}
\rho_k=y_k - \widetilde{Z}
\end{equation}
-->
<A NAME="eq:residuals"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:residuals"></A><IMG
WIDTH="92" HEIGHT="30" BORDER="0"
SRC="img180.png"
ALT="\begin{displaymath}
\rho_k=y_k - \widetilde{Z}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(B.<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
and the <EM>standard errors</EM>:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
<!-- MATH
\begin{equation}
\rho'_k=(y_k - \widetilde{Z})\sqrt{W_k}
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
WIDTH="146" HEIGHT="31" BORDER="0"
SRC="img181.png"
ALT="\begin{displaymath}
\rho'_k=(y_k - \widetilde{Z})\sqrt{W_k}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(B.<SPAN CLASS="arabic">6</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
The expected value of each standard error is 1. We can identify
possible outliers by their large standard errors. A simple way to
treat outliers is to just exclude from the fit any value that has a
standard error larger than a certain threshold. This has the
disadvantage that small changes in the values can cause large jumps in
the solution if an outlier just crosses the threshold. Instead, we
adjust the weights of the data points to reduce the outliers'
contribution to the solution:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
<!-- MATH
\begin{equation}
W'_k = \frac{W_k}{1 + \left({\rho'_k}\over{\alpha}\right)^\beta}
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
WIDTH="125" HEIGHT="62" BORDER="0"
SRC="img182.png"
ALT="\begin{displaymath}
W'_k = \frac{W_k}{1 + \left({\rho'_k}\over{\alpha}\right)^\beta}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(B.<SPAN CLASS="arabic">7</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
The weighting function reduces the weight of values that have residuals <SPAN CLASS="MATH"><IMG
WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img183.png"
ALT="$\alpha$"></SPAN> times larger
than expected to one half. Of course values with even larger residuals are downweighted even
more. The parameter <SPAN CLASS="MATH"><IMG
WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
SRC="img184.png"
ALT="$\beta$"></SPAN> tunes the ``sharpness'' of the weighting
function.<A NAME="tex2html69"
HREF="footnode.html#foot1264"><SUP>B.<SPAN CLASS="arabic">1</SPAN></SUP></A>A new estimate of the average is produced by:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
<!-- MATH
\begin{equation}
\widetilde{Z}=\sum_k(y_k-\widetilde{Z})W'_k
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
WIDTH="147" HEIGHT="48" BORDER="0"
SRC="img185.png"
ALT="\begin{displaymath}
\widetilde{Z}=\sum_k(y_k-\widetilde{Z})W'_k
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(B.<SPAN CLASS="arabic">8</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
The residual calculation, weighting and average estimating are
iterated until the estimate doesn't change.
<P>
Finally, the error for the estimated parameters is calculated.
the error of the zero point is:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
<!-- MATH
\begin{equation}
\epsilon_{\rm zp}^2 = \frac{\sum\rho_k^2W'_k}{\sum W'_k}
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
WIDTH="107" HEIGHT="50" BORDER="0"
SRC="img186.png"
ALT="\begin{displaymath}
\epsilon_{\rm zp}^2 = \frac{\sum\rho_k^2W'_k}{\sum W'_k}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(B.<SPAN CLASS="arabic">9</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
and the <EM>mean error of unit weight</EM> is:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
<!-- MATH
\begin{equation}
{\rm me1}^2 = \frac{\sum\rho_k^2W'_k}{N-1}
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
WIDTH="126" HEIGHT="46" BORDER="0"
SRC="img187.png"
ALT="\begin{displaymath}
{\rm me1}^2 = \frac{\sum\rho_k^2W'_k}{N-1}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(B.<SPAN CLASS="arabic">10</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where <SPAN CLASS="MATH"><IMG
WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img31.png"
ALT="$N$"></SPAN> is the number of standard stars. The mean error of unit weight
is 1 in the ideal case (when all the errors are estimated correctly). A significantly
larger value should raise doubts about the error estimates.
<P>
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