<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 98.1p1 release (March 2nd, 1998)
originally by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>Smoothing spline interpolation</TITLE>
<META NAME="description" CONTENT="Smoothing spline interpolation">
<META NAME="keywords" CONTENT="vol2">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
<LINK REL="STYLESHEET" HREF="vol2.css">
<LINK REL="next" HREF="node155.html">
<LINK REL="previous" HREF="node153.html">
<LINK REL="up" HREF="node152.html">
<LINK REL="next" HREF="node155.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html3367"
HREF="node155.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="icons.gif/next_motif.gif"></A>
<A NAME="tex2html3364"
HREF="node152.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="icons.gif/up_motif.gif"></A>
<A NAME="tex2html3358"
HREF="node153.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="icons.gif/previous_motif.gif"></A>
<A NAME="tex2html3366"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="icons.gif/contents_motif.gif"></A>
<BR>
<B> Next:</B> <A NAME="tex2html3368"
HREF="node155.html">Background estimate by filtering</A>
<B> Up:</B> <A NAME="tex2html3365"
HREF="node152.html">Background Definition</A>
<B> Previous:</B> <A NAME="tex2html3359"
HREF="node153.html">Bivariate polynomial interpolation</A>
<BR>
<BR>
<!--End of Navigation Panel-->
<H2><A NAME="SECTION001142000000000000000">
Smoothing spline interpolation</A>
</H2>
<P>
An alternative method performs the interpolation of interorder
background using smoothing spline polynomials. Spline interpolation
consists of the approximation of a function by means of series of
polynomials over adjacent intervals with continuous derivatives at the
end-point of the intervals. Smoothing spline interpolation enables to
control the variance of the residuals over the data set, as follows:
<P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\delta = \sum_{i=1}^{m} (y_i - \hat y_i)^{2}
\end{displaymath} -->
<IMG
WIDTH="154" HEIGHT="57"
SRC="img291.gif"
ALT="\begin{displaymath}\delta = \sum_{i=1}^{m} (y_i - \hat y_i)^{2} \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <I>y</I><SUB><I>i</I></SUB> is the <I>i</I><SUP><I>th</I></SUP> observed value and <IMG
WIDTH="48" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img292.gif"
ALT="\( \hat y_i \)">
the
<I>i</I><SUP><I>th</I></SUP> interpolated value is the sum of the squared residuals and the
smoothing spline algorithm will try to fit a solution such as:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\delta \le S*\alpha
\end{displaymath} -->
<IMG
WIDTH="87" HEIGHT="21"
SRC="img293.gif"
ALT="\begin{displaymath}\delta \le S*\alpha \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where S is the smoothing factor and
<!-- MATH: $\alpha=0.001$ -->
<IMG
WIDTH="52" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img294.gif"
ALT="\( \alpha=0.001 \)">
is the tolerance.
<P>
One must retain two particular values of S:
<UL>
<LI>S = 0. The interpolation pass through every observation value.
<LI>S very large. The interpolation consists of the one-piece
polynomial interpolation.
</UL>
<P>
The solution is estimated by an iterative process. Smoothing spline
interpolation is designed to smooth data sets which are mildly
contaminated with isolated errors. Convergence is not always secured
for this class of algorithms, which on the other hand enables to
control the residuals. The median of pixel values in a window surrounding
the background reference position is computed before spline interpolation.
The size of the window (session keyword <TT>BKGRAD</TT>) is defined
along the orders and along the columns of the raw spectrum.
<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html3367"
HREF="node155.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="icons.gif/next_motif.gif"></A>
<A NAME="tex2html3364"
HREF="node152.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="icons.gif/up_motif.gif"></A>
<A NAME="tex2html3358"
HREF="node153.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="icons.gif/previous_motif.gif"></A>
<A NAME="tex2html3366"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="icons.gif/contents_motif.gif"></A>
<BR>
<B> Next:</B> <A NAME="tex2html3368"
HREF="node155.html">Background estimate by filtering</A>
<B> Up:</B> <A NAME="tex2html3365"
HREF="node152.html">Background Definition</A>
<B> Previous:</B> <A NAME="tex2html3359"
HREF="node153.html">Bivariate polynomial interpolation</A>
<!--End of Navigation Panel-->
<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
</BODY>
</HTML>
