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<H3><A NAME="SECTION00081200000000000000">Gaussian kernel correlation integral</A></H3>
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The correlation sum Eq.(<A HREF="node30.html#eqdim2c2" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node30.html#eqdim2c2"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/cross_ref_motif.gif"></A>) can be regarded as an average density
of points where the local density is obtained by a kernel estimator with a step
kernel <IMG WIDTH=56 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7655" SRC="img152.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img152.gif">. A natural modification for small point sets is to
replace the sharp step kernel by a smooth kernel function of <EM>bandwidth</EM>
<IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img3.gif">. A particularly attractive case that has been studied in the
literature&nbsp;[<A HREF="citation.html#ghez1" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#ghez1">80</A>] is given by the Gaussian kernel, that is,
<IMG WIDTH=56 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7655" SRC="img152.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img152.gif"> is replaced by <IMG WIDTH=30 HEIGHT=22 ALIGN=BOTTOM ALT="tex2html_wrap_inline7661" SRC="img153.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img153.gif">.  The
resulting Gaussian kernel correlation sum <IMG WIDTH=39 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7663" SRC="img154.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img154.gif">
has the same scaling properties as the usual <IMG WIDTH=29 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7623" SRC="img142.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img142.gif">. It has been
observed in&nbsp;[<A HREF="citation.html#habil" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#habil">3</A>] that <IMG WIDTH=39 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7663" SRC="img154.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img154.gif"> can be
obtained from <IMG WIDTH=29 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7623" SRC="img142.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img142.gif"> via
<BR><A NAME="eqcg">&#160;</A><IMG WIDTH=500 HEIGHT=37 ALIGN=BOTTOM ALT="equation5759" SRC="img155.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img155.gif"><BR>
without having to repeat the whole computation. If <IMG WIDTH=29 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7623" SRC="img142.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img142.gif"> is given
at discrete values of <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img3.gif">, the integrals in Eq.(<A HREF="node32.html#eqcg" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node32.html#eqcg"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/cross_ref_motif.gif"></A>) can be
carried out numerically by interpolating <IMG WIDTH=29 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7623" SRC="img142.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img142.gif"> with pure power laws
. This is done in
<a href="../wuppertal/c2g.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/c2g.html">c2g</a> which uses a 15 point Gauss-Kronrod rule for the numerical integration.
<P>
<P><blockquote><A NAME="5947">&#160;</A><IMG WIDTH=343 HEIGHT=884 ALIGN=BOTTOM ALT="figure1650" SRC="img156.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img156.gif"><BR>
<STRONG>Figure:</STRONG> <A NAME="figdim2">&#160;</A>
   Dimension estimation for the (noise filtered) NMR laser data. Embedding
   dimensions 2 to 7 are shown. From above: <B>(a)</B> slopes are determined by
   straight line fits to the log-log plot of the correlation sum,
   Eq.&nbsp;(<A HREF="node30.html#eqdim2c2" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node30.html#eqdim2c2"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/cross_ref_motif.gif"></A>). <B>(b)</B> Takes-Theiler estimator of the same slope.
   <B>(c)</B> Slopes are obtained by straight line fits to the Gaussian kernel
   correlation sum, Eq.(<A HREF="node32.html#eqcg" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node32.html#eqcg"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/cross_ref_motif.gif"></A>). <B>(d)</B> Instead of the correlation
   dimension, it has been attempted to estimate the information dimension.<BR>
</blockquote><P><HR><A NAME="tex2html399" HREF="node33.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node33.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/next_motif.gif"></A> <A NAME="tex2html397" HREF="node30.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node30.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/up_motif.gif"></A> <A NAME="tex2html393" HREF="node31.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node31.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/previous_motif.gif"></A>   <BR>
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Wed Jan  6 15:38:27 CET 1999</I>
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