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<H2><A NAME="SECTION00081000000000000000">Correlation dimension</A></H2>
<A NAME="secdimc2"> </A>
Roughly speaking, the idea behind certain quantifiers of dimensions is that
the weight <IMG WIDTH=27 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7557" SRC="img131.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img131.gif"> of a typical <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">-ball covering part of the
invariant set scales with its diameter like <IMG WIDTH=65 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline7561" SRC="img132.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img132.gif">,
where the value for <I>D</I> depends also on the precise way one defines the
weight. Using the square of the probability <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6569" SRC="img20.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img20.gif"> to find a point of the set
inside the ball, the dimension is called the correlation dimension <IMG WIDTH=19 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7567" SRC="img133.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img133.gif">,
which is computed most efficiently by the correlation sum [<A HREF="citation.html#GP" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#GP">73</A>]:
<BR><A NAME="eqdim2c2"> </A><IMG WIDTH=500 HEIGHT=50 ALIGN=BOTTOM ALT="equation5740" SRC="img134.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img134.gif"><BR>
where <IMG WIDTH=11 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline7569" SRC="img135.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img135.gif"> are <I>m</I>-dimensional delay vectors, <IMG WIDTH=283 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7573" SRC="img136.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img136.gif"> the number of pairs of points covered by the sums,
<IMG WIDTH=11 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline6897" SRC="img52.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img52.gif"> is the Heaviside step function and <I>w</I> will be discussed below. On
sufficiently small length scales and when the embedding dimension <I>m</I> exceeds
the box-dimension of the attractor [<A HREF="citation.html#SauerYorke" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#SauerYorke">74</A>],
<BR><IMG WIDTH=500 HEIGHT=19 ALIGN=BOTTOM ALT="equation5742" SRC="img137.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img137.gif"><BR>
Since one does not know the box-dimension <I>a priori</I>, one checks for
convergence of the estimated values of <IMG WIDTH=19 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7567" SRC="img133.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img133.gif"> in <I>m</I>.
<P>
The literature on the correct and spurious estimation of the correlation
dimension is huge and this is certainly not the place to repeat all the
arguments. The relevant caveats and misconceptions are reviewed for example in
Refs. [<A HREF="citation.html#theiler_dim" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#theiler_dim">75</A>, <A HREF="citation.html#gss" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#gss">11</A>, <A HREF="citation.html#dim" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#dim">76</A>, <A HREF="citation.html#KantzSchreiber" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#KantzSchreiber">2</A>]. The most prominent precaution
is to exclude temporally correlated points from the pair counting by the so
called Theiler window <I>w</I> [<A HREF="citation.html#theiler_dim" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#theiler_dim">75</A>]. In order to become a consistent
estimator of the correlation <EM>integral</EM> (from which the dimension is
derived) the correlation <EM>sum</EM> should cover a random sample of points drawn
independently according to the invariant measure on the attractor. Successive
elements of a time series are not usually independent. In particular for highly
sampled flow data subsequent delay vectors are highly correlated. Theiler
suggested to remove this spurious effect by simply ignoring all pairs of points
in 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>) whose time indices differ by less than <I>w</I>, where <I>w</I>
should be chosen generously. With <i>O(N²)</i> pairs available, the loss of <I>O</I>(<I>N</I>)
pairs is not dramatic as long as <IMG WIDTH=50 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline7595" SRC="img138.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img138.gif">. At the very least, pairs with <I>j</I>=<I>k</I>
have to be excluded [<A HREF="citation.html#grass_finite" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#grass_finite">77</A>], since otherwise the strong bias to
<IMG WIDTH=49 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7599" SRC="img139.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img139.gif">, the mathematically correct value for a finite set of points, reduces
the scaling range drastically. Choosing <I>w</I>, the first zero of the
auto-correlation function, sometimes even the decay time of the autocorrelation
function, are not large enough since they reflect only overall linear
correlations [<A HREF="citation.html#theiler_dim" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#theiler_dim">75</A>, <A HREF="citation.html#dim" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#dim">76</A>]. The space-time-separation plot
(Sec. <A HREF="node15.html#secstp" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node15.html#secstp"><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>) provides a good means of determining a sufficient value
for <I>w</I>, as discussed for example in [<A HREF="citation.html#stp" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#stp">41</A>, <A HREF="citation.html#KantzSchreiber" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#KantzSchreiber">2</A>]. In some cases,
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