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<H2><A NAME="SECTION00082000000000000000">Information dimension</A></H2>
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Another way of attaching weight to <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">-balls, which is more natural, is
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"> itself. The resulting scaling exponent is called the
information dimension <IMG WIDTH=18 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7809" SRC="img157.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img157.gif">. Since the Kaplan-Yorke dimension of
Sec.<A HREF="node26.html#seclyap" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node26.html#seclyap"><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> is an approximation of <IMG WIDTH=18 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7809" SRC="img157.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img157.gif">, the computation of <IMG WIDTH=18 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7809" SRC="img157.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img157.gif">
through scaling properties is a relevant cross-check for highly deterministic
data. <IMG WIDTH=18 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7809" SRC="img157.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img157.gif"> can be computed from a modified correlation sum, where, however,
unpleasant systematic errors occur. The <EM>fixed mass</EM>
approach&nbsp;[<A HREF="citation.html#badiipoliti" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#badiipoliti">81</A>] circumvents these problems, so that, including
finite sample corrections&nbsp;[<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>], a rather robust estimator
exists. Instead of counting the number of points in a ball one asks here for
the diameter <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"> which a ball must have to contain a certain number <I>k</I>
of points when a time series of length <I>N</I> is given. Its scaling with <I>k</I> and
<I>N</I> yields the dimension in the limit of small length scales by
<BR><A NAME="eqdim1fixmass">&#160;</A><IMG WIDTH=500 HEIGHT=38 ALIGN=BOTTOM ALT="equation5952" SRC="img158.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img158.gif"><BR>
The routine <a href="../wuppertal/c1.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/c1.html">c1</a> computes the (geometric) mean length scale
<IMG WIDTH=107 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7827" SRC="img159.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img159.gif"> for which <I>k</I> neighbors are found in <I>N</I> data
points, as a function of <I>k</I>/<I>N</I>. Unlike the correlation sum, finite sample
corrections are necessary if <I>k</I> is small&nbsp;[<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>]. Essentially, the
<IMG WIDTH=20 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7837" SRC="img160.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img160.gif"> of <I>k</I> has to be replaced by the digamma function <IMG WIDTH=31 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7841" SRC="img161.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img161.gif">.  The
resulting expression is implemented in <a href="../wuppertal/c1.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/c1.html">c1</a>. Given <I>m</I> and <IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6553" SRC="img16.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img16.gif">, the
routine varies <I>k</I> and <I>N</I> such that the largest reasonable range of <I>k</I>/<I>N</I> is
covered with moderate computational effort. This means that for <IMG WIDTH=136 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7853" SRC="img162.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img162.gif"> (default: <I>K</I>=100), all <I>N</I> available points are searched for neighbors
and <I>k</I> is varied. For <IMG WIDTH=115 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7861" SRC="img163.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img163.gif">, <I>k</I>=<I>K</I> is kept fixed and <I>N</I> is
decreased.  The result for the NMR laser data is shown in
Fig.&nbsp;<A HREF="node32.html#figdim2" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node32.html#figdim2"><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>&nbsp;<B>(d)</B>, where a nice scaling with <IMG WIDTH=69 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7867" SRC="img164.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img164.gif"> can
be discerned. For comparability, the logarithmic derivative of <I>k</I>/<I>N</I> is plotted
versus <IMG WIDTH=107 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7871" SRC="img165.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img165.gif"> and not vice versa, although <I>k</I>/<I>N</I>
is the independent variable.  One easily detects again the violations of
scaling discussed before: Cut-off on the large scales, noise on small scales,
fluctuations on even smaller scales, and a scaling range in between. In this
example, <IMG WIDTH=18 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7809" SRC="img157.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img157.gif"> is close to <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">, and multifractality cannot be established
positively.
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Wed Jan  6 15:38:27 CET 1999</I>
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