node30.html

an analysis software with souce code for the time series with methods based on the theory of nonline

notably processes with inverse power law spectra, inspection requires <I>w</I> to be
of the order of the length of the time series. This indicates that the data
does not sample an invariant attractor sufficiently and the estimation of 
invariants like <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"> or Lyapunov exponents should be abandoned.
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Parameters in the routines <a href="../dresden/d2.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/dresden/d2.html">d2</a> and 
<a href="../wuppertal/c2naive.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/c2naive.html">c2naive</a> are as usual the 
embedding parameters <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 time delay, and the embedding
dimension, as well as the Theiler window.
<P>
Fast implementation of the correlation sum have been proposed by several
authors. At small length scales, the computation of pairs can be done in
<I>O(N</I>log<I>N)</I> or even <I>O</I>(<I>N</I>) time rather than
<i>O(N&#178;)</i>  without loosing any of
the precious pairs, see Ref.&nbsp;[<A HREF="citation.html#neigh" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#neigh">20</A>].  However, for intermediate size data
sets we also need the correlation sum at intermediate length scales where
neighbor searching becomes expensive. Many authors have tried to limit the use
of computational resources by restricting one of the sums 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>) to a fraction of the available points. By this practice,
however, one looses valuable statistics at the small length scales where points
are so scarce anyway that all pairs are needed for stable
results. In&nbsp;[<A HREF="citation.html#buzug" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#buzug">62</A>], buth approaches were combined for the first time by
using fast neighbor search for <IMG WIDTH=40 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline7619" SRC="img140.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img140.gif"> and restricting the sum
for <IMG WIDTH=40 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7621"
SRC="img141.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img141.gif">. The TISEAN implementation 
<a href="../dresden/d2.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/dresden/d2.html">d2</a>
goes one step further and selects the range for the sums individually for each
length scale to be processed. This turns out to give a major improvement in
speed. The user can specify a desired number of pairs which seems large enough
for a stable estimation of <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">, typically 1000 pairs will
suffice. Then the sums are extended to a range which guarantees that number of
pairs, or, if this cannot be achieved, to the whole time series. At the largest
length scales, this range may be rather small and the user may choose to give a
minimal number of reference points to ensure a representative average.  
In the program
<IMG WIDTH=16 HEIGHT=10 ALIGN=BOTTOM ALT="tex2html_wrap_inline7625"
SRC="img143.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img143.gif">, rather than restricting the range of the
sums, only a randomly selected subset is used. The randomization however
requires a sophisticated program structure in order to avoid an 
overhead in computation time.
<P>
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<UL><A NAME="CHILD_LINKS">&#160;</A>
<LI> <A NAME="tex2html381" HREF="node31.html#SECTION00081100000000000000" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node31.html#SECTION00081100000000000000">Takens-Theiler estimator</A>
<LI> <A NAME="tex2html382" HREF="node32.html#SECTION00081200000000000000" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node32.html#SECTION00081200000000000000">Gaussian kernel correlation integral</A>
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Wed Jan  6 15:38:27 CET 1999</I>
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