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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.2//EN"><!--Converted with LaTeX2HTML 96.1-h (September 30, 1996) by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds --><HTML><HEAD><TITLE>Correlation dimension</TITLE><META NAME="description" CONTENT="Correlation dimension"><META NAME="keywords" CONTENT="TiseanHTML"><META NAME="resource-type" CONTENT="document"><META NAME="distribution" CONTENT="global"><LINK REL=STYLESHEET HREF="TiseanHTML.css"></HEAD><BODY bgcolor=ffffff LANG="EN" > <A NAME="tex2html379" HREF="node31.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html377" HREF="node29.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html371" HREF="node29.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html380" HREF="node31.html">Takens-Theiler estimator</A><B>Up:</B> <A NAME="tex2html378" HREF="node29.html">Dimensions and entropies</A><B> Previous:</B> <A NAME="tex2html372" HREF="node29.html">Dimensions and entropies</A><BR> <P><H2><A NAME="SECTION00081000000000000000">Correlation dimension</A></H2><A NAME="secdimc2"> </A>Roughly speaking, the idea behind certain quantifiers of dimensions is thatthe weight <IMG WIDTH=27 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7557" SRC="img131.gif"> of a typical <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif">-ball covering part of theinvariant set scales with its diameter like <IMG WIDTH=65 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline7561" SRC="img132.gif">,where the value for <I>D</I> depends also on the precise way one defines theweight. Using the square of the probability <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6569" SRC="img20.gif"> to find a point of the setinside the ball, the dimension is called the correlation dimension <IMG WIDTH=19 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7567" SRC="img133.gif">,which is computed most efficiently by the correlation sum [<A HREF="citation.html#GP">73</A>]:<BR><A NAME="eqdim2c2"> </A><IMG WIDTH=500 HEIGHT=50 ALIGN=BOTTOM ALT="equation5740" SRC="img134.gif"><BR>where <IMG WIDTH=11 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline7569" SRC="img135.gif"> are <I>m</I>-dimensional delay vectors, <IMG WIDTH=283 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7573" SRC="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"> is the Heaviside step function and <I>w</I> will be discussed below. Onsufficiently small length scales and when the embedding dimension <I>m</I> exceedsthe box-dimension of the attractor [<A HREF="citation.html#SauerYorke">74</A>],<BR><IMG WIDTH=500 HEIGHT=19 ALIGN=BOTTOM ALT="equation5742" SRC="img137.gif"><BR>Since one does not know the box-dimension <I>a priori</I>, one checks forconvergence of the estimated values of <IMG WIDTH=19 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7567" SRC="img133.gif"> in <I>m</I>.<P>The literature on the correct and spurious estimation of the correlationdimension is huge and this is certainly not the place to repeat all thearguments. The relevant caveats and misconceptions are reviewed for example inRefs. [<A HREF="citation.html#theiler_dim">75</A>, <A HREF="citation.html#gss">11</A>, <A HREF="citation.html#dim">76</A>, <A HREF="citation.html#KantzSchreiber">2</A>]. The most prominent precautionis to exclude temporally correlated points from the pair counting by the socalled Theiler window <I>w</I> [<A HREF="citation.html#theiler_dim">75</A>]. In order to become a consistentestimator of the correlation <EM>integral</EM> (from which the dimension isderived) the correlation <EM>sum</EM> should cover a random sample of points drawnindependently according to the invariant measure on the attractor. Successiveelements of a time series are not usually independent. In particular for highlysampled flow data subsequent delay vectors are highly correlated. Theilersuggested to remove this spurious effect by simply ignoring all pairs of pointsin Eq.(<A HREF="node30.html#eqdim2c2"><IMG ALIGN=BOTTOM ALT="gif" SRC="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">. At the very least, pairs with <I>j</I>=<I>k</I>have to be excluded [<A HREF="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">, the mathematically correct value for a finite set of points, reducesthe scaling range drastically. Choosing <I>w</I>, the first zero of theauto-correlation function, sometimes even the decay time of the autocorrelationfunction, are not large enough since they reflect only overall linearcorrelations [<A HREF="citation.html#theiler_dim">75</A>, <A HREF="citation.html#dim">76</A>]. The space-time-separation plot(Sec. <A HREF="node15.html#secstp"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) provides a good means of determining a sufficient valuefor <I>w</I>, as discussed for example in [<A HREF="citation.html#stp">41</A>, <A HREF="citation.html#KantzSchreiber">2</A>]. In some cases,notably processes with inverse power law spectra, inspection requires <I>w</I> to beof the order of the length of the time series. This indicates that the datadoes 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"> or Lyapunov exponents should be abandoned.<P>Parameters in the routines <a href="../docs_c/d2.html">d2</a> and <a href="../docs_f/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">, the time delay, and the embeddingdimension, as well as the Theiler window.<P>Fast implementation of the correlation sum have been proposed by severalauthors. 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(N²)</i> without loosing any ofthe precious pairs, see Ref. [<A HREF="citation.html#neigh">20</A>]. However, for intermediate size datasets we also need the correlation sum at intermediate length scales whereneighbor searching becomes expensive. Many authors have tried to limit the useof computational resources by restricting one of the sums inEq.(<A HREF="node30.html#eqdim2c2"><IMG ALIGN=BOTTOM ALT="gif" SRC="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 pointsare so scarce anyway that all pairs are needed for stableresults. In [<A HREF="citation.html#buzug">62</A>], buth approaches were combined for the first time byusing fast neighbor search for <IMG WIDTH=40 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline7619" SRC="img140.gif"> and restricting the sumfor <IMG WIDTH=40 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7621"SRC="img141.gif">. The TISEAN implementation <a href="../docs_c/d2.html">d2</a>goes one step further and selects the range for the sums individually for eachlength scale to be processed. This turns out to give a major improvement inspeed. The user can specify a desired number of pairs which seems large enoughfor a stable estimation of <IMG WIDTH=29 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7623" SRC="img142.gif">, typically 1000 pairs willsuffice. Then the sums are extended to a range which guarantees that number ofpairs, or, if this cannot be achieved, to the whole time series. At the largestlength scales, this range may be rather small and the user may choose to give aminimal 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">, rather than restricting the range of thesums, only a randomly selected subset is used. The randomization howeverrequires a sophisticated program structure in order to avoid an overhead in computation time.<P><BR> <HR><UL><A NAME="CHILD_LINKS"> </A><LI> <A NAME="tex2html381" HREF="node31.html#SECTION00081100000000000000">Takens-Theiler estimator</A><LI> <A NAME="tex2html382" HREF="node32.html#SECTION00081200000000000000">Gaussian kernel correlation integral</A></UL><HR><A NAME="tex2html379" HREF="node31.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html377" HREF="node29.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html371" HREF="node29.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html380" HREF="node31.html">Takens-Theiler estimator</A><B>Up:</B> <A NAME="tex2html378" HREF="node29.html">Dimensions and entropies</A><B> Previous:</B> <A NAME="tex2html372" HREF="node29.html">Dimensions and entropies</A><P><ADDRESS><I>Thomas Schreiber <BR>Wed Jan 6 15:38:27 CET 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
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