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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>Takens-Theiler estimator</TITLE><META NAME="description" CONTENT="Takens-Theiler estimator"><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="tex2html391" HREF="node32.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html389" HREF="node30.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html383" HREF="node30.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html392" HREF="node32.html">Gaussian kernel correlation integral</A><B>Up:</B> <A NAME="tex2html390" HREF="node30.html">Correlation dimension</A><B> Previous:</B> <A NAME="tex2html384" HREF="node30.html">Correlation dimension</A><BR> <P><H3><A NAME="SECTION00081100000000000000">Takens-Theiler estimator</A></H3><P>Convergence to a finite correlation dimension can be checked by plotting scaledependent ``effective dimensions'' versus length scale for variousembeddings. The easiest way to proceed is to compute (numerically) thederivative of <IMG WIDTH=74 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7627" SRC="img144.gif"> with respect to <IMG WIDTH=29 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7629" SRC="img145.gif">, for exampleby fitting straight lines to the log-log plot of <IMG WIDTH=29 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7623" SRC="img142.gif">. InFig. <A HREF="node32.html#figdim2"><IMG ALIGN=BOTTOM ALT="gif"SRC="icons/cross_ref_motif.gif"></A> <B>(a)</B>we see the output of the routine <a href="../docs_f/c2.html">c2</a> actingon data from the NMR laser, processed by <a href="../docs_f/c2d.html">c2d</a> in order to obtain localslopes. By default, straight lines are fitted over one octave in <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif">,larger ranges give smoother results. We can see that on the large scales,self-similarity is broken due to the finite extension of the attractor, and onsmall but yet statistically significant scales we see the embedding dimensioninstead of a saturated, <I>m</I>-independent value. This is the effect of noise,which is infinite dimensional, and thus fills a volume in every embeddingspace. Only on the intermediate scales we see the desired <EM>plateau</EM> wherethe results are in good approximation independent of <I>m</I> and <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif">. Theregion where scaling is <EM>established</EM>, not just the range selected forstraight line fitting, is called the <EM>scaling range</EM>.<P>Since the statistical fluctuations in plots like Fig. <A HREF="node32.html#figdim2"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> <B>(a)</B>show characteristic (anti-)correlations, it has beensuggested [<A HREF="citation.html#takens_est">78</A>, <A HREF="citation.html#takens_theiler">79</A>] to apply a maximum likelihoodestimator to obtain optimal values for <IMG WIDTH=19 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7567" SRC="img133.gif">. The Takens-Theiler-estimatorreads<BR><A NAME="eqd2t"> </A><IMG WIDTH=500 HEIGHT=43 ALIGN=BOTTOM ALT="equation5753" SRC="img146.gif"><BR>and can be obtained by processing the output of <ahref="../docs_f/c2.html">c2</a> by <a href="../docs_f/c2t.html">c2t</a>. Since<IMG WIDTH=29 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7623" SRC="img142.gif"> is available only at discrete values<IMG WIDTH=111 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7645" SRC="img147.gif">, we interpolate it by a pure power law (or, equivalently, the log-log plot bystraight lines: <IMG WIDTH=152 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7647" SRC="img148.gif">) in between these. The resulting integrals can be solvedtrivially and summed: <BR><IMG WIDTH=500 HEIGHT=102 ALIGN=BOTTOM ALT="eqnarray5757" SRC="img149.gif"><BR>Plotting <IMG WIDTH=31 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7649" SRC="img150.gif"> versus <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif"> (Fig. <A HREF="node32.html#figdim2"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A><B>(b)</B>) is an interesting alternative to the usual local slopes plot,Fig. <A HREF="node32.html#figdim2"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> <B>(a)</B>. It is tempting to use such an ``estimator ofdimension'' as a black box to provide a number one might quote as a dimension.This would imply the unjustified assumption that all deviations from exactscaling behavior is due to the statistical fluctuations. Instead, one still hasto verify the existence of a scaling regime. Only then, <IMG WIDTH=50 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7653" SRC="img151.gif"> evaluated at the upper end of the scaling range is a reasonabledimension estimator.<P><HR><A NAME="tex2html391" HREF="node32.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html389" HREF="node30.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html383" HREF="node30.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html392" HREF="node32.html">Gaussian kernel correlation integral</A><B>Up:</B> <A NAME="tex2html390" HREF="node30.html">Correlation dimension</A><B> Previous:</B> <A NAME="tex2html384" HREF="node30.html">Correlation dimension</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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