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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>False nearest neighbors</TITLE><META NAME="description" CONTENT="False nearest neighbors"><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="tex2html162" HREF="node10.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html160" HREF="node7.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html156" HREF="node8.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html163" HREF="node10.html">Principal components</A><B>Up:</B> <A NAME="tex2html161" HREF="node7.html">Embedding parameters</A><B> Previous:</B> <A NAME="tex2html157" HREF="node8.html">Mutual information</A><BR> <P><H3><A NAME="SECTION00032200000000000000">False nearest neighbors</A></H3><P>A method to determine the minimal sufficient embedding dimension <I>m</I> wasproposed by Kennel et al. [<A HREF="citation.html#kennelFNN">28</A>]. It is called the <I>false nearestneighbor</I> method. The idea is quite intuitive. Suppose the minimal embeddingdimension for a given time series <IMG WIDTH=27 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6615" SRC="img23.gif"> is <IMG WIDTH=20 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6617" SRC="img24.gif">. This means that in a<IMG WIDTH=20 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6617" SRC="img24.gif">-dimensional delay space the reconstructed attractor is a one-to-one imageof the attractor in the original phase space. Especially, the topologicalproperties are preserved. Thus the neighbors of a given point are mapped ontoneighbors in the delay space. Due to the assumed smoothness of the dynamics,neighborhoods of the points are mapped onto neighborhoods again. Of course theshape and the diameter of the neighborhoods is changed according to theLyapunov exponents. But suppose now you embed in an <I>m</I>-dimensional space with<IMG WIDTH=55 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline6623" SRC="img25.gif">. Due to this projection the topological structure is no longerpreserved. Points are projected into neighborhoods of other points to whichthey wouldn't belong in higher dimensions. These points are called <I>falseneighbors</I>. If now the dynamics is applied, these false neighbors are nottypically mapped into the image of the neighborhood, but somewhere else, sothat the average ``diameter'' becomes quite large.<P>The idea of the algorithm <a href="../docs_c/false_nearest.html">false_nearest</a> is the following. For each point <IMG WIDTH=10 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6625" SRC="img26.gif">in the time series look for its nearest neighbor <IMG WIDTH=11 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline6627" SRC="img27.gif"> in a<I>m</I>-dimensional space. Calculate the distance <IMG WIDTH=59 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline6631" SRC="img28.gif">.Iterate both points and compute<BR><IMG WIDTH=500 HEIGHT=39 ALIGN=BOTTOM ALT="equation4487" SRC="img29.gif"><BR>If <IMG WIDTH=16 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline6633" SRC="img30.gif"> exceeds a given heuristic threshold <IMG WIDTH=16 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline6635" SRC="img31.gif">, this point is marked ashaving a false nearest neighbor [<A HREF="citation.html#kennelFNN">28</A>]. The criterion that theembedding dimension is high enough is that the fraction of points for which<IMG WIDTH=55 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline6637" SRC="img32.gif"> is zero, or at least sufficiently small. Two examples are shown inFig. <A HREF="node9.html#figfnn"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>. One is for the Lorenz system (crosses), one for the Hénonsystem (filled circles), and one for a Hénon time series corrupted by 10% ofGaussian white noise (open circles). One clearly sees that, as expected, <I>m</I>=2is sufficient for the Hénon and <I>m</I>=3 for the Lorenz system, whereas thesignature is less clear in the noisy case.<P><P><blockquote><A NAME="4482"> </A><IMG WIDTH=338 HEIGHT=214 ALIGN=BOTTOM ALT="figure346" SRC="img22.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figfnn"> </A> The fraction of false nearest neighbors as a function of the embedding dimension for noise free Lorenz (blue) and Hénon (red) time series, as well as a Hénon time series (green) corrupted by 10% of noise.<BR></blockquote><P>The introduction of the false nearest neighbors concept and other ad hocinstruments was partly a reaction to the finding that many results obtained forthe genuine invariants, like the correlation dimension, has been spuriousdue to caveats of the estimation procedure. In the latter case, serialcorrelations and small sample fluctuations can easily be mistaken for nonlinear determinism. It turns out, however, that the ad hoc quantitiesbasically suffer from the same problems - which can be cured by the sameprecautions. The implementation <a href="../docs_c/false_nearest.html">false_nearest</a> therefore allows to specify a minimaltemporal separation of valid neighbors.<P>Other software for the analysis of false nearest neighbors is available insource form from Kennel [<A HREF="citation.html#kennel">29</A>]. Or, if you prefer to pay for a license,from Ref. [<A HREF="citation.html#abla">30</A>].<P><HR><A NAME="tex2html162" HREF="node10.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html160" HREF="node7.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html156" HREF="node8.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html163" HREF="node10.html">Principal components</A><B>Up:</B> <A NAME="tex2html161" HREF="node7.html">Embedding parameters</A><B> Previous:</B> <A NAME="tex2html157" HREF="node8.html">Mutual information</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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