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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>Measuring weak nonlinearity</TITLE><META NAME="description" CONTENT="Measuring weak nonlinearity"><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="tex2html468" HREF="node40.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html466" HREF="node35.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html462" HREF="node38.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html469" HREF="node40.html">Conclusion and perspectives</A><B>Up:</B> <A NAME="tex2html467" HREF="node35.html">Testing for nonlinearity</A><B> Previous:</B> <A NAME="tex2html463" HREF="node38.html">General constrained randomization</A><BR> <P><H2><A NAME="SECTION00094000000000000000">Measuring weak nonlinearity</A></H2><P>When testing for nonlinearity, we would like to use quantifiers which areoptimized for the weak nonlinearity limit, which is not what most time seriesmethods of chaos theory have been designed for. The simple nonlinear predictionscheme (Sec. <A HREF="node18.html#seczeroth"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) has proven quite useful in this context. If usedas a comparative statistic, it should be noted that sometimes seeminglyinadequate embeddings or neighborhood sizes may lead to rather big errorswhich have, however, small fluctuations. The tradeoff between bias and variancemay be different from the situation where predictions are desired <EM>perse</EM>. The same rationale applies to quantities derived from the correlationsum. Neither the small scale limit, genuine scaling, or the Theiler correctionare formally necessary in a comparative test. However, any temptation tointerpret the results in terms like ``complexity'' or ``dimensionality'' shouldbe resisted, even though ``complexity'' doesn't seem to have an agreed-uponmeaning anyway. Apart from average prdiction errors, we have found thestabilities of short periodic orbits (see Sec. <A HREF="node19.html#secupo"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) useful for thedetectionof nonlinearity in surrogate data tests. As an alternative to thephase space based methods, more traditional measures of nonlinearity derivedfrom higher order autocorrelation functions ([<A HREF="citation.html#BI">86</A>]) may also be considered. If a time-reversal asymmetry ispresent, its statistical confirmation (routine <a href="../docs_f/timerev.html">timerev</a>) is a very powerfuldetector of nonlinearity [<A HREF="citation.html#diks2">87</A>]. Some measures of weak nonlinearity arecompared systematically in Ref. [<A HREF="citation.html#power">88</A>].<P><HR><A NAME="tex2html468" HREF="node40.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html466" HREF="node35.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html462" HREF="node38.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html469" HREF="node40.html">Conclusion and perspectives</A><B>Up:</B> <A NAME="tex2html467" HREF="node35.html">Testing for nonlinearity</A><B> Previous:</B> <A NAME="tex2html463" HREF="node38.html">General constrained randomization</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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