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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.0//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>Measures of nonlinearity</TITLE><META NAME="description" CONTENT="Measures of nonlinearity"><META NAME="keywords" CONTENT="Surrogates"><META NAME="resource-type" CONTENT="document"><META NAME="distribution" CONTENT="global"><LINK REL=STYLESHEET HREF="Surrogates.css"></HEAD><BODY bgcolor=#ffffff LANG="EN" > <A NAME="tex2html408" HREF="node34.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html406" HREF="node32.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html400" HREF="node32.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html409" HREF="node34.html">Iterative FFT surrogates</A><B>Up:</B> <A NAME="tex2html407" HREF="node32.html">The TISEAN implementation</A><B> Previous:</B> <A NAME="tex2html401" HREF="node32.html">The TISEAN implementation</A><BR> <P><H2><A NAME="SECTION000101000000000000000">Measures of nonlinearity</A></H2><P>A few programs in the package directly issue scalar quantities that can be usedin nonlinearity testing. These are the zeroth order nonlinear predictors (<a href="../docs_f/predict.html">predict</a> and <a href="../docs_c/zeroth.html">zeroth</a>) which implement Eq.(<A HREF="node4.html#eqerror">5</A>) and the timereversibility statistic (<a href="../docs_f/timerev.html>timerev</a>) implementing Eq.(<A HREF="node3.html#eqskew">3</A>). For acouple of other quantities, we have deliberately omitted a black box algorithmto turn the raw results into a single number. A typical example are theprograms for dimension estimation (<a href="../docs_c/d2.html">d2</a>,<a href="../docs_f/c2naive.html">c2naive</a>, and <a href="../docs_f/c1.html">c1</a>) which compute correlation sums for ranges of length scales <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline1920" SRC="img11.gif"> andembedding dimensions <I>m</I>. For dimension estimation, these curves have to beinterpreted with due care to establish scaling behaviour and convergence withincreasing <I>m</I>. Single numbers issued by black box routines have lead to toomany spurious results in the literature. Researchers often forget that suchnumbers are not interpretable as fractal dimensions at all but only useful forcomparison and classification. Without genuine scaling at small length scales,a data set that gives <IMG WIDTH=61 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline2450" SRC="img189.gif"> by some ad hoc method to estimate<IMG WIDTH=19 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline2452" SRC="img190.gif"> cannot be said to have more degrees of freedom, or be more``complex'' than one that yields <IMG WIDTH=61 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline2454" SRC="img191.gif">.<P>This said, users are welcome to write their own code to turn correlationintegrals, local slopes (<a href="../docs_f/c2d.html">c2d</a>), Takens' estimator (<a href="../docs_f/c2t.html">c2t</a>), or GaussianKernel correlation integrals (<a href="../docs_f/c2g.html">c2g</a>) into nonlinearity measures. The samesituation is found for Lyapunov exponents (<a href="../docs_c/lyap_k.html">lyap_k</a>, <a href="../docs_c/lyap_r.html">lyap_r</a>),entropies (<a href="../docs_c/boxcount.html">boxcount</a>) and other quantities. Since all of these havealready been described in Ref. [<A HREF="node36.html#tisean">9</A>], we refer the reader there forfurther details.<P><HR><A NAME="tex2html408" HREF="node34.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html406" HREF="node32.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html400" HREF="node32.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html409" HREF="node34.html">Iterative FFT surrogates</A><B>Up:</B> <A NAME="tex2html407" HREF="node32.html">The TISEAN implementation</A><B> Previous:</B> <A NAME="tex2html401" HREF="node32.html">The TISEAN implementation</A><P><ADDRESS><I>Thomas Schreiber <BR>Mon Aug 30 17:31:48 CEST 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
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