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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>Non-dynamic nonlinearity</TITLE><META NAME="description" CONTENT="Non-dynamic 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="tex2html357" HREF="node29.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html355" HREF="node27.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html349" HREF="node27.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html358" HREF="node29.html">Non-stationarity</A><B>Up:</B> <A NAME="tex2html356" HREF="node27.html">Questions of interpretation</A><B> Previous:</B> <A NAME="tex2html350" HREF="node27.html">Questions of interpretation</A><BR> <P><H2><A NAME="SECTION00071000000000000000">Non-dynamic nonlinearity</A></H2><A NAME="secrev"> </A>A non-invertible measurement function is with current methods indistinguishablefrom dynamic nonlinearity. The most common case is that the data are squaredmoduli of some underlying dynamical variable. This is supposed to be true forthe celebrated sunspot numbers. Sunspot activity is generally connected withmagnetic fields and is to first approximation proportional to the squared fieldstrength. Obviously, sunspot numbers are non-negative, but also the nullhypothesis of a monotonically rescaled Gaussian linear random process is to berejected since taking squares is not an invertible operation. Unfortunately,the framework of surrogate data does not currently provide a method to testagainst null hypothesis involving noninvertible measurement functions. Yetanother example is given by linearly filtered time series. Even if the nullhypothesis of a monotonically rescaled Gaussian linear random process is truefor the underlying signal, it is usually not true for filtered copies of it, inparticular sequences of first differences, see Prichard [<A HREF="node36.html#dean">50</A>] for adiscussion of this problem.<P>The catch is that nonlinear deterministic dynamical systems may produceirregular time evolution, or <EM>chaos</EM>, and the signals generated by suchprocesses will be easily found to be nonlinear by statistical methods. But manyauthors have confused cause and effect in this logic: deterministic chaos doesimply nonlinearity, but not vice versa. The confusion is partly due to theheavy use of methods inspired by chaos theory, leading to arguments like ``Ifthe fractal dimension algorithm has power to detect nonlinearity, the data musthave a fractal attractor!'' Let us give a very simple and commonplace examplewhere such a reasoning would lead the wrong way.<P>One of the most powerful [<A HREF="node36.html#power">13</A>, <A HREF="node36.html#theiler1">6</A>, <A HREF="node36.html#diks2">11</A>] indicators of nonlinearityin a time series is the change of statistical properties introduced by areversal of the time direction: Linear stochastic processes are fullycharacterised by their power spectrum which does not contain any information onthe direction of time. One of the simplest ways to measure time asymmetry isby taking the first differences of the series to some power, seeEq.(<A HREF="node3.html#eqskew">3</A>). Despite its high discrimination power, also for many butnot all dynamical nonlinearities, this statistic has not been very popular inrecent studies, probably since it is rather unspecific about the nature of thenonlinearity. Let us illustrate this apparent flaw by an example where timereversal asymmetry is generated by the measurement process.<P><blockquote><A NAME="984"> </A><IMG WIDTH=338 HEIGHT=252 ALIGN=BOTTOM ALT="figure1097" SRC="img182.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figar2"> </A>Upper panel: Output of the linear autoregressive process <IMG WIDTH=210 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2424" SRC="img180.gif">. Lower panel: the same after monotonic rescaling by <IMG WIDTH=71 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline2426" SRC="img181.gif">.<BR></blockquote><P><P><blockquote><A NAME="986"> </A><IMG WIDTH=338 HEIGHT=252 ALIGN=BOTTOM ALT="figure1098" SRC="img184.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figasym"> </A>Moving differences <IMG WIDTH=66 HEIGHT=13 ALIGN=MIDDLE ALT="tex2html_wrap_inline2428" SRC="img183.gif"> of the sequence shown in Fig. <A HREF="node28.html#figar2">21</A> (upper), and a surrogate time series (lower). A formal test shows that the nonlinearity is significant at the 99% level.<BR></blockquote><P>Consider a signal generated by a second order autoregressive (AR(2)) process<IMG WIDTH=210 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2424" SRC="img180.gif">. The sequence <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1912" SRC="img6.gif"> consists ofindependent Gaussian random numbers with a variance chosen such that the datahave unit variance. A typical output of 2000 samples is shown as the upperpanel in Fig. <A HREF="node28.html#figar2">21</A>. Let the measurement be such that the data isrescaled by the strictly monotonic function <IMG WIDTH=71 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline2426" SRC="img181.gif">, The resultingsequence (see the lower panel in Fig. <A HREF="node28.html#figar2">21</A>) still satisfies the nullhypothesis formulated above. This is no longer the case if we take differencesof this signal, a linear operation that superficially seems harmless for a``linear'' signal. Taking differences turns the up-down-asymmetry of the datainto a forward-backward asymmetry. As it has been pointed out byPrichard,[<A HREF="node36.html#dean">50</A>] the static nonlinearity and linear filtering are notinterchangeable with respect to the null hypothesis and the sequence<IMG WIDTH=246 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline2436" SRC="img185.gif"> must be considered nonlinear inthe sense that it violates the null hypothesis. Indeed, such a sequence (seethe upper panel in Fig. <A HREF="node28.html#figasym">22</A>) is found to be nonlinear at the 99% level of significance using the statistics given in Eq.(<A HREF="node3.html#eqskew">3</A>), butalso using nonlinear prediction errors. (Note that the nature of the statisticEq.(<A HREF="node3.html#eqskew">3</A>) requires a two-sided test.) A single surrogate series isshown in the lower panel of Fig. <A HREF="node28.html#figasym">22</A>. The tendency of the data toraise slowly but to fall fast is removed in the linear surrogate, as it should.<P><P><blockquote><A NAME="989"> </A><IMG WIDTH=338 HEIGHT=252 ALIGN=BOTTOM ALT="figure1099" SRC="img186.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figspike"> </A>A single spike is artificially introduced in an otherwise linear stochastic time sequence (upper). In the surrogate time series (lower), this leads to multiple short spikes. Although the surrogate data has the same frequency content and takes on the same set of values as the data, the remnants of the spike will lead to the detection of nonlinearity.<BR></blockquote><HR><A NAME="tex2html357" HREF="node29.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html355" HREF="node27.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html349" HREF="node27.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html358" HREF="node29.html">Non-stationarity</A><B>Up:</B> <A NAME="tex2html356" HREF="node27.html">Questions of interpretation</A><B> Previous:</B> <A NAME="tex2html350" HREF="node27.html">Questions of interpretation</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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