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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>Introduction</TITLE><META NAME="description" CONTENT="Introduction"><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="tex2html75" HREF="node2.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html73" HREF="Surrogates.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html67" HREF="Surrogates.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html76" HREF="node2.html">Detecting weak nonlinearity</A><B>Up:</B> <A NAME="tex2html74" HREF="Surrogates.html">Surrogate time series</A><B> Previous:</B> <A NAME="tex2html68" HREF="Surrogates.html">Surrogate time series</A><BR> <P><H1><A NAME="SECTION00010000000000000000">Introduction</A></H1><P>A nonlinear approach to analysing time seriesdata [<A HREF="node36.html#SFI">1</A>, <A HREF="node36.html#coping">2</A>, <A HREF="node36.html#abarbook">3</A>, <A HREF="node36.html#ourbook">4</A>, <A HREF="node36.html#habil">5</A>] can be motivated by two distinctreasons. One is intrinsic to the signal itself while the other is due toadditional knowledge we may have about the nature of the observedphenomenon. As for the first motivation, it might be that the arsenal of linearmethods has been exploited thoroughly but all the efforts left certainstructures in the time series unaccounted for. As for the second, a system maybe known to include nonlinear components and therefore a linear descriptionseems unsatisfactory in the first place. Such an argument is often heard forexample in brain research -- nobody expects for example the brain to be alinear device. In fact, there is ample evidence for nonlinearity in particularin small assemblies of neurons. Nevertheless, the latter reasoning is ratherdangerous. The fact that a system contains nonlinear components does not provethat this nonlinearity is also reflected in a specific signal we measure fromthat system. In particular, we do not know if it is of any practical use to gobeyond the linear approximation when analysing the signal. After all, we do notwant our data analysis to reflect our prejudice about the underlying system butto represent a fair account of the structures that are present in thedata. Consequently, the application of nonlinear time series methods has to bejustified by establishing nonlinearity in the time series.<P>Suppose we had measured the signal shown in Fig. <A HREF="node1.html#figarspikes">1</A> in somebiological setting. Visual inspection immediately reveals nontrivial structurein the serial correlations. The data fails a test for Gaussianity, thus rulingout a Gaussian linear stochastic process as its source. Depending on theassumptions we are willing to make on the underlying process, we might suggestdifferent origins for the observed strong ``spikyness'' of the dynamics.Superficially, low dimensional chaos seems unlikely due to the strongfluctuations, but maybe high dimensional dynamics? A large collection ofneurons could intermittently synchronise to give rise to the burst episodes. Infact, certain artificial neural network models show qualitatively similardynamics. The least interesting explanation, however, would be that all thespikyness comes from a distortion by the measurement procedure and all theserial correlations are due to linear stochastic dynamics. Occam's razor tellsus that we should be able to rule out such a simple explanation before weventure to construct more complicated models.<P>Surrogate data testing attempts to find the least interesting explanationthat cannot be ruled out based on the data. In the above example, the datashown in Fig. <A HREF="node1.html#figarspikes">1</A>, this would be the hypothesis that the datahas been generated by a stationary Gaussian linear stochastic process(equivalently, an <EM>autoregressive moving average</EM> or ARMA process) that isobserved through an invertible, static, but possible nonlinear observationfunction:<BR><IMG WIDTH=500 HEIGHT=16 ALIGN=BOTTOM ALT="equation1015" SRC="img1.gif"><BR> Neither the order <I>M</I>,<I>N</I>, the ARMA coefficients, nor the function <IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1908" SRC="img2.gif"> areassumed to be known. Without explicitly modeling these parameters, we stillknow that such a process would show characteristic linear correlations(reflecting the ARMA structure) and a characteristic single time probabilitydistribution (reflecting the action of <IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1908" SRC="img2.gif"> on the original Gaussiandistribution). Figure <A HREF="node1.html#figarspikes_surr">2</A> shows a surrogate time seriesthat is designed to have exactly these properties in common with the data butto be as random as possible otherwise. By a proper statistical test we can nowlook for additional structure that is present in the data but not in thesurrogates.<P><blockquote><A NAME="919"> </A><IMG WIDTH=360 HEIGHT=154 ALIGN=BOTTOM ALT="figure1017" SRC="img3.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figarspikes"> </A> A time series showing characteristic bursts.<BR></blockquote><P><blockquote><A NAME="921"> </A><IMG WIDTH=360 HEIGHT=154 ALIGN=BOTTOM ALT="figure1018" SRC="img4.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figarspikes_surr"> </A> A surrogate time series that has the same single time probability distribution and the same autocorrelation function as the sequence in Fig. <A HREF="node1.html#figarspikes">1</A>. The bursts are fully explained by these two properties.<BR></blockquote><P><P>In the case of the time series in Fig. <A HREF="node1.html#figarspikes">1</A>, there is noadditional structure since it has been generated by the rule<BR><IMG WIDTH=500 HEIGHT=19 ALIGN=BOTTOM ALT="equation1019" SRC="img5.gif"><BR>where <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1912" SRC="img6.gif"> are Gaussian independent increments and <IMG WIDTH=10 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline1914" SRC="img7.gif"> is chosenso that the data have unit variance.<A NAME="tex2html3" HREF="footnode.html#41"><IMG ALIGN=BOTTOM ALT="gif" SRC="foot_motif.gif"></A>This means that the strong nonlinearity that generates the bursts is due to thedistorted measurement that enhances ordinary fluctuations, generated by linearstochastic dynamics.<P>In order to systematically exclude simple explanations for time seriesobservations, this paper will discuss formal statistical tests fornonlinearity. We will formulate suitable null hypotheses for the underlyingprocess or for the observed structures themselves. In the former case, nullhypotheses will be extensions of the statement that the data were generated bya Gaussian linear stochastic processes. The latter situation may occur when itis difficult to properly define a class of possibly underlying processes but wewant to check if a particular set of observables gives a complete account ofthe statistics of the data. We will attempt to reject a null hypothesis bycomparing the value of a nonlinear parameter taken on by the data with itsprobability distribution. Since only exceptional cases allow for the exact orasymptotic derivation of this distribution unless strong additional assumptionsare made, we have to estimate it by a Monte Carlo resampling technique. Thisprocedure is known in the nonlinear time series literature as the method of<EM>surrogate data</EM>, see Refs. [<A HREF="node36.html#theiler1">6</A>, <A HREF="node36.html#theiler-sfi">7</A>, <A HREF="node36.html#fields">8</A>]. Most of thebody of this paper will be concerned with the problem of generating anappropriate Monte Carlo sample for a given null hypothesis.<P>We will also dwell on the proper interpretation of the outcome of such a test.Formally speaking, this is totally straightforward: A rejection at a givensignificance level means that if the null hypothesis is true, there is certainsmall probability to still see the structure we detected. Non-rejection meanseven less: either the null hypothesis is true, or the discriminating statisticswe are using fails to have power against the alternative realised in thedata. However, one is often tempted to go beyond this simple reasoning andspeculate either on the nature of the nonlinearity or non-stationarity thatlead to the rejection, or on the reason for the failure to reject.<P>Since the actual quantification of nonlinearity turns out to be the easiest --or in any case the least dangerous -- part of the problem, we will discuss itfirst. In principle, any nonlinear parameter can be employed for this purpose.They may however differ dramatically in their ability to detect different kindsof structures. Unfortunately, selecting the most suitable parameter has to bedone without making use of the data since that would render the test incorrect:If the measure of nonlinearity has been optimised formally or informally withrespect to the data, a fair comparison with surrogates is no longerpossible. Only information that is shared by data and surrogates, that is, forexample, linear correlations, may be considered for guidance. If multiple datasets are available, one could use some sequences for the selection of thenonlinearity parameter and others for the actual test. Otherwise, it isadvantageous to use one of the parameter free methods that can be set up withvery little detailed knowledge of the data.<P>Since we want to advocate to routinely use a nonlinearity test whenevernonlinear methods are planned to be applied, we feel that it is important tomake a practical implementation of such a test easily accessible. Therefore,one branch of the TISEAN free software package [<A HREF="node36.html#tisean">9</A>] is devoted tosurrogate data testing. Appendix <A HREF="node32.html#appA">A</A> will discuss the implementationalaspects necessary to understand what the programs in the package do.<P><HR><A NAME="tex2html75" HREF="node2.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html73" HREF="Surrogates.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html67" HREF="Surrogates.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html76" HREF="node2.html">Detecting weak nonlinearity</A><B>Up:</B> <A NAME="tex2html74" HREF="Surrogates.html">Surrogate time series</A><B> Previous:</B> <A NAME="tex2html68" HREF="Surrogates.html">Surrogate time series</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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