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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>Example: Southern oscillation index</TITLE><META NAME="description" CONTENT="Example: Southern oscillation index"><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="tex2html202" HREF="node14.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html200" HREF="node9.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html194" HREF="node12.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html203" HREF="node14.html">Periodicity artefacts</A><B>Up:</B> <A NAME="tex2html201" HREF="node9.html">Fourier based surrogates</A><B> Previous:</B> <A NAME="tex2html195" HREF="node12.html">Iteratively refined surrogates</A><BR> <P><H2><A NAME="SECTION00044000000000000000">Example: Southern oscillation index</A></H2><P>As an illustration let us perform a statistical test for nonlinearity on amonthly time series of the Southern Oscillation Index (SOI) from 1866 to 1994(1560 samples). For a reference on analysis of Southern Oscillation datasee Graham et al. [<A HREF="node36.html#Graham87">32</A>]. Since a discussion of this climatic phenomenon is not relevant to the issue at hand, let us just considerthe time series as an isolated data item. Our null hypothesis is that the datais adequately described by its single time probability distribution and itspower spectrum. This corresponds to the assumption that an autoregressivemoving average (ARMA) process is generating a sequence that is measured througha static monotonic, possibly nonlinear observation function.<P>For a test at the 99% level of significance (<IMG WIDTH=59 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline2052" SRC="img49.gif">), we generate acollection of <IMG WIDTH=89 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2054" SRC="img50.gif"> surrogate time series which share the single timesample probability distribution and the periodogram estimator with thedata. This is carried out using the iterative method described inSec. <A HREF="node12.html#seciterative">4.3</A> above (see also Ref. [<A HREF="node36.html#surrowe">30</A>]).Figure <A HREF="node13.html#figsoi">4</A> shows the data together with one of the 99 surrogates.<P>As a discriminating statistics we use a locally constant predictor in embeddingspace, using three dimensional delay coordinates at a delay time of onemonth. Neighbourhoods were selected at 0.2 times the rms amplitude of thedata. The test is set up in such a way that the null hypothesis may be rejectedwhen the prediction error is smaller for the data than for all of the 99surrogates. But, as we can see in Fig. <A HREF="node13.html#figsoipred">5</A>, this is not thecase. Predictability is not significantly reduced by destroying possiblenonlinear structure. This negative result can mean several things. Theprediction error statistics may just not have any power to detect the kind ofnonlinearity present. Alternatively, the underlying process may be linear andthe null hypothesis true. It could also be, and this seems the most likelyoption after all we know about the equations governing climate phenomena,that the process is nonlinear but the single time series at this samplingcovers such a poor fraction of the rich dynamics that it must appear linearstochastic to the analysis.<P><blockquote><A NAME="933"> </A><IMG WIDTH=360 HEIGHT=251 ALIGN=BOTTOM ALT="figure1044" SRC="img51.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figsoi"> </A> Monthly values of the Southern Oscillation Index (SOI) from 1866 to 1994 (upper trace) and a surrogate time series exhibiting the same auto-covariance function (lower trace). All linear properties of the fluctuations and oscillations are the same between both tracings. However, any possible nonlinear structure except for a static rescaling of the data is destroyed in the lower tracing by the randomisation procedure.<BR></blockquote><P><P><blockquote><A NAME="935"> </A><IMG WIDTH=319 HEIGHT=148 ALIGN=BOTTOM ALT="figure1045" SRC="img52.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figsoipred"> </A> Nonlinear prediction error measured for the SOI data set (see Fig. <A HREF="node13.html#figsoi">4</A>) and 99 surrogates. The value for the original data is plotted with a longer impulse. The mean and standard deviation of the statistic obtained from the surrogates is also represented by an error bar. It is evident that the data is not singled out by this property and we are unable to reject the null hypothesis of a linear stochastic stationary process, possibly rescaled by a nonlinear measurement function.<BR></blockquote><P><P>Of course, our test has been carried out disregarding any knowledge of the SOIsituation. It is very likely that more informed measures of nonlinearity may bemore successful in detecting structure. We would like to point out, however,that if such information is derived from the same data, or literature publishedon it, a bias is likely to occur. Similarly to the situation of multiple testson the same sample, the level of significance has to be adjusted properly.Otherwise, if many people try, someone will eventually, and maybe accidentally,find a measure that indicates nonlinear structure.<P><HR><A NAME="tex2html202" HREF="node14.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html200" HREF="node9.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html194" HREF="node12.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html203" HREF="node14.html">Periodicity artefacts</A><B>Up:</B> <A NAME="tex2html201" HREF="node9.html">Fourier based surrogates</A><B> Previous:</B> <A NAME="tex2html195" HREF="node12.html">Iteratively refined surrogates</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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