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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>Fourier based surrogates</TITLE><META NAME="description" CONTENT="Fourier based surrogates"><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="tex2html156" HREF="node10.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html154" HREF="Surrogates.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html148" HREF="node8.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html157" HREF="node10.html">Rescaled Gaussian linear process</A><B>Up:</B> <A NAME="tex2html155" HREF="Surrogates.html">Surrogate time series</A><B> Previous:</B> <A NAME="tex2html149" HREF="node8.html">Test design</A><BR> <P><H1><A NAME="SECTION00040000000000000000">Fourier based surrogates</A></H1><A NAME="secfourier"> </A>In this section, we will discuss a hierarchy of null hypotheses and theissues that arise when creating the corresponding surrogate data. The simplercases are discussed first in order to illustrate the reasoning. If we havefound serial correlations in a time series, that is, rejected the nullhypothesis of independence, we may ask of what nature these correlations are.The simplest possibility is to explain the observed structures bylinear two-point autocorrelations. A corresponding null hypothesis is that thedata have been generated by some linear stochastic process with Gaussianincrements. The most general univariate linear process is given by<BR><A NAME="eqarma"> </A><IMG WIDTH=500 HEIGHT=46 ALIGN=BOTTOM ALT="equation1027" SRC="img16.gif"><BR>where <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1912" SRC="img6.gif"> are Gaussian uncorrelated random increments. Thestatistical test is complicated by the fact that we do not want to test againstone particular linear process only (one specific choice of the <IMG WIDTH=12 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline1946" SRC="img17.gif"> and<IMG WIDTH=10 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline1948" SRC="img18.gif">), but against a whole class of processes. This is called a <EM>composite</EM> null hypothesis. The unknown values <IMG WIDTH=12 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline1946" SRC="img17.gif"> and <IMG WIDTH=10 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline1948" SRC="img18.gif"> are sometimesreferred to as <EM>nuisance parameters</EM>. There are basically three directionswe can take in this situation. First, we could try to make the discriminatingstatistic independent of the nuisance parameters. This approach has not beendemonstrated to be viable for any but some very simple statistics. Second, wecould determine which linear model is most likely realised in the data by a fitfor the coefficients <IMG WIDTH=12 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline1946" SRC="img17.gif"> and <IMG WIDTH=10 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline1948" SRC="img18.gif">, and then test against the hypothesis thatthe data has been generated by this particular model. Surrogates are simplycreated by running the fitted model. This <EM>typical realisations</EM> approachis the common choice in the bootstrap literature, see e.g. the classical bookby Efron [<A HREF="node36.html#efron">29</A>]. The main drawback is that we cannot recover the <EM>true</EM> underlying process by any fit procedure. Apart from problems associatedwith the choice of the correct model orders <I>M</I> and <I>N</I>, the data is byconstruction a very likely realisation of the fitted process. Otherrealisations will fluctuate <EM>around</EM> the data which induces a bias againstthe rejection of the null hypothesis. This issue is discussed thoroughly inRef. [<A HREF="node36.html#fields">8</A>], where also a calibration scheme is proposed.<P>The most attractive approach to testing for a composite null hypothesis seemsto be to create <EM>constrained realisations</EM> [<A HREF="node36.html#tp">25</A>]. Here it is useful tothink of the measurable properties of the time series rather than itsunderlying model equations. The null hypothesis of an underlying Gaussianlinear stochastic process can also be formulated by stating that all structureto be found in a time series is exhausted by computing first and second orderquantities, the mean, the variance and the auto-covariance function. Thismeans that a randomised sample can be obtained by creating sequences with thesame second order properties as the measured data, but which are otherwiserandom. When the linear properties are specified by the squared amplitudes ofthe (discrete) Fourier transform<BR><A NAME="eqpgram"> </A><IMG WIDTH=500 HEIGHT=53 ALIGN=BOTTOM ALT="equation1029" SRC="img19.gif"><BR>that is, the periodogram estimator of the power spectrum, surrogate time series<IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1962" SRC="img20.gif"> are readily created by multiplying the Fourier transformof the data by random phases and then transforming back to the time domain:<BR><A NAME="eqftsurro"> </A><IMG WIDTH=500 HEIGHT=47 ALIGN=BOTTOM ALT="equation1031" SRC="img21.gif"><BR>where <IMG WIDTH=85 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline1964" SRC="img22.gif"> are independent uniform random numbers.<P><BR> <HR><UL><A NAME="CHILD_LINKS"> </A><LI> <A NAME="tex2html158" HREF="node10.html#SECTION00041000000000000000">Rescaled Gaussian linear process</A><LI> <A NAME="tex2html159" HREF="node11.html#SECTION00042000000000000000">Flatness bias of AAFT surrogates</A><LI> <A NAME="tex2html160" HREF="node12.html#SECTION00043000000000000000">Iteratively refined surrogates</A><LI> <A NAME="tex2html161" HREF="node13.html#SECTION00044000000000000000">Example: Southern oscillation index</A><LI> <A NAME="tex2html162" HREF="node14.html#SECTION00045000000000000000">Periodicity artefacts</A><LI> <A NAME="tex2html163" HREF="node15.html#SECTION00046000000000000000">Iterative multivariate surrogates</A></UL><HR><A NAME="tex2html156" HREF="node10.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html154" HREF="Surrogates.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html148" HREF="node8.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html157" HREF="node10.html">Rescaled Gaussian linear process</A><B>Up:</B> <A NAME="tex2html155" HREF="Surrogates.html">Surrogate time series</A><B> Previous:</B> <A NAME="tex2html149" HREF="node8.html">Test design</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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