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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.2//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>Iterative Fourier transform method</TITLE><META NAME="description" CONTENT="Iterative Fourier transform method"><META NAME="keywords" CONTENT="TiseanHTML"><META NAME="resource-type" CONTENT="document"><META NAME="distribution" CONTENT="global"><LINK REL=STYLESHEET HREF="TiseanHTML.css"></HEAD><BODY bgcolor=ffffff LANG="EN" > <A NAME="tex2html450" HREF="node38.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html448" HREF="node35.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html442" HREF="node36.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html451" HREF="node38.html">General constrained randomization</A><B>Up:</B> <A NAME="tex2html449" HREF="node35.html">Testing for nonlinearity</A><B> Previous:</B> <A NAME="tex2html443" HREF="node36.html">The concept of surrogate </A><BR> <P><H2><A NAME="SECTION00092000000000000000">Iterative Fourier transform method</A></H2><P>Very few real time series which are suspected to show nonlinearity follow aGaussian single time distribution. Non-Gaussianity is the simplest kind ofnonlinear signature but it may have a trivial reason: The data may have beendistorted in the measurement process. Thus a possible null hypothesis would bethat there is a stationary Gaussian linear stochastic process that generates asequence <IMG WIDTH=32 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7961" SRC="img176.gif">, but the actual observations are <IMG WIDTH=73 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7963" SRC="img177.gif"> where<IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6547" SRC="img13.gif"> is a monotonic function. Constrained realizations of this nullhypothesis would require the generation of random sequences with the same powerspectrum (fully specifying the linear process) and the same single timedistribution (specifying the effect of the measurement function) as theobserved data. The <B>A</B>mplitude <B>A</B>djusted <B>F</B>ourier <B>T</B>ransform(AAFT) method proposed in [<A HREF="citation.html#theiler1">82</A>] attempts to invert the measurementfunction <IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6547" SRC="img13.gif"> by rescaling the data to a Gaussian distribution. Thenthe Fourier phases are randomized and the rescaling is inverted. As discussedin [<A HREF="citation.html#surrowe">83</A>], this procedure is biased towards a flatter spectrum sincethe inverse of <IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6547" SRC="img13.gif"> is not available exactly. In the same reference,a scheme is introduced that removes this bias by iteratively adjusting thespectrum and the distribution of the surrogates. Alternatingly, the surrogatesare rescaled to the exact values taken by the data and then the Fouriertransform is brought to the exact amplitudes obtained from the data.The discrepancy between both steps either converges to zero with the numberof iterations or to a finite inaccuracy which decreases with the length of thetime series. The program <a href="../docs_f/surrogates.html">surrogates</a> performs iterations until no furtherimprovement can be made. The last two stages are returned, one having the exactFourier amplitudes and one taking on the same values as the data. For not too exotic data these two versions should be almost identical. The relativediscrepancy is also printed.<P><P><blockquote><A NAME="6024"> </A><IMG WIDTH=214 HEIGHT=489 ALIGN=BOTTOM ALT="figure1881" SRC="img175.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figb_s"> </A> Upper: The human breath rate data from Fig. <A HREF="node23.html#figb"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>. Lower: the noise component extracted by the noise reduction scheme has been randomized in order to destroy correlations with the signal. The result appears slightly but significantly less structured than the original.<BR></blockquote><P>In Fig. <A HREF="node37.html#figb_s"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> we used this procedure to assess the hypothesis that the noise reduction on the breath data reported in Fig. <A HREF="node23.html#figb"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> removed an additive noise component which was independent of the signal. If thehypothesis were true, we could equally well add back on the noise sequenceor a randomized version of it which lacks any correlations to the signal.In the upper panel of Fig. <A HREF="node37.html#figb_s"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> we show the original data. In thelower panel we took the noise reduced version (c.f. Fig. <A HREF="node23.html#figb"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>, bottom)and added a surrogate of the supposed noise sequence. The result is similar butstill significantly different from the original to make the additivityassumption implausible.<P>Fourier based randomization schemes suffer from some caveats due to the theinherent assumption that the data constitutes one period of a periodic signal,which is not what we really expect. The possible artefacts are discussed forexample in [<A HREF="citation.html#theiler_sfi">84</A>] and can, in summary, lead to spurious rejectionof the null hypothesis. One precaution that should be taken when using<a href="../docs_f/surrogates.html">surrogates</a> is to make sure that the beginning and the end of the dataapproximately match in value and phase. Then, the periodicity assumption is nottoo far wrong and harmless. Usually, this amounts to the loss of a few pointsof the series. One should note, however, that the routine may truncate the databy a few points itself in order to be able to perform a <EM>fast</EM> Fouriertransform which requires the number of points to be factorizable by small primefactors.<P><HR><A NAME="tex2html450" HREF="node38.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html448" HREF="node35.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html442" HREF="node36.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html451" HREF="node38.html">General constrained randomization</A><B>Up:</B> <A NAME="tex2html449" HREF="node35.html">Testing for nonlinearity</A><B> Previous:</B> <A NAME="tex2html443" HREF="node36.html">The concept of surrogate </A><P><ADDRESS><I>Thomas Schreiber <BR>Wed Jan 6 15:38:27 CET 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
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