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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>Iterative multivariate surrogates</TITLE><META NAME="description" CONTENT="Iterative multivariate 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="tex2html220" HREF="node16.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html218" HREF="node9.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html214" HREF="node14.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html221" HREF="node16.html">General constrained randomisation</A><B>Up:</B> <A NAME="tex2html219" HREF="node9.html">Fourier based surrogates</A><B> Previous:</B> <A NAME="tex2html215" HREF="node14.html">Periodicity artefacts</A><BR> <P><H2><A NAME="SECTION00046000000000000000">Iterative multivariate surrogates</A></H2><A NAME="secmulti1"> </A><P><P>A natural generalisation of the null hypothesis of a Gaussian linear stochasticprocess is that of a multivariate process of the same kind. In this case, theprocess is determined by giving the cross-spectrum inaddition to the power spectrum of each of the channels. In Ref. [<A HREF="node36.html#multi">34</A>],it has been pointed out that phase randomised surrogates are readily producedby multiplying the Fourier phases of each of the channels by the same set of random phases since the cross-spectrum reflects relative phases only.The authors of Ref. [<A HREF="node36.html#multi">34</A>] did not discuss the possibility to combine multivariate phase randomisation with an amplitude adjustment step.The extension of the iterative refinement scheme introduced inSec. <A HREF="node12.html#seciterative">4.3</A> to the multivariate case is relativelystraightforward. Since deviations from a Gaussian distribution are very commonand may occur due to a simple invertible rescaling due to the measurementprocess, we want to give the algorithm here.<P>Recall that the iterative scheme consists of two procedures which are appliedin an alternating fashion until convergence to a fixed point is achieved. Theamplitude adjustment procedure by rank ordering (<A HREF="node12.html#eqstep2">13</A>) is readilyapplied to each channel individually. However, the spectral adjustment in theFourier domain has to be modified. Let us introduce a second index in order todenote the <I>M</I> different channels of a multivariate time series <IMG WIDTH=274 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2084" SRC="img71.gif">. The change that has to be appliedto the ``filter'' step, Eq.(<A HREF="node12.html#eqstep1">12</A>), is that the phases<IMG WIDTH=32 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline2086" SRC="img72.gif"> have to be replaced by phases <IMG WIDTH=30 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline2088" SRC="img73.gif"> with thefollowing properties. (We have dropped the superscript (<I>i</I>) for convenience.)The replacement should be minimal in the leastsquares sense, that is, it should minimise<BR><A NAME="eqsum"> </A><IMG WIDTH=500 HEIGHT=46 ALIGN=BOTTOM ALT="equation1058" SRC="img74.gif"><BR>Also, the new phases must implement the same phase differences exhibited by thecorresponding phases <IMG WIDTH=141 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline2092" SRC="img75.gif"> of the data:<BR><IMG WIDTH=500 HEIGHT=20 ALIGN=BOTTOM ALT="equation1060" SRC="img76.gif"><BR>The last equation can be fulfilled by setting <IMG WIDTH=119 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline2094" SRC="img77.gif">. With this, we have <IMG WIDTH=276 HEIGHT=33 ALIGN=MIDDLE ALT="tex2html_wrap_inline2096" SRC="img78.gif"> which is extremal when<BR><IMG WIDTH=500 HEIGHT=46 ALIGN=BOTTOM ALT="equation1062" SRC="img79.gif"><BR>The minimum is selected by taking <IMG WIDTH=16 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline2098" SRC="img80.gif"> in the correct quadrant.<P><P><blockquote><A NAME="944"> </A><IMG WIDTH=345 HEIGHT=388 ALIGN=BOTTOM ALT="figure1056" SRC="img69.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figbdat"> </A> Simultaneous surrogates for a bi-variate time series. The upper two panels show simultaneous recordings of the breath rate and the instantaneous heart rate of a human. The lower two panels show surrogate sequences that preserve the individual distributions and power spectra as well as the cross-correlation function between heart and breath rate. The most prominent difference between data and surrogates is the lack of coherence in the surrogate breath rate. <BR></blockquote><P><P><blockquote><A NAME="946"> </A><IMG WIDTH=364 HEIGHT=287 ALIGN=BOTTOM ALT="figure1057" SRC="img70.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figbx"> </A> Cross-correlation functions for the bi-variate data shown in Fig. <A HREF="node15.html#figbdat">8</A> (upper panel), and a surrogate that preserves the individual spectra and distributions as well as the relative Fourier phases (middle). The lower panel shows the same for surrogates prepared for each channel individually, that is, without explicitly preserving the cross-correlation structure.<BR></blockquote><P>As an example, let us generate a surrogate sequence for a simultaneousrecording of the breath rate and the instantaneous heart rate of ahuman during sleep. The data is again taken from data set B of theSanta Fe Institute time series contest [<A HREF="node36.html#gold">31</A>]. The 1944 datapoints are an end-point matched sub-sequence of the data used as amultivariate example in Ref. [<A HREF="node36.html#anneal">26</A>]. In the latter study, whichwill be commented on in Sec. <A HREF="node24.html#secmulti2">6.2</A> below, the breath ratesignal had been considered to be an input and therefore not beenrandomised. Here, we will randomise both channels under the conditionthat their individual spectra as well as their cross-correlationfunction are preserved as well as possible while matching theindividual distributions exactly. The iterative scheme introducedabove took 188 iterations to converge to a fixed point. The data anda bi-variate surrogate is shown in Fig. <A HREF="node15.html#figbdat">8</A>. InFig. <A HREF="node15.html#figbx">9</A>, the cross-correlation functions of the data and onesurrogate are plotted. Also, for comparison, the same for two individualsurrogates of the two channels. The most strikingdifference between data and surrogates is that the coherence of thebreath rate is lost. Thus, it is indeed reasonable to exclude thenonlinear structure in the breath dynamics from a further analysis ofthe heart rate by taking the breath rate as a given input signal.Such an analysis is however beyond the scope of the method discussedin this section. First of all, specifying the full cross-correlationfunction to a fixed signal plus the autocorrelation functionover-specifies the problem and there is no room for randomisation. InSec. <A HREF="node24.html#secmulti2">6.2</A> below, we will therefore revisit thisproblem. With the general constrained randomisation scheme to beintroduced below, it will be possible to specify a limited number oflags of the auto- and cross-correlation functions.<P><HR><A NAME="tex2html220" HREF="node16.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html218" HREF="node9.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html214" HREF="node14.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html221" HREF="node16.html">General constrained randomisation</A><B>Up:</B> <A NAME="tex2html219" HREF="node9.html">Fourier based surrogates</A><B> Previous:</B> <A NAME="tex2html215" HREF="node14.html">Periodicity artefacts</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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