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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>Flatness bias of AAFT surrogates</TITLE><META NAME="description" CONTENT="Flatness bias of AAFT 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="tex2html182" HREF="node12.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html180" HREF="node9.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html174" HREF="node10.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html183" HREF="node12.html">Iteratively refined surrogates</A><B>Up:</B> <A NAME="tex2html181" HREF="node9.html">Fourier based surrogates</A><B> Previous:</B> <A NAME="tex2html175" HREF="node10.html">Rescaled Gaussian linear process</A><BR> <P><H2><A NAME="SECTION00042000000000000000">Flatness bias of AAFT surrogates</A></H2><P>It is argued in Ref. [<A HREF="node36.html#surrowe">30</A>] that for short and strongly correlatedsequences, the AAFT algorithm can yield an incorrect test since it introduces abias towards a slightly flatter spectrum. In Fig. <A HREF="node11.html#figflat">3</A> we see powerspectral estimates of a clinical data set and of 19 AAFT surrogates. The datais taken from data set B of the Santa Fe Institute time seriescontest [<A HREF="node36.html#gold">31</A>]. It consists of 4096 samples of the breath rate of apatient with sleep apnoea. The sampling interval is 0.5 seconds. Thediscrepancy of the spectra is significant. A bias towards a white spectrum isnoted: power is taken away from the main peak to enhance the low and highfrequencies.<P><blockquote><A NAME="926"> </A><IMG WIDTH=349 HEIGHT=186 ALIGN=BOTTOM ALT="figure1037" SRC="img31.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figflat"> </A> Discrepancy of the power spectra of human breath rate data (solid line) and 19 AAFT surrogates (dashed lines). Here the power spectra have been computed with a square window of length 64.<BR></blockquote><P>Heuristically, the flatness bias can be understood as follows. Amplitudeadjustment attempts to invert the unknown measurement function <IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1908" SRC="img2.gif">empirically. The estimate <IMG WIDTH=39 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2000" SRC="img32.gif"> of the inverse obtained by therescaling of a finite sample to values drawn from a Gaussian distribution isexpected to be consistent but it is not exact for finite <I>N</I>. The samplingfluctuations of <IMG WIDTH=161 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2004" SRC="img33.gif"> will be essentiallyindependent of <I>n</I> and thus spectrally white. Consequently, Gaussian scalingamounts to adding a white component to the spectrum, which therefore tends tobecome flatter under the procedure. Since such a bias can lead to spuriousresults, surrogates have to be refined before a test can be performed.<P><HR><A NAME="tex2html182" HREF="node12.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html180" HREF="node9.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html174" HREF="node10.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html183" HREF="node12.html">Iteratively refined surrogates</A><B>Up:</B> <A NAME="tex2html181" HREF="node9.html">Fourier based surrogates</A><B> Previous:</B> <A NAME="tex2html175" HREF="node10.html">Rescaled Gaussian linear process</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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