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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>Iteratively refined surrogates</TITLE><META NAME="description" CONTENT="Iteratively refined 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="tex2html192" HREF="node13.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html190" HREF="node9.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html184" HREF="node11.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html193" HREF="node13.html">Example: Southern oscillation index</A><B>Up:</B> <A NAME="tex2html191" HREF="node9.html">Fourier based surrogates</A><B> Previous:</B> <A NAME="tex2html185" HREF="node11.html">Flatness bias of AAFT </A><BR> <P><H2><A NAME="SECTION00043000000000000000">Iteratively refined surrogates</A></H2><A NAME="seciterative"> </A>In Ref. [<A HREF="node36.html#surrowe">30</A>], we propose a method which iteratively correctsdeviations in spectrum and distribution from the goal set by the measureddata. In an alternating fashion, the surrogate is filtered towards the correctFourier amplitudes and rank-ordered to the correct distribution.<P>Let <IMG WIDTH=47 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2008" SRC="img34.gif"> be the Fourier amplitudes, Eq.(<A HREF="node9.html#eqpgram">7</A>), of thedata and <IMG WIDTH=28 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2010" SRC="img35.gif"> a copy of the data sorted by magnitude in ascending order.At each iteration stage (<I>i</I>), we have a sequence <IMG WIDTH=36 HEIGHT=36 ALIGN=MIDDLE ALT="tex2html_wrap_inline2014" SRC="img36.gif">that has the correct distribution (coincides with <IMG WIDTH=28 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2010" SRC="img35.gif"> when sorted), and asequence <IMG WIDTH=36 HEIGHT=36 ALIGN=MIDDLE ALT="tex2html_wrap_inline2018" SRC="img37.gif"> that has the correct Fourier amplitudesgiven by <IMG WIDTH=47 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2008" SRC="img34.gif">. One can start with <IMG WIDTH=38 HEIGHT=36 ALIGN=MIDDLE ALT="tex2html_wrap_inline2022" SRC="img38.gif">being either an AAFT surrogate, or simply a random shuffle of the data.<P>The step <IMG WIDTH=68 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2024" SRC="img39.gif"> is a very crude``filter'' in the Fourier domain: The Fourier amplitudes are simply <EM>replaced</EM> by the desired ones. First, take the (discrete) Fourier transform of<IMG WIDTH=36 HEIGHT=36 ALIGN=MIDDLE ALT="tex2html_wrap_inline2014" SRC="img36.gif">:<BR><IMG WIDTH=500 HEIGHT=46 ALIGN=BOTTOM ALT="equation1038" SRC="img40.gif"><BR>Then transform back, replacing the actual amplitudes by the desired ones, butkeeping the phases <IMG WIDTH=126 HEIGHT=38 ALIGN=MIDDLE ALT="tex2html_wrap_inline2028" SRC="img41.gif">:<BR><A NAME="eqstep1"> </A><IMG WIDTH=500 HEIGHT=47 ALIGN=BOTTOM ALT="equation1040" SRC="img42.gif"><BR>The step <IMG WIDTH=84 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2030" SRC="img43.gif"> proceeds by rankordering:<BR><A NAME="eqstep2"> </A><IMG WIDTH=500 HEIGHT=25 ALIGN=BOTTOM ALT="equation1042" SRC="img44.gif"><BR>It can be heuristically understood that the iteration scheme is attracted to afixed point <IMG WIDTH=81 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2032" SRC="img45.gif"> for large(<I>i</I>). Since the minimal possible change equals to the smallest nonzerodifference <IMG WIDTH=64 HEIGHT=13 ALIGN=MIDDLE ALT="tex2html_wrap_inline2036" SRC="img46.gif"> and is therefore finite for finite <I>N</I>, the fixedpoint is reached after a finite number of iterations. The remaining discrepancybetween <IMG WIDTH=29 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2040" SRC="img47.gif"> and <IMG WIDTH=29 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2042" SRC="img48.gif"> can betaken as a measure of the accuracy of the method. Whether the residual bias in<IMG WIDTH=29 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2040" SRC="img47.gif"> or <IMG WIDTH=29 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2042" SRC="img48.gif"> is more tolerabledepends on the data and the nonlinearity measure to be used. For coarselydigitised data,<A NAME="tex2html6" HREF="footnode.html#220"><IMG ALIGN=BOTTOM ALT="gif" SRC="foot_motif.gif"></A>deviations from the discrete distribution can lead to spurious resultswhence <IMG WIDTH=29 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2040" SRC="img47.gif"> is the safer choice. If linear correlationsare dominant, <IMG WIDTH=29 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2042" SRC="img48.gif"> can be more suitable.<P>The final accuracy that can be reached depends on the size and structure of thedata and is generally sufficient for hypothesis testing. In all the cases wehave studied so far, we have observed a substantial improvement over thestandard AAFT approach. Convergence properties are also discussedin [<A HREF="node36.html#surrowe">30</A>]. In Sec. <A HREF="node21.html#secaccuracy">5.5</A> below, we will say more about theremaining inaccuracies.<P><HR><A NAME="tex2html192" HREF="node13.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html190" HREF="node9.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html184" HREF="node11.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html193" HREF="node13.html">Example: Southern oscillation index</A><B>Up:</B> <A NAME="tex2html191" HREF="node9.html">Fourier based surrogates</A><B> Previous:</B> <A NAME="tex2html185" HREF="node11.html">Flatness bias of AAFT </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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