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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>Example: avoiding periodicity artefacts</TITLE><META NAME="description" CONTENT="Example: avoiding periodicity artefacts"><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="tex2html265" HREF="node20.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html263" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html257" HREF="node18.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html266" HREF="node20.html">Combinatorial minimisation and accuracy</A><B>Up:</B> <A NAME="tex2html264" HREF="node16.html">General constrained randomisation</A><B> Previous:</B> <A NAME="tex2html258" HREF="node18.html">Computational issues of simulated </A><BR> <P><H2><A NAME="SECTION00053000000000000000">Example: avoiding periodicity artefacts</A></H2><P><blockquote><A NAME="959"> </A><IMG WIDTH=338 HEIGHT=493 ALIGN=BOTTOM ALT="figure1075" SRC="img111.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figendanneal"> </A> Progressive stages of the simulated annealing scheme. The data used in Fig. <A HREF="node14.html#figend">6</A> is used to generate an annealed surrogate that minimises <IMG WIDTH=184 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline2218" SRC="img110.gif"> over all permutations of the data. From top to bottom, the values for <I>E</I> are: 0 (original data), 1.01 (random scramble), 0.51, 0.12, 0.015, and 0.00013.<BR></blockquote><P>Let us illustrate the use of the annealing method in the case of the standardnull hypothesis of a rescaled linear process. We will show how the periodicityartefact discussed in Sec. <A HREF="node14.html#secperiod">4.5</A> can be avoided by using a moresuitable cost function. We prepare a surrogate for the data shown inFig. <A HREF="node14.html#figend">6</A> (almost unstable AR(2) process) without truncating itslength. We minimise the cost function given by Eq.(<A HREF="node17.html#eqcost">23</A>), involvingall lags up to <IMG WIDTH=76 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline2222" SRC="img112.gif">. Also, we excluded the first andlast points from permutations as a cheap way of imposing the long rangecorrelation. In Fig. <A HREF="node19.html#figendanneal">11</A> we show progressive stages of theannealing procedure, starting from a random scramble. The temperature <I>T</I> isdecreased by 0.1% after either <IMG WIDTH=21 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline2226" SRC="img113.gif"> permutations have been tried or <IMG WIDTH=21 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline2228" SRC="img114.gif">have been successful. The final surrogate neither has spuriously matching endsnor the additional high frequency components we saw in Fig. <A HREF="node14.html#figend">6</A>. Theprice we had to pay was that the generation of one single surrogate took 6 h ofCPU time on a Pentium II PC at 350 MHz. If we had taken care of the long rangecorrelation by leaving the end points loose but taking <IMG WIDTH=93 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline2130" SRC="img93.gif">, convergence would have been prohibitively slow. Note that for aproper test, we would need at least 19 surrogates. We should stress that thisexample with its very slow decay of correlations is particularly nasty -- butstill feasible. Obviously, sacrificing 10% of the points to get rid of theend point mismatch is preferable here to spending several days of CPU timeon the annealing scheme. In other cases, however, we may not have such achoice.<P><HR><A NAME="tex2html265" HREF="node20.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html263" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html257" HREF="node18.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html266" HREF="node20.html">Combinatorial minimisation and accuracy</A><B>Up:</B> <A NAME="tex2html264" HREF="node16.html">General constrained randomisation</A><B> Previous:</B> <A NAME="tex2html258" HREF="node18.html">Computational issues of simulated </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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