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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>The curse of accuracy</TITLE><META NAME="description" CONTENT="The curse of accuracy"><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="tex2html283" HREF="node22.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html281" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html277" HREF="node20.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html284" HREF="node22.html">Various Examples</A><B>Up:</B> <A NAME="tex2html282" HREF="node16.html">General constrained randomisation</A><B> Previous:</B> <A NAME="tex2html278" HREF="node20.html">Combinatorial minimisation and accuracy</A><BR> <P><H2><A NAME="SECTION00055000000000000000">The curse of accuracy</A></H2><A NAME="secaccuracy"> </A>Strictly speaking, the concept of constrained realisations requires theconstraints to be fulfilled <EM>exactly</EM>, a practical impossibility. Most ofthe research efforts reported in this article have their origin in the attemptto increase the accuracy with which the constraints are implemented, that is,to minimise the bias resulting from any remaining discrepancy. Since mostmeasures of nonlinearity are also sensitive to linear correlations, a sideeffect of the reduced bias is a reduced variance of suchestimators. Paradoxically, thus the enhanced accuracy may result in falserejections of the null hypothesis on the ground of tiny differences in somenonlinear characteristics. This important point has been recently put forth byKugiumtzis [<A HREF="node36.html#dimitris">39</A>].<P>Consider the highly correlated autoregressive process <IMG WIDTH=292 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2272" SRC="img123.gif">, measured by the function<IMG WIDTH=138 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2274" SRC="img124.gif"> and then normalised to zero mean and unit variance. Thestrong correlation together with the rather strong static nonlinearity makesthis a very difficult data set for the generation ofsurrogates. Figure <A HREF="node21.html#figaccuracy">12</A> shows the bias and variance for a <EM>linear</EM> statistic, the unit lag autocorrelation <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2276" SRC="img125.gif">, Eq.(<A HREF="node14.html#eqcp">15</A>), ascompared to its goal value given by the data. The left part ofFig. <A HREF="node21.html#figaccuracy">12</A> shows <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2276" SRC="img125.gif"> versus the iteration count <I>i</I> for 200iterative surrogates, <I>i</I>=1 roughly corresponding to AAFT surrogates. Althoughthe mean accuracy increases dramatically compared to the first iterationstages, the data consistently remains outside a 2<IMG WIDTH=9 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline2284" SRC="img126.gif"> error bound. Sincenonlinear parameters will also pick up linear correlations, we have to expectspurious results in such a case. In the right part, annealed surrogates aregenerated with a cost function <IMG WIDTH=215 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2286" SRC="img127.gif">. The bias and variance of <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2276" SRC="img125.gif"> areplotted versus the cost <I>E</I>. Since the cost function involves <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2276" SRC="img125.gif">, it isnot surprising that we see good convergence of the bias. It is also noteworthythat the variance is in any event large enough to exclude spurious results dueto remaining discrepancy in the linear correlations.<P><blockquote><A NAME="961"> </A><IMG WIDTH=345 HEIGHT=189 ALIGN=BOTTOM ALT="figure1076" SRC="img128.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figaccuracy"> </A> Bias and variance of unit lag autocorrelation <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2276" SRC="img125.gif"> for ensembles of surrogates. Left part: <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2276" SRC="img125.gif"> plotted versus the iteration count <I>i</I> for 200 iterative surrogates. The AAFT method gives accuracies comparable to the value obtained for <I>i</I>=1. Right part: <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2276" SRC="img125.gif"> plotted versus the goal value of the cost function for 20 annealed surrogates. The horizontal line indicates the sample value for the data sequence. See text for discussion.<BR></blockquote><P>Kugiumtzis [<A HREF="node36.html#dimitris">39</A>] suggests to test the validity of the surrogatesample by performing a test using a linear statistic for normalisation. For thedata shown in Fig. <A HREF="node21.html#figaccuracy">12</A>, this would have detected the lack ofconvergence of the iterative surrogates. Currently, this seems to be the onlyway around the problem and we thus recommend to follow his suggestion. With themuch more accurate annealed surrogates, we haven't so far seen examples ofdangerous remaining inaccuracy, but we cannot exclude their possibility. Ifsuch a case occurs, it may be possible to generate unbiased ensembles ofsurrogates by specifying a cost function that explicitly minimises the bias.This would involve the whole collection of <I>M</I> surrogates at the same time,including extra terms like<BR><IMG WIDTH=500 HEIGHT=53 ALIGN=BOTTOM ALT="equation1077" SRC="img129.gif"><BR>Here, <IMG WIDTH=44 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline2306" SRC="img130.gif"> denotes the autocorrelation function of the<I>m</I>-th surrogate. In any event, this will be a very cumbersome procedure, interms of implementation and in terms of execution speed and it is questionableif it is worth the effort.<P><HR><A NAME="tex2html283" HREF="node22.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html281" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html277" HREF="node20.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html284" HREF="node22.html">Various Examples</A><B>Up:</B> <A NAME="tex2html282" HREF="node16.html">General constrained randomisation</A><B> Previous:</B> <A NAME="tex2html278" HREF="node20.html">Combinatorial minimisation and accuracy</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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