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<H2><A NAME="SECTION00055000000000000000">The curse of accuracy</A></H2>
<A NAME="secaccuracy">&#160;</A>
Strictly speaking, the concept of constrained realisations requires the
constraints to be fulfilled <EM>exactly</EM>, a practical impossibility. Most of
the research efforts reported in this article have their origin in the attempt
to increase the accuracy with which the constraints are implemented, that is,
to minimise the bias resulting from any remaining discrepancy. Since most
measures of nonlinearity are also sensitive to linear correlations, a side
effect of the reduced bias is a reduced variance of such
estimators. Paradoxically, thus the enhanced accuracy may result in false
rejections of the null hypothesis on the ground of tiny differences in some
nonlinear characteristics. This important point has been recently put forth by
Kugiumtzis&nbsp;[<A HREF="node36.html#dimitris" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img123.gif">, measured by the function
<IMG WIDTH=138 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2274" SRC="img124.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img124.gif"> and then normalised to zero mean and unit variance.  The
strong correlation together with the rather strong static nonlinearity makes
this a very difficult data set for the generation of
surrogates. Figure&nbsp;<A HREF="node21.html#figaccuracy" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img125.gif">, Eq.(<A HREF="node14.html#eqcp" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node14.html#eqcp">15</A>), as
compared to its goal value given by the data.  The left part of
Fig.&nbsp;<A HREF="node21.html#figaccuracy" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node21.html#figaccuracy">12</A> shows <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2276" SRC="img125.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img125.gif"> versus the iteration count <I>i</I> for 200
iterative surrogates, <I>i</I>=1 roughly corresponding to AAFT surrogates.  Although
the mean accuracy increases dramatically compared to the first iteration
stages, the data consistently remains outside a 2<IMG WIDTH=9 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline2284" SRC="img126.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img126.gif"> error bound.  Since
nonlinear parameters will also pick up linear correlations, we have to expect
spurious results in such a case. In the right part, annealed surrogates are
generated with a cost function <IMG WIDTH=215 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2286" SRC="img127.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img127.gif">. The bias and variance of <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2276" SRC="img125.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img125.gif"> are
plotted versus the cost&nbsp;<I>E</I>. Since the cost function involves <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2276" SRC="img125.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img125.gif">, it is
not surprising that we see good convergence of the bias. It is also noteworthy
that the variance is in any event large enough to exclude spurious results due
to remaining discrepancy in the linear correlations.
<P>
<blockquote><A NAME="961">&#160;</A><IMG WIDTH=345 HEIGHT=189 ALIGN=BOTTOM ALT="figure1076" SRC="img128.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img128.gif"><BR>
<STRONG>Figure:</STRONG> <A NAME="figaccuracy">&#160;</A>
   Bias and variance of unit lag autocorrelation <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2276" SRC="img125.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img125.gif"> for ensembles of
   surrogates.  Left part: <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2276" SRC="img125.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/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&nbsp;[<A HREF="node36.html#dimitris" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#dimitris">39</A>] suggests to test the validity of the surrogate
sample by performing a test using a linear statistic for normalisation. For the
data shown in Fig.&nbsp;<A HREF="node21.html#figaccuracy" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node21.html#figaccuracy">12</A>, this would have detected the lack of
convergence of the iterative surrogates.  Currently, this seems to be the only
way around the problem and we thus recommend to follow his suggestion. With the
much more accurate annealed surrogates, we haven't so far seen examples of
dangerous remaining inaccuracy, but we cannot exclude their possibility. If
such a case occurs, it may be possible to generate unbiased ensembles of
surrogates 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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img129.gif"><BR>
Here, <IMG WIDTH=44 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline2306" SRC="img130.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img130.gif"> denotes the autocorrelation function of the
<I>m</I>-th surrogate.  In any event, this will be a very cumbersome procedure, in
terms of implementation and in terms of execution speed and it is questionable
if it is worth the effort.
<P>
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Mon Aug 30 17:31:48 CEST 1999</I>
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