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<H2><A NAME="SECTION00054000000000000000">Combinatorial minimisation and accuracy</A></H2>
<P>
In principle, simulated annealing is able to reach arbitrary accuracy at the
expense of computer time. We should, however, remark on a few points.  Unlike
other minimisation problems, we are not really interested in the solutions that
put <I>E</I>=0 exactly. Most likely, these are the data set itself and a few simple
transformations of it that preserve the cost function (e.g. a time reversed
copy). On the other hand, combinatorics makes it very unlikely that we ever
reach one of these few of the <I>N</I>! permutations, unless <I>N</I> is really small or
the constraints grossly over-specify the problem.  This can be the case, for
example, if we include <EM>all</EM> possible lags of the autocorrelation function,
which gives as many (nonlinear) equations as unknowns, <I>I</I>=<I>N</I>. These may close
for small <I>N</I> in the space of permutations.  In such extreme situations, it is
possible to include extra cost terms penalising closeness to one of the trivial
transformations of the data.  Let us note that if the surrogates are ``too
similar'' to the data, this does not in itself affect the validity of the
test. Only the discrimination power may be severely reduced.
<P>
Now, if we don't want to reach <I>E</I>=0, how can we be sure that there are enough
independent realisations with <IMG WIDTH=42 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline2244" SRC="img115.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img115.gif">? The theoretical answer depends on
the form of the constraints in a complicated way and cannot be given in
general. We can, however, offer a heuristic argument that the number of
configurations with <I>E</I> smaller than some <IMG WIDTH=24 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline2248" SRC="img116.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img116.gif"> grows fast for large <I>N</I>.
Suppose that for large <I>N</I> the probability distribution of <I>E</I> converges to an
asymptotic form <I>p</I>(<I>E</I>). Assume further that <IMG WIDTH=288 HEIGHT=33 ALIGN=MIDDLE ALT="tex2html_wrap_inline2258" SRC="img117.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img117.gif"> is nonzero but maybe
very small. This is evidently true for autocorrelations, for example. While
thus the probability to find <IMG WIDTH=59 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline2260" SRC="img118.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img118.gif"> in a random draw from the
distribution of the data may be extremely small, say <IMG WIDTH=107 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2262" SRC="img119.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img119.gif"> at 10 sigmas from the mean energy, the total number of
permutations, figuring as the number of draws, grows as <IMG WIDTH=138 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline2264" SRC="img120.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img120.gif">, that is, much faster than exponentially. Thus, we expect
the number of permutations with <IMG WIDTH=59 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline2260" SRC="img118.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img118.gif"> to be <IMG WIDTH=79 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2268" SRC="img121.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img121.gif">.  For example, <IMG WIDTH=156 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2270" SRC="img122.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img122.gif">.
<P>
In any case, we can always monitor the convergence of the cost function to
avoid spurious results due to residual inaccuracy in the surrogates.  As we
will discuss below, it can also be a good idea to test the surrogates with a
linear test statistic before performing the actual nonlinearity test.
<P>
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Mon Aug 30 17:31:48 CEST 1999</I>
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