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<H2><A NAME="SECTION000103000000000000000">Annealed surrogates</A></H2>
<P>
For cases where the iterative scheme does not reach the necessary accuracy, or
whenever a more general null hypothesis is considered, the TISEAN package
offers an implementation of the constrained randomisation algorithm using a
cost function minimised by simulated annealing, as introduced in
Ref. [<A HREF="node36.html#anneal" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#anneal">26</A>] and described in Sec. <A HREF="node16.html#secanneal" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node16.html#secanneal">5</A>. Since one of the
main advantages of the approach is its flexibility, the implementation more
resembles a toolbox than a single program. The main driving routine <TT>
randomize</TT> takes care of the data input and output and operates the simulated
annealing procedure. It must be linked together with modules that implement a
cooling schedule, a cost function, and a permutation scheme. Within TISEAN,
several choices for each of these are already implemented but it is relatively
easy to add individual variants or completely different cost functions, cooling
or permutation schemes. With the development structure provided, the final
executables will then have names reflecting the components linked together, in
the form <font color=blue><TT>randomize_</TT><I>A</I>_<I>B</I>_<I>C</I></font>,
where <I>A</I> is a cost function module,
<I>B</I> a cooling scheme, and <I>C</I> a permutation scheme.
<P>
Currently, two permutation schemes are implemented. In general, one will use a
scheme <a href="../wuppertal/randomize_perm.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/randomize_perm.html">random</a> that selects a pair at random. It is, however, possible to
specify a list of points to be excluded from the permutations. This is useful
when the time series contains artifacts or some data points are missing and
have been replaced by dummy values. It is planned to add a
temperature-sensitive scheme that selects pairs close in magnitude at low
temperatures. For certain cost functions (e.g. the spike train spectrum), an
update can only be carried out efficiently if two consecutive points are
exchanged. This is implemented in an alternative permutation scheme <a href="../wuppertal/randomize_perm.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/randomize_perm.html">event</a>.
<P>
The only cooling scheme supported in the present version of TISEAN (2.0) is
exponential cooling (<a href="../wuppertal/randomize_cool.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/randomize_cool.html">exp</a>). This means that whenever a certain condition
is reached, the temperature is multiplied by a factor <IMG WIDTH=38 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2526" SRC="img206.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img206.gif">. Apart from
<IMG WIDTH=10 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline1914" SRC="img7.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img7.gif"> and the initial temperature <IMG WIDTH=15 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline2530" SRC="img207.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img207.gif">, two important parameters control
the cooling schedule. Cooling is performed either if a maximal total number of
trials <IMG WIDTH=34 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2532" SRC="img208.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img208.gif"> is exceeded, or if a maximal number
<IMG WIDTH=31 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2534" SRC="img209.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img209.gif"> of trials has been successfull since the last
cooling. Finally, a minimal number of successes <IMG WIDTH=30 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2536" SRC="img210.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img210.gif"> can be
specified below which the procedure is considered to be ``stuck''. All these
parameters can be specified explicitly. However, it is sometimes very
difficult to derive reasonable values except by trial and error. Slow cooling
is necessary if the desired accuracy of the constraint is high. It seems
reasonable to increase <IMG WIDTH=31 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2534" SRC="img209.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img209.gif"> and <IMG WIDTH=34 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2532" SRC="img208.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img208.gif">
with the system size, but also with the number of constraints incorporated in
the cost function. It can be convenient to use an automatic scheme that starts
with fast parameter settings and re-starts the procedure with slower settings
whenever it gets stuck, until a desired accuracy is reached. The initial
temperature can be selected automatically using the following algorithm. Start
with an arbitrary small initial temperature. Let the system evolve for
<IMG WIDTH=34 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2532" SRC="img208.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img208.gif"> steps (or <IMG WIDTH=31 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2534" SRC="img209.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img209.gif"> successes). If
less than 2/3 of the trials were successes, increase the initial temperature by
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