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a factor of ten to ``melt'' the system. This procedure is repeated until more
than 2/3 successes are reached. This ensures that we start with a temperature
that is high enough to leave all false minima. If the automatic scheme gets
stuck (the low temperature allows too few changes to take place), it re-starts
at the determined melting temperature. At the same time, the cooling rate is
decreased by <IMG WIDTH=59 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline2546" SRC="img211.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img211.gif">, and <IMG WIDTH=119 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline2548" SRC="img212.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img212.gif">.  We suggest to create one surrogate with
the automatic scheme and then use the final values of <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">, <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
<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"> for subsequent runs. Of course, other more
sophisticated cooling schemes may be suitable depending on the specific
situation. The reader is referred to the standard literature&nbsp;[<A HREF="node36.html#annealbook" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#annealbook">38</A>].
<P>
Several cost functions are currently implemented in TISEAN. Each of them is of
the general form (<A HREF="node17.html#eqE" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node17.html#eqE">22</A>) and the constraints can be matched
in either the <IMG WIDTH=16 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline2112" SRC="img86.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img86.gif">, <IMG WIDTH=17 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline2108" SRC="img85.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img85.gif">, or the <IMG WIDTH=23 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline2560" SRC="img213.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img213.gif"> (or maximum) norms.
In the <IMG WIDTH=16 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline2112" SRC="img86.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img86.gif"> and <IMG WIDTH=17 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline2108" SRC="img85.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img85.gif"> norms, autocorrelations are weighted by
<IMG WIDTH=64 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2566" SRC="img214.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img214.gif"> and frequencies by <IMG WIDTH=68 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2568" SRC="img215.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img215.gif">.
<P>
Autocorrelations (<a href="../wuppertal/randomize_auto.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/randomize_auto.html">auto</a>, or a periodic version <a href="../wuppertal/randomize_auto.html#autop" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/randomize_auto.html#autop">autop</a>) are the most
common constraints available. Apart from the type of
average, one has to specify the maximal lag <IMG WIDTH=29 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline2128" SRC="img92.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img92.gif"> (see
e.g. Eq.(<A HREF="node17.html#eqcost" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node17.html#eqcost">23</A>)). This can save a substantial fraction of the
computation time if only short range correlations are present.  For each
update, only <IMG WIDTH=54 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2572" SRC="img216.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img216.gif"> terms have to be updated.
<P>
For unevenly sampled data (see Sec.&nbsp;<A HREF="node25.html#secuneven" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node25.html#secuneven">6.3</A>), the cost function <TT>
uneven</TT> implements binned autocorrelations as defined by
Eq.(<A HREF="node25.html#eqcdelta" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node25.html#eqcdelta">26</A>). The update of the histogram at each annealing step takes
a number of steps proportional to the number of bins. The user has to specify
the bin size <IMG WIDTH=12 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline2330" SRC="img140.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img140.gif"> and the total lag time covered contiguously by the bins.
<P>
For surrogate spike trains, either the spike train peridogram
Eq.(<A HREF="node26.html#eqspower" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node26.html#eqspower">28</A>) or binned correlations Eq.(<A HREF="node26.html#eqcspike" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node26.html#eqcspike">29</A>) can be used.
In the former case, the cost function is coded in <TT>spikespec</TT>.  The user
has to give the total number of frequencies and the frequency
resolution. Internally, the event times <IMG WIDTH=13 HEIGHT=20 ALIGN=MIDDLE ALT="tex2html_wrap_inline2576" SRC="img217.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img217.gif"> are used. A computationally
feasible update is only possible if two consecutive intervals <IMG WIDTH=62 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline2578" SRC="img218.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img218.gif"> and
<IMG WIDTH=63 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline2580" SRC="img219.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img219.gif"> are exchanged by <IMG WIDTH=153 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline2582" SRC="img220.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img220.gif"> (done by the
permutation scheme <TT>event</TT>).  As a consequence, coverage of permutation
space is quite inefficient. With binned autocorrelations <TT>spikeauto</TT>,
intervals are kept internally and any two intervals may be swapped, using the
standard permutation scheme <TT>random</TT>.
<P>
The documentation distributed with the TISEAN package describes how to add
further cost functions. Essentially, one needs to provide cost function
specific option parsing and input/output functions, a module that computes the
full cost function and one that performs an update upon permutation. The latter
should be coded very carefully. First it is the single spot that uses most of
the computation time and second, it must keep the cost function consistent for
all possible permutations. It is advisable to make extensive tests against
freshly computed cost functions before entering production.
<P>
In future releases of TISEAN, it is planned to include routines for cross
correlations in mutivariate data, multivariate spike trains, and mixed signals.
We hope that users take the present modular implementation as a starting point
for the implementation of other null hypotheses.
<P>
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Mon Aug 30 17:31:48 CEST 1999</I>
</ADDRESS>
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