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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>Annealed surrogates</TITLE><META NAME="description" CONTENT="Annealed surrogates"><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="tex2html426" HREF="node36.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html424" HREF="node32.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html420" HREF="node34.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html427" HREF="node36.html">References</A><B>Up:</B> <A NAME="tex2html425" HREF="node32.html">The TISEAN implementation</A><B> Previous:</B> <A NAME="tex2html421" HREF="node34.html">Iterative FFT surrogates</A><BR> <P><H2><A NAME="SECTION000103000000000000000">Annealed surrogates</A></H2><P>For cases where the iterative scheme does not reach the necessary accuracy, orwhenever a more general null hypothesis is considered, the TISEAN packageoffers an implementation of the constrained randomisation algorithm using acost function minimised by simulated annealing, as introduced inRef. [<A HREF="node36.html#anneal">26</A>] and described in Sec. <A HREF="node16.html#secanneal">5</A>. Since one of themain advantages of the approach is its flexibility, the implementation moreresembles a toolbox than a single program. The main driving routine <TT>randomize</TT> takes care of the data input and output and operates the simulatedannealing procedure. It must be linked together with modules that implement acooling schedule, a cost function, and a permutation scheme. Within TISEAN,several choices for each of these are already implemented but it is relativelyeasy to add individual variants or completely different cost functions, coolingor permutation schemes. With the development structure provided, the finalexecutables will then have names reflecting the components linked together, inthe 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 ascheme <a href="../docs_f/randomize_perm.html">random</a> that selects a pair at random. It is, however, possible tospecify a list of points to be excluded from the permutations. This is usefulwhen the time series contains artifacts or some data points are missing andhave been replaced by dummy values. It is planned to add atemperature-sensitive scheme that selects pairs close in magnitude at lowtemperatures. For certain cost functions (e.g. the spike train spectrum), anupdate can only be carried out efficiently if two consecutive points areexchanged. This is implemented in an alternative permutation scheme <a href="../docs_f/randomize_perm.html">event</a>.<P>The only cooling scheme supported in the present version of TISEAN (2.0) isexponential cooling (<a href="../docs_f/randomize_cool.html">exp</a>). This means that whenever a certain conditionis reached, the temperature is multiplied by a factor <IMG WIDTH=38 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2526" SRC="img206.gif">. Apart from<IMG WIDTH=10 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline1914" SRC="img7.gif"> and the initial temperature <IMG WIDTH=15 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline2530" SRC="img207.gif">, two important parameters controlthe cooling schedule. Cooling is performed either if a maximal total number oftrials <IMG WIDTH=34 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2532" SRC="img208.gif"> is exceeded, or if a maximal number<IMG WIDTH=31 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2534" SRC="img209.gif"> of trials has been successfull since the lastcooling. Finally, a minimal number of successes <IMG WIDTH=30 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2536" SRC="img210.gif"> can bespecified below which the procedure is considered to be ``stuck''. All theseparameters can be specified explicitly. However, it is sometimes verydifficult to derive reasonable values except by trial and error. Slow coolingis necessary if the desired accuracy of the constraint is high. It seemsreasonable to increase <IMG WIDTH=31 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2534" SRC="img209.gif"> and <IMG WIDTH=34 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2532" SRC="img208.gif">with the system size, but also with the number of constraints incorporated inthe cost function. It can be convenient to use an automatic scheme that startswith fast parameter settings and re-starts the procedure with slower settingswhenever it gets stuck, until a desired accuracy is reached. The initialtemperature can be selected automatically using the following algorithm. Startwith an arbitrary small initial temperature. Let the system evolve for<IMG WIDTH=34 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2532" SRC="img208.gif"> steps (or <IMG WIDTH=31 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2534" SRC="img209.gif"> successes). Ifless than 2/3 of the trials were successes, increase the initial temperature bya factor of ten to ``melt'' the system. This procedure is repeated until morethan 2/3 successes are reached. This ensures that we start with a temperaturethat is high enough to leave all false minima. If the automatic scheme getsstuck (the low temperature allows too few changes to take place), it re-startsat the determined melting temperature. At the same time, the cooling rate isdecreased by <IMG WIDTH=59 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline2546" SRC="img211.gif">, and <IMG WIDTH=119 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline2548" SRC="img212.gif">. We suggest to create one surrogate withthe automatic scheme and then use the final values of <IMG WIDTH=15 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline2530" SRC="img207.gif">, <IMG WIDTH=10 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline1914" SRC="img7.gif"> and<IMG WIDTH=34 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2532" SRC="img208.gif"> for subsequent runs. Of course, other moresophisticated cooling schemes may be suitable depending on the specificsituation. The reader is referred to the standard literature [<A HREF="node36.html#annealbook">38</A>].<P>Several cost functions are currently implemented in TISEAN. Each of them is ofthe general form (<A HREF="node17.html#eqE">22</A>) and the constraints can be matchedin either the <IMG WIDTH=16 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline2112" SRC="img86.gif">, <IMG WIDTH=17 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline2108" SRC="img85.gif">, or the <IMG WIDTH=23 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline2560" SRC="img213.gif"> (or maximum) norms.In the <IMG WIDTH=16 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline2112" SRC="img86.gif"> and <IMG WIDTH=17 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline2108" SRC="img85.gif"> norms, autocorrelations are weighted by<IMG WIDTH=64 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2566" SRC="img214.gif"> and frequencies by <IMG WIDTH=68 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2568" SRC="img215.gif">.<P>Autocorrelations (<a href="../docs_f/randomize_auto.html">auto</a>, or a periodic version <a href="../docs_f/randomize_auto.html#autop">autop</a>) are the mostcommon constraints available. Apart from the type ofaverage, one has to specify the maximal lag <IMG WIDTH=29 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline2128" SRC="img92.gif"> (seee.g. Eq.(<A HREF="node17.html#eqcost">23</A>)). This can save a substantial fraction of thecomputation time if only short range correlations are present. For eachupdate, only <IMG WIDTH=54 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2572" SRC="img216.gif"> terms have to be updated.<P>For unevenly sampled data (see Sec. <A HREF="node25.html#secuneven">6.3</A>), the cost function <TT>uneven</TT> implements binned autocorrelations as defined byEq.(<A HREF="node25.html#eqcdelta">26</A>). The update of the histogram at each annealing step takesa number of steps proportional to the number of bins. The user has to specifythe bin size <IMG WIDTH=12 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline2330" SRC="img140.gif"> and the total lag time covered contiguously by the bins.<P>For surrogate spike trains, either the spike train peridogramEq.(<A HREF="node26.html#eqspower">28</A>) or binned correlations Eq.(<A HREF="node26.html#eqcspike">29</A>) can be used.In the former case, the cost function is coded in <TT>spikespec</TT>. The userhas to give the total number of frequencies and the frequencyresolution. Internally, the event times <IMG WIDTH=13 HEIGHT=20 ALIGN=MIDDLE ALT="tex2html_wrap_inline2576" SRC="img217.gif"> are used. A computationallyfeasible update is only possible if two consecutive intervals <IMG WIDTH=62 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline2578" SRC="img218.gif"> and<IMG WIDTH=63 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline2580" SRC="img219.gif"> are exchanged by <IMG WIDTH=153 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline2582" SRC="img220.gif"> (done by thepermutation scheme <TT>event</TT>). As a consequence, coverage of permutationspace is quite inefficient. With binned autocorrelations <TT>spikeauto</TT>,intervals are kept internally and any two intervals may be swapped, using thestandard permutation scheme <TT>random</TT>.<P>The documentation distributed with the TISEAN package describes how to addfurther cost functions. Essentially, one needs to provide cost functionspecific option parsing and input/output functions, a module that computes thefull cost function and one that performs an update upon permutation. The lattershould be coded very carefully. First it is the single spot that uses most ofthe computation time and second, it must keep the cost function consistent forall possible permutations. It is advisable to make extensive tests againstfreshly computed cost functions before entering production.<P>In future releases of TISEAN, it is planned to include routines for crosscorrelations in mutivariate data, multivariate spike trains, and mixed signals.We hope that users take the present modular implementation as a starting pointfor the implementation of other null hypotheses.<P><HR><A NAME="tex2html426" HREF="node36.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html424" HREF="node32.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html420" HREF="node34.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html427" HREF="node36.html">References</A><B>Up:</B> <A NAME="tex2html425" HREF="node32.html">The TISEAN implementation</A><B> Previous:</B> <A NAME="tex2html421" HREF="node34.html">Iterative FFT surrogates</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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