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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>Null hypotheses, constraints, and cost functions</TITLE><META NAME="description" CONTENT="Null hypotheses, constraints, and cost functions"><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="tex2html245" HREF="node18.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html243" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html237" HREF="node16.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A>   <BR><B> Next:</B> <A NAME="tex2html246" HREF="node18.html">Computational issues of simulated </A><B>Up:</B> <A NAME="tex2html244" HREF="node16.html">General constrained randomisation</A><B> Previous:</B> <A NAME="tex2html238" HREF="node16.html">General constrained randomisation</A><BR> <P><H2><A NAME="SECTION00051000000000000000">Null hypotheses, constraints, and cost functions</A></H2><P>As we have discussed previously, we will often have to specify a nullhypothesis in terms of a complete set of observable properties of the data.Only in specific cases (e.g. the two point autocorrelation function), there isa one-to-one correspondence to a class of models (here the ARMA process).  Inany case, if <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1962" SRC="img20.gif"> denotes a surrogate time series, theconstraints will most often be of (or can be brought into) the form<BR><A NAME="eqF">&#160;</A><IMG WIDTH=500 HEIGHT=16 ALIGN=BOTTOM ALT="equation1064" SRC="img81.gif"><BR>Such constraints can always be turned into a cost function<BR><A NAME="eqE">&#160;</A><IMG WIDTH=500 HEIGHT=54 ALIGN=BOTTOM ALT="equation1066" SRC="img82.gif"><BR>The fact that <IMG WIDTH=56 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2102" SRC="img83.gif"> has a global minimum when the constraintsare fulfilled is unaffected by the choice of the weights <IMG WIDTH=46 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline2104" SRC="img84.gif"> and theorder <I>q</I> of the average.  The least squares or <IMG WIDTH=17 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline2108" SRC="img85.gif"> average is obtained at<I>q</I>=2, <IMG WIDTH=16 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline2112" SRC="img86.gif"> at <I>q</I>=1 and the maximum distance when <IMG WIDTH=47 HEIGHT=16 ALIGN=MIDDLE ALT="tex2html_wrap_inline2116" SRC="img87.gif">. Geometricaveraging is also possible (and can be formally obtained by taking the limit<IMG WIDTH=40 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline2118" SRC="img88.gif"> in a proper way).  We have experimented with different choices of <I>q</I>but we haven't found a choice that is uniformly superior to others. It seemsplausible to give either uniform weights or to enhance those constraints whichare particularly difficult to fulfil. Again, conclusive empirical results arestill lacking.<P>Consider as an example the constraint that the sample autocorrelation functionof the surrogate <IMG WIDTH=114 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline2122" SRC="img89.gif">(data rescaled to zero mean and unit variance) are the same as those of thedata, <IMG WIDTH=113 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2124" SRC="img90.gif">. This is done by specifying zerodiscrepancy as a constraint <IMG WIDTH=296 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline2126" SRC="img91.gif">. Ifthe correlations decay fast, <IMG WIDTH=29 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline2128" SRC="img92.gif"> can be restricted,otherwise <IMG WIDTH=93 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline2130" SRC="img93.gif"> (the largest available lag).  Thus, apossible cost function could read<BR><A NAME="eqcost">&#160;</A><IMG WIDTH=500 HEIGHT=21 ALIGN=BOTTOM ALT="equation1072" SRC="img94.gif"><BR>Other choices of <I>q</I> and the weights are of course also possible.<P>In all the cases considered in this paper, one constraint will be that thesurrogates take on the same values as the data but in different time order.This ensures that data and surrogates can equally likely be drawn from the same(unknown) single time probability distribution. This particular constraint isnot included in the cost function but identically fulfilled by considering onlypermutations without replacement of the data for minimisation.<P>By introducing a cost function, we have turned a difficult nonlinear, highdimensional root finding problem (<A HREF="node17.html#eqF">21</A>) into a minimisation problem(<A HREF="node17.html#eqE">22</A>). This leads to extremely many false minima whence such a strategyis discouraged for general root finding problems&nbsp;[<A HREF="node36.html#Press92">42</A>]. Here, thesituation is somewhat different since we need to solve Eq.(<A HREF="node17.html#eqF">21</A>) onlyover the set of all permutations of <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1972" SRC="img24.gif">. Although this set is big, it isstill discrete and powerful combinatorial minimisation algorithms areavailable that can deal with false minima very well. We choose to minimise<IMG WIDTH=56 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2102" SRC="img83.gif"> among all permutations <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1962" SRC="img20.gif"> of theoriginal time series <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1972" SRC="img24.gif"> using the method of <EM>simulated annealing</EM>.Configurations are updated by exchanging pairs in <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1962" SRC="img20.gif">. Theannealing scheme will decide which changes to accept and which to reject. Withan appropriate cooling scheme, the annealing procedure can reach any desiredaccuracy. Apart from simulated annealing, genetic algorithms&nbsp;[<A HREF="node36.html#genetic">35</A>]have become very popular for this kind of problems and there is no reason whythey couldn't be used for the present purpose as well.<P><HR><A NAME="tex2html245" HREF="node18.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html243" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html237" HREF="node16.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A>   <BR><B> Next:</B> <A NAME="tex2html246" HREF="node18.html">Computational issues of simulated </A><B>Up:</B> <A NAME="tex2html244" HREF="node16.html">General constrained randomisation</A><B> Previous:</B> <A NAME="tex2html238" HREF="node16.html">General constrained randomisation</A><P><ADDRESS><I>Thomas Schreiber <BR>Mon Aug 30 17:31:48 CEST 1999</I></ADDRESS></BODY></HTML>
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