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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>Typical vs. constrained realisations</TITLE><META NAME="description" CONTENT="Typical vs. constrained realisations"><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="tex2html128" HREF="node7.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html126" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html120" HREF="node5.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html129" HREF="node7.html">The null hypothesis: model </A><B>Up:</B> <A NAME="tex2html127" HREF="node5.html">Surrogate data testing</A><B> Previous:</B> <A NAME="tex2html121" HREF="node5.html">Surrogate data testing</A><BR> <P><H2><A NAME="SECTION00031000000000000000">Typical vs. constrained realisations</A></H2><P>Traditional bootstrap methods use explicit model equations that have to beextracted from the data and are then run to produce Monte Carlo samples.This <EM>typical realisations</EM> approach can be very powerful for thecomputation of confidence intervals, provided the model equations can beextracted successfully. The latter requirement is very delicate. Ambiguities inselecting the proper model class and order, as well as the parameter estimationproblem have to be addressed. Whenever the null hypothesis involves an unknown<EM>function</EM> (rather than just a few parameters) these problems becomeprofound. A recent example of a <EM>typical realisations</EM> approach to creatingsurrogates in the dynamical systems context is given by Ref. [<A HREF="node36.html#witt">24</A>].There, a Markov model is fitted to a coarse-grained dynamics obtained bybinning the two dimensional delay vector distribution of a time series.Then, essentially the transfer matrix is iterated to yield surrogatesequences. We will offer some discussion of that work later inSec. <A HREF="node27.html#secinterpret">7</A>.<P>As discussed by Theiler and Prichard [<A HREF="node36.html#tp">25</A>], thealternative approach of <EM>constrained realisations</EM> is more suitable for thepurpose of hypothesis testing we are interested in here. It avoids the fittingof model equations by directly imposing the desired structures onto therandomised time series. However, the choice of possible null hypothesis islimited by the difficulty of imposing arbitrary structures on otherwise randomsequences. In the following, we will discuss a number of null hypotheses andalgorithms to provide the adequately constrained realisations. The most generalmethod to generate constrained randomisations of time series [<A HREF="node36.html#anneal">26</A>] is described in Sec. <A HREF="node16.html#secanneal">5</A>.<P>Consider as a toy example the null hypothesis that the data consists ofindependent draws from a fixed probability distribution. Surrogate time seriescan be simply obtained by randomly shuffling the measured data. If we findsignificantly different serial correlations in the data and the shuffles, wecan reject the hypothesis of independence. Constrained realisations areobtained by creating permutations <EM>without replacement</EM>. The surrogates areconstrained to take on exactly the same values as the data, just in randomtemporal order. We could also have used the data to infer the probabilitydistribution and drawn new time series from it. These permutations <EM>withreplacement</EM> would then be what we called typical realisations.<P>Obviously, independence is not an interesting null hypothesis for most timeseries problems. It becomes relevant when the residual errors of a time seriesmodel are evaluated. For example in the BDS test fornonlinearity [<A HREF="node36.html#brockpaper1">27</A>], an ARMA model is fitted to the data. If thedata are linear, then the residuals are expected to be independent.It has been pointed out, however, that the resulting test is not particularlypowerful for chaotic data [<A HREF="node36.html#bleach">28</A>].<P><HR><A NAME="tex2html128" HREF="node7.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html126" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html120" HREF="node5.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html129" HREF="node7.html">The null hypothesis: model </A><B>Up:</B> <A NAME="tex2html127" HREF="node5.html">Surrogate data testing</A><B> Previous:</B> <A NAME="tex2html121" HREF="node5.html">Surrogate data testing</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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