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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>The null hypothesis: model class vs. properties</TITLE><META NAME="description" CONTENT="The null hypothesis: model class vs. properties"><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="tex2html138" HREF="node8.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html136" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html130" HREF="node6.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html139" HREF="node8.html">Test design</A><B>Up:</B> <A NAME="tex2html137" HREF="node5.html">Surrogate data testing</A><B> Previous:</B> <A NAME="tex2html131" HREF="node6.html">Typical vs. constrained realisations</A><BR> <P><H2><A NAME="SECTION00032000000000000000">The null hypothesis: model class vs. properties</A></H2><P>From the bootstrap literature we are used to defining null hypothesis for timeseries in terms of a class of processes that is assumed to contain the specificprocess that generated the data. For most of the literature on surrogate data,this situation hasn't changed. One very common null hypothesis goes back toTheiler and coworkers [<A HREF="node36.html#theiler1">6</A>] and states that the data have beengenerated by a Gaussian linear stochastic process with constant coefficients.Constrained realisations are created by requiring that the surrogate timeseries have the same Fourier amplitudes as the data. We can clearly see in thisexample that what is needed for the constrained realisations approach is a setof observable properties that is known to fully specify the process. Theprocess itself is not reconstructed. But this example is also exceptional. Weknow that the class of processes defined by the null hypothesis is fullyparametrised by the set of ARMA(<I>M</I>,<I>N</I>) models (autoregressive moving average,see Eq.(<A HREF="node9.html#eqarma">6</A>) below). If we allow for arbitrary orders <I>M</I> and <I>N</I>,there is a one-to-one correspondence between the ARMA coefficients and thepower spectrum. The power spectrum is here estimated by the Fourieramplitudes. The Wiener-Khinchin theorem relates it to the autocorrelationfunction by a simple Fourier transformation. Consequently, specifying eitherthe class of processes or the set of constraints are two ways to achieve thesame goal. The only generalisation of this favourable situation that has beenfound so far is the null hypothesis that the ARMA output may have been observedby a static, invertible measurement function. In that case, constraining thesingle time probability distribution and the Fourier amplitudes is sufficient.<P>If we want to go beyond this hypothesis, all we can do in general is to specifythe set of constraints we will impose. We cannot usually say which class ofprocesses this choice corresponds to. We will have to be content withstatements that a given set of statistical parameters exhaustively describesthe statistical properties of a signal. Hypotheses in terms of a model classare usually more informative but specifying sets of observables gives us muchmore flexibility.<P><HR><A NAME="tex2html138" HREF="node8.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html136" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html130" HREF="node6.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html139" HREF="node8.html">Test design</A><B>Up:</B> <A NAME="tex2html137" HREF="node5.html">Surrogate data testing</A><B> Previous:</B> <A NAME="tex2html131" HREF="node6.html">Typical vs. constrained realisations</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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