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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>Surrogate data testing</TITLE><META NAME="description" CONTENT="Surrogate data testing"><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="tex2html115" HREF="node6.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html113" HREF="Surrogates.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html107" HREF="node4.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html116" HREF="node6.html">Typical vs. constrained realisations</A><B>Up:</B> <A NAME="tex2html114" HREF="Surrogates.html">Surrogate time series</A><B> Previous:</B> <A NAME="tex2html108" HREF="node4.html">Phase space observables</A><BR> <P><H1><A NAME="SECTION00030000000000000000">Surrogate data testing</A></H1><P>All of the measures of nonlinearity mentioned above share a common property.Their probability distribution on finite data sets is not known analytically -except maybe when strong additional assumptions about the data are made. Someauthors have tried to give error bars for measures like predictabilities (e.g.Barahona and Poon [<A HREF="node36.html#volterra">21</A>]) or averages of pointwise dimensions (e.g.Skinner et al. [<A HREF="node36.html#skinner">22</A>]) based on the observation that these quantitiesare averages (mean values or medians) of many individual terms, in which casethe variance (or quartile points) of the individual values yield an errorestimate. This reasoning is however only valid if the individual terms areindependent, which is usually not the case for time series data. In fact, it isfound empirically that nonlinearity measures often do not even follow aGaussian distribution. Also the standard error given byRoulston [<A HREF="node36.html#roulston">23</A>] for the mutual information is fully correct only foruniformly distributed data. His derivation assumes a smooth rescaling touniformity. In practice, however, we have to rescale either to <EM>exact</EM>uniformity or by rank-ordering uniform variates. Both transformations are ingeneral non-smooth and introduce a bias in the joint probabilities. In view ofthe serious difficulties encountered when deriving confidence limits orprobability distributions of nonlinear statistics with analytical methods, itis highly preferable to use a Monte Carlo resampling technique for thispurpose.<P><BR> <HR><UL><A NAME="CHILD_LINKS"> </A><LI> <A NAME="tex2html117" HREF="node6.html#SECTION00031000000000000000">Typical vs. constrained realisations</A><LI> <A NAME="tex2html118" HREF="node7.html#SECTION00032000000000000000">The null hypothesis: model class vs. properties</A><LI> <A NAME="tex2html119" HREF="node8.html#SECTION00033000000000000000">Test design</A></UL><HR><A NAME="tex2html115" HREF="node6.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html113" HREF="Surrogates.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html107" HREF="node4.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html116" HREF="node6.html">Typical vs. constrained realisations</A><B>Up:</B> <A NAME="tex2html114" HREF="Surrogates.html">Surrogate time series</A><B> Previous:</B> <A NAME="tex2html108" HREF="node4.html">Phase space observables</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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