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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.2//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 concept of surrogate data</TITLE><META NAME="description" CONTENT="The concept of surrogate data"><META NAME="keywords" CONTENT="TiseanHTML"><META NAME="resource-type" CONTENT="document"><META NAME="distribution" CONTENT="global"><LINK REL=STYLESHEET HREF="TiseanHTML.css"></HEAD><BODY bgcolor=ffffff LANG="EN" > <A NAME="tex2html440" HREF="node37.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html438" HREF="node35.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html432" HREF="node35.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html441" HREF="node37.html">Iterative Fourier transform method</A><B>Up:</B> <A NAME="tex2html439" HREF="node35.html">Testing for nonlinearity</A><B> Previous:</B> <A NAME="tex2html433" HREF="node35.html">Testing for nonlinearity</A><BR> <P><H2><A NAME="SECTION00091000000000000000">The concept of surrogate data</A></H2><P>The degree of nonlinearity can be measured in several ways. But how muchnonlinear predictability, say, is necessary to exclude more trivialexplanations? All quantifiers of nonlinearity show fluctuations but thedistributions, or error bars if you wish, are not available analytically. Itis therefore necessary to use Monte Carlo techniques to assess the significanceof results. One important method in this context is the method of surrogatedata [<A HREF="citation.html#theiler1">82</A>]. A null hypothesis is formulated, for example that thedata has been created by a stationary Gaussian linear process, and then it isattempted to reject this hypothesis by comparing results for the data toappropriate realizations of the null hypothesis. Since the null assumption isnot a simple one but leaves room for free parameters, the Monte Carlo samplehas to take these into account. One approach is to construct <EM>constrainedrealizations</EM> of the null hypothesis. The idea is that the free parameters leftby the null are reflected by specific properties of the data. For example theunknown coefficients of an autoregressive process are reflected in theautocorrelation function. Constrained realizations are obtained by randomizingthe data subject to the constraint that an appropriate set of parametersremains fixed. For example, random data with a given periodogram can be made byassuming random phases and taking the inverse Fourier transform of the givenperiodogram. Random data with the same distribution as a given data set can begenerated by permuting the data randomly without replacement. Asking for agiven spectrum and a given distribution at the same time poses already a muchmore difficult question.<P><HR><A NAME="tex2html440" HREF="node37.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html438" HREF="node35.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html432" HREF="node35.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html441" HREF="node37.html">Iterative Fourier transform method</A><B>Up:</B> <A NAME="tex2html439" HREF="node35.html">Testing for nonlinearity</A><B> Previous:</B> <A NAME="tex2html433" HREF="node35.html">Testing for nonlinearity</A><P><ADDRESS><I>Thomas Schreiber <BR>Wed Jan 6 15:38:27 CET 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
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