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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>Test design</TITLE><META NAME="description" CONTENT="Test design"><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="tex2html146" HREF="node9.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html144" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html140" HREF="node7.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html147" HREF="node9.html">Fourier based surrogates</A><B>Up:</B> <A NAME="tex2html145" HREF="node5.html">Surrogate data testing</A><B> Previous:</B> <A NAME="tex2html141" HREF="node7.html">The null hypothesis: model </A><BR> <P><H2><A NAME="SECTION00033000000000000000">Test design</A></H2><P>Before we go into detail about the generation of surrogate samples, let usoutline how an actual test can be carried out. Many examples are known ofnonlinearity measures that aren't even approximately normally distributed. Ithas therefore been advocated since the early days [<A HREF="node36.html#theiler1">6</A>] to use robuststatistics rather than parametric methods for the actual statistical test. Inother words, we discourage the common practice to represent the distribution ofthe nonlinearity measure by an error bar and deriving the significance from thenumber of ``sigmas'' the data lies outside these bounds. Such a reasoningimplicitly assumes a Gaussian distribution.<P>Instead, we follow Theiler et al. [<A HREF="node36.html#theiler1">6</A>] by using a rank-order test.First, we select a residual probability <IMG WIDTH=10 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline1914" SRC="img7.gif"> of a false rejection,corresponding to a level of significance <IMG WIDTH=104 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1932" SRC="img12.gif">. Then, for aone-sided test (e.g. looking for <EM>small</EM> prediction errors only), wegenerate <IMG WIDTH=90 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1934" SRC="img13.gif"> surrogate sequences. Thus, including the data itself,we have <IMG WIDTH=25 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1936" SRC="img14.gif"> sets. Therefore, the probability that the data bycoincidence has the smallest, say, prediction error is exactly <IMG WIDTH=10 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline1914" SRC="img7.gif">, asdesired. For a two-sided test (e.g. for time asymmetry which can go bothways), we would generate <IMG WIDTH=90 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1940" SRC="img15.gif"> surrogates, resulting in a probability<IMG WIDTH=10 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline1914" SRC="img7.gif"> that the data gives <EM>either</EM> the smallest <EM>or</EM> the largestvalue.<P>For a minimal significance requirement of 95% , we thus need at least 19 or 39surrogate time series for one- and two-sided tests, respectively. Theconditions for rank based tests with more samples can be easily worked out.Using more surrogates can increase the discrimination power.<P><HR><A NAME="tex2html146" HREF="node9.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html144" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html140" HREF="node7.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html147" HREF="node9.html">Fourier based surrogates</A><B>Up:</B> <A NAME="tex2html145" HREF="node5.html">Surrogate data testing</A><B> Previous:</B> <A NAME="tex2html141" HREF="node7.html">The null hypothesis: model </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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