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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>Iterative FFT surrogates</TITLE><META NAME="description" CONTENT="Iterative FFT surrogates"><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="tex2html418" HREF="node35.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html416" HREF="node32.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html410" HREF="node33.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html419" HREF="node35.html">Annealed surrogates</A><B>Up:</B> <A NAME="tex2html417" HREF="node32.html">The TISEAN implementation</A><B> Previous:</B> <A NAME="tex2html411" HREF="node33.html">Measures of nonlinearity</A><BR> <P><H2><A NAME="SECTION000102000000000000000">Iterative FFT surrogates</A></H2><P>The workhorse for the generation of surrogate data within the TISEAN package isthe program <a href="../docs_f/surrogates.html">surrogates</a>. It implements the iterative Fourier based schemeintroduced in Ref. [<A HREF="node36.html#surrowe">30</A>] and discussed in Sec. <A HREF="node12.html#seciterative">4.3</A>. Ithas been extended to be able to handle multivariate data as discussed inSec. <A HREF="node15.html#secmulti1">4.6</A>. An FFT routine is used that can handle data sets of <I>N</I>points if <I>N</I> can be factorised using prime factors 2, 3, and 5 only. Routinesthat take arbitrary <I>N</I> will end up doing a slow Fourier transform if <I>N</I> isnot factorisable with small factors. Occasionally, the length restrictionresults in the loss of a few points.<P>The routine starts with a random scramble as <IMG WIDTH=38 HEIGHT=36 ALIGN=MIDDLE ALT="tex2html_wrap_inline2022" SRC="img38.gif">,performs as many iterates as necessary to reach a fixed point and then printsout <IMG WIDTH=29 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2040" SRC="img47.gif"> or <IMG WIDTH=29 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2042" SRC="img48.gif">, asdesired. Further, the number of iterations is shown and the residual root meansquared discrepancy between <IMG WIDTH=29 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2040" SRC="img47.gif"> and<IMG WIDTH=29 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2042" SRC="img48.gif">. The number of iterations can be limited by anoption. In particular, <I>i</I>=0 gives the initial scramble as<IMG WIDTH=38 HEIGHT=36 ALIGN=MIDDLE ALT="tex2html_wrap_inline2022" SRC="img38.gif"> or a non-rescaled FFT surrogate as<IMG WIDTH=38 HEIGHT=36 ALIGN=MIDDLE ALT="tex2html_wrap_inline2478" SRC="img192.gif">. The first iterate, <IMG WIDTH=38 HEIGHT=36 ALIGN=MIDDLE ALT="tex2html_wrap_inline2480" SRC="img193.gif">, isapproximately (but not quite) equivalent to an AAFT surrogate. It is advisableto evaluate the residual discrepancy whenever the algorithm took more than afew iterations. In cases of doubt if the accuracy is sufficient, it may beuseful to plot the autocorrelation function (<TT>corr</TT> or <TT>autocor</TT>)of the data and <IMG WIDTH=29 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2040" SRC="img47.gif">, and, in the multivariate case,the cross-correlation function (<TT>xcor</TT>) between the channels.The routine can generate up to 999 surrogates in one call.<P><h4><a name="endtoend">End point effects</a></h4>Since the periodicity artefact discussed in Sec. <A HREF="node14.html#secperiod">4.5</A> can lead tospurious test results, we need to select a suitable sub-sequence of the databefore making surrogates. For this purpose, TISEAN contains the program <a href="../docs_f/endtoend.html">endtoend</a>. Let <IMG WIDTH=107 HEIGHT=36 ALIGN=MIDDLE ALT="tex2html_wrap_inline2484" SRC="img194.gif"> be a sub-sequence of length<IMG WIDTH=14 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline2486" SRC="img195.gif"> and offset <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline2488" SRC="img196.gif">. The program then computes the contribution of theend-to-end mismatch <IMG WIDTH=101 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline2490" SRC="img197.gif"> to the total powerin the sub-sequence:<BR><IMG WIDTH=500 HEIGHT=52 ALIGN=BOTTOM ALT="equation1100" SRC="img198.gif"><BR>as well as the contribution of the mismatch in the first derivative<BR><IMG WIDTH=500 HEIGHT=53 ALIGN=BOTTOM ALT="equation1106" SRC="img199.gif"><BR>and the weighted average <BR><IMG WIDTH=500 HEIGHT=26 ALIGN=BOTTOM ALT="equation1112" SRC="img200.gif"><BR> The weight <I>w</I> can be selected by the user and is set to 0.5 by default.For multivariate data with <I>M</I> channels, <IMG WIDTH=141 HEIGHT=39 ALIGN=MIDDLE ALT="tex2html_wrap_inline2496" SRC="img201.gif"> is used.<P>Now the program goes through a sequence of decreasing <IMG WIDTH=173 HEIGHT=32 ALIGN=MIDDLE ALT="tex2html_wrap_inline2498" SRC="img202.gif">, and for each <IMG WIDTH=14 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline2486" SRC="img195.gif"> determines <IMG WIDTH=15 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2502" SRC="img203.gif"> such that<IMG WIDTH=47 HEIGHT=36 ALIGN=MIDDLE ALT="tex2html_wrap_inline2504" SRC="img204.gif"> is minimal. The values of <IMG WIDTH=14 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline2486" SRC="img195.gif">, <IMG WIDTH=15 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2502" SRC="img203.gif">,and <IMG WIDTH=47 HEIGHT=36 ALIGN=MIDDLE ALT="tex2html_wrap_inline2504" SRC="img204.gif"> are printed whenever <IMG WIDTH=9 HEIGHT=16 ALIGN=MIDDLE ALT="tex2html_wrap_inline2512" SRC="img205.gif"> hasdecreased. One can thus easily find a sub-sequence that achieves negligible endpoint mismatch with the minimal loss of data.<P><HR><A NAME="tex2html418" HREF="node35.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html416" HREF="node32.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html410" HREF="node33.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html419" HREF="node35.html">Annealed surrogates</A><B>Up:</B> <A NAME="tex2html417" HREF="node32.html">The TISEAN implementation</A><B> Previous:</B> <A NAME="tex2html411" HREF="node33.html">Measures of nonlinearity</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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