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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>Rescaled Gaussian linear process</TITLE><META NAME="description" CONTENT="Rescaled Gaussian linear process"><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="tex2html172" HREF="node11.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html170" HREF="node9.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html164" HREF="node9.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A>   <BR><B> Next:</B> <A NAME="tex2html173" HREF="node11.html">Flatness bias of AAFT </A><B>Up:</B> <A NAME="tex2html171" HREF="node9.html">Fourier based surrogates</A><B> Previous:</B> <A NAME="tex2html165" HREF="node9.html">Fourier based surrogates</A><BR> <P><H2><A NAME="SECTION00041000000000000000">Rescaled Gaussian linear process</A></H2><P>The two null hypotheses discussed so far (independent random numbers andGaussian linear processes) are not what we want to test against in mostrealistic situations. In particular, the most obvious deviation from theGaussian linear process is usually that the data do not follow a Gaussiansingle time probability distribution. This is quite obvious for data obtainedby measuring intervals between events, e.g. heart beats since intervals arestrictly positive. There is however a simple generalisation of the nullhypothesis that explains deviations from the normal distribution by the actionof an invertible, static measurement function: <BR><A NAME="eqdistort">&#160;</A><IMG WIDTH=500 HEIGHT=46 ALIGN=BOTTOM ALT="equation1033" SRC="img23.gif"><BR> We want to regard a time series from such a process as essentially linear sincethe only nonlinearity is contained in the -- in principle invertible --measurement function <IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1908" SRC="img2.gif">.<P>Let us mention right away that the restriction that <IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1908" SRC="img2.gif"> must beinvertible is quite severe and often undesired. The reason why we have toimpose it is that otherwise we couldn't give a complete specification of theprocess in terms of observables and constraints. The problem is furtherillustrated in Sec.&nbsp;<A HREF="node28.html#secrev">7.1</A> below.<P>The most common method to create surrogate data sets for this null hypothesisessentially attempts to invert <IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1908" SRC="img2.gif"> by rescaling the time series<IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1972" SRC="img24.gif"> to conform with a Gaussian distribution. The rescaled version isthen phase randomised (conserving Gaussianity on average) and the result isrescaled to the empirical distribution of <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1972" SRC="img24.gif">. The rescaling is done bysimple rank ordering. Suppose we want to rescale the sequence <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1972" SRC="img24.gif"> so thatthe rescaled sequence <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1978" SRC="img25.gif"> takes on the same values as some referencesequence <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1980" SRC="img26.gif"> (e.g. draws from a Gaussian distribution). Let <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1980" SRC="img26.gif"> besorted in ascending order and <IMG WIDTH=58 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1984" SRC="img27.gif"> denote the ascending rank of<IMG WIDTH=14 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline1986" SRC="img28.gif">, e.g. <IMG WIDTH=89 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1988" SRC="img29.gif"> if <IMG WIDTH=14 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline1986" SRC="img28.gif"> is the 3rd smallest element of <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1972" SRC="img24.gif">. Then the rescaled sequence is given by<BR><A NAME="eqrank">&#160;</A><IMG WIDTH=500 HEIGHT=18 ALIGN=BOTTOM ALT="equation1035" SRC="img30.gif"><BR>The <EM>amplitude adjusted Fourier transform</EM> (AAFT) method has beenoriginally proposed by Theiler et al.&nbsp;[<A HREF="node36.html#theiler1">6</A>]. It results in acorrect test when <I>N</I> is large, the correlation in the data is not too strongand <IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1908" SRC="img2.gif"> is close to the identity. Otherwise, there is a certain biastowards a too flat spectrum, to be discussed in the following section.<P><HR><A NAME="tex2html172" HREF="node11.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html170" HREF="node9.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html164" HREF="node9.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A>   <BR><B> Next:</B> <A NAME="tex2html173" HREF="node11.html">Flatness bias of AAFT </A><B>Up:</B> <A NAME="tex2html171" HREF="node9.html">Fourier based surrogates</A><B> Previous:</B> <A NAME="tex2html165" HREF="node9.html">Fourier based surrogates</A><P><ADDRESS><I>Thomas Schreiber <BR>Mon Aug 30 17:31:48 CEST 1999</I></ADDRESS></BODY></HTML>
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