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<H2><A NAME="SECTION00041000000000000000">Rescaled Gaussian linear process</A></H2>
<P>
The two null hypotheses discussed so far (independent random numbers and
Gaussian linear processes) are not what we want to test against in most
realistic situations. In particular, the most obvious deviation from the
Gaussian linear process is usually that the data do not follow a Gaussian
single time probability distribution. This is quite obvious for data obtained
by measuring intervals between events, e.g. heart beats since intervals are
strictly positive. There is however a simple generalisation of the null
hypothesis that explains deviations from the normal distribution by the action
of an invertible, static measurement function: 
<BR><A NAME="eqdistort">&#160;</A><IMG WIDTH=500 HEIGHT=46 ALIGN=BOTTOM ALT="equation1033" SRC="img23.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img23.gif"><BR> 
We want to regard a time series from such a process as essentially linear since
the 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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img2.gif"> must be
invertible is quite severe and often undesired. The reason why we have to
impose it is that otherwise we couldn't give a complete specification of the
process in terms of observables and constraints. The problem is further
illustrated in Sec.&nbsp;<A HREF="node28.html#secrev" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node28.html#secrev">7.1</A> below.
<P>
The most common method to create surrogate data sets for this null hypothesis
essentially attempts to invert <IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1908" SRC="img2.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img2.gif"> by rescaling the time series
<IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1972" SRC="img24.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img24.gif"> to conform with a Gaussian distribution. The rescaled version is
then phase randomised (conserving Gaussianity on average) and the result is
rescaled to the empirical distribution of <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1972" SRC="img24.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img24.gif">. The rescaling is done by
simple rank ordering. Suppose we want to rescale the sequence <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1972" SRC="img24.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img24.gif"> so that
the rescaled sequence <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1978" SRC="img25.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img25.gif"> takes on the same values as some reference
sequence <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1980" SRC="img26.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img26.gif"> (e.g. draws from a Gaussian distribution). Let <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1980" SRC="img26.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img26.gif"> be
sorted in ascending order and <IMG WIDTH=58 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1984" SRC="img27.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img27.gif"> denote the ascending rank of
<IMG WIDTH=14 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline1986" SRC="img28.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img28.gif">, e.g. <IMG WIDTH=89 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1988" SRC="img29.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img29.gif"> if <IMG WIDTH=14 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline1986" SRC="img28.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img28.gif"> is the 3rd smallest element of 
<IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1972" SRC="img24.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img30.gif"><BR>
The <EM>amplitude adjusted Fourier transform</EM> (AAFT) method has been
originally proposed by Theiler et al.&nbsp;[<A HREF="node36.html#theiler1" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#theiler1">6</A>]. It results in a
correct test when <I>N</I> is large, the correlation in the data is not too strong
and <IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline1908" SRC="img2.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img2.gif"> is close to the identity. Otherwise, there is a certain bias
towards a too flat spectrum, to be discussed in the following section.
<P>
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Mon Aug 30 17:31:48 CEST 1999</I>
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