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<H1><A NAME="SECTION00010000000000000000">Introduction</A></H1>
<P>
A nonlinear approach to analysing time series
data [<A HREF="node36.html#SFI" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#SFI">1</A>, <A HREF="node36.html#coping" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#coping">2</A>, <A HREF="node36.html#abarbook" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#abarbook">3</A>, <A HREF="node36.html#ourbook" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#ourbook">4</A>, <A HREF="node36.html#habil" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#habil">5</A>] can be motivated by two distinct
reasons. One is intrinsic to the signal itself while the other is due to
additional knowledge we may have about the nature of the observed
phenomenon. As for the first motivation, it might be that the arsenal of linear
methods has been exploited thoroughly but all the efforts left certain
structures in the time series unaccounted for. As for the second, a system may
be known to include nonlinear components and therefore a linear description
seems unsatisfactory in the first place. Such an argument is often heard for
example in brain research -- nobody expects for example the brain to be a
linear device. In fact, there is ample evidence for nonlinearity in particular
in small assemblies of neurons. Nevertheless, the latter reasoning is rather
dangerous. The fact that a system contains nonlinear components does not prove
that this nonlinearity is also reflected in a specific signal we measure from
that system. In particular, we do not know if it is of any practical use to go
beyond the linear approximation when analysing the signal. After all, we do not
want our data analysis to reflect our prejudice about the underlying system but
to represent a fair account of the structures that are present in the
data. Consequently, the application of nonlinear time series methods has to be
justified by establishing nonlinearity in the time series.
<P>
Suppose we had measured the signal shown in Fig. <A HREF="node1.html#figarspikes" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node1.html#figarspikes">1</A> in some
biological setting. Visual inspection immediately reveals nontrivial structure
in the serial correlations. The data fails a test for Gaussianity, thus ruling
out a Gaussian linear stochastic process as its source. Depending on the
assumptions we are willing to make on the underlying process, we might suggest
different origins for the observed strong ``spikyness'' of the dynamics.
Superficially, low dimensional chaos seems unlikely due to the strong
fluctuations, but maybe high dimensional dynamics? A large collection of
neurons could intermittently synchronise to give rise to the burst episodes. In
fact, certain artificial neural network models show qualitatively similar
dynamics. The least interesting explanation, however, would be that all the
spikyness comes from a distortion by the measurement procedure and all the
serial correlations are due to linear stochastic dynamics. Occam's razor tells
us that we should be able to rule out such a simple explanation before we
venture to construct more complicated models.
<P>
Surrogate data testing attempts to find the least interesting explanation
that cannot be ruled out based on the data. In the above example, the data
shown in Fig. <A HREF="node1.html#figarspikes" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node1.html#figarspikes">1</A>, this would be the hypothesis that the data
has been generated by a stationary Gaussian linear stochastic process
(equivalently, an <EM>autoregressive moving average</EM> or ARMA process) that is
observed through an invertible, static, but possible nonlinear observation
function:
<BR><IMG WIDTH=500 HEIGHT=16 ALIGN=BOTTOM ALT="equation1015" SRC="img1.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img1.gif"><BR>
Neither the order <I>M</I>,<I>N</I>, the ARMA coefficients, nor the 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"> are
assumed to be known. Without explicitly modeling these parameters, we still
know that such a process would show characteristic linear correlations
(reflecting the ARMA structure) and a characteristic single time probability
distribution (reflecting the action of <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"> on the original Gaussian
distribution). Figure <A HREF="node1.html#figarspikes_surr" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node1.html#figarspikes_surr">2</A> shows a surrogate time series
that is designed to have exactly these properties in common with the data but
to be as random as possible otherwise. By a proper statistical test we can now
look for additional structure that is present in the data but not in the
surrogates.
<P>
<blockquote><A NAME="919"> </A><IMG WIDTH=360 HEIGHT=154 ALIGN=BOTTOM ALT="figure1017" SRC="img3.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img3.gif"><BR>
<STRONG>Figure:</STRONG> <A NAME="figarspikes"> </A>
A time series showing characteristic bursts.<BR>
</blockquote>
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