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<H1><A NAME="SECTION00030000000000000000">Surrogate data testing</A></H1>
<P>
All of the measures of nonlinearity mentioned above share a common property.
Their probability distribution on finite data sets is not known analytically -
except maybe when strong additional assumptions about the data are made. Some
authors have tried to give error bars for measures like predictabilities (e.g.
Barahona and Poon&nbsp;[<A HREF="node36.html#volterra" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#volterra">21</A>]) or averages of pointwise dimensions (e.g.
Skinner et al.&nbsp;[<A HREF="node36.html#skinner" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#skinner">22</A>]) based on the observation that these quantities
are averages (mean values or medians) of many individual terms, in which case
the variance (or quartile points) of the individual values yield an error
estimate. This reasoning is however only valid if the individual terms are
independent, which is usually not the case for time series data. In fact, it is
found empirically that nonlinearity measures often do not even follow a
Gaussian distribution.  Also the standard error given by
Roulston&nbsp;[<A HREF="node36.html#roulston" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#roulston">23</A>] for the mutual information is fully correct only for
uniformly distributed data. His derivation assumes a smooth rescaling to
uniformity. In practice, however, we have to rescale either to <EM>exact</EM>
uniformity or by rank-ordering uniform variates. Both transformations are in
general non-smooth and introduce a bias in the joint probabilities.  In view of
the serious difficulties encountered when deriving confidence limits or
probability distributions of nonlinear statistics with analytical methods, it
is highly preferable to use a Monte Carlo resampling technique for this
purpose.
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<I>Thomas Schreiber <BR>
Mon Aug 30 17:31:48 CEST 1999</I>
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