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<H2><A NAME="SECTION00022000000000000000">Phase space observables</A></H2>
<P>
When a nonlinearity test is performed with the question in mind if nonlinear
deterministic modeling of the signal may be useful, it seems most appropriate
to use a test statistic that is related to a nonlinear deterministic approach.
We have to keep in mind, however, that a positive test result only indicates
nonlinearity, not necessarily determinism. Since nonlinearity tests are usually
performed on data sets which do not show unambiguous signatures of
low-dimensional determinism (like clear scaling over several orders of
magnitude), one cannot simply estimate one of the quantitative indicators of
chaos, like the fractal dimension or the Lyapunov exponent. The formal answer
would almost always be that both are probably infinite. Still, some useful test
statistics are at least inspired by these quantities. Usually, some effective
value at a finite length scale has to be computed without establishing scaling
region or attempting to approximate the proper limits.
<P>
In order to define an observable in <I>m</I>-dimensional phase space, we first have
to reconstruct that space from a scalar time series, for example by the method
of delays:
<BR><A NAME="eqdelay">&#160;</A><IMG WIDTH=500 HEIGHT=18 ALIGN=BOTTOM ALT="equation1023" SRC="img9.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img9.gif"><BR>
One of the more robust choices of phase space observable is a nonlinear
prediction error with respect to a locally constant predictor
<I>F</I> that can be defined by
<BR><A NAME="eqerror">&#160;</A><IMG WIDTH=500 HEIGHT=43 ALIGN=BOTTOM ALT="equation1025" SRC="img10.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img10.gif"><BR>
The prediction over one time step is performed by averaging over the future
values of all neighbouring delay vectors closer than <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline1920" SRC="img11.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img11.gif"> in <I>m</I>
dimensions.
<P>
We have to consider the limiting case that the deterministic signature to be
detected is weak. In that case, the major limiting factor for the performance
of a statistical indicator is its variance since possible differences between
two samples may be hidden among the statistical fluctuations. In
Ref.&nbsp;[<A HREF="node36.html#power" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#power">13</A>], a number of popular measures of nonlinearity are compared
quantitatively. The results can be summarised by stating that in the presence
of time-reversal asymmetry, the particular quantity Eq.(<A HREF="node3.html#eqskew" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node3.html#eqskew">3</A>) that
derives from the three-point autocorrelation function gives very reliable
results. However, many nonlinear evolution equations produce little or no
time-reversal asymmetry in the statistical properties of the signal. In these
cases, simple measures like a prediction error of a locally constant phase
space predictor, Eq.(<A HREF="node4.html#eqerror" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node4.html#eqerror">5</A>), performed best.  It was found to be
advantageous to choose embedding and other parameters in order to obtain a
quantity that has a small spread of values for different realisations of the
same process, even if at these parameters no valid embedding could be expected.
<P>
Of course, prediction errors are not the only class of nonlinearity measures
that has been optimised for robustness. Notable other examples are
coarse-grained redundancies&nbsp;[<A HREF="node36.html#milan2" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#milan2">14</A>, <A HREF="node36.html#pompe" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#pompe">15</A>, <A HREF="node36.html#pt" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#pt">16</A>], and, at an even higher
level of coarse-graining, symbolic methods&nbsp;[<A HREF="node36.html#hao" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#hao">17</A>]. The very popular method
of <EM>false nearest neighbours</EM>&nbsp;[<A HREF="node36.html#FNN" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#FNN">18</A>] can be easily modified to yield a
scalar quantity suitable for nonlinearity testing. The same is true for the
concept of <EM>unstable periodic orbits</EM> (UPOs)&nbsp;[<A HREF="node36.html#PM" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#PM">19</A>, <A HREF="node36.html#soso" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#soso">20</A>].
<P>
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Mon Aug 30 17:31:48 CEST 1999</I>
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