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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>Phase space observables</TITLE><META NAME="description" CONTENT="Phase space observables"><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="tex2html105" HREF="node5.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html103" HREF="node2.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html99" HREF="node3.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html106" HREF="node5.html">Surrogate data testing</A><B>Up:</B> <A NAME="tex2html104" HREF="node2.html">Detecting weak nonlinearity</A><B> Previous:</B> <A NAME="tex2html100" HREF="node3.html">Higher order statistics</A><BR> <P><H2><A NAME="SECTION00022000000000000000">Phase space observables</A></H2><P>When a nonlinearity test is performed with the question in mind if nonlineardeterministic modeling of the signal may be useful, it seems most appropriateto 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 indicatesnonlinearity, not necessarily determinism. Since nonlinearity tests are usuallyperformed on data sets which do not show unambiguous signatures oflow-dimensional determinism (like clear scaling over several orders ofmagnitude), one cannot simply estimate one of the quantitative indicators ofchaos, like the fractal dimension or the Lyapunov exponent. The formal answerwould almost always be that both are probably infinite. Still, some useful teststatistics are at least inspired by these quantities. Usually, some effectivevalue at a finite length scale has to be computed without establishing scalingregion or attempting to approximate the proper limits.<P>In order to define an observable in <I>m</I>-dimensional phase space, we first haveto reconstruct that space from a scalar time series, for example by the methodof delays:<BR><A NAME="eqdelay"> </A><IMG WIDTH=500 HEIGHT=18 ALIGN=BOTTOM ALT="equation1023" SRC="img9.gif"><BR>One of the more robust choices of phase space observable is a nonlinearprediction error with respect to a locally constant predictor<I>F</I> that can be defined by<BR><A NAME="eqerror"> </A><IMG WIDTH=500 HEIGHT=43 ALIGN=BOTTOM ALT="equation1025" SRC="img10.gif"><BR>The prediction over one time step is performed by averaging over the futurevalues of all neighbouring delay vectors closer than <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline1920" SRC="img11.gif"> in <I>m</I>dimensions.<P>We have to consider the limiting case that the deterministic signature to bedetected is weak. In that case, the major limiting factor for the performanceof a statistical indicator is its variance since possible differences betweentwo samples may be hidden among the statistical fluctuations. InRef. [<A HREF="node36.html#power">13</A>], a number of popular measures of nonlinearity are comparedquantitatively. The results can be summarised by stating that in the presenceof time-reversal asymmetry, the particular quantity Eq.(<A HREF="node3.html#eqskew">3</A>) thatderives from the three-point autocorrelation function gives very reliableresults. However, many nonlinear evolution equations produce little or notime-reversal asymmetry in the statistical properties of the signal. In thesecases, simple measures like a prediction error of a locally constant phasespace predictor, Eq.(<A HREF="node4.html#eqerror">5</A>), performed best. It was found to beadvantageous to choose embedding and other parameters in order to obtain aquantity that has a small spread of values for different realisations of thesame process, even if at these parameters no valid embedding could be expected.<P>Of course, prediction errors are not the only class of nonlinearity measuresthat has been optimised for robustness. Notable other examples arecoarse-grained redundancies [<A HREF="node36.html#milan2">14</A>, <A HREF="node36.html#pompe">15</A>, <A HREF="node36.html#pt">16</A>], and, at an even higherlevel of coarse-graining, symbolic methods [<A HREF="node36.html#hao">17</A>]. The very popular methodof <EM>false nearest neighbours</EM> [<A HREF="node36.html#FNN">18</A>] can be easily modified to yield ascalar quantity suitable for nonlinearity testing. The same is true for theconcept of <EM>unstable periodic orbits</EM> (UPOs) [<A HREF="node36.html#PM">19</A>, <A HREF="node36.html#soso">20</A>].<P><HR><A NAME="tex2html105" HREF="node5.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html103" HREF="node2.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html99" HREF="node3.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html106" HREF="node5.html">Surrogate data testing</A><B>Up:</B> <A NAME="tex2html104" HREF="node2.html">Detecting weak nonlinearity</A><B> Previous:</B> <A NAME="tex2html100" HREF="node3.html">Higher order statistics</A><P><ADDRESS><I>Thomas Schreiber <BR>Mon Aug 30 17:31:48 CEST 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
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