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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>Questions of interpretation</TITLE><META NAME="description" CONTENT="Questions of interpretation"><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="tex2html345" HREF="node28.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html343" HREF="Surrogates.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html337" HREF="node26.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html346" HREF="node28.html">Non-dynamic nonlinearity</A><B>Up:</B> <A NAME="tex2html344" HREF="Surrogates.html">Surrogate time series</A><B> Previous:</B> <A NAME="tex2html338" HREF="node26.html">Spike trains</A><BR> <P><H1><A NAME="SECTION00070000000000000000">Questions of interpretation</A></H1><A NAME="secinterpret"> </A>Having set up all the ingredients for a statistical hypothesis test ofnonlinearity, we may ask what we can learn from the outcome of such a test.The formal answer is of course that we have, or have not, rejected a specifichypothesis at a given level of significance. How interesting this informationis, however, depends on the null hypothesis we have chosen. The test is mostmeaningful if the null hypothesis is plausible enough so that we are preparedto believe it in the lack of evidence against it. If this is not the case, wemay be tempted to go beyond what is justified by the test in ourinterpretation. Take as a simple example a recording of hormone concentrationin a human. We can test for the null hypothesis of a stationary Gaussian linearrandom process by comparing the data to phase randomised Fourier surrogates.Without any test, we know that the hypothesis cannot be true since hormoneconcentration, unlike Gaussian variates, is strictly non-negative. If wefailed to reject the null hypothesis by a statistical argument, we willtherefore go ahead and reject it anyway by common sense, and the test waspointless. If we did reject the null hypothesis by finding a coarse-grained``dimension'' which is significantly lower in the data than in the surrogates,the result formally does not give any new information but we might be temptedto speculate on the possible interpretation of the ``nonlinearity'' detected.<P>This example is maybe too obvious, it was meant only to illustrate that thehypothesis we test against is often not what we would actually accept to betrue. Other, less obvious and more common, examples include signals which areknown (or found by inspection) to be non-stationary (which is not covered bymost null hypotheses), or signals which are likely to be measured in a staticnonlinear, but non-invertible way. Before we discuss these two specific caveatsin some more detail, let us illustrate the delicacy of these questions with areal data example.<P><blockquote><A NAME="977"> </A><IMG WIDTH=360 HEIGHT=252 ALIGN=BOTTOM ALT="figure1094" SRC="img175.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figseizure"> </A>Intracranial neuronal potential recording during an epileptic seizure (upper) and a surrogate data set with the same linear correlations and the same amplitude distribution (lower). The data was kindly provided by K. Lehnertz and C. Elger.<BR></blockquote><P><P><blockquote><A NAME="979"> </A><IMG WIDTH=328 HEIGHT=148 ALIGN=BOTTOM ALT="figure1095" SRC="img176.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figbarcode"> </A> Surrogate data test for the data shown in Fig.<A HREF="node27.html#figseizure">18</A>. Since the prediction error is lower for the data (longer line) than for 99 surrogates (shorter lines), the null hypothesis may be rejected at the 99% level of significance. The error bar indicates the mean and standard deviation of the statistic computed for the surrogates.<BR></blockquote><P><P>Figure <A HREF="node27.html#figseizure">18</A> shows as an intra-cranial recording of the neuronalelectric field during an epileptic seizure, together with one iterativelygenerated surrogate data set [<A HREF="node36.html#surrowe">30</A>] that has the same amplitudedistribution and the same linear correlations or frequency content as the data.We have eliminated the end-point mismatch by truncating the series to 1875samples. A test was scheduled at the 99% level of significance, usingnonlinear prediction errors (see Eq.(<A HREF="node4.html#eqerror">5</A>), <I>m</I>=3, <IMG WIDTH=37 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline2416" SRC="img177.gif">,<IMG WIDTH=47 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline2418" SRC="img178.gif">) as a discriminating statistics. The nonlinear correlations weare looking for should enhance predictability and we can thus perform aone-sided test for a significantly <EM>smaller</EM> error. In a test with one dataset and 99 surrogates, the likelihood that the data would yield the smallesterror by mere coincidence is exactly 1 in 100. Indeed, as can be seen inFig. <A HREF="node27.html#figbarcode">19</A>, the test just rejects the null hypothesis.<P><P><blockquote><A NAME="981"> </A><IMG WIDTH=356 HEIGHT=201 ALIGN=BOTTOM ALT="figure1096" SRC="img179.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figseizdel"> </A> Left: Same data set as in Fig. <A HREF="node27.html#figseizure">18</A>, rendered in time delay coordinates. Right: A surrogate data set plotted in the same way.<BR></blockquote><P><P>Unfortunately, the test itself does not give any guidance as to what kind ofnonlinearity is present and we have to face notoriously ill-defined questionslike what is the most <EM>natural</EM> interpretation. Similar spike-and-wavedynamics as in the present example has been previously reported [<A HREF="node36.html#FEEG">47</A>] aschaotic, but these findings have been questioned [<A HREF="node36.html#TEEG">48</A>].Hernández and coworkers [<A HREF="node36.html#cuba">49</A>] have suggested a stochastic limit cycleas a simple way of generating spike-and-wave-like dynamics.<P>If we represent the data in time delay coordinates -- which is what we wouldusually do with chaotic systems -- the nonlinearity is reflected by the``hole'' in the centre (left panel in Fig. <A HREF="node27.html#figseizdel">20</A>). A linearstochastic process could equally well show oscillations, but its amplitudewould fluctuate in a different way, as we can see in the right panel of thesame figure for an iso-spectral surrogate. It is difficult to answer thequestion if the nonlinearity could have been generated by a static mechanismlike the measurement process (beyond the invertible rescaling allowed by thenull hypothesis). Deterministic chaos in a narrower sense seemsrather unlikely if we regard the prediction errors shown inFig. <A HREF="node27.html#figbarcode">19</A>: Although significantly lower than that of thesurrogates, the absolute value of the nonlinear prediction error is still morethan 50% of the rms amplitude of the data (which had been rescaled to unitvariance). Not surprisingly, the correlation integral (not shown here) does notshow any proper scaling region either. Thus, all we can hand back to theclinical researchers is a solid statistical result but the insight into whatprocess <EM>is</EM> generating the oscillations is limited.<P>A recent suggestion for surrogates for the validation of <EM>unstable periodicorbits</EM> (UPOs) may serve as an example for the difficulty in interpretingresults for more fancy null hypothesis. Dolan and coworkers [<A HREF="node36.html#witt">24</A>]coarse-grain amplitude adjusted data in order to extract a transfer matrix thatcan be iterated to yield typical realisations of a Markov chain.<A NAME="tex2html25" HREF="footnode.html#982"><IMG ALIGN=BOTTOM ALT="gif" SRC="foot_motif.gif"></A>The rationale there is to test if the finding of a certain number of UPOs couldbe coincidental, that is, not generated by dynamical structure. Testingagainst an order <I>D</I> Markov model removes dynamical structure beyond the``attractor shape'' (AS) in <I>D</I>+1 dimensions. It is not clear to us what theinterpretation of such a test would be. In the case of a rejection, they wouldinfer a dynamical nature of the UPOs found. But that would most probably meanthat in some higher dimensional space, the dynamics could be successfullyapproximated by a Markov chain acting on a sufficiently fine mesh. This is atleast true for finite dimensional dynamical systems. In other words, we cannotsee what sort of dynamical structure would generate UPOs but not show itssignature in some higher order Markov approximation.<P><BR> <HR><UL><A NAME="CHILD_LINKS"> </A><LI> <A NAME="tex2html347" HREF="node28.html#SECTION00071000000000000000">Non-dynamic nonlinearity</A><LI> <A NAME="tex2html348" HREF="node29.html#SECTION00072000000000000000">Non-stationarity</A></UL><HR><A NAME="tex2html345" HREF="node28.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html343" HREF="Surrogates.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html337" HREF="node26.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html346" HREF="node28.html">Non-dynamic nonlinearity</A><B>Up:</B> <A NAME="tex2html344" HREF="Surrogates.html">Surrogate time series</A><B> Previous:</B> <A NAME="tex2html338" HREF="node26.html">Spike trains</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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