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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>Non-stationarity</TITLE><META NAME="description" CONTENT="Non-stationarity"><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="tex2html365" HREF="node30.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html363" HREF="node27.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html359" HREF="node28.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html366" HREF="node30.html">Conclusions: Testing a Hypothesis </A><B>Up:</B> <A NAME="tex2html364" HREF="node27.html">Questions of interpretation</A><B> Previous:</B> <A NAME="tex2html360" HREF="node28.html">Non-dynamic nonlinearity</A><BR> <P><H2><A NAME="SECTION00072000000000000000">Non-stationarity</A></H2><A NAME="secnonstat"> </A>It is quite common in bio-medical time series (and elsewhere) that otherwiseharmless looking data once in a while are interrupted by a singular event, forexample a spike. It is now debatable whether such spikes can be generated by alinear process by nonlinear rescaling. We do not want to enter such adiscussion here but merely state that a time series that covers only one or afew such events is not suitable for the statistical study of the spikegeneration process. The best working assumption is that the spike comes in bysome external process, thus rendering the time series non-stationary. In anycase, the null hypotheses we are usually testing against are not likely togenerate such singular events autonomously. Thus, typically, a series with asingle spike will be found to violate the null hypothesis, but, arguably, thecause is non-stationarity rather than non-linearity. Let us discuss as a simpleexample the same AR(2) process considered previously, this time without anyrescaling. Only at a single instant, <I>n</I>=1900, the system is kicked by a largeimpulse instead of the Gaussian variate <IMG WIDTH=32 HEIGHT=16 ALIGN=MIDDLE ALT="tex2html_wrap_inline2440" SRC="img187.gif">. This impulse leads tothe formation of a rather large spike. Such a sequence is shown inFig. <A HREF="node28.html#figspike">23</A>. Note that due to the correlations in the process, thespike covers more than a single measurement.<P>When we generate surrogate data, the first observation we make is that it takesthe algorithm more than 400 iterations in order to converge to a reasonabletradeoff between the correct spectrum and the required distribution ofpoints. Nevertheless, the accuracy is quite good -- the spectrum is correctwithin 0.1% of the rms amplitude. Visual inspection of the lower panel ofFig. <A HREF="node28.html#figspike">23</A> shows that the spectral content -- and the assumed values-- during the single spike are represented in the surrogates by a large numberof shorter spikes. The surrogates cannot know of an external kick. The visualresult can be confirmed by a statistical test with several surrogates, equallywell (99% significance) by a time asymmetry statistic or a nonlinearprediction error.<P>If non-stationarity is known to be present, it is necessary to include it inthe null hypothesis explicitly. This is in general very difficult but can beundertaken in some well behaved cases. In Sec. <A HREF="node23.html#secincluding">6.1</A> we discussedthe simplest situation of a slow drift in the calibration of the data. It hasbeen shown empirically [<A HREF="node36.html#poster">52</A>] that a slow drift in system parameters isnot as harmful as expected [<A HREF="node36.html#Timmer2">53</A>]. It is possible to generatesurrogates for sliding windows and restrict the discriminating statistics toexclude the points at the window boundaries. It is quite obvious that specialcare has to be taken in such an analysis.<P><HR><A NAME="tex2html365" HREF="node30.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html363" HREF="node27.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html359" HREF="node28.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html366" HREF="node30.html">Conclusions: Testing a Hypothesis </A><B>Up:</B> <A NAME="tex2html364" HREF="node27.html">Questions of interpretation</A><B> Previous:</B> <A NAME="tex2html360" HREF="node28.html">Non-dynamic nonlinearity</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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