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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>Including non-stationarity</TITLE><META NAME="description" CONTENT="Including 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="tex2html307" HREF="node24.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html305" HREF="node22.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html299" HREF="node22.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html308" HREF="node24.html">Multivariate data</A><B>Up:</B> <A NAME="tex2html306" HREF="node22.html">Various Examples</A><B> Previous:</B> <A NAME="tex2html300" HREF="node22.html">Various Examples</A><BR> <P><H2><A NAME="SECTION00061000000000000000">Including non-stationarity</A></H2><A NAME="secincluding"> </A><P>Constrained randomisation using combinatorial minimisation is a very flexiblemethod since in principle arbitrary constraints can be realised. Although itis seldom possible to specify a formal null hypothesis for more generalconstraints, it can be quite useful to be able to incorporate into thesurrogates any feature of the data that is understood already or that isuninteresting. Non-stationarity has been excluded so far by requiring theequations defining the null hypothesis to remain constant in time. This has atwo-fold consequence. First, and most importantly, we must keep in mind thatthe test will have discrimination power against non-stationary signals as avalid alternative to the null hypothesis. Thus a rejection can be due tononlinearity or non-stationarity equally well.<P><blockquote><A NAME="964"> </A><IMG WIDTH=359 HEIGHT=287 ALIGN=BOTTOM ALT="figure1079" SRC="img131.gif"><BR><STRONG>Figure:</STRONG> Non-stationary financial time series (BUND Future returns, top) and a surrogate (bottom) preserving the non-stationary structure quantified by running window estimates of the local mean and variance (middle).<A NAME="figbund"> </A><BR></blockquote>Second, if we do want to include non-stationarity in the null hypothesis wehave to do so explicitly. Let us illustrate how this can be done with anexample from finance. The time series consists of 1500 daily returns (until theend of 1996) of the <EM>BUND Future</EM>, a derived German financial instrument.The data were kindly provided by Thomas Schürmann, WGZ-Bank Düsseldorf. Ascan be seen in the upper panel of Fig. <A HREF="node23.html#figbund">13</A>, the sequence isnon-stationary in the sense that the local variance and to a lesser extent alsothe local mean undergo changes on a time scale that is long compared to thefluctuations of the series itself. This property is known in the statisticalliterature as <EM>heteroscedasticity</EM> and modelled by the so-calledGARCH [<A HREF="node36.html#garch">40</A>] and related models. Here, we want to avoid the constructionof an explicit model from the data but rather ask the question if the data iscompatible with the null hypothesis of a correlated linear stochastic processwith time dependent local mean and variance. We can answer this question in astatistical sense by creating surrogate time series that show the same linearcorrelations and the same time dependence of the running mean and runningvariance as the data and comparing a nonlinear statistic between data andsurrogates. The lower panel in Fig. <A HREF="node23.html#figbund">13</A> shows a surrogate timeseries generated using the annealing method. The cost function was set up tomatch the autocorrelation function up to five days and the moving mean andvariance in sliding windows of 100 days duration. In Fig. <A HREF="node23.html#figbund">13</A> therunning mean and variance are shown as points and error bars, respectively, inthe middle trace. The deviation of these between data and surrogate has beenminimised to such a degree that it can no longer be resolved. A comparison ofthe time-asymmetry statistic Eq.(<A HREF="node3.html#eqskew">3</A>) for the data and 19 surrogatesdid not reveal any discrepancy, and the null hypothesis could not be rejected.<P><HR><A NAME="tex2html307" HREF="node24.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html305" HREF="node22.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html299" HREF="node22.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html308" HREF="node24.html">Multivariate data</A><B>Up:</B> <A NAME="tex2html306" HREF="node22.html">Various Examples</A><B> Previous:</B> <A NAME="tex2html300" HREF="node22.html">Various Examples</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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