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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.2//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>Locally linear prediction</TITLE><META NAME="description" CONTENT="Locally linear prediction"><META NAME="keywords" CONTENT="TiseanHTML"><META NAME="resource-type" CONTENT="document"><META NAME="distribution" CONTENT="global"><LINK REL=STYLESHEET HREF="TiseanHTML.css"></HEAD><BODY bgcolor=ffffff LANG="EN" > <A NAME="tex2html275" HREF="node21.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html273" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html267" HREF="node19.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html276" HREF="node21.html">Global function fits</A><B>Up:</B> <A NAME="tex2html274" HREF="node16.html">Nonlinear prediction</A><B> Previous:</B> <A NAME="tex2html268" HREF="node19.html">Finding unstable periodic orbits</A><BR> <P><H2><A NAME="SECTION00054000000000000000">Locally linear prediction</A></H2><P>If there is a good reason to assume that the relation <IMG WIDTH=89 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6971" SRC="img59.gif"> isfulfilled by the experimental data in good approximation (say, within 5%) forsome unknown <I>f</I> and that <I>f</I> is smooth, predictions can be improved byfitting local linear models. They can be considered as the local Taylorexpansion of the unknown <I>f</I>, and are easily determined by minimizing<BR><A NAME="eqpredictloclin1"> </A><IMG WIDTH=500 HEIGHT=37 ALIGN=BOTTOM ALT="equation4924" SRC="img60.gif"><BR>with respect to <IMG WIDTH=17 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6979" SRC="img61.gif"> and <IMG WIDTH=14 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline6981" SRC="img62.gif">, where <IMG WIDTH=18 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline6983" SRC="img63.gif"> is the<IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif">-neighborhood of <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6691" SRC="img38.gif">, excluding <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6691" SRC="img38.gif"> itself, as before. Then,the prediction is <IMG WIDTH=120 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline6991" SRC="img64.gif">. The minimization problem canbe solved through a set of coupled linear equations, a standard linear algebraproblem. This scheme is implemented in <a href="../docs_c/onestep.html">onestep</a>. For moderate noise levelsand time series lengths this can give a reasonable improvement over <a href="../docs_c/zeroth.html">zeroth</a>and <a href="../docs_f/predict.html">predict</a>. Moreover, as discussed in Sec.<A HREF="node26.html#seclyap"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>, these linear mapsare needed for the computation of the Lyapunov spectrum. Locally linearapproximation was introduced in [<A HREF="citation.html#Eckmann">45</A>, <A HREF="citation.html#fsid0">46</A>]. We should note that thestraight least squares solution of Eq.(<A HREF="node20.html#eqpredictloclin1"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) is not alwaysoptimal and a number of strategies are available to regularize the problem if the matrix becomes nearly singular and to remove the bias due to the errors inthe ``independent'' variables. These strategies have in common that anypossible improvement is bought with considerable complication of the procedure,requiring subtle parameter adjustments. We refer the reader toRefs. [<A HREF="citation.html#kugiLL">51</A>, <A HREF="citation.html#jaeger">52</A>] for advanced material.<P>In Fig. <A HREF="node20.html#figpredictINOnstep"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> we show iterated predictions of thePoincaré map data from the CO<IMG WIDTH=6 HEIGHT=11 ALIGN=MIDDLE ALT="tex2html_wrap_inline6701" SRC="img39.gif"> laser (Fig. <A HREF="node11.html#figdelayPoincare"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) in adelay representation (using <a href="../docs_c/nstep.html">nstep</a> in two dimensions). The resulting data donot only have the correct marginal distribution and power spectrum, but alsoform a perfect skeleton of the original noisy attractor. There areof course artefacts due to noise and the roughness of this approach,but there are good reasons to assume that the line-like substructurereflects fractality of the unperturbed system.<P><P><blockquote><A NAME="4961"> </A><IMG WIDTH=235 HEIGHT=223 ALIGN=BOTTOM ALT="figure836" SRC="img65.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figpredictINOnstep"> </A> Time delay representation of 5000 iterations of the local linear predictor <a href="../docs_c/nstep.html">nstep</a> in two dimensions, starting from the last delay vector of Fig. <A HREF="node11.html#figdelayPoincare"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>.<BR></blockquote><P><P><a name="casdagli"></a>Casdagli [<A HREF="citation.html#Casdagli_royal">53</A>] suggested to use local linear models as a test for nonlinearity: He computed the average forecast error as afunction of the neighborhood size on which the fit for <IMG WIDTH=17 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6979" SRC="img61.gif"> and <IMG WIDTH=14 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline6981" SRC="img62.gif"> isperformed. If the optimum occurs at large neighborhood sizes, the data are (inthis embedding space) best described by a linear stochastic process, whereas anoptimum at rather small sizes supports the idea of the existence of a nonlinearalmost deterministic equation of motion. This protocol is implemented in theroutine <a href="../docs_c/ll-ar.html">ll-ar</a>, see Fig. <A HREF="node20.html#figpredictcasdagli"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>.<P><P><blockquote><A NAME="5011"> </A><IMG WIDTH=334 HEIGHT=250 ALIGN=BOTTOM ALT="figure901" SRC="img66.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figpredictcasdagli"> </A> The Casdagli test for nonlinearity: The rms prediction error of local linear models as a function of the neighborhood size <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif">. Lower (green) curve: The CO<IMG WIDTH=6 HEIGHT=11 ALIGN=MIDDLE ALT="tex2html_wrap_inline6701" SRC="img39.gif"> laser data. These data are obviously highly deterministic in <I>m</I>=4 dimensions and with lag <IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6553" SRC="img16.gif">=6. Central (blue) curve: The breath rate data shown in Fig. <A HREF="node23.html#figb"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> with <I>m</I>=4 and <IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6553" SRC="img16.gif">=1. Determinism is weaker (presumably due to a much higher noise level), but still the nonlinear structure is dominant. Upper (red) curve: Numerically generated data of an AR(5) process, a linearly correlated random process (<I>m</I>=5, <IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6553" SRC="img16.gif">=1).<BR></blockquote><P><HR><A NAME="tex2html275" HREF="node21.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html273" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html267" HREF="node19.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html276" HREF="node21.html">Global function fits</A><B>Up:</B> <A NAME="tex2html274" HREF="node16.html">Nonlinear prediction</A><B> Previous:</B> <A NAME="tex2html268" HREF="node19.html">Finding unstable periodic orbits</A><P><ADDRESS><I>Thomas Schreiber <BR>Wed Jan 6 15:38:27 CET 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
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