📄 node18.html
字号:
<!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>Simple nonlinear prediction</TITLE><META NAME="description" CONTENT="Simple nonlinear 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="tex2html255" HREF="node19.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html253" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html247" HREF="node17.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html256" HREF="node19.html">Finding unstable periodic orbits</A><B>Up:</B> <A NAME="tex2html254" HREF="node16.html">Nonlinear prediction</A><B> Previous:</B> <A NAME="tex2html248" HREF="node17.html">Model validation</A><BR> <P><H2><A NAME="SECTION00052000000000000000">Simple nonlinear prediction</A></H2><A NAME="seczeroth"> </A>Conventional linear prediction schemes average over all locations in phasespace when they extract the correlations they exploit for predictability.Tong [<A HREF="citation.html#TAR">42</A>] promoted an extension that fits different linear models if thecurrent state is below or above a given threshold (TAR, <B>T</B>hreshold <B>A</B>utoregressive <B>M</B>odel). If we expect more than a slight nonlinearcomponent to be present, it is preferable to make the approximation as local inphase space as possible. There have been many similar suggestions in theliterature how to exploit local structure, seee.g. [<A HREF="citation.html#Arkady">43</A>, <A HREF="citation.html#sugimay">44</A>, <A HREF="citation.html#Eckmann">45</A>, <A HREF="citation.html#fsid0">46</A>]. The simplest approach is to make theapproximation local but only keep the zeroth order, that is, approximate thedynamics locally by a constant. In the TISEAN package we include such arobust and simple method: In a delay embedding space, all neighbors of <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6691" SRC="img38.gif">are saught, if we want to predict the measurements at time <I>n</I>+<I>k</I>. The forecastis then simply<BR><A NAME="eqpredictlazy"> </A><IMG WIDTH=500 HEIGHT=44 ALIGN=BOTTOM ALT="equation4804" SRC="img48.gif"><BR>i.e. the average over the ``futures'' of the neighbors. The average forecasterrors obtained with the routine <ahref="../docs_c/zeroth.html">zeroth</a> (<a href="../docs_f/predict.html">predict</a> would give similarresults) for the laser output data used in Fig. <A HREF="node11.html#figdelayPoincare"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> as afunction of the number <I>k</I> of steps ahead the predictions are made is shown inFig. <A HREF="node18.html#figpredictINOzero"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>. One can also iterate the predictions by usingthe time series as a data base.<P><P><blockquote><A NAME="4798"> </A><IMG WIDTH=345 HEIGHT=247 ALIGN=BOTTOM ALT="figure657" SRC="img47.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figpredictINOzero"> </A> Predictions <I>k</I> time steps ahead (no iterated predictions) using the program <a href="../docs_c/zeroth.html">zeroth</a>. Top (red) curve: embedding dimension two is insufficient, since these flow data fill a (2+<IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif">)-dimensional attractor. Second from top (blue): Although embedding dimension four should in theory be a good embedding, <IMG WIDTH=37 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline6837" SRC="img46.gif"> suppresses structure perpendicular to the diagonal so that the predictions are as bad as in <I>m</I>=2! Lower (green, pink) curves: <I>m</I>=3 and 4 with a delay of about 4-8 time units serve well. <BR></blockquote><P>Apart from the embedding parameters, all that has to be specified for zerothorder predictions is the size of the neighborhoods. Since the diffusivemotion below the noise level cannot be predicted anyway, it makes sense toselect neighborhoods which are at least as large as the noise level, maybetwo or three times larger. For fairly clean time series, this guideline mayresult in neighborhoods with very few points. Therefore <ahref="../docs_c/zeroth.html">zeroth</a> alsopermits to specify the minimal number of neighbors to base the predictionson.<P>A relevant modification of this method is to extend the neighborhood <IMG WIDTH=11 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline6891" SRC="img49.gif"> to infinity, but to introduce a distance dependent weight,<BR><A NAME="eqpredictkernel"> </A><IMG WIDTH=500 HEIGHT=44 ALIGN=BOTTOM ALT="equation4809" SRC="img50.gif"><BR>where <I>w</I> is called the kernel. For <IMG WIDTH=111 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6895" SRC="img51.gif"> where <IMG WIDTH=11 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline6897" SRC="img52.gif"> isthe Heaviside step function, we return to Eq.(<A HREF="node18.html#eqpredictlazy"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>).<P><HR><A NAME="tex2html255" HREF="node19.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html253" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html247" HREF="node17.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html256" HREF="node19.html">Finding unstable periodic orbits</A><B>Up:</B> <A NAME="tex2html254" HREF="node16.html">Nonlinear prediction</A><B> Previous:</B> <A NAME="tex2html248" HREF="node17.html">Model validation</A><P><ADDRESS><I>Thomas Schreiber <BR>Wed Jan 6 15:38:27 CET 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -