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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>Global function fits</TITLE><META NAME="description" CONTENT="Global function fits"><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="tex2html283" HREF="node22.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html281" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html277" HREF="node20.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html284" HREF="node22.html">Nonlinear noise reduction</A><B>Up:</B> <A NAME="tex2html282" HREF="node16.html">Nonlinear prediction</A><B> Previous:</B> <A NAME="tex2html278" HREF="node20.html">Locally linear prediction</A><BR> <P><H2><A NAME="SECTION00055000000000000000">Global function fits</A></H2><P>The local linear fits are very flexible, but can go wrong on parts of thephase space where the points do not span the available space dimensions andwhere the inverse of the matrix involved in the solution of the minimizationdoes not exist. Moreover, very often a large set of different linear maps isunsatisfying. Therefore many authors suggested to fit global nonlinearfunctions to the data, i.e. to solve<BR><A NAME="eqpredictglobal"> </A><IMG WIDTH=500 HEIGHT=33 ALIGN=BOTTOM ALT="equation5013" SRC="img67.gif"><BR>where <IMG WIDTH=14 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline7063" SRC="img68.gif"> is now a nonlinear function in closed form with parameters <I>p</I>,with respect to which the minimization is done. Polynomials, radial basisfunctions, neural nets, orthogonal polynomials, and many other approaches havebeen used for this purpose. The results depend on how far the chosen ansatz<IMG WIDTH=14 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline7063" SRC="img68.gif"> is suited to model the unknown nonlinear function, and on how well thedata are deterministic at all. We included the routines <ahref="../docs_c/rbf.html">rbf</a> and <a href="../docs_c/polynom.html">polynom</a>in the TISEAN package, where <IMG WIDTH=14 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline7063" SRC="img68.gif"> is modeled by radial basisfunctions [<A HREF="citation.html#rbf">54</A>, <A HREF="citation.html#lenny_rbf">55</A>] and polynomials [<A HREF="citation.html#Casdagli_pred">56</A>],respectively. The advantage of these two models is that the parameters <I>p</I>occur linearly in the function <I>f</I> and can thus be determined by simple linear algebra, and the solution is unique. Both features are lost formodels where the parameters enter nonlinearly.<P>In order to make global nonlinear predictions, one has to supply the embeddingdimension and time delay as usual. Further, for <a href="../docs_c/polynom.html">polynom</a> the order of thepolynomial has to be given. The program returns the coefficients of the model.In <a href="../docs_c/rbf.html">rbf</a> one has to specify the number of basis functions to be distributedon the data. The width of the radial basis functions (Lorentzians in ourprogram) is another parameter, but since the minimization is so fast, theprogram runs many trial values and returns parameters for thebest. Figure <A HREF="node21.html#figpredictINOrbf"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> shows the result of a fit to the CO<IMG WIDTH=6 HEIGHT=11 ALIGN=MIDDLE ALT="tex2html_wrap_inline6701" SRC="img39.gif">laser time series (Fig. <A HREF="node11.html#figdelayPoincare"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) with radial basis functions.<P><P><blockquote><A NAME="5085"> </A><IMG WIDTH=235 HEIGHT=223 ALIGN=BOTTOM ALT="figure948" SRC="img69.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figpredictINOrbf"> </A> Attractor obtained by iterating the model that has been obtained by a fit with 40 radial basis functions in two dimensions to the time series shown in Fig. <A HREF="node11.html#figdelayPoincare"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>. Compare also Fig. <A HREF="node20.html#figpredictINOnstep"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>.<BR></blockquote><P><P>If global models are desired in order to infer the structure and properties ofthe underlying system, they should be tested by iterating them. The predictionerrors, although small in size, could be systematic and thus repel theiterated trajectory from the range where the original data are located. Itcan be useful to study a dependence of the size or the sign of the predictionerrors on the position in the embedding space, since systematic errors can bereduced by a different model. Global models are attractive because they yieldclosed expressions for the full dynamics. One must not forget, however, thatthese models describe the observed process only in regions of the space whichhave been visited by the data. Outside this area, the shape of the modeldepends exclusively on the chosen ansatz. In particular, polynomials divergeoutside the range of the data and hence can be unstable under iteration.<P><HR><A NAME="tex2html283" HREF="node22.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html281" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html277" HREF="node20.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html284" HREF="node22.html">Nonlinear noise reduction</A><B>Up:</B> <A NAME="tex2html282" HREF="node16.html">Nonlinear prediction</A><B> Previous:</B> <A NAME="tex2html278" HREF="node20.html">Locally linear prediction</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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