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<H2><A NAME="SECTION00055000000000000000">Global function fits</A></H2>
<P>
The local linear fits are very flexible, but can go wrong on parts of the
phase space where the points do not span the available space dimensions and
where the inverse of the matrix involved in the solution of the minimization
does not exist.  Moreover, very often a large set of different linear maps is
unsatisfying. Therefore many authors suggested to fit global nonlinear
functions to the data, i.e. to solve
<BR><A NAME="eqpredictglobal">&#160;</A><IMG WIDTH=500 HEIGHT=33 ALIGN=BOTTOM ALT="equation5013" SRC="img67.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img67.gif"><BR>
where <IMG WIDTH=14 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline7063" SRC="img68.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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 basis
functions, neural nets, orthogonal polynomials, and many other approaches have
been 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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img68.gif"> is suited to model the unknown nonlinear function, and on how well the
data are deterministic at all. We included the routines <a
href="../dresden/rbf.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/dresden/rbf.html">rbf</a> and <a href="../dresden/polynom.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/dresden/polynom.html">polynom</a>
in the TISEAN package, where <IMG WIDTH=14 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline7063" SRC="img68.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img68.gif"> is modeled by radial basis
functions&nbsp;[<A HREF="citation.html#rbf" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#rbf">54</A>, <A HREF="citation.html#lenny_rbf" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#lenny_rbf">55</A>] and polynomials&nbsp;[<A HREF="citation.html#Casdagli_pred" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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 for
models where the parameters enter nonlinearly.
<P>
In order to make global nonlinear predictions, one has to supply the embedding
dimension and time delay as usual. Further, for <a href="../dresden/polynom.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/dresden/polynom.html">polynom</a> the order of the
polynomial has to be given. The program returns the coefficients of the model.
In <a href="../dresden/rbf.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/dresden/rbf.html">rbf</a> one has to specify the number of basis functions to be distributed
on the data.  The width of the radial basis functions (Lorentzians in our
program) is another parameter, but since the minimization is so fast, the
program runs many trial values and returns parameters for the
best. Figure&nbsp;<A HREF="node21.html#figpredictINOrbf" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node21.html#figpredictINOrbf"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img39.gif">
laser time series (Fig.&nbsp;<A HREF="node11.html#figdelayPoincare" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node11.html#figdelayPoincare"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/cross_ref_motif.gif"></A>) with radial basis functions.
<P>
<P><blockquote><A NAME="5085">&#160;</A><IMG WIDTH=235 HEIGHT=223 ALIGN=BOTTOM ALT="figure948" SRC="img69.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img69.gif"><BR>
<STRONG>Figure:</STRONG> <A NAME="figpredictINOrbf">&#160;</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.&nbsp;<A HREF="node11.html#figdelayPoincare" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node11.html#figdelayPoincare"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/cross_ref_motif.gif"></A>. Compare also 
   Fig.&nbsp;<A HREF="node20.html#figpredictINOnstep" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node20.html#figpredictINOnstep"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/cross_ref_motif.gif"></A>.<BR>
</blockquote><P>
<P>
If global models are desired in order to infer the structure and properties of
the underlying system, they should be tested by iterating them. The prediction
errors, although small in size, could be systematic and thus repel the
iterated trajectory from the range where the original data are located.  It
can be useful to study a dependence of the size or the sign of the prediction
errors on the position in the embedding space, since systematic errors can be
reduced by a different model.  Global models are attractive because they yield
closed expressions for the full dynamics. One must not forget, however, that
these models describe the observed process only in regions of the space which
have been visited by the data. Outside this area, the shape of the model
depends exclusively on the chosen ansatz. In particular, polynomials diverge
outside the range of the data and hence can be unstable under iteration.
<P>
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Wed Jan  6 15:38:27 CET 1999</I>
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