From: guiller@CICA.ES (Guillermo Heredia)
Newsgroups: bit.listserv.frac-l
Subject: RE: DOCUMENTS
Date: 6 Nov 92 20:24:00 GMT
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        Hello, Roby:

>For example, speaking about the search for determinism and
>chaos in time series, what do you think of the different
>methods to choose the two parameters:
>
>1) LAG TIME (tau)
>2) EMBEDDING DIMENSION (m)
>
>used in the time-delayed phase reconstruction (Takens method)
>
>I'm aware of the methods by Liebert, Pawelzik and Schuster, 1991
>the one of Matt Kennel et al. (false nearest neighbours), and
>the one by Fraser and Swinney (mutual information and similar
>things).
>
>Have some of you used and perhaps compared them?
>

        That is a very interesting question. I use a method
due to Th. Buzug anf G. Pfister. It is a dual method: it uses
two algorithms, the 'Fill Factor' and the 'Integral Local
Deformation' (ILD). And best of all, it can detect optimal
delay time AND embedding dimension.

        The basic idea of the fill factor method is to request
an optimum utilization of the embedding space yielding the
maximum separation of trajectories.

        The ILD method considers that points on neighboring
trajectories in the original phase space remain neighboring
points for small evolution times, and are only separated
according to a positive Lyapunov exponent in the chaotic case.
In bad reconstructions, trajectories can cross other
trajectories, and flows of contrary directions aproach each
other. That is what ILD measures.

        I have tested these algorithms on standard systems
(Lorenz, Rossler chaos, Rossler hyperchaos, ...) and works
well. It is relatively fast compared to the other methods.

        There are other methods, but I have not tested them. I
can tell you what Buzug and Pfister say about Factor / ILD .

        - Fraser and Swinney's 'mutual information': it is
VERY time consuming, and becomes impracticable for embedding
dimensions greater than four. Only determines delay time.

        - Liebert et al's 'wavering product': they say it is
very sensitive to noise. It determines both delay time and
embedding dimensions.

        I have also seen the 'false neighbours method' (Kennel
et al.) but I have not tested it. Only determines embedding
dimension.

Reference:

Th. Buzug and G. Pfister. Optimal delay time and embedding
dimension for delay-time coordinates by analysis of global
static and local dynamical behavior of strange attractors.
Phys. Rev. A, 45 7073 (15 May 1992)

        I am also working on Lyapunov exponents (Wolf method)
and fractal dimension (correlation dimension).

        I am very interested in your opinion of those methods.


        Guillermo Heredia
        guiller@cica.es
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU