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Are there any techniques that do not dependent on the choice of the variable ?

Christophe LETELLIER
16/09/2009
Challenge for an open question (to complete)

Since Takens’ theorem, it is known that it is possible to reconstruct a phase portrait from the measurement of a single scalar time series. Such a phase portrait is diffeomorphically equivalent to the original phase portrait when the dimension of the reconstructed space is sufficiently large (typically 2D+1 where D is ideally the Hausdorff dimension). In general, this is what is retained from Takens’ theorem. But there is an important assumption : the measurement function has to be generic. This is not necessarily true in every practical situation, particularly when there are symmetry properties. Moreover, the measured time series is always limited in various ways : data length, accuracy and resolution. As a consequence, many analyses provide results that are variable dependent, that is, they would not provide the same answer if performed using different variables of the system. This is more or less always implicitly taken into account in the literature, but without any explanation.

Therefore, it would be helpful to have tools to help practitioners understand and interpret such limitations. One possible way of ranking the different variables of a dynamical system is based on concepts adapted from observability theory. Observability is directly related to the quality of the phase portrait that may be reconstructed from the considered variable and it can be quantified by coefficients derived from concepts inherited from control theory [1] [2]. This means that, in practice, some variables can turn out to be much more convenient for analysis than others, a fact which is not perceived from Takens’ theorem. It is hoped that other tools be developed to highlight and quantify this phenomenon.

For instance, it is very often written that dynamical invariants are not affected by the choice of the variable used to reconstruct the phase space. But this is not necessarily true since Lefranc and his co-workers expressed a dependence on the observable in the reliability of dimension estimation . During the last decade over which the observability coefficients were introduced and used in the context of nonlinear dynamics, it was shown that global modeling was very sensitive to the choice of the variables. For instance, it is easy to obtain a global model from variable y of the Rössler system but nearly impossible from its variable z. It is more or less expected that synchronization and control are also dependent on the choice of the variable. It is also known that recurrence plots are not the same when computed from different continuous time series [3]. Such a feature is every now and then implicitly mentioned, sometimes reported but quite rarely explained. It is therefore an open question as to how can such an “unexpected performance” be understood and explained.

The main objective of this controversy topic is to try to establish which techniques are robust to the choice of the observable – if there are any – and how this dependence could be avoided, either in preprocessing the time series or by modifying the technique itself. Contributors are therefore invited to test whether their preferred techniques provide results that are dependent on the choice of observable in the context of some of the systems proposed in this “controversy topic”. It should be clear that the techniques tested are not only techniques for analysing (characterizing) the underlying dynamics but also techniques for controlling, synchronizing, etc. some dynamical systems. Thus, contributors are invited to clearly explain why they use one variable rather than another in their investigation.

Chaos would invite submission that address one or more of the following questions :

- Is every technique dependent on the choice of the observable ?
- In case of an affirmative answer, how can such a feature be explained ?
- How could the technique be modified to avoid such dependence ?
- Would it be possible to pre-process the time series to eliminate (reduce) such dependence ?
- In which ways can the observability of a system be quantified ?
- Can observability be assessed from data only ?
- It is known that a multivariate analysis is somewhat more complicated than the scalar counterpart, mainly because there is a large choice of possible embeddings. Is it possible to identify which subset of variables is more appropriate to investigate the underlying dynamics ?

Using the technique of their choice, the contributors will show whether their results are dependent on the choice of the variable(s) used or not. It should be clear that the invitation to address the present problem is not a competition but rather a call to investigate and report, in an explicit way, the dependence of results on the variable(s) used. There are many possible avenues by which to address this problem : dynamics modelling and characterization, synchronization, control, to mention a few. This open question is mainly concerned with data analysis ; however contributions that would address this problem in terms of more theoretical issues (e.g. alternative ways of rewriting the system) are also welcome. This is a particularly important question for developing control loops. In addressing these questions, contributors should be sure to give operational definitions of the terms used. The contributors should not attempt a lengthy review of the literature, but if the conclusions disagree with conclusions reached by others, they can feel free to pinpoint the sources of the discrepancy.

A set of different systems with a time series for each of its variables could be provided by the organizers along with a ranking of such variables from an observability point of view. This would serve as a benchmark to new ways of quantifying variable-dependence. A possible set of systems is :

  • A Chua system (experimental data from a system with an inversion symmetry)

Contributors would be invited to mainly use the provided systems but other cases are also welcome. Our purpose is to investigate and document the way in which analysis may depend on the choice of variables by inviting contributors to explicitly address this question with their preferred techniques. Indeed, we suspect that most of the data analyses are variable dependent.

Christophe Letellier (University of Rouen, France)
Luis A. Aguirre (Universidade Federal de Minas Gerais, Brazil)

[1] C. Letellier, J. Maquet, L. Le Sceller, G. Gouesbet & L. A. Aguirre, On the non-equivalence of observables in phase space reconstructions from recorded time series, Journal of Physics A, 31, 7913-7927, 1998.

[2] C. Letellier & L. Aguirre, Investigating nonlinear dynamics from time series : the influence of symmetries and the choice of observables, Chaos, 12, 549-558, 2002.

[3] C. Letellier, Estimating the Shannon entropy : recurrence plots versus symbolic dynamics, Physical Review Letters, 96, 254102, 2006.

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