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12
On Selecting Models for Nonlinear Time Series
 Physica D
, 1995
"... Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintainin ..."
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Cited by 49 (14 self)
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Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintaining good robustness against noise. The models that are built are close to the simplest possible according to a description length criterion. The method will deliver a linear model if that has shorter description length than a nonlinear model. We show how our models can be used for prediction, smoothing and interpolation in the usual way. We also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series. 1 The Model Selection Problem As our understanding of chaotic and other nonlinear phenomena has grown, it has become apparent that linear models are inadequate to model most dynamical processes. Nevertheless, linear models...
Estimating Physical Invariant Measures and Space Averages of Dynamical Systems Indicators
, 1996
"... We consider discrete, differentiable dynamical systems T : M \Gamma \Delta oe where M is a smooth ddimensional manifold embedded in Euclidean space, and shall be concerned with ergodic averages of realvalued functions g : M ! R. Such averages may be performed by arithmetically averaging g along ..."
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Cited by 8 (2 self)
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We consider discrete, differentiable dynamical systems T : M \Gamma \Delta oe where M is a smooth ddimensional manifold embedded in Euclidean space, and shall be concerned with ergodic averages of realvalued functions g : M ! R. Such averages may be performed by arithmetically averaging g along an infinitely long single orbit (time averaging) or by integrating g with respect to an ergodic invariant measure (space averaging). We are particularly interested in the situation where these two methods yield identical answers for a large number of orbits, as in this situation the invariant measure has some physical significance. A dynamical indicator that arises as an ergodic average are the Lyapunov exponents of T . These quantities describe asymptotic rates of local stretching (or contraction) of phase space under T . Chapter 1 of this thesis describes in detail a new method of computing Lyapunov exponents from either an experimental set of data or a known map T , using a spatial aver...
Constructing Invariant Measures from Data
, 1995
"... We present a method of approximating an invariant measure of a dynamical system from a finite set of experimental data. Our reconstruction technique automatically provides us with a partition of phase space, and we assign each set in the partition a certain weight. By refining the partition, we may ..."
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Cited by 5 (3 self)
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We present a method of approximating an invariant measure of a dynamical system from a finite set of experimental data. Our reconstruction technique automatically provides us with a partition of phase space, and we assign each set in the partition a certain weight. By refining the partition, we may make our approximation to an invariant measure of the reconstructed system as accurate as we wish. Our method provides us with both a singular and an absolutely continuous approximation, so that the most suitable representation may be chosen for a particular problem. 1 Introduction Let M be a finitedimensional smooth compact manifold and f : M ! M a C 2 map. The map f describes a discrete dynamical system on M . We assume that the orbits of f eventually lie near some limiting set ae M . Whether is an attractor strictly contained in M , or is the whole of M , we do not care. A further assumption is that the system is transitive; this is to give some reason for hoping that almost all orb...
Lyapunov Exponents and Triangulations
, 1993
"... this paper, all systems will be assumed to be ergodic. We will be interested in a version of Oseledec's Multiplicative Ergodic Theorem [1, 2] for differentiable maps. There are two 2 versions, one for general differentiable maps and one for diffeomorphisms. To avoid confusion, we present only ..."
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Cited by 3 (1 self)
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this paper, all systems will be assumed to be ergodic. We will be interested in a version of Oseledec's Multiplicative Ergodic Theorem [1, 2] for differentiable maps. There are two 2 versions, one for general differentiable maps and one for diffeomorphisms. To avoid confusion, we present only the diffeomorphic case which has stronger results.
Models of knowing and the investigation of dynamical systems
 for the RBF1 and L 10 Predictions of the NMR laser
, 1999
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Physica D 164 (2002) 187–201 Applying the method of surrogate data to cyclic time series
, 2001
"... The surrogate data methodology is used to test a given time series for membership of specific classes of dynamical systems. Currently, there are three algorithms that are widely applied in the literature. The most general of these tests the hypothesis of nonlinearly scaled linearly filtered noise. H ..."
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The surrogate data methodology is used to test a given time series for membership of specific classes of dynamical systems. Currently, there are three algorithms that are widely applied in the literature. The most general of these tests the hypothesis of nonlinearly scaled linearly filtered noise. However, these tests and the many extensions of them that have been suggested are inappropriate for data exhibiting strong cyclic components. For such data it is more natural to ask if there exist any long term (of period longer than the data cycle length) determinism. In this paper we discuss existing techniques that attempt to address this hypothesis and introduce a new approach. This new approach generates surrogates that are constrained (i.e., they look like the data) and for cyclic time series tests the null hypothesis of a periodic orbit with uncorrelated noise. We examine various alternative implementations of this algorithm, applying it to a variety of known test systems and experimental time
Physica D 194 (2004) 283–296 Optimal embedding parameters: a modelling paradigm
, 2004
"... The reconstruction of a dynamical system from a time series requires the selection of two parameters: the embedding dimension de and the embedding lag τ. Many competing criteria to select these parameters exist, and all are heuristic. Within the context of modelling the evolution operator of the und ..."
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The reconstruction of a dynamical system from a time series requires the selection of two parameters: the embedding dimension de and the embedding lag τ. Many competing criteria to select these parameters exist, and all are heuristic. Within the context of modelling the evolution operator of the underlying dynamical system, we show that one only need be concerned with the product deτ. We introduce an information theoretic criterion for the optimal selection of the embedding window dw = deτ. For infinitely long time series, this method is equivalent to selecting the embedding lag that minimises the nonlinear model prediction error. For short and noisy time series, we find that the results of this new algorithm are datadependent and are superior to estimation of embedding parameters with the standard techniques.
Triangulating Noisy Dynamical Systems
"... Triangulation and tesselation methods have proven successful for reconstructing dynamical systems which have approximately the same dynamics as given trajectories or time series. Such reconstructions can produce other trajectories with similar dynamics, so giving a system on which one can conduct ..."
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Triangulation and tesselation methods have proven successful for reconstructing dynamical systems which have approximately the same dynamics as given trajectories or time series. Such reconstructions can produce other trajectories with similar dynamics, so giving a system on which one can conduct experiments, but can also be used to locate equilibria and determine their types, carry out bifurcation studies, estimate state manifolds and so on. In the past, reconstruction by triangulation and tesselation methods has been restricted to lownoise or nonoise cases. This paper shows how to construct triangulation models for noisy systems; the models can then be used in the same ways as models of noisefree systems. 1 Introduction An important part of controlling systems is modeling them. The system identification and modeling problem is wellstudied for linear systems, but there seem to be few results for strongly nonlinear systems. In this paper we present an approach that gener...
Physica D 132 (1999) 133–149 Models of knowing and the investigation of dynamical systems
"... We present three distinct concepts of what constitutes a scientific understanding of a dynamical system. The development of each of these paradigms has resulted in a significant expansion in the kind of system that can be investigated. In particular, the recentlydeveloped ‘algorithmic modelling par ..."
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We present three distinct concepts of what constitutes a scientific understanding of a dynamical system. The development of each of these paradigms has resulted in a significant expansion in the kind of system that can be investigated. In particular, the recentlydeveloped ‘algorithmic modelling paradigm ’ has allowed us to enlarge the domain of discourse to include complex realworld processes that cannot necessarily be described by conventional differential equations. ©1999 Elsevier Science B.V. All rights reserved.