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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 39 (11 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 7 (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 4 (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 the d ..."
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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
 Physica D
, 1999
"... . We present three distinct concepts of what constitutes a scienti. ..c understanding of a dynamical system . The development of each of these paradigms has resulted in a signi...cant expansion in the kind of system that can be investigated. In particular, the recentlydeveloped "algorithm ic mod ..."
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Cited by 1 (0 self)
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. We present three distinct concepts of what constitutes a scienti. ..c understanding of a dynamical system . The development of each of these paradigms has resulted in a signi...cant expansion in the kind of system that can be investigated. In particular, the recentlydeveloped "algorithm ic modelling paradigm" has allowed us to enlarge the domain of discourse to include complex realworld processes that cannot be necessarily be described by conventional dierential equations. 1. Introduction What do we mean when we say that we understand a dynamical system? In this essay, we identify three distinct paradigms for scienti...c understanding of dynamical systems. These paradigms are the models of knowing of the title. The introduction of new models of knowing has resulted in a signi...cant expansion in the kinds of systems that can be investigated scienti...cally. The ...rst paradigm, which we shall refer to as the Newtonian 1 , was established in the seventeenth century. Accor...
Natural Trees  NeighbourhoodLocation in a Nut Shell
"... We present the notion of a natural tree as an efficient method for storing spatial information for quick access. A natural tree is a representation of spatial adjacency, organised to allow efficient addition of new data, access to existing data, or deletions. The nodes of a natural tree are compound ..."
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We present the notion of a natural tree as an efficient method for storing spatial information for quick access. A natural tree is a representation of spatial adjacency, organised to allow efficient addition of new data, access to existing data, or deletions. The nodes of a natural tree are compound elements obtained by a particular Delaunay triangulation algorithm. Improvements to that algorithm allow both the construction of the triangulation and subsequent access to neighbourhood information to be O(N log N ). Applications include geographical information systems, contouring, and dynamical systems reconstruction. 1. Introduction The operation of neighbourhoodlocation is central to the working of geographical information systems (GIS) as Okabe et al. (1994) have emphasized. The growing importance of GIS and the increasing sizes of their associated databases (Milne et al. 1993) implies that the efficiency of neighbourhood location will become a crucial factor. In this paper we intr...
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.
Using a Fuzzy Piecewise Regression Analysis to Predict the Nonlinear TimeSeries of Turbulent Flows with Automatic ChangePoint Detection
, 2001
"... Abstract. Research has already shown that turbulent flow consists of some coherent time and spaceorganized vortical structures. Some dynamic systems and experimental models are employed to understand the turbulent generation mechanism. However, these approaches still cannot provide a good nonlinea ..."
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Abstract. Research has already shown that turbulent flow consists of some coherent time and spaceorganized vortical structures. Some dynamic systems and experimental models are employed to understand the turbulent generation mechanism. However, these approaches still cannot provide a good nonlinear analysis of turbulent timeseries. In the real turbulent flow, very complicated nonlinear behaviors, which are affected by many vague factors are present. Based on the nonlinear behavior and the results of from this traditional research, we introduce multivariate statistical analysis of an experimental study to explain practical phenomenon. In this paper, a new approach of fuzzy piecewise regression analysis with automatic changepoint detection is proposed to predict the nonlinear timeseries of turbulent flows. In order to show the practicality and usefulness of this model, we present an example of predicting the nearwall turbulence timeseries as a verifiable model. The results of practical applications show that the proposed method is appropriate and appears to be useful in nonlinear analysis and in fuzzy environments to predict the turbulence timeseries.