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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...
Detecting Nonlinearity in Experimental Data
 International Journal of Bifurcation and Chaos Submitted
, 1997
"... The technique of surrogate data has been used as a method to test for membership of particular classes of linear systems. We suggest an obvious extension of this to classes of nonlinear parametric models and demonstrate our methods with respiratory data from sleeping human infants. Although our data ..."
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Cited by 5 (5 self)
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The technique of surrogate data has been used as a method to test for membership of particular classes of linear systems. We suggest an obvious extension of this to classes of nonlinear parametric models and demonstrate our methods with respiratory data from sleeping human infants. Although our data are clearly distinct from the different classes of linear systems we are unable to distinguish between our data and surrogates generated by nonlinear models. Hence we conclude that human respiration is likely to be a nonlinear system with more than 2 degrees of freedom with a limit cycle that is driven by high dimensional dynamics or noise.
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...
Fractional Derivatives Applied to Phase Space Reconstructions 1
"... The concept and application of phasespace reconstructions are reviewed. Fractional derivatives are then proposed for the purpose of reconstructing dynamics from a single observed time history. A procedure is presented in which the fractional derivatives of time series data are obtained in the frequ ..."
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Cited by 1 (1 self)
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The concept and application of phasespace reconstructions are reviewed. Fractional derivatives are then proposed for the purpose of reconstructing dynamics from a single observed time history. A procedure is presented in which the fractional derivatives of time series data are obtained in the frequency domain. The method is applied to the Lorenz system. The ability of the method to unfold the data is assessed by the method of global false nearest neighbors. The reconstructed data is used to compute recurrences and correlation dimensions. The reconstruction is compared to the commonly used method of delays in order to assess the choice of reconstruction parameters, and also the quality of results. 1
Testing Time Series for Nonlinearity
, 1999
"... The technique of surrogate data analysis may be employed to test the hypothesis that an observed data set was generated by one of several specific classes of dynamical system. Current algorithms for surrogate data analysis enable one, in a generic way, to test for membership of the following thre ..."
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Cited by 1 (0 self)
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The technique of surrogate data analysis may be employed to test the hypothesis that an observed data set was generated by one of several specific classes of dynamical system. Current algorithms for surrogate data analysis enable one, in a generic way, to test for membership of the following three classes of dynamical system: (0) independent and identically distributed noise, (1) linearly filtered noise, and (2) a monotonic nonlinear transformation of linearly filtered noise. We show that one may apply statistics from nonlinear dynamical systems theory, in particular those derived from the correlation integral, as test statistics for the hypothesis that an observed time series is consistent with each of these three linear classes of dynamical system. Using statistics based on the correlation integral we show that it is also possible to test much broader (and not necessarily linear) hypotheses. We illustrate these methods with radial basis models and an algorithm to estimate t...
Linear and Nonlinear Dynamical Systems . . .
, 1996
"... This work presents a methodology for analyzing developmental and physiological time series from the perspective of dynamical systems. An overview of recent advances in nonlinear techniques for time series analysis is presented. Methods for generating a nonlinear dynamical systems analog to a covaria ..."
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This work presents a methodology for analyzing developmental and physiological time series from the perspective of dynamical systems. An overview of recent advances in nonlinear techniques for time series analysis is presented. Methods for generating a nonlinear dynamical systems analog to a covariance matrix are proposed. A novel application of structural equation modeling is proposed in which structural expectations can be fit to these nonlinear dependency matrices. A data set has been selected to demonstrate an application of some of these linear and nonlinear descriptive analyses, a suurogate data null hypothesis test, and nonlinear dependency analysis. The dynamical systems methods are evaluated in the light of (a) whether the techniques can be successfully applied to the example data and if so, (b) whether the results of these analyses provide insight into the processes under study which was not provided by other analyses.
On chaotic nature of speech signals
, 812
"... Various phonemes are considered in terms of nonlinear dynamics. Phase portraits of the signals in the embedded space, correlation dimension estimate and the largest Lyapunov exponent are analyzed. It is shown that the speech signals have comparatively small dimension and the positive largest Lyapuno ..."
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Various phonemes are considered in terms of nonlinear dynamics. Phase portraits of the signals in the embedded space, correlation dimension estimate and the largest Lyapunov exponent are analyzed. It is shown that the speech signals have comparatively small dimension and the positive largest Lyapunov exponent 1
Fractional Derivative Reconstruction of Forced Oscillators ∗
"... Fractional derivatives are applied in the reconstruction, from a single observable, of the dynamics of a Duffing oscillator and a twowell experiment. The fractional derivatives of time series data are obtained in the frequency domain. The derivative fraction is evaluated using the average mutual in ..."
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Fractional derivatives are applied in the reconstruction, from a single observable, of the dynamics of a Duffing oscillator and a twowell experiment. The fractional derivatives of time series data are obtained in the frequency domain. The derivative fraction is evaluated using the average mutual information between the observable and its fractional derivative. The ability of this reconstruction method to unfold the data is assessed by the method of global false nearest neighbors. The reconstructed data is used to compute recurrences and fractal dimensions. The reconstruction is compared to the true phase space and the delay reconstruction in order to assess the reconstruction parameters and the quality of results.