Markov models from data by simple nonlinear time series predictors in delay embedding spaces

Ragwitz, Mario; Kantz, Holger

doi:10.1103/physreve.65.056201

Cited by 129 publications

(120 citation statements)

References 12 publications

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“…The dimension of the embedding was set using the Cao criterion [25], while the embedding delay time was set as the autocorrelation decay time. Other criteria to obtain embedding parameters, such as described in [19], provide similar results. Furthermore, each time series was time-delayed, so that they had maximal mutual information with the destination of the flow.…”

Section: Tests On Simulated and Experimental Datasupporting

confidence: 65%

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Assessing Coupling Dynamics from an Ensemble of Time Series

Gómez-Herrero

Rutanen

et al. 2015

Entropy

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Finding interdependency relations between time series provides valuable knowledge about the processes that generated the signals. Information theory sets a natural framework for important classes of statistical dependencies. However, a reliable estimation from information-theoretic functionals is hampered when the dependency to be assessed is brief or evolves in time. Here, we show that these limitations can be partly alleviated when we have access to an ensemble of independent repetitions of the time series. In particular, we gear a data-efficient estimator of probability densities to make use of the full structure of trial-based measures. By doing so, we can obtain time-resolved estimates for a family of entropy combinations (including mutual information, transfer entropy and their conditional counterparts), which are more accurate than the simple average of individual estimates over trials. We show with simulated and real data generated by coupled electronic circuits that Entropy 2015, 17 1959 the proposed approach allows one to recover the time-resolved dynamics of the coupling between different subsystems.

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Section: Tests On Simulated and Experimental Datasupporting

confidence: 65%

“…We consider three simultaneously measured time series generated from stochastic processes X, Y and Z, which can be approximated as stationary Markov processes [19] of finite order. The state space of X can then be reconstructed using the delay embedded vectors x(n) = (x(n), ..., x(n − d x + 1)) for n = 1, .…”

Section: Entropy Combinationsmentioning

confidence: 99%

Assessing Coupling Dynamics from an Ensemble of Time Series

Gómez-Herrero

Rutanen

et al. 2015

Entropy

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“…For a chaotic deterministic system the delay should be of the order of the first minimum of the time delayed mutual information [22]. For a dynamical system of Langevin type the delay should be smaller than this, depending on the noise level [17]. We use the delay that minimizes the mean prediction error.…”

Section: Resultsmentioning

confidence: 99%

“…A variety of tools have been developed for this case [16]. Recently, it has been shown that models originally proposed for deterministic chaotic systems also apply if the underlying dynamics is governed by a Markov process [17]. Let us outline first how phase space models are constructed for deterministic chaotic systems, and then we will show why these algorithms work for Markovian processes as well.…”

Section: Nonlinear Modelmentioning

confidence: 99%

Nonlinear determinism in time series measurements of two-dimensional turbulence

Ragwitz

Baroud

Plapp

et al. 2002

Physica D: Nonlinear Phenomena

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Experiments on two-dimensional and three-dimensional turbulent flows in a rotating annulus are analyzed using linear and nonlinear time series predictors. The models are used to predict the time series, a time S ahead and to calculate the velocity increment u S = u(s n + S) − u(s n ) between the current value of the time series s n and a point time S apart. For two-dimensional flow, the nonlinear model provides superior predictions to the linear model for u S positive and large. In contrast, for three-dimensional turbulence the prediction of the nonlinear model is no better than the linear model. For two-dimensional turbulence, the probability density functions for the predicted u S from the linear model have an exponential tail, while for the nonlinear model the tail exhibits power law decay. The scaling exponent of this power law can be explained using arguments of the Kolmogorov 1941 theory. Our findings contradict the common assumption that two-dimensional turbulence shares the unpredictability properties of three-dimensional turbulent flows.

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“…Such processes are called hidden Markov processes (HMP) [25]. An HMP can also be understood as a Markov process with noisy observations, providing a nonlinear stochastic approach that is more appropriate for signals with noisy components [26].…”

Section: Hmm-based Entropy Measuresmentioning

confidence: 99%

Entropies from Markov Models as Complexity Measures of Embedded Attractors

Arias-Londoño

Godino-Llorente

2015

Entropy

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This paper addresses the problem of measuring complexity from embedded attractors as a way to characterize changes in the dynamical behavior of different types of systems with a quasi-periodic behavior by observing their outputs. With the aim of measuring the stability of the trajectories of the attractor along time, this paper proposes three new estimations of entropy that are derived from a Markov model of the embedded attractor. The proposed estimators are compared with traditional nonparametric entropy measures, such as approximate entropy, sample entropy and fuzzy entropy, which only take into account the spatial dimension of the trajectory. The method proposes the use of an unsupervised algorithm to find the principal curve, which is considered as the "profile trajectory", that will serve to adjust the Markov model. The new entropy measures are evaluated using three synthetic experiments and three datasets of physiological signals. In terms of consistency and discrimination capabilities, the results show that the proposed measures perform better than the other entropy measures used for comparison purposes.

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Markov models from data by simple nonlinear time series predictors in delay embedding spaces

Cited by 129 publications

References 12 publications

Assessing Coupling Dynamics from an Ensemble of Time Series

Assessing Coupling Dynamics from an Ensemble of Time Series

Nonlinear determinism in time series measurements of two-dimensional turbulence

Entropies from Markov Models as Complexity Measures of Embedded Attractors

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