Non-Markovian processes with long-range correlations: fractal dimension analysis

Cáceres, Manuel O.

doi:10.1590/s0103-97331999000100011

Cited by 24 publications

(13 citation statements)

References 5 publications

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“…A variety of anomalous systems exist for which the powerful Boltzmann-Gibbs statistics formalism exhibits serious difficulties, as the long-range interacting systems [30], non-Markovian processes [31], among others. To deal with these systems, an attempt has been made in [32] by postulating a nonextensive entropy (Tsallis entropy).…”

Section: Brief Review On Permutation Entropy Renyi Permutation Entromentioning

confidence: 99%

Classification of Normal and Pre-Ictal EEG Signals Using Permutation Entropies and a Generalized Linear Model as a Classifier

Redelico

Traversaro

García

et al. 2017

Entropy

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Abstract:In this contribution, a comparison between different permutation entropies as classifiers of electroencephalogram (EEG) records corresponding to normal and pre-ictal states is made. A discrete probability distribution function derived from symbolization techniques applied to the EEG signal is used to calculate the Tsallis entropy, Shannon Entropy, Renyi Entropy, and Min Entropy, and they are used separately as the only independent variable in a logistic regression model in order to evaluate its capacity as a classification variable in a inferential manner. The area under the Receiver Operating Characteristic (ROC) curve, along with the accuracy, sensitivity, and specificity are used to compare the models. All the permutation entropies are excellent classifiers, with an accuracy greater than 94.5% in every case, and a sensitivity greater than 97%. Accounting for the amplitude in the symbolization technique retains more information of the signal than its counterparts, and it could be a good candidate for automatic classification of EEG signals.

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Section: Brief Review On Permutation Entropy Renyi Permutation Entromentioning

confidence: 99%

Classification of Normal and Pre-Ictal EEG Signals Using Permutation Entropies and a Generalized Linear Model as a Classifier

Redelico

Traversaro

García

et al. 2017

Entropy

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show abstract

“…Moreover, Eq. (25) can also be used to describe superdiffusion or subdiffusion processes [51,52,53], whose asymptotic mean square displacements grow as t α , α = 1 [59]. In particular, Eq.…”

Section: Non-markovian Continuous Diffusion Modelsmentioning

confidence: 99%

Relativistic diffusion processes and random walk models

2007

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The nonrelativistic standard model for a continuous, one-parameter diffusion process in position space is the Wiener process. As well-known, the Gaussian transition probability density function (PDF) of this process is in conflict with special relativity, as it permits particles to propagate faster than the speed of light. A frequently considered alternative is provided by the telegraph equation, whose solutions avoid superluminal propagation speeds but suffer from singular (non-continuous) diffusion fronts on the light cone, which are unlikely to exist for massive particles. It is therefore advisable to explore other alternatives as well. In this paper, a generalized Wiener process is proposed that is continuous, avoids superluminal propagation, and reduces to the standard Wiener process in the non-relativistic limit. The corresponding relativistic diffusion propagator is obtained directly from the nonrelativistic Wiener propagator, by rewriting the latter in terms of an integral over actions. The resulting relativistic process is non-Markovian, in accordance with the known fact that nontrivial continuous, relativistic Markov processes in position space cannot exist. Hence, the proposed process defines a consistent relativistic diffusion model for massive particles and provides a viable alternative to the solutions of the telegraph equation.

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“…A useful and well known property of α‐stable Lévy motions is their paths are fractals. If one takes the fractal dimension to be the box dimension, D b [ Falconer , 2003; Cáceres , 1999], then the fractal dimension of a 3‐dimensional α‐stable Levy process is D b = max{1, 4 − 1/α}. If one takes the fractal dimension to be the divider dimension, D d , then D d = α.…”

Section: Lévy Motion and Upscalingmentioning

confidence: 99%

Renormalizing chaotic dynamics in fractal porous media with application to microbe motility

Park

Kleinfelter

Cushman

2006

Geophysical Research Letters

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Motivated by the need to understand the movement of microbes in natural porous systems and the evolution of their genetic information, a renormalization procedure for motile particles in media with fractal functionality between upper and lower cutoffs is developed and applied to Lévy particles. On the micro scale, particle trajectories are the solution to an integrated stochastic ordinary differential equation (SODE) with Markov, stationary, ergodic drift subject to Lévy diffusion. The Lévy diffusion allows for self‐motile particles. On the meso scale, the trajectory is the solution to an integrated SODE with Lévy drift and diffusion arising from the microscale asymptotics. Lévy drift is associated with the fractal character of the Lagrangian velocity. On the macro scale, the process is driven by the asymptotics of the mesoscale drift without additional diffusion. Renormalized dispersion equations are presented on the meso and macro scales.

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Non-Markovian processes with long-range correlations: fractal dimension analysis

Cited by 24 publications

References 5 publications

Classification of Normal and Pre-Ictal EEG Signals Using Permutation Entropies and a Generalized Linear Model as a Classifier

Classification of Normal and Pre-Ictal EEG Signals Using Permutation Entropies and a Generalized Linear Model as a Classifier

Relativistic diffusion processes and random walk models

Renormalizing chaotic dynamics in fractal porous media with application to microbe motility

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