Embedology

Sauer, Tim; Yorke, James A.; Casdagli, Martin

doi:10.1007/bf01053745

Cited by 2,117 publications

(1,641 citation statements)

References 15 publications

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“…State space based synchrony (or "generalized synchronization") evaluates synchrony by analyzing the interdependence between the signals in a state space reconstructed domain (see (Sauer et al, 2005) for a review). The central hypothesis behind this approach is that the signals at hand are generated by an (unknown) deterministic, potentially high-dimensional, non-linear dynamical system:…”

Section: State Space Based Synchronymentioning

confidence: 99%

“…If τ and m are appropriately chosen, and the signals are generated by a deterministic dynamical system with a smooth attractor, the delay vectors lie on a smooth manifold ("mapping") in R m . The principle of state space reconstruction can be generalized from smooth attractors to generic sets of arbitrary box-counting dimension, including fractal attractors (Sauer et al, 2005).…”

Section: State Space Based Synchronymentioning

confidence: 99%

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A comparative study of synchrony measures for the early diagnosis of Alzheimer's disease based on EEG

Dauwels

Vialatte

Musha³

et al. 2010

NeuroImage

371

384

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It is well known that EEG signals of Alzheimer's disease (AD) patients are generally less synchronous than in age-matched control subjects. However, this effect is not always easily detectable. This is especially the case for patients in the pre-symptomatic phase, commonly referred to as Mild Cognitive Impairment (MCI), during which neuronal degeneration is occurring prior to the clinical symptoms appearance. In this paper, various synchrony measures are studied in the context of AD diagnosis, including the correlation coefficient, mean-square and phase coherence, Granger causality, phase synchrony indices, information-theoretic divergence measures, state space based measures, and the recently proposed stochastic event synchrony measures. Experiments with EEG data show that many of those measures are strongly correlated (or anti-correlated) with the correlation coefficient, and hence, provide little complementary information about EEG synchrony. Measures that are only weakly correlated with the correlation coefficient include the phase synchrony indices, Granger causality measures, and stochastic-event synchrony measures. In addition, those three families of synchrony measures are mutually uncorrelated, and therefore, they each seem to capture a specific kind of interdependence. For the data set at hand, only two synchrony measures are able to convincingly distinguish AD patients from age-matched control patients, i.e., Granger causality (in particular, full-frequency directed transfer function) and stochastic event synchrony. Those two measures are used as features to distinguish MCI patients from age-matched control subjects, yielding a leave-one-out classification rate of 83%. The classification performance may be further improved by adding complementary features from EEG; this approach may eventually lead to a reliable EEG-based diagnostic tool for MCI and AD.

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Section: State Space Based Synchronymentioning

confidence: 99%

Section: State Space Based Synchronymentioning

confidence: 99%

A comparative study of synchrony measures for the early diagnosis of Alzheimer's disease based on EEG

Dauwels

Vialatte

Musha³

et al. 2010

NeuroImage

371

384

View full text Add to dashboard Cite

show abstract

“…In the formulation of Hunt and Kaloshin [33] (see also [55]), if the attractor has dimension d then for a prevalent set of Lipschitz functions h : R n → R and all T sufficiently small, the map …”

Section: Other Embedding Resultsmentioning

confidence: 99%

Parametrising the attractor of the two-dimensional Navier–Stokes equations with a finite number of nodal values

Friz

Robinson

2001

Physica D: Nonlinear Phenomena

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We consider the solutions lying on the global attractor of the two-dimensional Navier-Stokes equations with periodic boundary conditions and analytic forcing. We show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely, proving a conjecture due to Foias and Temam. Our results also hold for the complex Ginzburg-Landau equation, the Kuramoto-Sivashinsky equation, and reaction-diffusion equations with analytic nonlinearities.

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“…We accomplished this by taking the measured time series, s(t), and creating d E dimensional vectors by using the well-known time delay method (Takens, 1981;Eckmann and Ruelle, 1985;Eckmann et al, 1986;Sauer et al, 1991). Theory states that by doing this construction for a deterministic process, the resulting picture is equivalent (topologically) to measuring all independent variables, such as state space variables and their time derivatives.…”

Section: Local Expansion Ratesmentioning

confidence: 99%