Linear Filters and Non-Linear Systems

Broomhead, D. S.; Huke, J. P.; Muldoon, Mark

doi:10.1111/j.2517-6161.1992.tb01887.x

Cited by 69 publications

(55 citation statements)

References 18 publications

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“…Some investigators (Badii and Politi 1986;Badii et al 1988) have predicted an increase in dimension with cutoff frequency, related to the Lyapunov exponent spectrum of the attractor and due to the fact that the filter adds a state variable to the system dynamics. Other work (Mitschke 1990;Broomhead et al 1992) postulated that changes in dimension are due to corruption of the phase structure of the data, and that a linear-phase filter (FIR filter or forward-reverse filtering) would not affect the dimension. Our result is more difficult to interpret.…”

Section: Filteringmentioning

confidence: 98%

On the correlation dimension of optokinetic nystagmus eye movements: computational parameters, filtering, nonstationarity, and surrogate data

Shelhamer

1997

Biological Cybernetics

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We discuss the estimation of the correlation dimension of optokinetic nystagmus (OKN), a type of reflexive eye movement. Parameters of the time-delay reconstruction of the attractor are investigated, including the number of data points, the time delay, the window duration, and the duration of the signal being analyzed. Adequate values are recommended. Digital low-pass filtering causes the dimension to increase as the filter cutoff frequency decreases, in accord with a previously published prediction. The stationarity of the correlation dimension is examined; the dimension appears to decrease over the course of 120 s of continuous stimulation. Implications for the reliable estimation of the dimension are considered. Several surrogate data sets are constructed, based on both early (0-30 s) and late (100-130 s) OKN segments. Most of the surrogate data sets randomize some aspect of the original OKN, while maintaining other aspects. Dimensions are found for all surrogates and for the original OKN. Evidence is found that is consistent with some amount of deterministic and nonlinear dynamics in OKN. When this structure is randomized in the surrogate, the dimension changes or the dimension algorithm ceases to converge to a finite value. Implications for further analysis and modeling of OKN are discussed.

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Section: Filteringmentioning

confidence: 98%

On the correlation dimension of optokinetic nystagmus eye movements: computational parameters, filtering, nonstationarity, and surrogate data

Shelhamer

1997

Biological Cybernetics

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“…After these preprocessing procedures were completed, each data set was analyzed to test whether the preprocessing had altered the underlying dynamics. [57][58][59][60][61][62] This was done by statistically testing for the continuity and differentiability between the original and preprocessed data embedding, as described in Section VI.…”

Section: A Preprocessingmentioning

confidence: 99%

“…Here again, care must be taken in choosing the smoothing filters to make sure that the filtering does not affect the underlying dynamics of the system which produced the time series. [57][58][59][60][61][62] We used two kinds of filtering to reduce the effect of experimental noise. In the first method, we used a smoothing filter of the form x (n)ϭ ͚ iϭ1 3 a i x(nϩiϪ2) with a i ϭ1/3 to smooth the raw clutter data.…”

Section: B Filteringmentioning

confidence: 99%

Chaotic dynamics of sea clutter

Haykin

Puthusserypady

1997

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The notion that a deterministic nonlinear dynamical system (with relatively few degrees of freedom) can display aperiodic behavior has a strong bearing on sea clutter characterization: random-looking sea clutter may be the outcome of a chaotic process. This new approach envisages deterministic rules for the underlying sea clutter dynamics, in contrast to the stochastic approach where sea clutter is viewed as a random process with a large number of degrees of freedom. In this paper, we demonstrate, convincingly for the first time, the chaotic dynamics of sea clutter. We say so on the basis of results obtained using radar data collected from a series of extensive and thorough experiments, which have been carried out with ground-truthed sea clutter data sets at three different sites. The study includes correlation dimension analysis (based on the maximum likelihood principle) and Lyapunov spectrum analysis. The Lyapunov (Kaplan-Yorke) dimension, which is a byproduct of Lyapunov spectrum analysis, shows that it is indeed a good estimator of the correlation dimension. The Lyapunov spectrum also reveals that sea clutter is produced by a coupled system of nonlinear differential equations of order five or six. (c) 1997 American Institute of Physics.

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“…Quantities obtained using embedding techniques, in particular the correlation dimension d, remain invariant when such filters are applied to large noiseless data series [28], i.e. in the conditions of Takens theorem.…”

Section: The -Signals To Be Analysed Processed Versus Non-processed mentioning

confidence: 99%

Episodes of low-dimensional self-organized dynamics from electroencephalographic α-signals

Cerf

Daoudi

Hénoune

et al. 1997

Biological Cybernetics

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Self-organized neuronal dynamics revealed by cortical alpha-rhythms occur as episodes, which are rarely observed without extraction of the alpha-band from the other spectral components. Three episodes of an unusually long duration of 10 s, two with no signal processing after data recording at the clinic, are described and show evidence of low-dimensional alpha-dynamics. The evidence is gained from an analysis of scaled structures appearing in families of slope curves of the correlation integrals and is checked against time reparametrization. The data for the two unprocessed 10-s episodes are used for a test of the methodology, as well as a re-examination of the adequacy of the model of an autonomous dynamic system in steady state and of the concept of an attractor in brain dynamics investigations. Striking evidence for the model's inadequacy is provided by the episode of subject S1. In this example five consecutive overlapping 6-s sections do show evidence for low-dimensional dynamics, whereas the 10-s section containing those sections does not. The episode of subject 1 provides an example of alpha-activity which may involve self-organized dynamics extending down to low frequencies. The system (the neuronal network) showing episodes of attractor-ruled dynamics, under conditions of blurred and smoothly fading out evidence that it stays on an attractor, is designated as being ruled by a 'shadow-attractor'. This concept is compared with that of a 'quasi-attractor' introduced by H. Haken in studies of physiological systems. One possible mechanism for the observed episodes is proposed, based on a time-dependent number of enslaved sub-systems.

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Linear Filters and Non-Linear Systems

Cited by 69 publications

References 18 publications

On the correlation dimension of optokinetic nystagmus eye movements: computational parameters, filtering, nonstationarity, and surrogate data

On the correlation dimension of optokinetic nystagmus eye movements: computational parameters, filtering, nonstationarity, and surrogate data

Chaotic dynamics of sea clutter

Episodes of low-dimensional self-organized dynamics from electroencephalographic α-signals

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