Correlation dimension and systematic geometric effects

Nerenberg, M. A. H.; Essex, Christopher

doi:10.1103/physreva.42.7065

Cited by 169 publications

(86 citation statements)

References 13 publications

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“…For the chaos detection algorithm, the original data set was divided into 10 subsets of 1000 peak intervals each. The original data set size of Ϸ12 000 agrees well with estimates of the minimal number of data points necessary to identify nonlinear structures [21][22][23][24] : 10 2ϩ0.4d or 10 d , where d is the dimension of the structure under study.…”

Section: Datasupporting

confidence: 54%

Dynamic Analysis of Renal Nerve Activity Responses to Baroreceptor Denervation in Hypertensive Rats

DiBona

Jones

2001

Hypertension

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Abstract-Sinoaortic and cardiac baroreflexes exert important control over renal sympathetic nerve activity. Alterations in these reflex mechanisms contribute to renal sympathoexcitation in hypertension. Nonlinear dynamic analysis was used to examine the chaotic behavior of renal sympathetic nerve activity in normotensive Sprague-Dawley and Wistar-Kyoto rats and spontaneously hypertensive rats before and after complete baroreceptor denervation (sinoaortic and cardiac baroreceptor denervation). The peak interval sequence of synchronized renal sympathetic nerve discharge was extracted and used for analysis. In all rat strains, this yielded systems whose correlation dimensions converged to similar low values over the embedding dimension range of 10 to 15 and whose greatest Lyapunov exponents were positive. Key Words: nonlinear dynamics Ⅲ denervation Ⅲ baroreceptors Ⅲ renal nerves T he sinoaortic and cardiac baroreceptor reflexes exert important control over sympathetic nerve activity. The details of these control mechanisms have been studied by observing steady state changes in mean arterial pressure (MAP), heart rate (HR), and sympathetic nerve activity at various times after complete disruption of these reflexes.After sinoaortic baroreceptor denervation, MAP, HR, and renal sympathetic nerve activity (RSNA) were increased on day 1 but returned to control levels on day 14. 1 On day 1, variability of MAP was increased, while that of HR and RSNA was decreased. On day 14, variability of MAP remained increased, while that of HR and RSNA returned to control levels. MAP and RSNA were strongly (Ϸ90%) negatively correlated before sinoaortic baroreceptor denervation but only 30% negatively correlated on days 1 and 14 and 25% positively correlated on days 1 and 14. These results indicate that low MAP variability results from sinoaortic baroreflex-mediated fluctuations in HR and RSNA that are inversely related and that high MAP variability after sinoaortic baroreceptor denervation is infrequently positively correlated with RSNA. Because MAP variability can be reduced by interventions that block the sympathetic nervous system, 2,3 it appears that MAP variability associated with sinoaortic baroreceptor denervation is mediated largely by a permissive role of peripheral sympathetic nervous system activity. This is especially prominent in the conscious state, in which MAP, HR, and RSNA responses to environmental alerting stimuli are exaggerated.Time series of normal heartbeat (ie, R-R intervals), arterial pressure, and peak intervals of synchronized RSNA display complex nonlinear dynamics, including deterministic chaos. In normal animals subjected to sinoaortic and cardiac baroreceptor denervation, the regulation of arterial pressure 4 -6 (dogs) and RSNA 7 (rats) became more simple, with significant reduction in 2 indices of chaotic behavior, the correlation dimension and greatest Lyapunov exponent. Similarly, the heartbeat 8 of patients and the RSNA 7 of rats with congestive heart failure showed marked reduction in chaotic behavio...

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Section: Datasupporting

confidence: 54%

Dynamic Analysis of Renal Nerve Activity Responses to Baroreceptor Denervation in Hypertensive Rats

DiBona

Jones

2001

Hypertension

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“…One of the most common criticisms on the use of correlation dimension method (especially the GrassbergerProcaccia algorithm) for hydrologic (and other real) time series is that it significantly underestimates the dimension when the data size is small (e.g. Nerenberg and Essex, 1990;Schertzer et al, 2002). Many studies have already addressed this issue through various means (e.g.…”

Section: Discussionmentioning

confidence: 99%

Hydrologic system complexity and nonlinear dynamic concepts for a catchment classification framework

Sivakumar

Singh

2012

Hydrol. Earth Syst. Sci.

132

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Abstract. The absence of a generic modeling framework in hydrology has long been recognized. With our current practice of developing more and more complex models for specific individual situations, there is an increasing emphasis and urgency on this issue. There have been some attempts to provide guidelines for a catchment classification framework, but research in this area is still in a state of infancy. To move forward on this classification framework, identification of an appropriate basis and development of a suitable methodology for its representation are vital. The present study argues that hydrologic system complexity is an appropriate basis for this classification framework and nonlinear dynamic concepts constitute a suitable methodology. The study employs a popular nonlinear dynamic method for identification of the level of complexity of streamflow and for its classification. The correlation dimension method, which has its base on data reconstruction and nearest neighbor concepts, is applied to monthly streamflow time series from a large network of 117 gaging stations across 11 states in the western United States (US). The dimensionality of the time series forms the basis for identification of system complexity and, accordingly, streamflows are classified into four major categories: low-dimensional, medium-dimensional, highdimensional, and unidentifiable. The dimension estimates show some "homogeneity" in flow complexity within certain regions of the western US, but there are also strong exceptions.

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“…Each series has 1000 data points, which is greater than the number of points required (10 2+0.4 * m =650, where m=2 for the Lorenz time series) for the correct estimation of the correlation exponent of the Lorenz time series (e.g. Nerenberg and Essex, 1990). The plot of Ln C(r) vs. Ln r described in Sect.…”

Section: Results With Simulated Datamentioning

confidence: 99%

“…A large scaling region may be better delineated when a large data set is used which, in turn, results in a better estimation of the slope. Nerenberg and Essex (1990) suggested that the minimum number of data points required for the correct estimation of the correlation exponent is N min =10 2+0.4m , where m is the embedding dimension. The presence of noise may affect the scaling behavior and may tend to make the slope of the Ln C(r) vs. Ln r plot larger for small values of r resulting in an overestimation of the correlation exponent.…”

Section: Correlation Dimensionmentioning

confidence: 99%

Detection and predictive modeling of chaos in finite hydrological time series

Khan

Ganguly

Saigal

2005

Nonlin. Processes Geophys.

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Abstract. The ability to detect the chaotic signal from a finite time series observation of hydrologic systems is addressed in this paper. The presence of random and seasonal components in hydrological time series, like rainfall or runoff, makes the detection process challenging.Tests with simulated data demonstrate the presence of thresholds, in terms of noise to chaotic-signal and seasonality to chaoticsignal ratios, beyond which the set of currently available tools is not able to detect the chaotic component. The investigations also indicate that the decomposition of a simulated time series into the corresponding random, seasonal and chaotic components is possible from finite data. Real streamflow data from the Arkansas and Colorado rivers are used to validate these results. Neither of the raw time series exhibits chaos. While a chaotic component can be extracted from the Arkansas data, such a component is either not present or can not be extracted from the Colorado data. This indicates that real hydrologic data may or may not have a detectable chaotic component. The strengths and limitations of the existing set of tools for the detection and modeling of chaos are also studied.

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Correlation dimension and systematic geometric effects

Cited by 169 publications

References 13 publications

Dynamic Analysis of Renal Nerve Activity Responses to Baroreceptor Denervation in Hypertensive Rats

Dynamic Analysis of Renal Nerve Activity Responses to Baroreceptor Denervation in Hypertensive Rats

Hydrologic system complexity and nonlinear dynamic concepts for a catchment classification framework

Detection and predictive modeling of chaos in finite hydrological time series

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