Biomedical subjects

Holger Kantz

Publications and source records attributed to Holger Kantz.

13 recordsLinked to original sources

PubMed2003

Effects of random noise on a simple class of growing network models.

We investigate the effects of random noise on network systems. In particular, we consider a simple class of growing network models whose topological structure is determined by the preferred attachment A(k). We introduce a noise-induced attachment A(k) which includes fluctuations in the number of links of individual nodes due to a random noise. We carry out the numerical simulations to show that the topological structure of the networks is determined not only by A(k) but also by the strength of the noise. Analytic and numerical solutions are also presented to support this observation. In addition, we study the stability of networks against attacks under the noisy condition. Similarly, we introduce a noise-induced preferred deletion B(k), and show that noise is an essential feature to determine the stability of networks.

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PubMed2003

Noise-induced fluctuations of period lengths of stable periodic orbits.

We discuss a class of one-dimensional maps, which possesses a globally attracting stable periodic orbit. Despite a strongly negative Lyapunov exponent, a small amount of noise can introduce fluctuations of the period length. It is shown that this is a reasonable model for the observed dynamics of a bubble formation experiment in a heated capillary embedded in boiling water.

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PubMed2002

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

We analyze prediction schemes for stochastic time series data. We propose that under certain conditions, a scalar time series, obtained from a vector-valued Markov process can be modeled as a finite memory Markov process in the observable. The transition rules of the process are easily computed using simple nonlinear time series predictors originally proposed for deterministic chaotic signals. The optimal time lag entering the embedding procedure is shown to be significantly smaller than the deterministic case. The concept is illustrated for simulated data and for surface wind velocity data, for which the deterministic part of the dynamics is shown to be nonlinear.

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PubMed2002

Unexpected robustness against noise of a class of nonhyperbolic chaotic attractors.

Chaotic attractors arising in physical systems are often nonhyperbolic. We compare two sources of nonhyperbolicity: (1) tangencies between stable and unstable manifolds, and (2) unstable dimension variability. We study the effects of noise on chaotic attractors with these nonhyperbolic behaviors by investigating the scaling laws for the Hausdorff distance between the noisy and the deterministic attractors. Whereas in the presence of tangencies, interactive noise yields attractor deformations, attractors with only dimension variability are robust, despite the fact that shadowing is grossly violated.

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PubMed2002

Optimizing of recurrence plots for noise reduction.

We propose a way to automatically detect the best neighborhood size for a local projective noise reduction filter, where a typical problem is the proper identification of the noise level. Here we make use of concepts from the recurrence quantification analysis in order to adaptively tune the filter along the incoming time series. We define an index, to be computed via recurrence plots, whose minimum gives a clear indication of the best size of the neighborhood in the embedding space. Comparison of the local projective noise reduction filter using this optimization scheme with the state of the art is also provided.

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PubMed1999

Practical implementation of nonlinear time series methods: The TISEAN package.

We describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters. Computer programs that implement the resulting strategies are publicly available as the TISEAN software package. The use of each algorithm will be illustrated with a typical application. As to the theoretical background, we will essentially give pointers to the literature. (c) 1999 American Institute of Physics.

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PubMed1998

Dynamical properties of a ferroelectric capacitor observed through nonlinear time series analysis.

By data analysis the ordinary differential equation for the description of an experimental electric resonance circuit with nonlinear capacitor is derived. Triglycine sulfate (TGS) was used as nonlinear dielectric material. This is the most thoroughly investigated ferroelectric with a second order phase transition. Its static dielectric small signal behavior is well described in the framework of the Landau theory, yielding a Duffing-type ordinary differential equation as a model equation of the circuit. Data analysis allows us to check carefully the validity of this model and to determine required corrections of this simplified equation. (c) 1998 American Institute of Physics.

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PubMed1997

Scalar observations from a class of high-dimensional chaotic systems: Limitations of the time delay embedding.

The time delay embedding for the reconstruction of a state space from scalar data introduces strong folding of the smooth manifold in which a chaotic attractor is embedded, which is absent in some more natural state space. In order to observe the deterministic nature of data, the typical length scale related to this folding has to be resolved. Above this length scale the data appear to be random. For a particular model class we prove these statements and we derive analytically the dependence of this length scale on the complexity of the system. We show that the number of scalar observations required to observe determinism increases exponentially in the product of the system's entropy and dimension. (c) 1997 American Institute of Physics.

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PubMed1996

Enlarged scaling ranges for the KS-entropy and the information dimension.

Numerical estimates of the Kolmogorov-Sinai entropy based on a finite amount of data decay towards zero in the relevant limits. Rewriting differences of block entropies as averages over decay rates, and ignoring all parts of the sample where these rates are uncomputable because of the lack of neighbours, yields improved entropy estimates. In the same way, the scaling range for estimates of the information dimension can be extended considerably. The improvement is demonstrated for experimental data. (c) 1996 American Institute of Physics.

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PubMed1996

Unbiased reconstruction of the dynamics underlying a noisy chaotic time series.

Refined methods for the construction of a deterministic dynamical system which can consistently reproduce observed aperiodic data are discussed. The determination of the dynamics underlying a noisy chaotic time series suffers strongly from two systematic errors: One is a consequence of the so-called "error-in-variables problem." Standard least-squares fits implicitly assume that the independent variables are noise free and that the dependent variable is noisy. We show that due to the violation of this assumption one receives considerably wrong results for moderate noise levels. A straightforward modification of the cost function solves this problem. The second problem consists in a mutual inconsistency between the images of a point under the model dynamics and the corresponding observed values. For an improved fit we therefore introduce a multistep prediction error which exploits the information stored in the time series in a better way. The performance is demonstrated by several examples, including experimental data. (c) 1996 American Institute of Physics.

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PubMed1995

Noise in chaotic data: Diagnosis and treatment.

A prominent limiting factor in the analysis of chaotic time series are measurement errors in the data. We show that this influence can be quite severe, depending on the nature of the noise, the complexity of the signal, and on the application one has in mind. Theoretical considerations yield general upper bounds on the tolerable noise level for dimension, entropy and Lyapunov estimates. We discuss methods to detect and analyze the noise present in a measured data set. We show how the situation can be improved by nonlinear noise reduction. (c) 1995 American Institute of Physics.

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PubMed1995

Dimension estimates and physiological data.

Dimension estimates for data from physiological systems are notoriously difficult since the data are far from ideal in the sense of deterministic dynamical systems. Possible pitfalls and necessary precautions are pointed out and a recipe is given which is viable for those researchers who want to use the Grassberger-Procaccia algorithm but who are not familiar with the vast existing literature on dimension estimates. The relevance of dimension estimates for the characterization of physiological data is discussed, where both the cases of finding and not finding a low dimension are considered. (c) 1995 American Institute of Physics.

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PubMed1993

On noise reduction methods for chaotic data.

Recently proposed noise reduction methods for nonlinear chaotic time sequences with additive noise are analyzed and generalized. All these methods have in common that they work iteratively, and that in each step of the iteration the noise is suppressed by requiring locally linear relations among the delay coordinates, i.e., by moving the delay vectors towards some smooth manifold. The different methods can be compared unambiguously in the case of strictly hyperbolic systems corrupted by measurement noise of infinitesimally low level. It was found that all proposed methods converge in this ideal case, but not equally fast. Different problems arise if the system is not hyperbolic, and at higher noise levels. A new scheme which seems to avoid most of these problems is proposed and tested, and seems to give the best noise reduction so far. Moreover, large improvements are possible within the new scheme and the previous schemes if their parameters are not kept fixed during the iteration, and if corrections are included which take into account the curvature of the attracting manifold. Finally, the fact that comparison with simple low-pass filters tends to overestimate the relative achievements of these nonlinear noise reduction schemes is stressed, and it is suggested that they should be compared to Wiener-type filters.

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