A practical method for calculating largest Lyapunov exponents from small data sets
TL;DR: A new method for calculating the largest Lyapunov exponent from an experimental time series is presented that is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level.
About: This article is published in Physica D: Nonlinear Phenomena.The article was published on 1993-05-15. It has received 2942 citations till now. The article focuses on the topics: Lyapunov exponent & Correlation dimension.
Citations
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TL;DR: The various applications of HRV and different linear, frequency domain, wavelet domain, nonlinear techniques used for the analysis of the HRV are discussed.
Abstract: Heart rate variability (HRV) is a reliable reflection of the many physiological factors modulating the normal rhythm of the heart. In fact, they provide a powerful means of observing the interplay between the sympathetic and parasympathetic nervous systems. It shows that the structure generating the signal is not only simply linear, but also involves nonlinear contributions. Heart rate (HR) is a nonstationary signal; its variation may contain indicators of current disease, or warnings about impending cardiac diseases. The indicators may be present at all times or may occur at random-during certain intervals of the day. It is strenuous and time consuming to study and pinpoint abnormalities in voluminous data collected over several hours. Hence, HR variation analysis (instantaneous HR against time axis) has become a popular noninvasive tool for assessing the activities of the autonomic nervous system. Computer based analytical tools for in-depth study of data over daylong intervals can be very useful in diagnostics. Therefore, the HRV signal parameters, extracted and analyzed using computers, are highly useful in diagnostics. In this paper, we have discussed the various applications of HRV and different linear, frequency domain, wavelet domain, nonlinear techniques used for the analysis of the HRV.
2,344 citations
Cites methods from "A practical method for calculating ..."
...Several methods have been proposed: Lyapunov exponents [105], 1/f slope [64], approximate entropy (ApEn) [93] and detrended fluctuation analysis (DFA) [91]....
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TL;DR: In this paper, the authors describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos and present a variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation.
Abstract: 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.
1,381 citations
TL;DR: 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.
Abstract: Nonlinear time series analysis is becoming a more and more reliable tool for the study of complicated dynamics from measurements. The concept of low-dimensional chaos has proven to be fruitful in the understanding of many complex phenomena despite the fact that very few natural systems have actually been found to be low dimensional deterministic in the sense of the theory. In order to evaluate the long term usefulness of the nonlinear time series approach as inspired by chaos theory, it will be important that the corresponding methods become more widely accessible. This paper, while not a proper review on nonlinear time series analysis, tries to make a contribution to this process by describing the actual implementation of the algorithms, and their proper usage. Most of the methods require the choice of certain parameters for each specific time series application. We will try to give guidance in this respect. The scope and selection of topics in this article, as well as the implementational choices that have been made, correspond to the contents of the software package TISEAN which is publicly available from this http URL . In fact, this paper can be seen as an extended manual for the TISEAN programs. It fills the gap between the technical documentation and the existing literature, providing the necessary entry points for a more thorough study of the theoretical background.
1,356 citations
Cites background from "A practical method for calculating ..."
...A reliable characterization requires that the independence of embedding parameters and the exponential law for the growth of distances are checked [69,70] explicitly....
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...[70] where only the closest neighbor is followed for each reference point....
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TL;DR: Interpretation of results in terms of 'functional sources' and 'functional networks' allows the identification of three basic patterns of brain dynamics: normal, ongoing dynamics during a no-task, resting state in healthy subjects, and hypersynchronous, highly nonlinear dynamics of epileptic seizures and degenerative encephalopathies.
1,226 citations
Cites methods from "A practical method for calculating ..."
...Later simpler and faster algorithms to compute the largest Lyapunov exponent were introduced by Kantz and by Rosenstein et al. (Kantz, 1994; Rosenstein et al., 1993)....
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TL;DR: A critically discuss the literature on seizure prediction and address some of the problems and pitfalls involved in the designing and testing of seizure-prediction algorithms, and point towards possible future developments and propose methodological guidelines for future studies on seizure predictions.
Abstract: The sudden and apparently unpredictable nature of seizures is one of the most disabling aspects of the disease epilepsy. A method capable of predicting the occurrence of seizures from the electroencephalogram (EEG) of epilepsy patients would open new therapeutic possibilities. Since the 1970s investigations on the predictability of seizures have advanced from preliminary descriptions of seizure precursors to controlled studies applying prediction algorithms to continuous multi-day EEG recordings. While most of the studies published in the 1990s and around the turn of the millennium yielded rather promising results, more recent evaluations could not reproduce these optimistic findings, thus raising a debate about the validity and reliability of previous investigations. In this review, we will critically discuss the literature on seizure prediction and address some of the problems and pitfalls involved in the designing and testing of seizure-prediction algorithms. We will give an account of the current state of this research field, point towards possible future developments and propose methodological guidelines for future studies on seizure prediction.
1,018 citations
Cites background or methods from "A practical method for calculating ..."
...In order to avoid these shortcomings, a combination of improved algorithms can be used (Rosenstein et al., 1993; Kantz, 1994b) according to which the Lmax can be estimated from djðiÞ Cje,...
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...Moreover, it strongly depends on parameters used for the state space reconstruction and is computationally highly expensive (Rosenstein et al., 1993)....
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References
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TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.
Abstract: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.
16,554 citations
Additional excerpts
...21 [24] E....
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...Lorenz [24] ẋ = σ(y − x) σ = 16....
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TL;DR: In this article, the authors present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series, which provide a qualitative and quantitative characterization of dynamical behavior.
8,128 citations
TL;DR: A review of the main mathematical ideas and their concrete implementation in analyzing experiments can be found in this paper, where the main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions).
Abstract: Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.
4,619 citations
TL;DR: In this article, a measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable, and the relation of this measure to fractal dimension and information-theoretic entropy is discussed.
Abstract: A new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable. The relation of this new measure to fractal dimension and information-theoretic entropy is discussed.
4,323 citations
"A practical method for calculating ..." refers background or methods in this paper
...In particular, methods exist for calculating correlation dimension ( D2) [20], Kolmogorov entropy [21], and Lyapunov characteristic exponents [15, 17, 32, 39]....
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...The Grassberger-Procaccia algorithm [20] estimates dimension by examining the scaling properties of the correlation sum, Cm (r) ....
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...Hence, the practical significance of the GPA is questionable, and the Lyapunov exponents may provide a more useful characterization of chaotic systems....
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...However, the GPA is sensitive to variations in its parameters, e.g., number of data points [28], embedding dimension [28], reconstruction delay [3], and it is usually unreliable except for long, noise-free time series....
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...The Grassberger-Procaccia algorithm (GPA) [20] appears to be the most popular method used to quantify chaos....
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