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18
Finding Chaos in Noisy Systems
, 1991
"... In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is comp ..."
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Cited by 50 (1 self)
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In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is compared for two similar initial conditions, this exponent is related to the rate at which the subsequent trajectories diverge. A bounded system with a positive LE is one operational definition of chaotic behavior. Most methods for determining the LE have assumed thousands of observations generated from carefully controlled physical experiments. Less attention has been given to estimating the LE for biological and economic systems that are subjected to random perturbations and observed over a limited amount of time. Using nonparametric regression techniques (Neural Networks and Thin Plate Splines) it is possible to consistently estimate the LE. The properties of these methods have been studied using simulated data and are applied to a biological time series: marten fur returns for the Hudson Bay Company (18201900). Based on a nonparametric analysis there is little evidence for lowdimensional chaos in these data. Although these methods appear to work well for systems perturbed by small amounts of noise, finding chaos in a system with a significant stochastic component may be difficult.
Generalized Redundancies for Time Series Analysis
 Physica D
, 1995
"... Extensions to various informationtheoretic quantities (such as entropy, redundancy, and mutual information) are discussed in the context of their role in nonlinear time series analysis. We also discuss "linearized" versions of these quantities and their use as benchmarks in tests for nonlinearity. ..."
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Cited by 27 (0 self)
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Extensions to various informationtheoretic quantities (such as entropy, redundancy, and mutual information) are discussed in the context of their role in nonlinear time series analysis. We also discuss "linearized" versions of these quantities and their use as benchmarks in tests for nonlinearity. Many of these quantities can be expressed in terms of the generalized correlation integral, and this expression permits us to more clearly exhibit the relationships of these quantities to each other and to other commonly used nonlinear statistics (such as the BDS and GreenSavit statistics). Further, numerical estimation of these quantities is found to be more accurate and more efficient when the the correlation integral is employed in the computation. Finally, we consider several "local" versions of these quantities, including a local KolmogorovSinai entropy, which gives an estimate of variability of the shortterm predictability. 1 Introduction In Shaw's influential (and prizewinning)...
Don't Bleach Chaotic Data
, 1993
"... this paper, that observation is extended. Even when the bleaching is constrained to relatively low order (by the Akaike criterion, for instance), and even for tasks other than detecting nonlinear structure, we find that the effect of bleaching on chaotic data can be detrimental. On the other hand, b ..."
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Cited by 11 (1 self)
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this paper, that observation is extended. Even when the bleaching is constrained to relatively low order (by the Akaike criterion, for instance), and even for tasks other than detecting nonlinear structure, we find that the effect of bleaching on chaotic data can be detrimental. On the other hand, bleaching
Chaotic time series Part I: Estimation of some invariant properties in state space
 Modeling, Identification and Control, 15(4):205  224
, 1995
"... Certain deterministic nonlinear systems may show chaotic behaviour. Time series derived from such systems seem stochastic when analyzed with linear techniques. However, uncovering the deterministic structure is important because it allows constructing more realistic and better models and thus impro ..."
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Cited by 8 (5 self)
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Certain deterministic nonlinear systems may show chaotic behaviour. Time series derived from such systems seem stochastic when analyzed with linear techniques. However, uncovering the deterministic structure is important because it allows constructing more realistic and better models and thus improved predictive capabilities. This paper provides a review of two main key features of chaotic systems, the dimensions of their strange attractors and the Lyapunov exponents. The emphasis is on state space reconstruction techniques that are used to estimate these properties, given scalar observations. Data generated from equations known to display chaotic behaviour are used for illustration. A compilation of applications to real data from widely different fields is given. If chaos is found to be present, one may proceed to build nonlinear models, which is the topic of the second paper in this series.
On the Genetic "memory" of Chaotic Attractor of the Barotropic Ocean Model.
"... The structure of the attractor of the barotropic ocean model is studied in this paper. This structure is partially explained by the sequence of bifurcation the system is subjected by variations of the leading parameters. The principal feature of the studied system is the existence of two "almost inv ..."
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Cited by 5 (0 self)
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The structure of the attractor of the barotropic ocean model is studied in this paper. This structure is partially explained by the sequence of bifurcation the system is subjected by variations of the leading parameters. The principal feature of the studied system is the existence of two "almost invariant" basins of chaotic attractor with very rare transitions between them. This fact related to the rise of the couple of nonsymmetric stable stationary solutions in the model with symmetric forcing. The "memory" of chaos appears also in the presence of maxima in the spectrum of energy. These maxima correspond either to the principal frequency of the limit cycle arose in the Hopf bifurcation, or to the frequencies of the Feigenbaum phenomenon.
Local Lyapunov exponents of the quasigeostrophic ocean dynamics.
, 1996
"... The predictability of the quasigeostrophic ocean model is considered in this paper. This is a simple dynamical model which assumes the ocean depth to be divided into n layers of different water density. This model is driven by the wind stress on the surface, it includes effects of lateral and bott ..."
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Cited by 5 (4 self)
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The predictability of the quasigeostrophic ocean model is considered in this paper. This is a simple dynamical model which assumes the ocean depth to be divided into n layers of different water density. This model is driven by the wind stress on the surface, it includes effects of lateral and bottom friction, of the Earth rotation and the nonlinear interaction between adjacent layers. Internal instability of the system leads to the rapid divergence of its trajectories limiting time of deterministic prediction of the system. To estimate the divergence rate of trajectories we compute Lyapunov exponents of this system. These exponents show us the average growth rate of a small possible perturbation of initial data on infinite time scale. Along with Lyapunov exponents, their generalization, local Lyapunov exponents, are computed as a measure of divergence rate on finite time scales. These exponent provide us with the information about principal mechanisms of local instability. They can ...
Estimating Lyapunov Exponents In Chaotic Time Series With Locally Weighted Regression
, 1994
"... Nonlinear dynamical systems often exhibit chaos, which is characterized by sensitive dependence on initial values or more precisely by a positive Lyapunov exponent. Recognizing and quantifying chaos in time series represents an important step toward understanding the nature of random behavior and re ..."
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Cited by 4 (1 self)
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Nonlinear dynamical systems often exhibit chaos, which is characterized by sensitive dependence on initial values or more precisely by a positive Lyapunov exponent. Recognizing and quantifying chaos in time series represents an important step toward understanding the nature of random behavior and revealing the extent to which shortterm forecasts may be improved. We will focus on the statistical problem of quantifying chaos and nonlinearity via Lyapunov exponents. Predicting the future or determining Lyapunov exponents requires estimation of an autoregressive function or its partial derivatives from time series. The multivariate locally weighted polynomial fit is studied for this purpose. In the nonparametric regression context, explicit asymptotic expansions for the conditional bias and conditional covariance matrix of the regression and partial derivative estimators are derived for both the local linear fit and the local quadratic fit. These results are then generalized to the time s...
The bootstrap and Lyapunov exponents in deterministic chaos
, 1999
"... Inasmuch as Lyapunov exponents provide a necessary condition for chaos in a dynamical system, confidence bounds on estimated Lyapunov exponents are of great interest. Estimates derived either from observations or from numerical integrations are limited to trajectories of finite length, and it is the ..."
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Cited by 3 (2 self)
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Inasmuch as Lyapunov exponents provide a necessary condition for chaos in a dynamical system, confidence bounds on estimated Lyapunov exponents are of great interest. Estimates derived either from observations or from numerical integrations are limited to trajectories of finite length, and it is the uncertainties in (the distribution of) these finite time Lyapunov exponents which are of interest. Within this context a bootstrap algorithm for quantifying sampling uncertainties is shown to be inappropriate for multiplicativeergodic statistics of deterministic chaos. This result remains unchanged in the presence of observational noise. As originally proposed, the algorithm is also inappropriate for general nonlinear stochastic processes, a modified version is presented which may prove of value in the case of stochastic dynamics. A new approach towards quantifying the minimum duration of observations required to estimate global Lyapunov exponents is suggested and is explored in a companio...
Seizure Detection in EEG signals: A Comparison of Different Approaches
"... Abstract — In this paper, the performance of traditional variancebased method for detection of epileptic seizures in EEG signals are compared with various methods based on nonlinear time series analysis, entropies, logistic regression, discrete wavelet transform and time frequency distributions. We ..."
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Abstract — In this paper, the performance of traditional variancebased method for detection of epileptic seizures in EEG signals are compared with various methods based on nonlinear time series analysis, entropies, logistic regression, discrete wavelet transform and time frequency distributions. We noted that variancebased method in compare to the mentioned methods had the best result (100%) applied on the same database.
Statistics over Lyapunov Exponents for Feature Extraction: Electroencephalographic Changes Detection Case
"... Abstract – A new approach based on the consideration that electroencephalogram (EEG) signals are chaotic signals was presented for automated diagnosis of electroencephalographic changes. This consideration was tested successfully using the nonlinear dynamics tools, like the computation of Lyapunov e ..."
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Abstract – A new approach based on the consideration that electroencephalogram (EEG) signals are chaotic signals was presented for automated diagnosis of electroencephalographic changes. This consideration was tested successfully using the nonlinear dynamics tools, like the computation of Lyapunov exponents. This paper presented the usage of statistics over the set of the Lyapunov exponents in order to reduce the dimensionality of the extracted feature vectors. Since classification is more accurate when the pattern is simplified through representation by important features, feature extraction and selection play an important role in classifying systems such as neural networks. Multilayer perceptron neural network (MLPNN) architectures were formulated and used as basis for detection of electroencephalographic changes. Three types of EEG signals (EEG signals recorded from healthy volunteers with eyes open, epilepsy patients in the epileptogenic zone during a seizurefree interval, and epilepsy patients during epileptic seizures) were classified. The selected Lyapunov exponents of the EEG signals were used as inputs of the MLPNN trained with LevenbergMarquardt algorithm. The classification results confirmed that the proposed MLPNN has potential in detecting the electroencephalographic changes.