Results 1  10
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17
A practical method for calculating largest Lyapunov exponents from small data sets
 PHYSICA D
, 1993
"... Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close statespace trajectories and estimate the amount of chaos in a system. We present a new m ..."
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Cited by 62 (0 self)
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Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close statespace trajectories and estimate the amount of chaos in a system. We present a new method for calculating the largest Lyapunov exponent from an experimental time series. The method follows directly from the definition of the largest Lyapunov exponent and is accurate because it takes advantage of all the available data. We show that the algorithm is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. Furthermore, one may use the algorithm to calculate simultaneously the correlation dimension. Thus, one sequence of computations will yield an estimate of both the level of chaos and the system complexity.
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.
ConstrainedRealization MonteCarlo method for Hypothesis Testing
 Physica D
"... : We compare two theoretically distinct approaches to generating artificial (or "surrogate") data for testing hypotheses about a given data set. The first and more straightforward approach is to fit a single "best" model to the original data, and then to generate surrogate data sets that are "typica ..."
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Cited by 42 (1 self)
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: We compare two theoretically distinct approaches to generating artificial (or "surrogate") data for testing hypotheses about a given data set. The first and more straightforward approach is to fit a single "best" model to the original data, and then to generate surrogate data sets that are "typical realizations" of that model. The second approach concentrates not on the model but directly on the original data; it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics. Examples of these two approaches are provided for two simple cases: a test for deviations from a gaussian distribution, and a test for serial dependence in a time series. Additionally, we consider tests for nonlinearity in time series based on a Fourier transform (FT) method and on more conventional autoregressive movingaverage (ARMA) fits to the data. The comparative performance of hypothesis testing schemes based on these two approaches...
A SingleBlind Controlled Competition Among Tests for Nonlinearity and Chaos
 Journal of Econometrics
, 1997
"... Abstract: Interest has been growing in testing for nonlinearity or chaos in economic data, but much controversy has arisen about the available results. This paper explores the reasons for these empirical difficulties. We designed and ran a singleblind controlled competition among five highly regard ..."
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Cited by 39 (5 self)
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Abstract: Interest has been growing in testing for nonlinearity or chaos in economic data, but much controversy has arisen about the available results. This paper explores the reasons for these empirical difficulties. We designed and ran a singleblind controlled competition among five highly regarded tests for nonlinearity or chaos with ten simulated data series. The data generating mechanisms include linear processes, chaotic recursions, and nonchaotic stochastic processes; and both large and small samples were included in the experiment. The data series were produced in a single blind manner by the competition manager and sent by email, without identifying information, to the experiment participants. Each such participant is an acknowledged expert in one of the tests and has a possible vested interest in producing the best possible results with that one test. The results of this competition provide much surprising information about the power functions of some of the best regarded tests for nonlinearity or noisy chaos.
Lyapunov Exponents From Random Fibonacci Sequences To The Lorenz Equations
 Department of Computer Science, Cornell University
, 1998
"... this paper (Mathematical Reviews:29 #648) with the words "This is a profound memoir." 9 will show in Chapter 3, there are simple algorithms for bounding the Lyapunov exponents in this setting. The advanced state of the theory for random matrix products is a peculiar situation because deterministic ..."
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Cited by 12 (1 self)
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this paper (Mathematical Reviews:29 #648) with the words "This is a profound memoir." 9 will show in Chapter 3, there are simple algorithms for bounding the Lyapunov exponents in this setting. The advanced state of the theory for random matrix products is a peculiar situation because deterministic matrix products that govern sensitive dependence on initial conditions are barely understood; it is as if the strong law of large numbers were well understood without a satisfactory theory of convergence of infinite series. The elements of the theory of random matrix products are carefully explained in the beautiful monograph by Bougerol [16]. The basic result about Lyapunov exponents, lim
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.
Nonparametric Neural Network Estimation of Lyapunov Exponents and a Direct Test for Chaos
, 2000
"... This paper derives the asymptotic distribution of the nonparametric neural network estimator of the Lyapunov exponent in a noisy system. Positivity of the Lyapunov exponent is an operational definition of chaos. We introduce a statistical framework for testing the chaotic hypothesis based on the est ..."
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Cited by 8 (2 self)
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This paper derives the asymptotic distribution of the nonparametric neural network estimator of the Lyapunov exponent in a noisy system. Positivity of the Lyapunov exponent is an operational definition of chaos. We introduce a statistical framework for testing the chaotic hypothesis based on the estimated Lyapunov exponents and a consistent variance estimator. A simulation study to evaluate small sample performance is reported. We also apply our procedures to daily stock return data. In most cases, the hypothesis of chaos in the stock return series is rejected at the 1 % level with an exception in some higher power transformed absolute returns.
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...
Approved by: Advisor
, 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 ..."
Abstract
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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 series context. The joint asymptotic normality of the estimators is established under general shortrange dependence conditions, where the asymptotic