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The Maintenance of Uncertainty
 in Control Systems
, 1997
"... It is important to remain uncertain, of observation, model and law. For the Fermi Summer School, Criticisms Requested email : lenny@maths.ox.ac.uk, Contents 1 ..."
Abstract

Cited by 27 (6 self)
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It is important to remain uncertain, of observation, model and law. For the Fermi Summer School, Criticisms Requested email : lenny@maths.ox.ac.uk, Contents 1
Monte Carlo Based Ensemble Forecasting
 Statistics and Computing
, 1998
"... Ensemble forecasting involves the use of several integrations of a numerical model. Even if this model is assumed to be known, ensembles are needed due to uncertainty in initial conditions. The ideas discussed in this paper incorporate aspects of both analytic model approximations and Monte Carlo ar ..."
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Cited by 6 (0 self)
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Ensemble forecasting involves the use of several integrations of a numerical model. Even if this model is assumed to be known, ensembles are needed due to uncertainty in initial conditions. The ideas discussed in this paper incorporate aspects of both analytic model approximations and Monte Carlo arguments to gain some efficiency in the generation and use of ensembles. Efficiency is gained through the use of importance sampling Monte Carlo. Once ensemble members are generated, suggestions for their use, involving both approximation and statistical notions such as kernel density estimation and mixture modeling are discussed. Fully deterministic procedures derived from the Monte Carlo analysis are also described. Examples using the threedimensional Lorenz system are described. Address: Mark Berliner Department of Statistics Ohio State University 1958 Neil Ave. Columbus, OH 432101247 USA email: mb@stat.ohiostate.edu Keywords and Phrases: Chaos, Importance sampling, Kernel density es...
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...
Improving Climate Prediction Using Seasonal SpaceTime Models
 Department of Statistics, Florida State University
, 1996
"... In this paper a class of seasonal spacetime models is introduced for general lattice systems. Covariance properties of spatial firstorder models, including stationarity conditions, are studied. Procedures for examining spatial independence and symmetry of the models are developed. Estimation appro ..."
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Cited by 1 (1 self)
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In this paper a class of seasonal spacetime models is introduced for general lattice systems. Covariance properties of spatial firstorder models, including stationarity conditions, are studied. Procedures for examining spatial independence and symmetry of the models are developed. Estimation approaches in time series analysis are adopted, and forecasting techniques using the seasonal spacetime models are discussed. The models are applied to 516 consecutive maps of monthlyaveraged 500 mb geopotential heights over a 10 \Theta 10 lattice in the extratropical Northern Hemisphere for the purpose of improving climate prediction. It is found that spacetime models with instantaneous spatial component give better fit than other models in terms of maximizing the conditional likelihood function, but their forecast ability is poor because of inverse problems. On the other hand, spacetime models without instantaneous spatial component provide more accurate forecast values than univariate time...
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
Can noise induce chaos?
, 2003
"... An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended ..."
Abstract
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An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a longterm average over the deterministic attractor while the SLE is the longterm average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE’s should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that ‘‘chaos’ ’ should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.