Ebook: Nonlinear Dynamics and Statistics

All models are lies. "The Earth orbits the sun in an ellipse with the sun at one focus" is false, but accurate enough for almost all purposes. This book describes the current state of the art of telling useful lies about time-varying systems in the real world. Specifically, it is about trying to "understand" (that is, tell useful lies about) dynamical systems directly from observa­ tions, either because they are too complex to model in the conventional way or because they are simply ill-understood. B(:cause it overlaps with conventional time-series analysis, building mod­ els of nonlinear dynamical systems directly from data has been seen by some observers as a somewhat ill-informed attempt to reinvent time-series analysis. The truth is distinctly less trivial. It is surely impossible, except in a few special cases, to re-create Newton's astonishing feat of writing a short equation that is an excellent description of real-world phenomena. Real systems are connected to the rest of the world; they are noisy, non­ stationary, and have high-dimensional dynamics; even when the dynamics contains lower-dimensional attractors there is almost never a coordinate system available in which these at tractors have a conventionally simple description.

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Recently, a great deal of progress has been made in the modeling and understanding of processes with nonlinear dynamics, even when only time series data are available. Modern reconstruction theory deals with creating nonlinear dynamical models from data and is at the heart of this improved understanding. Most of the work has been done by dynamicists, but for the subject to reach maturity, statisticians and signal processing engineers need to provide input both to the theory and to the practice. The book brings together different approaches to nonlinear time series analysis in order to begin a synthesis that will lead to better theory and practice in all the related areas. This book describes the state of the art in nonlinear dynamical reconstruction theory. The chapters are based upon a workshop held at the Isaac Newton Institute, Cambridge University, UK, in late 1998. The book's chapters present theory and methods topics by leading researchers in applied and theoretical nonlinear dynamics, statistics, probability, and systems theory. Features and topics: * disentangling uncertainty and error: the predictability of nonlinear systems * achieving good nonlinear models * delay reconstructions: dynamics vs. statistics * introduction to Monte Carlo Methods for Bayesian Data Analysis * latest results in extracting dynamical behavior via Markov Models * data compression, dynamics and stationarity Professionals, researchers, and advanced graduates in nonlinear dynamics, probability, optimization, and systems theory will find the book a useful resource and guide to current developments in the subject.

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Recently, a great deal of progress has been made in the modeling and understanding of processes with nonlinear dynamics, even when only time series data are available. Modern reconstruction theory deals with creating nonlinear dynamical models from data and is at the heart of this improved understanding. Most of the work has been done by dynamicists, but for the subject to reach maturity, statisticians and signal processing engineers need to provide input both to the theory and to the practice. The book brings together different approaches to nonlinear time series analysis in order to begin a synthesis that will lead to better theory and practice in all the related areas. This book describes the state of the art in nonlinear dynamical reconstruction theory. The chapters are based upon a workshop held at the Isaac Newton Institute, Cambridge University, UK, in late 1998. The book's chapters present theory and methods topics by leading researchers in applied and theoretical nonlinear dynamics, statistics, probability, and systems theory. Features and topics: * disentangling uncertainty and error: the predictability of nonlinear systems * achieving good nonlinear models * delay reconstructions: dynamics vs. statistics * introduction to Monte Carlo Methods for Bayesian Data Analysis * latest results in extracting dynamical behavior via Markov Models * data compression, dynamics and stationarity Professionals, researchers, and advanced graduates in nonlinear dynamics, probability, optimization, and systems theory will find the book a useful resource and guide to current developments in the subject.
Content:
Front Matter....Pages i-xxii
Front Matter....Pages 1-1
Challenges in Modeling Nonlinear Systems: A Worked Example....Pages 3-29
Disentangling Uncertainty and Error: On the Predictability of Nonlinear Systems....Pages 31-64
Achieving Good Nonlinear Models: Keep It Simple, Vary the Embedding, and Get the Dynamics Right....Pages 65-80
Delay Reconstruction: Dynamics versus Statistics....Pages 81-103
Some Remarks on the Statistical Modeling of Chaotic Systems....Pages 105-126
The Identification and Estimation of Nonlinear Stochastic Systems....Pages 127-166
Front Matter....Pages 167-167
An Introduction to Monte Carlo Methods for Bayesian Data Analysis....Pages 169-217
Constrained Randomization of Time Series for Nonlinearity Tests....Pages 219-232
Removing the Noise from Chaos Plus Noise....Pages 233-244
Embedding Theorems, Scaling Structures, and Determinism in Time Series....Pages 245-265
Consistent Estimation of a Dynamical Map....Pages 267-280
Extracting Dynamical Behavior via Markov Models....Pages 281-321
Formulas for the Eckmann-Ruelle Matrix....Pages 323-336
Front Matter....Pages 337-337
Noise and Nonlinearity in an Ecological System....Pages 339-364
Cluster-Weighted Modeling: Probabilistic Time Series Prediction, Characterization, and Synthesis....Pages 365-385
Data Compression, Dynamics, and Stationarity....Pages 387-412
Analyzing Nonlinear Dynamical Systems with Nonparametric Regression....Pages 413-434
Optimization of Embedding Parameters for Prediction of Seizure Onset with Mutual Information....Pages 435-451
Detection of a Nonlinear Oscillator Underlying Experimental Time Series: The Sunspot Cycle....Pages 453-473

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Recently, a great deal of progress has been made in the modeling and understanding of processes with nonlinear dynamics, even when only time series data are available. Modern reconstruction theory deals with creating nonlinear dynamical models from data and is at the heart of this improved understanding. Most of the work has been done by dynamicists, but for the subject to reach maturity, statisticians and signal processing engineers need to provide input both to the theory and to the practice. The book brings together different approaches to nonlinear time series analysis in order to begin a synthesis that will lead to better theory and practice in all the related areas. This book describes the state of the art in nonlinear dynamical reconstruction theory. The chapters are based upon a workshop held at the Isaac Newton Institute, Cambridge University, UK, in late 1998. The book's chapters present theory and methods topics by leading researchers in applied and theoretical nonlinear dynamics, statistics, probability, and systems theory. Features and topics: * disentangling uncertainty and error: the predictability of nonlinear systems * achieving good nonlinear models * delay reconstructions: dynamics vs. statistics * introduction to Monte Carlo Methods for Bayesian Data Analysis * latest results in extracting dynamical behavior via Markov Models * data compression, dynamics and stationarity Professionals, researchers, and advanced graduates in nonlinear dynamics, probability, optimization, and systems theory will find the book a useful resource and guide to current developments in the subject.
Content:
Front Matter....Pages i-xxii
Front Matter....Pages 1-1
Challenges in Modeling Nonlinear Systems: A Worked Example....Pages 3-29
Disentangling Uncertainty and Error: On the Predictability of Nonlinear Systems....Pages 31-64
Achieving Good Nonlinear Models: Keep It Simple, Vary the Embedding, and Get the Dynamics Right....Pages 65-80
Delay Reconstruction: Dynamics versus Statistics....Pages 81-103
Some Remarks on the Statistical Modeling of Chaotic Systems....Pages 105-126
The Identification and Estimation of Nonlinear Stochastic Systems....Pages 127-166
Front Matter....Pages 167-167
An Introduction to Monte Carlo Methods for Bayesian Data Analysis....Pages 169-217
Constrained Randomization of Time Series for Nonlinearity Tests....Pages 219-232
Removing the Noise from Chaos Plus Noise....Pages 233-244
Embedding Theorems, Scaling Structures, and Determinism in Time Series....Pages 245-265
Consistent Estimation of a Dynamical Map....Pages 267-280
Extracting Dynamical Behavior via Markov Models....Pages 281-321
Formulas for the Eckmann-Ruelle Matrix....Pages 323-336
Front Matter....Pages 337-337
Noise and Nonlinearity in an Ecological System....Pages 339-364
Cluster-Weighted Modeling: Probabilistic Time Series Prediction, Characterization, and Synthesis....Pages 365-385
Data Compression, Dynamics, and Stationarity....Pages 387-412
Analyzing Nonlinear Dynamical Systems with Nonparametric Regression....Pages 413-434
Optimization of Embedding Parameters for Prediction of Seizure Onset with Mutual Information....Pages 435-451
Detection of a Nonlinear Oscillator Underlying Experimental Time Series: The Sunspot Cycle....Pages 453-473
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