Nonlinear Dynamics and Statistics

1. Front Matter

   Pages i-xxii

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2. Issues in Reconstructing Dynamics

   1. Front Matter

      Pages 1-1

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   2. Challenges in Modeling Nonlinear Systems: A Worked Example

      • Henry D. I. Abarbanel

      Pages 3-29

   3. Disentangling Uncertainty and Error: On the Predictability of Nonlinear Systems

      • Leonard A. Smith

      Pages 31-64

   4. Achieving Good Nonlinear Models: Keep It Simple, Vary the Embedding, and Get the Dynamics Right

      • Kevin Judd, Michael Small, Alistair I. Mees

      Pages 65-80

   5. Delay Reconstruction: Dynamics versus Statistics

      • Jaroslav Stark

      Pages 81-103

   6. Some Remarks on the Statistical Modeling of Chaotic Systems

      • Dominique Guegan

      Pages 105-126

   7. The Identification and Estimation of Nonlinear Stochastic Systems

      • Peter Young

      Pages 127-166

3. Fundamentals

   1. Front Matter

      Pages 167-167

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   2. An Introduction to Monte Carlo Methods for Bayesian Data Analysis

      • Christophe Andrieu, Arnaud Doucet, William J. Fitzgerald

      Pages 169-217

   3. Constrained Randomization of Time Series for Nonlinearity Tests

      • Thomas Schreiber, Andreas Schmitz

      Pages 219-232

   4. Removing the Noise from Chaos Plus Noise

      • Steven P. Lalley

      Pages 233-244

   5. Embedding Theorems, Scaling Structures, and Determinism in Time Series

      • Colleen D. Cutler

      Pages 245-265

   6. Consistent Estimation of a Dynamical Map

      • Andrew Nobel

      Pages 267-280

   7. Extracting Dynamical Behavior via Markov Models

      • Gary Froyland

      Pages 281-321

   8. Formulas for the Eckmann-Ruelle Matrix

      • Timothy D. Sauer

      Pages 323-336

4. Methods and Applications

   1. Front Matter

      Pages 337-337

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   2. Noise and Nonlinearity in an Ecological System

      • Paul A. Dixon, Maria J. Milicich, George Sugihara

      Pages 339-364

   3. Cluster-Weighted Modeling: Probabilistic Time Series Prediction, Characterization, and Synthesis

      • Bernd Schoner, Neil Gershenfeld

      Pages 365-385

   4. Data Compression, Dynamics, and Stationarity

      • Matthew B. Kennel, Alistair I. Mees

      Pages 387-412

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.

• Monte Carlo method
• Time series
• behavior
• data analysis
• dynamical systems
• modeling
• nonlinear dynamics
• optimization
• uncertainty

• Centre for Applied Dynamics and Optimization, University of Western Australia, Perth, Australia

  Alistair I. Mees

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