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.

Table of Contents:
I Issues in Reconstructing Dynamics.- 1 Challenges in Modeling Nonlinear Systems: A Worked Example.- 2 Disentangling Uncertainty and Error: On the Predictability of Nonlinear Systems.- 3 Achieving Good Nonlinear Models: Keep It Simple, Vary the Embedding, and Get the Dynamics Right.- 4 Delay Reconstruction: Dynamics versus Statistics.- 5 Some Remarks on the Statistical Modeling of Chaotic Systems.- 6 The Identification and Estimation of Nonlinear Stochastic Systems.- II Fundamentals.- 7 An Introduction to Monte Carlo Methods for Bayesian Data Analysis.- 8 Constrained Randomization of Time Series for Nonlinearity Tests.- 9 Removing the Noise from Chaos Plus Noise.- 10 Embedding Theorems, Scaling Structures, and Determinism in Time Series.- 11 Consistent Estimation of a Dynamical Map.- 12 Extracting Dynamical Behavior via Markov Models.- 13 Formulas for the Eckmann-Ruelle Matrix.- III Methods and Applications.- 14 Noise and Nonlinearity in an Ecological System.- 15 Cluster-Weighted Modeling: Probabilistic Time Series Prediction, Characterization, and Synthesis.- 16 Data Compression, Dynamics, and Stationarity.- 17 Analyzing Nonlinear Dynamical Systems with Nonparametric Regression.- 18 Optimization of Embedding Parameters for Prediction of Seizure Onset with Mutual Information.- 19 Detection of a Nonlinear Oscillator Underlying Experimental Time Series: The Sunspot Cycle.