Nonlinear Time Series Analysis

The paradigm of deterministic chaos has influenced thinking in many fields of science. Chaotic systems show rich and surprising mathematical structures. In the applied sciences, deterministic chaos provides a striking explanation for irregular behaviour and anomalies in systems which do not seem to be inherently stochastic. The most direct link between chaos theory and the real world is the analysis of time series from real systems in terms of nonlinear dynamics. Experimental technique and data analysis have seen such dramatic progress that, by now, most fundamental properties of nonlinear dynamical systems have been observed in the laboratory. Great efforts are being made to exploit ideas from chaos theory wherever the data displays more structure than can be captured by traditional methods. Problems of this kind are typical in biology and physiology but also in geophysics, economics, and many other sciences. Greatly updated edition of a book that sold more than 4000 copies and had terrific reviews Unique in its use of real-world examples Broad scope of applications across the sciences and social sciences Preface to the first edition Preface to the second edition Acknowledgements Part I Basic topics Chapter 1 Introduction: why nonlinear methods? Further reading Chapter 2 Linear tools and general considerations 2.1 Stationarity and sampling 2.2 Testing for stationarity 2.3 Linear correlations and the power spectrum 2.3.1 Stationarity and the low-frequency component in the power spectrum 2.4 Linear filters 2.5 Linear predictions Further reading Exercises Chapter 3 Phase space methods 3.1 Determinism: uniqueness in phase space 3.2 Delay reconstruction 3.3 Finding a good embedding 3.3.1 False neighbours 3.3.2 The time lag 3.4 Visual inspection of data 3.5 Poincare surface of section 3.6 Recurrence plots Further reading Exercises Chapter 4 Determinism and predictability 4.1 Sources of predictability 4.2 Simple nonlinear prediction algorithm 4.3 Verification of successful prediction 4.4 Cross-prediction errors: probing stationarity 4.5 Simple nonlinear noise reduction Further reading Excercises Chapter 5 Instability: Lyapunov exponents 5.1 Sensitive dependence on initial conditions 5.2 Exponential divergence 5.3 Measuring the maximal exponent from data Further reading Exercises Chapter 6 Self -similarity: dimensions 6.1 Attractor geometry and fractals 6.2 Correlation dimension 6.3 Correlation sum from a time series 6.4 Interpretation and pitfalls 6.5 Temporal correlations, non-stationarity; and space time separation plots 6.6 Practical considerations 6.7 A useful application: determination of the noise level using the correlation integral 6.8 Multi-scale or self-similar signals 6.8.1 Scaling laws 6.8.2 Detrendedfluctuation analysis Further reading Exercises Chapter 7 Using nonlinear methods when determinism is weak 7.1 Testing for nonlinearity with surrogate data 7.1.1 The null hypothesis 7.1.2 How to make surrogate data sets 7.1.3 Which statistics to use 7.1.4 What can go wrong 7.1.5 What we have learned 7.2 Nonlinear statistics for system discrimination 7.3 Extracting qualitative information from a time series Further reading Exercises Chapter 8 Selected nonlinear phenomena 8.1 Robustness and limit cycles 8.2 Coexistence of attractors 8.3 Transients 8.4 Intermittency 8.5 Structural stability 8.6 Bifurcations 8.7 Quasi-periodicity Further reading Part II Advanced topics Chapter 9 Advanced embedding methods 9.1 Embedding theorems 9.1.1 Whitney's embedding theorem 9.1.2 Takens's delay embedding theorem 9.2 The time lag 9.3 Filtered delay embeddings 9.3.1 Derivative coordinates 9.3.2 Principal component analysis 9.4 Fluctuating time intervals 9.5 Multichannel measurements 9.5.1 Equivalent variables at different positions 9.5.2 Variables with different physical meanings 9.5.3 Distributed systems 9.6 Embedding of interspike intervals 9.7 High dimensional chaos and the limitations of the time delay embedding 9.8 Embedding for systems with time delayed feedback Further reading Exercises Chapter 10 Chaotic data and noise 10.1 Measurement noise and dynamical noise 10.2 Effects of noise 10.3 Nonlinear noise reduction 10.3.1 Noise reduction by gradient descent 10.3.2 Local projective noise reduction 10.3.3 Implementation of locally projective noise reduction 10.3.4 How much noise is taken out? 10.3.5 Consistency tests 10.4 An application: foetal ECG extraction Further reading Exercises Chapter 11 More about invariant quantities 11.1 Ergodicity and strange attractors 11.2 Lyapunov exponents II 11.2.1 The spectrum of Lyapunov exponents and invariant manifolds 11.2.2 Flows versus maps 11.2.3 Tangent space method 11.2.4 Spurious exponents 11.2.5 Almm;t two dimensional flow ... 11.3 Dimensions II 11.3.1 Generalised dimensions, multi-fractals 11.3.2 Information dimension from a time series 11.4 Entropies 11.4.1 Chaos and the .flow of information 11.4.2 Entropies of a static distribution 11.4.3 The Kolmogorov-Sinai entropy 11.4.4 TheE-entropy per unit time 11.4.5 Entropies from time series data 11.5 How things are related 11.5.1 Pesin's identity 11.5.2 Kaplan-Yorke conjecture Further reading Exercises Chapter 12 Modelling and forecasting 12.1 Linear stochastic models and filters 12.1.1 Linear filters 12.1.2 Nonlinear filters 12.2 Deterministic dynamics 12.3 Local methods in phase space 12.3.1 Almost model free methods 12.3.2 Local linear fits 12.4 Global nonlinear models 12.4.1 Polynomials 12.4.2 Radial basis functions 12.4.3 Neural networks 12.4.4 What to do in practice 12.5 Improved cost functions 12.5.1 Overjitting and model costs 12.5.2 The e"ors-in-variables problem 12.5.3 Modelling versus prediction 12.6 Model verification 12.7 Nonlinear stochastic processes from data 12.7.1 Fokker-Planck equations from data 12.7.2 Markov chains in embedding space 12.7.3 No embedding theorem for Markov chains 12. 7.4 Predictions for Markov chain data 12. 7.5 Modelling Markov chain data 12. 7.6 Choosing embedding parameters for Markov chains 12. 7. 7 Application: prediction of surface wind velocities 12.8 Predicting prediction errors 12.8.1 Predictability map 12.8.2 Individual error prediction 12.9 Multi-step predictions versus iterated one-step predictions Further reading Exercises Chapter 13 Non-stationary signals 13.1 Detecting non-stationarity 13.1.1 Making non-stationary data stationary 13.2 Over-embedding 13.2.1 Deterministic systems with parameter drift 13.2.2 Markov chain with parameter drift 13.2.3 Data analysis in over-embedding spaces 13.2.4 Application: noise reduction for human voice 13.3 Parameter spaces from data Exercises Chapter 14 Coupling and synchronisation of nonlinear systems 14.1 Measures for interdependence 14.2 Transfer entropy 14.3 Synchronisation Further reading Exercises Chapter 15 Chaos control 15.1 Unstable periodic orbits and their invariant manifolds 15.1.1 Locating periodic orbits 15.1.2 Stable/unstable manifolds from data 15.2 OGY-control and derivates 15.3 Variants of OGY-control 15.4 Delayed feedback 15.5 Tracking 15.6 Related aspects Further reading Exercises Appendix A Using the TISEAN programs A.1 Information relevant to most of the routines A.1.1 Efficient neighbour searching A.1.2 Re-occurring command options A.1.2.2 The help option A.1.2.2 Input data A.1.2.2 Embedding space A.1.2.2 Defining neighbourhoods A.1.2.2 Output data A.2 Second-order statistics and linear models A.3 Phase space tools A.4 Prediction and modelling A.4.1 Locally constant predictor A.4.2 Locally linear prediction A.4.3 Global nonlinear models A.5 Lyapunov exponents A.6 Dimensions and entropies A.6.1 The correlation sum A.6.2 Information dimension, fixed mass algorithm A.6.3 Entropies A.7 Surrogate data and test statistics A.8 Noise reduction A.9 Finding unstable periodic orbits A.10 Multivariate data Appendix B Description of the experimental data sets B.1 Lorenz-like chaos in an NH3 laser B.2 Chaos in a periodically modulated NMR laser B.3 Vibrating string B.4 Taylor-Couette flow B.5 Multichannel physiological data B.6 Heart rate during atrial fibrillation B.7 Human electrocardiogram (ECG) B.8 Phonation data B.9 Postural control data B.10 Autonomous C02 laser with feedback B.11 Nonlinear electric resonance circuit B.12 Frequency doubling solid state laser B.13 Surface wind velocities References Index The time variability of many natural and social phenomena is not well described by standard methods of data analysis. However, nonlinear time series analysis uses chaos theory and nonlinear dynamics to understand seemingly unpredictable behavior. The results are applied to real data from physics, biology, medicine, and engineering in this volume. Researchers from all experimental disciplines, including physics, the life sciences, and the economy, will find the work helpful in the analysis of real world systems. First Edition Hb (1997): 0-521-55144-7 First Edition Pb (1997): 0-521-65387-8 "This book represents a modern approach to time series analysis which is based on the theory of dynamical systems. It starts from a sound outline of the underlying theory to arrive at very practical issues, which are illustrated using a large number of empirical data sets taken from various fields. This book will hence be highly useful for scientists and engineers from all disciplines who study time variable signals, including the earth, life and social sciences."--Jacket The time variability of many natural and social phenomena is not well described by standard methods of data analysis. Nonlinear time series analysis uses chaos theory and nonlinear dynamics to understand such seemingly unpredictable behaviour. Results are applied to real data from physics, biology, medicine, and engineering