Kantz H., Schreiber T. Nonlinear Time Series Analysis

Second Edition. — Cambridge: Cambridge University Press, 2003. — 369 p. — ISBN: 0521821509.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.Basic topics.
Introduction: why nonlinear methods?
Linear tools and general considerations.
Stationarity and sampling.
Testing for stationarity.
Linear correlations and the power spectrum.
Stationarity and the low-frequency component in the power spectrum.
Linear filters.
Linear predictions.
Phase space methods.
Determinism: uniqueness in phase space.
Delay reconstruction.
Finding a good embedding.
False neighbours.
The time lag.
Visual inspection of data.
Poincare surface of section.
Recurrence plots.
Determinism and predictability.
Sources of predictability.
Simple nonlinear prediction algorithm.
Verification of successful prediction.
Cross-prediction errors: probing stationarity.
Simple nonlinear noise reduction.
Instability: Lyapunov exponents.
Sensitive dependence on initial conditions.
Exponential divergence.
Measuring the maximal exponent from data.
Self-similarity: dimensions.
Attractor geometry and fractals.
Correlation dimension.
Correlation sum from a time series.
Interpretation and pitfalls.
Temporal correlations, non-stationarity; and space time separation plots.
Practical considerations.
A useful application: determination of the noise level using the correlation integral.
Multi-scale or self-similar signals.
Scaling laws.
Detrendedfluctuation analysis.
Using nonlinear methods when determinism is weak.
Testing for nonlinearity with surrogate data.
The null hypothesis.
How to make surrogate data sets.
Which statistics to use.
What can go wrong.
What we have learned.
Nonlinear statistics for system discrimination.
Extracting qualitative information from a time series.
Selected nonlinear phenomena.
Robustness and limit cycles.
Coexistence of attractors.
Transients.
Intermittency.
Structural stability.
Bifurcations.
Quasi-periodicity.
Advanced topics.
Advanced embedding methods.
Embedding theorems.
Whitney's embedding theorem.
Takens's delay embedding theorem.
The time lag.
Filtered delay embeddings.
Derivative coordinates.
Principal component analysis.
Fluctuating time intervals.
Multichannel measurements.
Equivalent variables at different positions.
Variables with different physical meanings.
Distributed systems.
Embedding of interspike intervals.
High dimensional chaos and the limitations of the time delay embedding.
Embedding for systems with time delayed feedback.
Chaotic data and noise.
Measurement noise and dynamical noise.
Effects of noise.
Nonlinear noise reduction.
Noise reduction by gradient descent.
Local projective noise reduction.
Implementation of locally projective noise reduction.
How much noise is taken out?
Consistency tests.
An application: foetal ECG extraction.
More about invariant quantities.
Ergodicity and strange attractors.
Lyapunov exponents II.
The spectrum of Lyapunov exponents and invariant manifolds.
Flows versus maps.
Tangent space method.
Spurious exponents.
Almost two dimensional flow.
Dimensions II.
Generalised dimensions, multi-fractals.
Information dimension from a time series.
Entropies.
Chaos and the flow of information.
Entropies of a static distribution.
The Kolmogorov-Sinai entropy.
The ε-entropy per unit time.
Entropies from time series data.
How things are related.
Pesin's identity.
Kaplan-Yorke conjecture.
Modelling and forecasting.
Linear stochastic models and filters.
Linear filters.
Nonlinear filters.
Deterministic dynamics.
Local methods in phase space.
Almost model free methods.
Local linear fits.
Global nonlinear models.
Polynomials.
Radial basis functions.
Neural networks.
What to do in practice.
Improved cost functions.
Overjitting and model costs.
The errors-in-variables problem.
Modelling versus prediction.
Model verification.
Nonlinear stochastic processes from data.
Fokker-Planck equations from data.
Markov chains in embedding space.
No embedding theorem for Markov chains.
Predictions for Markov chain data.
Modelling Markov chain data.
Choosing embedding parameters for Markov chains.
Application: prediction of surface wind velocities.
Predicting prediction errors.
Predictability map.
Individual error prediction.
Multi-step predictions versus iterated one-step predictions.
Non-stationary signals.
Detecting non-stationarity.
Making non-stationary data stationary.
Over-embedding.
Deterministic systems with parameter drift.
Markov chain with parameter drift.
Data analysis in over-embedding spaces.
Application: noise reduction for human voice.
Parameter spaces from data.
Coupling and synchronisation of nonlinear systems.
Measures for interdependence.
Transfer entropy.
Synchronisation.
Chaos control.
Unstable periodic orbits and their invariant manifolds.
Locating periodic orbits.
Stable/unstable manifolds from data.
OGY-control and derivates.
Variants of OGY-control.
Delayed feedback.
Tracking.
Related aspects.
Appendix Using the TISEAN programs.
Information relevant to most of the routines.
Efficient neighbour searching.
Re-occurring command options.
The help option.
Input data.
Embedding space.
Defining neighbourhoods.
Output data.
Second-order statistics and linear models.
Phase space tools.
Prediction and modelling.
Locally constant predictor.
Locally linear prediction.
Global nonlinear models.
Lyapunov exponents.
Dimensions and entropies.
The correlation sum.
Information dimension, fixed mass algorithm.
Entropies.
Surrogate data and test statistics.
Noise reduction.
Finding unstable periodic orbits.
Multivariate data.
Appendix Description of the experimental data sets.
Lorenz-like chaos in an NH₃ laser.
Chaos in a periodically modulated NMR laser.
Vibrating string.
Taylor-Couette flow.
Multichannel physiological data.
Heart rate during atrial fibrillation.
Human electrocardiogram (ECG).
Phonation data.
Postural control data.
Autonomous CO₂ laser with feedback.
Nonlinear electric resonance circuit.
Frequency doubling solid state laser.
Surface wind velocities.