### Introduction

- Linearity and the beginning of time series analysis
- Irregular time series and determinism
- The objective of nonlinear time series analysis
- Outline of the organisation of the present study

### Dynamical systems, time series and attractors

- Overview
- Dynamical systems and state spaces
- Measurements and time series
- Deterministic dynamical systems
- Attractors
- Linear systems
- Invariant measures
- Sensitive dependence on initial conditions
- Maps and discretised flows
- Some important maps
- Some important flows

- Stochastic dynamical systems
- Pure noise time series
- Noise in dynamical systems
- Linear stochastic systems

- Nonstationarity
- Experimental and observational time series
- Electroencephalograms

### Linear methods

- Overview
- Linear autocorrelation
- Fourier spectrum estimation
- Discrete Fourier transform and power spectrum
- Practical application of Fourier spectrum estimation

- Linear prediction and linear filtering

### State Space Reconstruction: Theoretical foundations

- Overview
- The reconstruction problem
- Definition of an embedding
- Measures of the distortion due to embedding
- The embedding theorem of Whitney and its generalisation
- Time-delay embedding
- The embedding theorem of Takens and its generalisation
- Some historical remarks
- Filtered time-delay embedding
- Derivatives and Legendre coordinates
- Principal components: definition and properties
- Principal components: applications

- Other reconstruction methods
- Interspike intervals

### State space reconstruction: Practical application

- Overview
- The effect of noise on state space reconstruction
- The choice of the time delay
- In search of optimal embedding parameters
- The Fillfactor algorithm
- Comparing different reconstructions by PCA
- The Integral Local Deformation (ILD) algorithm
- Other algorithms for the estimation of optimal embedding parameters

### Dimensions: Basic definitions

- Overview
- Why estimate dimensions?
- Topological dimension
- Hausdorff dimension
- Capacity dimension
- Generalisation of the Hausdorff dimension
- Generalisation of capacity dimension
- Information dimension
- Continuous definition of generalised dimensions
- Pointwise dimension
- Invariance of dimension under reconstruction
- Invariance of dimension under filtering
- Methods for the calculation of dimensions
- Box-counting algorithm
- Pairwise-distance algorithm

### Lyapunov exponents and entropies

- Overview
- Lyapunov exponents
- Estimation of Lyapunov exponents from time series
- Kaplan-Yorke dimension
- Generalised entropies
- Correlation entropy for time-delay embeddings
- Pesin's theorem and partial dimensions

### Numerical estimation of the correlation dimension

- Overview
- Correlation dimension as a tail parameter
- Estimation of the correlation integral
- Efficient implementations
- The choice of metric
- Typical behaviour of
*C(r)* - Dynamical range of
*C(r)* - Dimension estimation in the case of unknown embedding dimension
- Global least squares approach
- Chord estimator
- Local slopes approach
- Implementation of the local slopes approach
- Typical behaviour of the local slopes approach

- Maximum-likelihood estimators
- The Takens estimator
- Extensions to the Takens estimator
- The binomial estimator
- The algorithm of Judd

- Intrinsic dimension and nearest-neighbour algorithms

### Sources of error and data set size requirements

- Overview
- Classification of errors
- Edge effects and singularities
- Hypercubes with uniform measure
- Underestimation due to edge effect
- Data set size requirements for avoiding edge effects
- Distributions with singularities

- Lacunarity
- Additive measurement noise
- Finite-resolution error
- Autocorrelation error
- Periodic-sampling error
- Circles
- Trajectory bias and temporal autocorrelation
- Space time separation plots
- Quasiperiodic signals
- Topological structure of
*N*-tori - Autocorrelations in
*N*-tori - Noise with power-law spectrum
- Unrepresentativity error

- Statistical error
- Other estimates of data set size requirements

### Monte Carlo analysis of dimension estimation

- Overview
- Calibration systems
- Mackey-Glass system
- Gaussian white noise
- Filtered noise

*N*-spheres- Analytical estimation of statistical error
- Minimum data set size for
*N*-spheres - Monte Carlo analysis of statistical error
- Limited number of reference points
- Comparison between GPA and JA
- Results for maximum metric

- Multiple Lorenz systems: True state space
- Monte Carlo analysis of statistical error
- Comparison between GPA and JA
- Results for maximum metric

- Multiple Lorenz systems: Reconstructed state space
- Exact derivative coordinates
- Time-delay coordinates
- Hybrid coordinates

### Surrogate data tests

- Overview
- Null hypotheses for surrogate data testing
- Creation of surrogate data sets
- Typical-realisation surrogates
- Constrained-realisation surrogates
- Surrogates with non-gaussian distribution

- Refinements of constrained-realisation surrogate data set creation procedures
- Improved AAPR surrogates
- The wraparound artifact
- Noisy sine waves
- Limited phase randomisation
- Remedies against the wraparound artifact

- Evaluating the results of surrogate data tests
- Interpretation of the results of surrogate data tests
- Choice of the test statistic for surrogate data tests
- Application of surrogate data testing to correlation dimension estimation

### Dimension analysis of the human EEG

- Overview
- The beginning of dimension analysis of the EEG
- Application of dimension analysis to cerebral diseases and psychiatric disorders
- EEG recordings from epileptic patients
- EEG recordings from human sleep

- Scepticism against finite dimension estimates from EEG recordings
- Application of GPA to an EEG time series from sleep stage IV
- Interpretation of the finite estimates found in the literature

- Dimension analysis using moving windows
- Application to nonstationary time series
- Application to stationary time series
- Application to a nonstationary EEG time series

- Dimension analysis of EEG time series: Valuable or impractical?

### Testing for determinism in time series

- Overview
- The BDS-statistic
- The dependence parameters
*delta_m*by Savit & Green- Generalisations of the delta_m
- Predictability parameters and the relationship between the
*delta_m*and entropies

- Testing for determinism and minimum embedding dimension
- Continuous versus discrete data sets
- Reduction of EEG time series to discrete phase information
- Savit-Green analysis of ISI series from multiple Lorenz systems
- Distribution of the dependence parameters
*delta_m(r)* - Surrogate data testing applied to the predictability parameters
*S_m(r)*

- Distribution of the dependence parameters
- Savit-Green analysis of ISI series from nonstationary time series
- Savit-Green analysis of ISI series from EEG time series
- Analysis of an EEG time series from sleep stage IV
- Analysis of a nonstationary EEG time series

- Surrogate data testing of differenced time series

### Conclusion

Andreas Galka 28.12.1998 / 23.7.2003