Topics in nonlinear time series analysis

Table of contents

1. Introduction

   1. Linearity and the beginning of time series analysis
   2. Irregular time series and determinism
   3. The objective of nonlinear time series analysis
   4. Outline of the organisation of the present study

2. Dynamical systems, time series and attractors

   1. Overview
   2. Dynamical systems and state spaces
   3. Measurements and time series
   4. Deterministic dynamical systems
      1. Attractors
      2. Linear systems
      3. Invariant measures
      4. Sensitive dependence on initial conditions
      5. Maps and discretised flows
      6. Some important maps
      7. Some important flows
   5. Stochastic dynamical systems
      1. Pure noise time series
      2. Noise in dynamical systems
      3. Linear stochastic systems
   6. Nonstationarity
   7. Experimental and observational time series
      1. Electroencephalograms

3. Linear methods

   1. Overview
   2. Linear autocorrelation
   3. Fourier spectrum estimation
      1. Discrete Fourier transform and power spectrum
      2. Practical application of Fourier spectrum estimation
   4. Linear prediction and linear filtering

4. State Space Reconstruction: Theoretical foundations

   1. Overview
   2. The reconstruction problem
   3. Definition of an embedding
   4. Measures of the distortion due to embedding
   5. The embedding theorem of Whitney and its generalisation
   6. Time-delay embedding
   7. The embedding theorem of Takens and its generalisation
   8. Some historical remarks
   9. Filtered time-delay embedding
      1. Derivatives and Legendre coordinates
      2. Principal components: definition and properties
      3. Principal components: applications
   10. Other reconstruction methods
   11. Interspike intervals

5. State space reconstruction: Practical application

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

6. Dimensions: Basic definitions

   1. Overview
   2. Why estimate dimensions?
   3. Topological dimension
   4. Hausdorff dimension
   5. Capacity dimension
   6. Generalisation of the Hausdorff dimension
   7. Generalisation of capacity dimension
   8. Information dimension
   9. Continuous definition of generalised dimensions
   10. Pointwise dimension
   11. Invariance of dimension under reconstruction
   12. Invariance of dimension under filtering
   13. Methods for the calculation of dimensions
       1. Box-counting algorithm
       2. Pairwise-distance algorithm

7. Lyapunov exponents and entropies

   1. Overview
   2. Lyapunov exponents
   3. Estimation of Lyapunov exponents from time series
   4. Kaplan-Yorke dimension
   5. Generalised entropies
   6. Correlation entropy for time-delay embeddings
   7. Pesin's theorem and partial dimensions

8. Numerical estimation of the correlation dimension

   1. Overview
   2. Correlation dimension as a tail parameter
   3. Estimation of the correlation integral
   4. Efficient implementations
   5. The choice of metric
   6. Typical behaviour of C(r)
   7. Dynamical range of C(r)
   8. Dimension estimation in the case of unknown embedding dimension
   9. Global least squares approach
   10. Chord estimator
   11. Local slopes approach
       1. Implementation of the local slopes approach
       2. Typical behaviour of the local slopes approach
   12. Maximum-likelihood estimators
       1. The Takens estimator
       2. Extensions to the Takens estimator
       3. The binomial estimator
       4. The algorithm of Judd
   13. Intrinsic dimension and nearest-neighbour algorithms

9. Sources of error and data set size requirements

   1. Overview
   2. Classification of errors
   3. Edge effects and singularities
      1. Hypercubes with uniform measure
      2. Underestimation due to edge effect
      3. Data set size requirements for avoiding edge effects
      4. Distributions with singularities
   4. Lacunarity
   5. Additive measurement noise
   6. Finite-resolution error
   7. Autocorrelation error
      1. Periodic-sampling error
      2. Circles
      3. Trajectory bias and temporal autocorrelation
      4. Space time separation plots
      5. Quasiperiodic signals
      6. Topological structure of N-tori
      7. Autocorrelations in N-tori
      8. Noise with power-law spectrum
      9. Unrepresentativity error
   8. Statistical error
   9. Other estimates of data set size requirements

10. Monte Carlo analysis of dimension estimation

    1. Overview
    2. Calibration systems
       1. Mackey-Glass system
       2. Gaussian white noise
       3. Filtered noise
    3. N-spheres
       1. Analytical estimation of statistical error
       2. Minimum data set size for N-spheres
       3. Monte Carlo analysis of statistical error
       4. Limited number of reference points
       5. Comparison between GPA and JA
       6. Results for maximum metric
    4. Multiple Lorenz systems: True state space
       1. Monte Carlo analysis of statistical error
       2. Comparison between GPA and JA
       3. Results for maximum metric
    5. Multiple Lorenz systems: Reconstructed state space
       1. Exact derivative coordinates
       2. Time-delay coordinates
       3. Hybrid coordinates

11. Surrogate data tests

    1. Overview
    2. Null hypotheses for surrogate data testing
    3. Creation of surrogate data sets
       1. Typical-realisation surrogates
       2. Constrained-realisation surrogates
       3. Surrogates with non-gaussian distribution
    4. Refinements of constrained-realisation surrogate data set creation procedures
       1. Improved AAPR surrogates
       2. The wraparound artifact
       3. Noisy sine waves
       4. Limited phase randomisation
       5. Remedies against the wraparound artifact
    5. Evaluating the results of surrogate data tests
    6. Interpretation of the results of surrogate data tests
    7. Choice of the test statistic for surrogate data tests
    8. Application of surrogate data testing to correlation dimension estimation

12. Dimension analysis of the human EEG

    1. Overview
    2. The beginning of dimension analysis of the EEG
    3. Application of dimension analysis to cerebral diseases and psychiatric disorders
       1. EEG recordings from epileptic patients
       2. EEG recordings from human sleep
    4. Scepticism against finite dimension estimates from EEG recordings
       1. Application of GPA to an EEG time series from sleep stage IV
       2. Interpretation of the finite estimates found in the literature
    5. Dimension analysis using moving windows
       1. Application to nonstationary time series
       2. Application to stationary time series
       3. Application to a nonstationary EEG time series
    6. Dimension analysis of EEG time series: Valuable or impractical?

13. Testing for determinism in time series

    1. Overview
    2. The BDS-statistic
    3. The dependence parameters delta_m by Savit & Green
       1. Generalisations of the delta_m
       2. Predictability parameters and the relationship between the delta_m and entropies
    4. Testing for determinism and minimum embedding dimension
    5. Continuous versus discrete data sets
    6. Reduction of EEG time series to discrete phase information
    7. Savit-Green analysis of ISI series from multiple Lorenz systems
       1. Distribution of the dependence parameters delta_m(r)
       2. Surrogate data testing applied to the predictability parameters S_m(r)
    8. Savit-Green analysis of ISI series from nonstationary time series
    9. Savit-Green analysis of ISI series from EEG time series
       1. Analysis of an EEG time series from sleep stage IV
       2. Analysis of a nonstationary EEG time series
    10. Surrogate data testing of differenced time series

14. Conclusion