Topics in Nonlinear Time Series Analysis: With Implications for EEG Analysis

Contents 10
Preface 8
Chapter 1  Introduction 18
 1.1 Linearity and the beginning of time series analysis 18
 1.2 Irregular time series and determinism 20
 1.3 The objective of nonlinear time series analysis 21
 1.4 Outline of the organisation of the present study 22
Chapter 2  Dynamical systems, time series and attractors 26
 2.1 Overview 26
 2.2 Dynamical systems and state spaces 26
 2.3 Measurements and time series 27
 2.4 Deterministic dynamical systems 29
  2.4.1 Attractors 29
  2.4.2 Linear systems 32
  2.4.3 Invariant measures 32
  2.4.4 Sensitive dependence on initial conditions 35
  2.4.5 Maps and discretised flows 36
  2.4.6 Some important maps 38
  2.4.7 Some important flows 41
 2.5 Stochastic dynamical systems 45
  2.5.1 Pure noise time series 46
  2.5.2 Noise in dynamical systems 47
  2.5.3 Linear stochastic systems 48
 2.6 Nonstationarity 50
 2.7 Experimental and observational time series 52
  2.7.1 Electroencephalograms 53
Chapter 3  Linear methods 56
 3.1 Overview 56
 3.2 Linear autocorrelation 56
 3.3 Fourier spectrum estimation 57
  3.3.1 Discrete Fourier transform and power spectrum 57
  3.3.2 Practical application of Fourier spectrum estimation 59
 3.4 Linear prediction and linear filtering 63
Chapter 4  State Space Reconstruction: Theoretical foundations 66
 4.1 Overview 66
 4.2 The reconstruction problem 66
 4.3 Definition of an embedding 68
 4.4 Measures of the distortion due to embedding 69
 4.5 The embedding theorem of Whitney and its generalisation 70
 4.6 Time-delay embedding 72
 4.7 The embedding theorem of Takens and its generalisation 74
 4.8 Some historical remarks 76
 4.9 Filtered time-delay embedding 76
  4.9.1 Derivatives and Legendre coordinates 77
  4.9.2 Principal components: definition and properties 81
  4.9.3 Principal components: applications 84
 4.10 Other reconstruction methods 86
 4.11 Interspike intervals 87
Chapter 5  State space reconstruction: Practical application 90
 5.1 Overview 90
 5.2 The effect of noise on state space reconstruction 90
 5.3 The choice of the time delay 92
 5.4 In search of optimal embedding parameters 95
  5.4.1 The Fillfactor algorithm 96
  5.4.2 Comparing different reconstructions by PCA 99
  5.4.3 The Integral Local Deformation (ILD) algorithm 101
  5.4.4 Other algorithms for the estimation of optimal embedding parameters 104
Chapter 6  Dimensions: Basic definitions 110
 6.1 Overview 110
 6.2 Why estimate dimensions? 111
 6.3 Topological dimension 112
 6.4 Hausdorff dimension 112
 6.5 Capacity dimension 114
 6.6 Generalisation of the Hausdorff dimension 115
 6.7 Generalisation of capacity dimension 117
 6.8 Information dimension 119
 6.9 Continuous definition of generalised dimensions 120
 6.10 Pointwise dimension 120
 6.11 Invariance of dimension under reconstruction 121
 6.12 Invariance of dimension under filtering 123
 6.13 Methods for the calculation of dimensions 124
  6.13.1 Box-counting algorithm 124
  6.13.2 Pairwise-distance algorithm 126
Chapter 7  Lyapunov exponents and entropies 130
 7.1 Overview 130
 7.2 Lyapunov exponents 130
 7.3 Estimation of Lyapunov exponents from time series 132
 7.4 Kaplan-Yorke dimension 133
 7.5 Generalised entropies 134
 7.6 Correlation entropy for time-delay embeddings 136
 7.7 Pesin's theorem and partial dimensions 137
Chapter 8  Numerical estimation of the correlation dimension 140
 8.1 Overview 140
 8.2 Correlation dimension as a tail parameter 140
 8.3 Estimation of the correlation integral 141
 8.4 Efficient implementations 143
 8.5 The choice of metric 144
 8.6 Typical behaviour of C(r) 145
 8.7 Dynamical range of C(r) 148
 8.8 Dimension estimation in the case of unknown embedding dimension 150
 8.9 Global least squares approach 151
 8.10 Chord estimator 153
 8.11 Local-slopes approach 154
  8.11.1 Implementation of the local-slopes approach 155
  8.11.2 Typical behaviour of the local-slopes approach 155
 8.12 Maximum-likelihood estimators 158
  8.12.1 The Takens estimator 158
  8.12.2 Extensions to the Takens estimator 160
  8.12.3 The binomial estimator 161
  8.12.4 The algorithm of Judd 162
 8.13 Intrinsic dimension and nearest-neighbour algorithms 164
Chapter 9  Sources of error and data set size requirements 166
 9.1 Overview 166
 9.2 Classification of errors 166
 9.3 Edge effects and singularities 168
  9.3.1 Hypercubes with uniform measure 168
  9.3.2 Underestimation due to edge effect 169
  9.3.3 Data set size requirements for avoiding edge effects 170
  9.3.4 Distributions with singularities 172
 9.4 Lacunarity 173
 9.5 Additive measurement noise 175
 9.6 Finite-resolution error 176
 9.7 Autocorrelation error 177
  9.7.1 Periodic-sampling error 178
  9.7.2 Circles 181
  9.7.3 Trajectory bias and temporal autocorrelation 183
  9.7.4 Space-time separation plots 186
  9.7.5 Quasiperiodic signals 186
  9.7.6 Topological structure of Nt-tori 189
  9.7.7 Autocorrelations in Nt-tori 190
  9.7.8 Noise with power-law spectrum 192
  9.7.9 Unrepresentativity error 195
 9.8 Statistical error 195
 9.9 Other estimates of data set size requirements 197
Chapter 10  Monte Carlo analysis of dimension estimation 200
 10.1 Overview 200
 10.2 Calibration systems 201
  10.2.1 Mackey-Glass system 201
  10.2.2 Gaussian white noise 203
  10.2.3 Filtered noise 205
 10.3 Ns-spheres 205
  10.3.1 Analytical estimation of statistical error 206
  10.3.2 Minimum data set size for Ns-spheres 209
  10.3.3 Monte Carlo analysis of statistical error 211
  10.3.4 Limited number of reference points 214
  10.3.5 Comparison between GPA and JA 215
  10.3.6 Results for maximum metric 217
 10.4 Multiple Lorenz systems: True state space 219
  10.4.1 Monte Carlo analysis of statistical error 220
  10.4.2 Comparison between GPA and JA 223
  10.4.3 Results for maximum metric 225
 10.5 Multiple Lorenz systems: Reconstructed state space 226
  10.5.1 Exact derivative coordinates 227
  10.5.2 Time-delay coordinates 229
  10.5.3 Hybrid coordinates 235
Chapter 11  Surrogate data tests 238
 11.1 Overview 238
 11.2 Null hypotheses for surrogate data testing 239
 11.3 Creation of surrogate data sets 241
  11.3.1 Typical-realisation surrogates 241
  11.3.2 Constrained-realisation surrogates 243
  11.3.3 Surrogates with non-gaussian distribution 247
 11.4 Refinements of constrained-realisation surrogate data set creation procedures 249
  11.4.1 Improved AAPR surrogates 249
  11.4.2 The wraparound artifact 251
  11.4.3 Noisy sine waves 252
  11.4.4 Limited phase randomisation 255
  11.4.5 Remedies against the wraparound artifact 257
 11.5 Evaluating the results of surrogate data tests 259
 11.6 Interpretation of the results of surrogate data tests 261
 11.7 Choice of the test statistic for surrogate data tests 262
 11.8 Application of surrogate data testing to correlation dimension estimation 263
Chapter 12  Dimension analysis of the human EEG 266
 12.1 Overview 266
 12.2 The beginning of dimension analysis of the EEG 267
 12.3 Application of dimension analysis to cerebral diseases and psychiatric disorders 268
  12.3.1 EEG recordings from epileptic patients 269
  12.3.2 EEG recordings from human sleep 269
 12.4 Scepticism against finite dimension estimates from EEG recordings 271
  12.4.1 Application of GPA to an EEG time series from sleep stage IV 272
  12.4.2 Interpretation of the finite estimates found in the literature 275
 12.5 Dimension analysis using moving windows 278
  12.5.1 Application to nonstationary time series 279
  12.5.2 Application to stationary time series 282
  12.5.3 Application to a nonstationary EEG time series 284
 12.6 Dimension analysis of EEG time series: Valuable or impractical? 288
Chapter 13  Testing for determinism in time series 290
 13.1 Overview 290
 13.2 The BDS-statistic 291
 13.3 The dependence parameters δm by Savit & Green 294
  13.3.1 Generalisations of the δm 297
  13.3.2 Predictability parameters and the relationship between the δm and entropies 298
 13.4 Testing for determinism and minimum embedding dimension 300
 13.5 Continuous versus discrete data sets 303
 13.6 Reduction of EEG time series to discrete phase information 304
 13.7 Savit-Green analysis of ISI series from multiple Lorenz systems 308
  13.7.1 Distribution of the dependence parameters δm (r) 308
  13.7.2 Surrogate data testing applied to the predictability parameters Šm(r) 310
 13.8 Savit-Green analysis of ISI series from nonstationary time series 313
 13.9 Savit-Green analysis of ISI series from EEG time series 315
  13.9.1 Analysis of an EEG time series from sleep stage IV 316
  13.9.2 Analysis of a nonstationary EEG time series 318
 13.10 Surrogate data testing of differenced time series 321
Chapter 14  Conclusion 326
Table of notation 332
Bibliography 338
Index 354