非线性时间序列分析 第2版 英文PDF电子书下载

Ⅰ Basic topics 1

1 Introduction:why nonlinear methods? 3

2 Linear tools and general considerations 13

2.1 Stationarity and sampling 13

2.2 Testing for stationarity 15

2.3 Linear correlations and the power spectrum 18

2.3.1 Stationarity and the low-frequency component in the power spectrum 23

2.4 Linear filters 24

2.5 Linear predictions 27

3 Phase space methods 30

3.1 Determinism:uniqueness in phase space 30

3.2 Delay reconstruction 35

3.3 Finding a good embedding 36

3.3.1 False neighbours 37

3.3.2 The time lag 39

3.4 Visual inspection of data 39

3.5 Poincaré surface of section 41

3.6 Recurrence plots 43

4 Determinism and predictability 48

4.1 Sources of predictability 48

4.2 Simple nonlinear prediction algorithm 50

4.3 Verification of successful prediction 53

4.4 Cross-prediction errors:probing stationarity 56

4.5 Simple nonlinear noise reduction 58

5 Instability:Lyapunov exponents 65

5.1 Sensitive dependence on initial conditions 65

5.2 Exponential divergence 66

5.3 Measuring the maximal exponent from data 69

6 Self-similarity:dimensions 75

6.1 Attractor geometry and fractals 75

6.2 Correlation dimension 77

6.3 Correlation sum from a time series 78

6.4 Interpretation and pitfalls 82

6.5 Temporal correlations,non-stationarity,and space time separation plots 87

6.6 Practical considerations 91

6.7 A useful application:determination of the noise level using the correlation integral 92

6.8 Multi-scale or self-similar signals 95

6.8.1 Scaling laws 96

6.8.2 Detrended fluctuation analysis 100

7 Using nonlinear methods when determinism is weak 105

7.1 Testing for nonlinearity with surrogate data 107

7.1.1 The null hypothesis 109

7.1.2 How to make surrogate data sets 110

7.1.3 Which statistics to use 113

7.1.4 What can go wrong 115

7.1.5 What we have learned 117

7.2 Nonlinear statistics for system discrimination 118

7.3 Extracting qualitative information from a time series 121

8 Selected nonlinear phenomena 126

8.1 Robustness and limit cycles 126

8.2 Coexistence of attractors 128

8.3 Transients 128

8.4 Intermittency 129

8.5 Structural stability 133

8.6 Bifurcations 135

8.7 Quasi-periodicity 139

Ⅱ Advanced topics 141

9 Advanced embedding methods 143

9.1 Embedding theorems 143

9.1.1 Whitney's embedding theorem 144

9.1.2 Takens's delay embedding theorem 146

9.2 The time lag 148

9.3 Filtered delay embeddings 152

9.3.1 Derivative coordinates 152

9.3.2 Principal component analysis 154

9.4 Fluctuating time intervals 158

9.5 Multichannel measurements 159

9.5.1 Equivalent variables at different positions 160

9.5.2 Variables with different physical meanings 161

9.5.3 Distributed systems 161

9.6 Embedding of interspike intervals 162

9.7 High dimensional chaos and the limitations of the time delay embedding 165

9.8 Embedding for systems with time delayed feedback 171

10 Chaotic data and noise 174

10.1 Measurement noise and dynamical noise 174

10.2 Effects of noise 175

10.3 Nonlinear noise reduction 178

10.3.1 Noise reduction by gradient descent 179

10.3.2 Local projective noise reduction 180

10.3.3 Implementation of locally projective noise reduction 183

10.3.4 How much noise is taken out? 186

10.3.5 Consistency tests 191

10.4 An application:foetal ECG extraction 193

11 More about invariant quantities 197

11.1 Ergodicity and strange attractors 197

11.2 Lyapunov exponents Ⅱ 199

11.2.1 The spectrum of Lyapunov exponents and invariant manifolds 200

11.2.2 Flows versus maps 202

11.2.3 Tangent space method 203

11.2.4 Spurious exponents 205

11.2.5 Almost two dimensional flows 211

11.3 Dimensions Ⅱ 212

11.3.1 Generalised dimensions,multi-fractals 213

11.3.2 Information dimension from a time series 215

11.4 Entropies 217

11.4.1 Chaos and the flow of information 217

11.4.2 Entropies of a static distribution 218

11.4.3 The Kolmogorov-Sinai entropy 220

11.4.4 The ∈-entropy per unit time 222

11.4.5 Entropies from time series data 226

11.5 How things are related 229

11.5.1 Pesin's identity 229

11.5.2 Kaplan-Yorke conjecture 231

12 Modelling and forecasting 234

12.1 Linear stochastic models and filters 236

12.1.1 Linear filters 237

12.1.2 Nonlinear filters 239

12.2 Deterministic dynamics 240

12.3 Local methods in phase space 241

12.3.1 Almost model free methods 241

12.3.2 Local linear fits 242

12.4 Global nonlinear models 244

12.4.1 Polynomials 244

12.4.2 Radial basis functions 245

12.4.3 Neural networks 246

12.4.4 What to do in practice 248

12.5 Improved cost functions 249

12.5.1 Overfitting and model costs 249

12.5.2 The errors-in-variables problem 251

12.5.3 Modelling versus prediction 253

12.6 Model verification 253

12.7 Nonlinear stochastic processes from data 256

12.7.1 Fokker—Planck equations from data 257

12.7.2 Markov chains in embedding space 259

12.7.3 No embedding theorem for Markov chains 260

12.7.4 Predictions for Markov chain data 261

12.7.5 Modelling Markov chain data 262

12.7.6 Choosing embedding parameters for Markov chains 263

12.7.7 Application:prediction of surface wind velocities 264

12.8 Predicting prediction errors 267

12.8.1 Predictability map 267

12.8.2 Individual error prediction 268

12.9 Multi-step predictions versus iterated one-step predictions 271

13 Non-stationary signals 275

13.1 Detecting non-stationarity 276

13.1.1 Making non-stationary data stationary 279

13.2 Over-embedding 280

13.2.1 Deterministic systems with parameter drift 280

13.2.2 Markov chain with parameter drift 281

13.2.3 Data analysis in over-embedding spaces 283

13.2.4 Application:noise reduction for human voice 286

13.3 Parameter spaces from data 288

14 Coupling and synchronisation of nonlinear systems 292

14.1 Measures for interdependence 292

14.2 Transfer entropy 297

14.3 Synchronisation 299

15 Chaos control 304

15.1 Unstable periodic orbits and their invariant manifolds 306

15.1.1 Locating periodic orbits 306

15.1.2 Stable/unstable manifolds from data 312

15.2 OGY-control and derivates 313

15.3 Variants of OGY-control 316

15.4 Delayed feedback 317

15.5 Tracking 318

15.6 Related aspects 319

A Using the TISEAN programs 321

A.1 Information relevant to most of the routines 322

A.1.1 Efficient neighbour searching 322

A.1.2 Re-occurring command options 325

A.2 Second-order statistics and linear models 326

A.3 Phase space tools 327

A.4 Prediction and modelling 329

A.4.1 Locally constant predictor 329

A.4.2 Locally linear prediction 329

A.4.3 Global nonlinear models 330

A.5 Lyapunov exponents 331

A.6 Dimensions and entropies 331

A.6.1 The correlation sum 331

A.6.2 Information dimension,fixed mass algorithm 332

A.6.3 Entropies 333

A.7 Surrogate data and test statistics 334

A.8 Noise reduction 335

A.9 Finding unstable periodic orbits 336

A.10 Multivariate data 336

B Description of the experimental data sets 338

B.1 Lorenz-like chaos in an NH3 laser 338

B.2 Chaos in a periodically modulated NMR laser 340

B.3 Vibrating string 342

B.4 Taylor-Couette flow 342

B.5 Multichannel physiological data 343

B.6 Heart rate during atrial fibrillation 343

B.7 Human electrocardiogram(ECG) 344

B.8 Phonation data 345

B.9 Postural control data 345

B.10 Autonomous CO2 laser with feedback 345

B.11 Nonlinear electric resonance circuit 346

B.12 Frequency doubling solid state laser 348

B.13 Surface wind velocities 349

References 350

Index 365