## Nonlinear Time Series AnalysisThe 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. |

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### Contents

Introduction why nonlinear methods? | 3 |

Chapter 2 Linear tools and general considerations | 13 |

22 Testing for stationarity | 15 |

23 Linear correlations and the power spectrum | 18 |

231 Stationarity and the lowfrequency component in the power spectrum | 23 |

24 Linear filters | 24 |

25 Linear predictions | 27 |

Phase space methods | 30 |

1131 Generalised dimensions multifractals | 213 |

1132 Information dimension from a time series | 215 |

114 Entropies | 217 |

1142 Entropies of a static distribution | 218 |

1143 The KolmogorovSinai entropy | 220 |

1144 The entropy per unit time | 222 |

1145 Entropies from time series data | 226 |

115 How things are related | 229 |

32 Delay reconstruction | 35 |

33 Finding a good embedding | 36 |

331 False neighbours | 37 |

332 The time lag | 38 |

34 Visual inspection of data | 39 |

35 Poincaré surface of section | 41 |

36 Recurrence plots | 44 |

Determinism and predictability | 48 |

42 Simple nonlinear prediction algorithm | 50 |

43 Verification of successful prediction | 53 |

probing stationarity | 56 |

45 Simple nonlinear noise reduction | 58 |

Lyapunov exponents | 65 |

52 Exponential divergence | 66 |

53 Measuring the maximal exponent from data | 69 |

Selfsimilarity dimensions | 75 |

62 Correlation dimension | 77 |

63 Correlation sum from a time series | 78 |

64 Interpretation and pitfalls | 82 |

65 Temporal correlations nonstationarity and space time separation plots | 87 |

66 Practical considerations | 91 |

determination of the noise level using the correlation integral | 92 |

68 Multiscale or selfsimilar signals | 95 |

681 Scaling laws | 96 |

682 Detrendedfluctuation analysis | 100 |

Using nonlinear methods when determinism is weak | 105 |

71 Testing for nonlinearity with surrogate data | 107 |

711 The null hypothesis | 109 |

712 How to make surrogate data sets | 110 |

713 Which statistics to use | 113 |

714 What can go wrong | 115 |

715 What we have learned | 117 |

72 Nonlinear statistics for system discrimination | 118 |

73 Extracting qualitative information from a time series | 121 |

Selected nonlinear phenomena | 126 |

82 Coexistence of attractors | 128 |

84 Intermittency | 129 |

85 Structural stability | 133 |

86 Bifurcations | 135 |

87 Quasiperiodicity | 139 |

Advanced topics | 141 |

Advanced embedding methods | 143 |

911 Whitneys embedding theorem | 144 |

912 Takenss delay embedding theorem | 146 |

92 The time lag | 148 |

93 Filtered delay embeddings | 152 |

932 Principal component analysis | 154 |

94 Fluctuating time intervals | 158 |

95 Multichannel measurements | 159 |

951 Equivalent variables at different positions | 160 |

952 Variables with different physical meanings | 161 |

96 Embedding of interspike intervals | 162 |

97 High dimensional chaos and the limitations of the time delay embedding | 165 |

98 Embedding for systems with time delayed feedback | 171 |

Chaotic data and noise | 174 |

102 Effects of noise | 175 |

103 Nonlinear noise reduction | 178 |

1031 Noise reduction by gradient descent | 179 |

1032 Local protective noise reduction | 180 |

1033 Implementation of locally projective noise reduction | 183 |

1034 How much noise is taken out? | 187 |

1035 Consistency tests | 191 |

foetal ECG extraction | 193 |

More about invariant quantities | 197 |

112 Lyapunov exponents II | 199 |

1121 The spectrum of Lyapunov exponents and invariant manifolds | 200 |

1122 Flows versus maps | 202 |

1123 Tangent space method | 203 |

1124 Spurious exponents | 205 |

1125 Almost two dimensional flows | 211 |

113 Dimensions II | 212 |

1152 KaplanYorke conjecture | 231 |

Modelling and forecasting | 234 |

121 Linear stochastic models and filters | 236 |

1211 Linear filters | 237 |

1212 Nonlinear filters | 239 |

122 Deterministic dynamics | 240 |

123 Local methods in phase space | 241 |

1232 Local linear fits | 242 |

124 Global nonlinear models | 244 |

1242 Radial basis functions | 245 |

1243 Neural networks | 246 |

1244 What to do in practice | 248 |

125 Improved cost functions | 249 |

1252 The errorsinvariables problem | 251 |

1253 Modelling versus prediction | 253 |

127 Nonlinear stochastic processes from data | 256 |

1271 FokkerPlanck equations from data | 257 |

1272 Markov chains in embedding space | 259 |

1273 No embedding theorem for Markov chains | 260 |

1274 Predictions for Markov chain data | 261 |

1275 Modelling Markov chain data | 262 |

1276 Choosing embedding parameters for Markov chains | 263 |

prediction of surface wind velocities | 264 |

128 Predicting prediction errors | 267 |

1282 Individual error prediction | 268 |

129 Multistep predictions versus iterated onestep predictions | 271 |

Nonstationary signals | 275 |

131 Detecting nonstationarity | 276 |

1311 Making nonstationary data stationary | 279 |

132 Overembedding | 280 |

1322 Markov chain with parameter drift | 281 |

1323 Data analysis in overembedding spaces | 283 |

noise reduction for human voice | 286 |

133 Parameter spaces from data | 288 |

Coupling and synchronisation of nonlinear systems | 292 |

142 Transfer entropy | 297 |

143 Synchronisation | 299 |

Chaos control | 304 |

151 Unstable periodic orbits and their invariant manifolds | 306 |

1512 Stableunstable manifolds from data | 312 |

152 OGYcontrol and derivates | 313 |

153 Variants of OGYcontrol | 316 |

154 Delayed feedback | 317 |

155 Tracking | 318 |

156 Related aspects | 319 |

Using the TISEAN programs | 321 |

A1 Information relevant to most of the routines | 322 |

A12 Reoccurring command options | 325 |

A2 Secondorder statistics and linear models | 326 |

A3 Phase space tools | 327 |

A4 Prediction and modelling | 329 |

A43 Global nonlinear models | 330 |

A5 Lyapunov exponents | 331 |

A62 Information dimension fixed mass algorithm | 332 |

A63 Entropies | 333 |

A7 Surrogate data and test statistics | 334 |

A8 Noise reduction | 335 |

A9 Finding unstable periodic orbits | 336 |

Description of the experimental data sets | 338 |

B2 Chaos in a periodically modulated NMR laser | 340 |

B3 Vibrating string | 342 |

B5 Multichannel physiological data | 343 |

B7 Human electrocardiogram ECG | 344 |

B8 Phonation data | 345 |

B11 Nonlinear electric resonance circuit | 346 |

B12 Frequency doubling solid state laser | 348 |

B13 Surface wind velocities | 349 |

350 | |

365 | |

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### Common terms and phrases

algorithm amplitude Appendix B.2 approximation attractor dimension autocorrelation function average behaviour bifurcation chaos chaotic attractor chaotic systems Chapter component compute correlation dimension correlation integral correlation sum curves data set delay vectors dependence determined discussed distribution dynamical systems electric resonance embedding dimension entropy Example experimental exponential filter finite fixed point flow data fluctuations fractal frequency Gaussian global Hence Henon map initial conditions input interval iteration Kolmogorov-Sinai entropy length scales limit cycle linear low dimensional Lyapunov exponents manifold Markov chain maximal Lyapunov exponent mean method minimisation motion neighbourhood neighbours NMR laser data noise level non-stationary nonlinear noise reduction observed oscillations output parameters periodic orbits phase space plot power spectrum prediction error problem quantities random numbers reconstruction recurrence plot sampling self-similarity series analysis signal stable stationary statistical stochastic processes surface of section temporal correlations trajectory typical unstable variables variance yields zero