Nonlinear Time Series Analysis

Front Cover
Cambridge University Press, 2004 - Mathematics - 369 pages
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The 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
References
350
Index
365
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