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

by Galka, Andreas

• ISBN13: 9789810241483
• ISBN10: 9810241488
• Format: Hardcover
• Copyright: 2000-05-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

Provides a thorough review of a class of powerful algorithms fro the numerical analysis of complex time series data which were obtained from dynamical systems.

Table of Contents

Preface	vii
Introduction	1	(8)
Linearity and the beginning of time series analysis	1	(2)
Irregular time series and determinism	3	(1)
The objective of nonlinear time series analysis	4	(1)
Outline of the organisation of the present study	5	(4)
Dynamical systems, time series and attractors	9	(30)
Overview	9	(1)
Dynamical systems and state spaces	9	(1)
Measurements and time series	10	(2)
Deterministic dynamical systems	12	(16)
Attractors	12	(3)
Linear systems	15	(1)
Invariant measures	15	(3)
Sensitive dependence on initial conditions	18	(1)
Maps and discretised flows	19	(2)
Some important maps	21	(3)
Some important flows	24	(4)
Stochastic dynamical systems	28	(5)
Pure noise time series	29	(1)
Noise in dynamical systems	30	(1)
Linear stochastic systems	31	(2)
Nonstationarity	33	(2)
Experimental and observational time series	35	(4)
Electroencephalograms	36	(3)
Linear methods	39	(10)
Overview	39	(1)
Linear autocorrelation	39	(1)
Fourier spectrum estimation	40	(6)
Discrete Fourier transform and power spectrum	40	(2)
Practical application of Fourier spectrum estimation	42	(4)
Linear prediction and linear filtering	46	(3)
State Space Reconstruction: Theoretical foundations	49	(24)
Overview	49	(1)
The reconstruction problem	49	(2)
Definition of an embedding	51	(1)
Measures of the distortion due to embedding	52	(1)
The embedding theorem of Whitney and its generalisation	53	(2)
Time-delay embedding	55	(2)
The embedding theorem of Takens and its generalisation	57	(2)
Some historical remarks	59	(1)
Filtered time-delay embedding	59	(10)
Derivatives and Legendre coordinates	60	(4)
Principal components: definition and properties	64	(3)
Principal components: applications	67	(2)
Other reconstruction methods	69	(1)
Interspike intervals	70	(3)
State space reconstruction: Practical application	73	(20)
Overview	73	(1)
The effect of noise on state space reconstruction	73	(2)
The choice of the time delay	75	(3)
In search of optimal embedding parameters	78	(15)
The Fillfactor algorithm	79	(3)
Comparing different reconstructions by PCA	82	(2)
The Integral Local Deformation (ILD) algorithm	84	(3)
Other algorithms for the estimation of optimal embedding parameters	87	(6)
Dimensions: Basic definitions	93	(20)
Overview	93	(1)
Why estimate dimensions?	94	(1)
Topological dimension	95	(1)
Hausdorff dimension	95	(2)
Capacity dimension	97	(1)
Generalisation of the Hausdorff dimension	98	(2)
Generalisation of capacity dimension	100	(2)
Information dimension	102	(1)
Continuous definition of generalised dimensions	103	(1)
Pointwise dimension	103	(1)
Invariance of dimension under reconstruction	104	(2)
Invariance of dimension under filtering	106	(1)
Methods for the calculation of dimensions	107	(6)
Box-counting algorithm	107	(2)
Pairwise-distance algorithm	109	(4)
Lyapunov exponents and entropies	113	(10)
Overview	113	(1)
Lyapunov exponents	113	(2)
Estimation of Lyapunov exponents from time series	115	(1)
Kaplan-Yorke dimension	116	(1)
Generalised entropies	117	(2)
Correlation entropy for time-delay embeddings	119	(1)
Pesin's theorem and partial dimensions	120	(3)
Numerical estimation of the correlation dimension	123	(26)
Overview	123	(1)
Correlation dimension as a tail parameter	123	(1)
Estimation of the correlation integral	124	(2)
Efficient implementations	126	(1)
The choice of metric	127	(1)
Typical behaviour of C(r)	128	(3)
Dynamical range of C(r)	131	(2)
Dimension estimation in the case of unknown embedding dimension	133	(1)
Global least squares approach	134	(2)
Chord estimator	136	(1)
Local-slopes approach	137	(4)
Implementation of the local-slopes approach	138	(1)
Typical behaviour of the local-slopes approach	138	(3)
Maximum-likelihood estimators	141	(6)
The Takens estimator	141	(2)
Extensions to the Taken Estimator	143	(1)
The binomial estimator	144	(1)
The algorithm of Judd	145	(2)
Intrinsic dimension and nearest-neighbour algorithms	147	(2)
Sources of error and data set size requirements	149	(34)
Overview	149	(1)
Classification of errors	149	(2)
Edge effects and singularities	151	(5)
Hypercubes with uniform measure	151	(1)
Underestimation due to edge effect	152	(1)
Data set size requirements for avoiding edge effects	153	(2)
Distributions with singularities	155	(1)
Lacunarity	156	(2)
Additive measurement noise	158	(1)
Finit-resolution error	159	(1)
Autocorrelation error	160	(18)
Periodic-sampling error	161	(3)
Circles	164	(2)
Trajectory bias and temporal autocorrelation	166	(3)
Space-time separation plots	169	(1)
Quasiperiodic signals	169	(3)
Topological structure of Nt-tori	172	(1)
Autocorrelations in Nt-tori	173	(2)
Noise with power-law spectrum	175	(3)
Unrepresentativity error	178	(1)
Statistical error	178	(2)
Other estimates of data set size requirements	180	(3)
Monte Carlo analysis of dimension estimation	183	(38)
Overview	183	(1)
Calibration systems	184	(4)
Mackey-Glass system	184	(2)
Gaussian white noise	186	(2)
Filtered noise	188	(1)
Ns-spheres	188	(14)
Analytical estimation of statistical error	189	(3)
Minimum data set size for Ns-spheres	192	(2)
Monte Carlo analysis of statistical error	194	(3)
Limited number of reference points	197	(1)
Comparison between GPA and JA	198	(2)
Results for maximum metric	200	(2)
Multiple Lorenz systems: True state space	202	(7)
Monte Carlo analysis of statistical error	203	(3)
Comparison between GPA and JA	206	(2)
Results for maximum metric	208	(1)
Multiple Lorenz systems: Reconstructed state space	209	(12)
Exact derivative coordinates	210	(2)
Time-delay coordinates	212	(6)
Hybrid coordinates	218	(3)
Surrogate data tests	221	(28)
Overview	221	(1)
Null hypotheses for surrogate data testing	222	(2)
Creation of surrogate data sets	224	(8)
Typical-realisation surrogates	224	(2)
Constrained-realisation surrogates	226	(4)
Surrogates with non-gaussian distribution	230	(2)
Refinements of constrained-realisation surrogate data set creation procedures	232	(10)
Improved AAPR surrogates	232	(2)
The wraparound artifact	234	(1)
Noisy sine waves	235	(3)
Limited phase randomisation	238	(2)
Remedies against the wraparound artifact	240	(2)
Evaluating the results of surrogate data tests	242	(2)
Interpretation of the results of surrogate data tests	244	(1)
Choice of the test statistic for surrogate data tests	245	(1)
Application of surrogate data testing to correlation dimension estimation	246	(3)
Dimension analysis of the human EEG	249	(24)
Overview	249	(1)
The beginning of dimension analysis of the EEG	250	(1)
Application of dimension analysis to cerebral diseases and psychiatric disorders	251	(3)
EEG recordings from epileptic patients	252	(1)
EEG recordings from human sleep	252	(2)
Scepticism against finite dimension estimates from EEG recordings	254	(7)
Application of GPA to an EEG time series from sleep stage IV	255	(3)
Interpretation of the finite estimates found in the literature	258	(3)
Dimension analysis using moving windows	261	(10)
Application to nonstationary time series	262	(3)
Application to stationary time series	265	(2)
Application to a nonstationary EEG time series	267	(4)
Dimension analysis of EEG time series: Valuable or impractical?	271	(2)
Testing for determinism in time series	273	(36)
Overview	273	(1)
The BDS-statistic	274	(3)
The dependence parameters δm by Savit & Green	277	(6)
Generalisations of the δm	280	(1)
Predictability parameters and the relationship between the δm and entropies	281	(2)
Testing for determinism and minimum embedding dimension	283	(3)
Continuous versus discrete data sets	286	(1)
Reduction of EEG time series to discrete phase information	287	(4)
Savit-Green analysis of ISI series from multiple Lorenz systems	291	(5)
Distribution of the dependence parameters δm(r)	291	(2)
Surrogate data testing applied to the predictability parameters Sm(r)	293	(3)
Savit-Green analysis of ISI series from nonstationary time series	296	(2)
Savit-Green analysis of ISI series from EEG time series	298	(6)
Analysis of an EEG time series from sleep stage IV	299	(2)
Analysis of a nonstationary EEG time series	301	(3)
Surrogate data testing of differenced time series	304	(5)
Conclusion	309	(6)
Table of notation	315	(6)
Bibliography	321	(16)
Index	337