Applied Nonlinear Time Series Analysis : Applications in Physics, Physiology and Finance

by Small, Michael

• ISBN13: 9789812561176
• ISBN10: 981256117X
• Format: Hardcover
• Copyright: 2005-03-28
• Publisher: World Scientific Pub Co Inc

Summary

Nonlinear time series methods have developed rapidly over a quarter of a century and have reached an advanced state of maturity during the last decade. Implementations of these methods for experimental data are now widely accepted and fairly routine; however, genuinely useful applications remain rare. This book focuses on the practice of applying these methods to solve real problems. To illustrate the usefulness of these methods, a wide variety of physical and physiological systems are considered. The technical tools utilized in this book fall into three distinct, but interconnected areas: quantitative measures of nonlinear dynamics. Monte-Carlo statistical hypothesis testing, and nonlinear modeling. Ten highly detailed applications serve as case studies of fruitful applications and illustrate the mathematical techniques described in the text.

Table of Contents

Preface	vii
1. Time series embedding and reconstruction	1	(46)
1.1 Stochasticity and determinism: Why should we bother?	2	(3)
1.2 Embedding dimension	5	(5)
1.2.1 False Nearest Neighbours	6	(1)
1.2.2 False strands and so on	7	(1)
1.2.3 Embed, embed and then embed	8	(1)
1.2.4 Embed and model, and then embed again	9	(1)
1.3 Embedding lag	10	(4)
1.3.1 Autocorrelation	10	(1)
1.3.2 Mutual information	11	(1)
1.3.3 Approximate period	11	(1)
1.3.4 Generalised embedding lags	12	(2)
1.4 Which comes first?	14	(1)
1.5 An embedding zoo	15	(4)
1.6 Irregular embeddings	19	(9)
1.6.1 Finding irregular embeddings	21	(7)
1.7 Embedding window	28	(13)
1.7.1 A modelling paradigm	30	(4)
1.7.2 Examples	34	(7)
1.8 Application: Sunspots and chaotic laser dynamics: Improved modelling and superior dynamics	41	(3)
1.9 Summary	44	(3)
2. Dynamic measures and topological invariants	47	(38)
2.1 Correlation dimension	48	(6)
2.2 Entropy, complexity and information	54	(15)
2.2.1 Entropy	54	(4)
2.2.2 Complexity	58	(2)
2.2.3 Alternative encoding schemes	60	(9)
2.3 Application: Detecting ventricular arrhythmia	69	(5)
2.4 Lyapunov exponents and nonlinear prediction error	74	(6)
2.5 Application: Potential predictability in financial time series	80	(2)
2.6 Summary	82	(3)
3. Estimation of correlation dimension	85	(30)
3.1 Preamble	86	(1)
3.2 Box-counting and the Grassberger-Procaccia algorithm	87	(3)
3.3 Judd's algorithm	90	(5)
3.4 Application: Distinguishing sleep states by monitoring respiration	95	(7)
3.5 The Gaussian Kernel algorithm	102	(3)
3.6 Application: Categorising cardiac dynamics from measured ECG	105	(6)
3.7 Even more algorithms	111	(4)
4. The method of surrogate data	115	(34)
4.1 The rationale and language of surrogate data	116	(4)
4.2 Linear surrogates	120	(5)
4.2.1 Algorithm 0 and its analogues	121	(1)
4.2.2 Algorithm 1 and its applications	122	(1)
4.2.3 Algorithm 2 and its problems	123	(2)
4.3 Cycle shuffled surrogates	125	(4)
4.4 Test statistics	129	(4)
4.4.1 The Kolmogorov-Smirnov test	131	(1)
4.4.2 The χ² test	131	(1)
4.4.3 Noise dimension	132	(1)
4.4.4 Moments of the data	132	(1)
4.5 Correlation dimension: A pivotal test statistic - linear hypotheses	133	(10)
4.5.1 The linear hypotheses	135	(1)
4.5.2 Calculations	136	(6)
4.5.3 Results	142	(1)
4.6 Application: Are financial time series deterministic?	143	(4)
4.7 Summary	147	(2)
5. Non-standard and non-linear surrogates	149	(30)
5.1 Generalised nonlinear null hypotheses: The hypothesis is the model	150	(5)
5.1.1 The "pivotalness" of dynamic measures	152	(1)
5.1.2 Correlation dimension: A pivotal test statistic - non-linear hypothesis	153	(2)
5.2 Application: Infant sleep apnea	155	(2)
5.3 Pseudo-periodic surrogates	157	(9)
5.3.1 Shadowing surrogates	158	(3)
5.3.2 The parameters of the algorithm	161	(2)
5.3.3 Linear noise and chaos	163	(3)
5.4 Application: Mimicking human vocalisation patterns	166	(2)
5.5 Application: Are financial time series really deterministic?	168	(6)
5.6 Simulated annealing and other computational methods	174	(2)
5.7 Summary	176	(3)
6. Identifying the dynamics	179	(44)
6.1 Phenomenological and ontological models	180	(1)
6.2 Application: Severe Acute Respiratory Syndrome: Assessing governmental control strategies during the SARS outbreak in Hong Kong	181	(14)
6.3 Local models	195	(3)
6.4 The importance of embedding for modelling	198	(2)
6.5 Semi-local models	200	(8)
6.5.1 Radial basis functions	200	(1)
6.5.2 Minimum description length principle	201	(4)
6.5.3 Pseudo linear models	205	(2)
6.5.4 Cylindrical basis models	207	(1)
6.6 Application: Predicting onset of Ventricular Fibrillation, and evaluating time since onset	208	(15)
7. Applications	223	(6)
Bibliography	229	(12)
Index	241