Chaos

by Chan, Kung-Sik; Tong, Howell H.

• ISBN13: 9780387952802
• ISBN10: 0387952802
• Format: Hardcover
• Copyright: 2001-10-01
• Publisher: Springer Verlag

Summary

This book discusses dynamical systems that are typically driven by stochastic dynamic noise. It is written by two statisticians essentially for the statistically inclined readers, although readers whose primary interests are in determinate systems will find some of the methodology explained in this book of interest.The statistical approach adopted in this book differs in many ways from the deterministic approach to dynamical systems. Even the very basic notion of initial-value sensitivity requires careful development in the new setting provided. This book covers, in varying depth, many of the contributions made by the statisticians in the past twenty years or so towards our understanding of estimation, the Lyapunov-like index, the nonparametric regression, and many others, many of which are motivated by their dynamical system counterparts but have now acquired a distinct statistical flavour.Kung-Sik Chan is a professor at the University of Iowa, Department of Statistics and Actuarial Science. He is an elected member of the International Statistical Institute. He has served on the editorial boards of the Journal of Business and Economic Statistics and Statistica Sinica. He received a Faculty Scholar Award from the University of Iowa in 1996.Howell Tong holds the Chair of Statistics at the London School of Economics and the University of Hong Kong. He is a foreign member of the Norwegian Academy of Science and Letters, an elected member of the International Statistical Institute and a Council member of its Bernoulli Society, an elected fellow of the Institute of Mathematical Statistics, and an honorary fellow of the Institute of Actuaries (London). He was the Founding Dean of the Graduate School and sometimes the Acting Pro-Vice Chancellor (Research) at the University of Hong Kong. He has served on the editorial boards of several international journals, including Biometrika, Journal of Royal Statistical Society (Series B), Statistica Sinica, and others. He is a guest professor of the Academy of Mathematical and System Sciences of the Chinese Academy of Sciences and received a National Natural Science Prize (China) in the category of Mathematics and Mechanics (Class II) in 2001. He has also held visiting professorships at various universities, including the Imperial College in London, the ETH in Zurich, the Fourier University in Grenoble, the Wall Institute at the University of British Columbia, Vancouver, and the Chinese University of Hong Kong.

Table of Contents

Preface	vii
Acknowledgements	ix
Notation	xv
Introduction and Case Studies	1	(16)
Why Chaos?	1	(1)
Our Aims	2	(1)
Disclaimers	3	(1)
Case Studies	3	(1)
Case 1	3	(4)
Case 2	7	(3)
Case 3	10	(3)
Case 4	13	(4)
Deterministic Chaos	17	(12)
Introduction	17	(2)
Natural Measures and Ergodicity	19	(2)
Lyapunov Exponents	21	(1)
Correlation Dimension	22	(4)
Invariance	26	(1)
Delay Coordinates and Dynamics Reconstruction	26	(2)
Notes	28	(1)
Chaos and Stochastic Systems	29	(32)
A Stochastic Dynamical System	29	(1)
Stochastic Difference Equations	30	(1)
Ergodicity of Stochastic Difference Equations	31	(9)
Markov Chain Techniques	31	(4)
Ergodicity and Stability	35	(5)
Shadowing	40	(3)
Introduction	40	(1)
Noise Reduction	41	(2)
Initial-value Sensitivity in a Stochastic Dynamical System	43	(8)
Identical-noise-realisation Approach	43	(4)
An alternative	47	(2)
Two Examples	49	(2)
Dimensions of State Vector	51	(6)
Projections and Cylinder Sets	51	(4)
A Quasi-Distance Function and Embedding Dimension	55	(2)
Observation Noise	57	(3)
Notes	60	(1)
Statistical Analysis I	61	(52)
Introduction	61	(1)
Plots	61	(7)
Reverse Time Plots	61	(1)
Scatter Plots	62	(1)
Lagged Regression Plots	63	(1)
The Grassberger-Procaccia Plots	64	(1)
DVS Plots	64	(2)
Recurrence Plots	66	(1)
Some Comments	67	(1)
Principal Component Analysis	68	(3)
Tests for Nonlinearity and the Method of Surrogate Data	71	(16)
Introduction	71	(7)
Some Properties of the Surrogate Data	78	(3)
Validity of the Method of Surrogate Data	81	(6)
Estimation of Embedding Dimension	87	(21)
Introduction	87	(2)
Nonparametric Stochastic Regression	89	(1)
Cross-validatory Selection	90	(4)
Examples	94	(5)
Sample Size	99	(3)
Simulations	102	(5)
Estimation of the Local Dimension	107	(1)
Detection for Operational Determinism	108	(5)
Statistical Analysis II	113	(66)
Correlation Dimension	113	(17)
Introduction	113	(1)
Sample Correlation Integral	114	(2)
Grassberger-Procaccia Estimator	116	(4)
Takens Estimator	120	(2)
Unknown Embedding Dimension	122	(1)
A Real Application	123	(2)
Miscellanea	125	(2)
BDS Statistic	127	(3)
Estimation of Measures of Initial-Value Sensitivity	130	(26)
The Noise-free Case	131	(1)
Noisy Data	132	(1)
A Neural Network Approach for Identical Noise Realisations	132	(6)
A Spline Approach for Identical Noise Realisations	138	(1)
An example	139	(3)
A Radically Different Approach	142	(13)
Theoretical Justification	155	(1)
Map Reconstruction	156	(23)
Global Function Approximations	157	(4)
Local Function Approximations	161	(1)
The Threshold Principle	161	(1)
The General Threshold Model	161	(5)
Some Recent Developments	166	(4)
Semi-local Function Approximations	170	(1)
Introduction	170	(1)
The Radial Basis Function Approach	171	(1)
The Kernel Smoothing Approach	172	(2)
A Semi-Parametric Approach	174	(5)
Nonlinear Least-Square Prediction	179	(34)
Introduction	179	(1)
Non-Uniform Noise Amplification in a Stochastic System	179	(2)
A Decomposition Theorem	181	(2)
Estimation of λm(x) and Σ2m(x)	183	(2)
Examples	185	(7)
A Selective Summary of Recent Approaches and Remarks	192	(11)
Nonlinear Autoregressive Approach	193	(1)
The Nearest Neighbour Approach	194	(3)
Miscellaneous Remarks	197	(6)
Interval Predictors and Predictive Distributions	203	(10)
Asymmetric Loss Functions	203	(2)
Estimation of Expectiles	205	(2)
Estimation of Quantiles	207	(1)
Two Examples	207	(4)
Theoretical Justification	211	(2)
Miscellaneous Topics	213	(22)
Chaos-Driven Systems	213	(2)
Chaotic Maps and Their Time Reversals	215	(4)
Point Processes	219	(2)
Controlling Chaos	221	(8)
Synchronisation	229	(6)
Appendix A Deterministic Chaos	235	(30)
A.1 Introduction	235	(1)
A.2 Attracting Sets	236	(1)
A.3 Another Look At the Logistic Maps	237	(3)
A.4 Attractors	240	(1)
A.5 Two Approaches to Studying Chaos	240	(1)
A.6 Invariant and Ergodic Distributions	241	(2)
A.7 Lyapunov Exponents	243	(6)
A.8 Natural Measures	249	(1)
A.9 Dimensions of an Attractor	249	(3)
A.9.1 Box-Counting Dimension	249	(2)
A.9.2 Correlation Dimension	251	(1)
A.10 Map Reconstruction	252	(1)
A.11 Some Elements of Differentiable Manifolds	253	(4)
A.12 Hyperbolic Sets	257	(5)
A.13 Notes	262	(3)
Appendix B Supplements to Chapter 3	265	(10)
B.1 Criteria for Ergodicity	265	(4)
B.1.1 Notes	269	(1)
B.2 Proofs of Two Theorems in §3.3.2	269	(3)
B.3 Shadowing and Hyperbolic Attractors	272	(3)
Appendix C Data Sets and Software	275	(2)
References	277	(14)
Subject Index	291	(6)
Author Index	297