The problem of detecting abrupt changes in the behavior of an observed signal or time series arises in a variety of fields, including climate modeling, finance, image analysis, and security. Quickest detection refers to real-time detection of such changes as quickly as possible after they occur. Using the framework of optimal stopping theory, this book describes the fundamentals underpinning the field, providing the background necessary to design, analyze, and understand quickest detection algorithms. For the first time the authors bring together results which were previously scattered across disparate disciplines, and provide a unified treatment of several different approaches to the quickest detection problem. This book is essential reading for anyone who wants to understand the basic statistical procedures for change detection from a fundamental viewpoint, and for those interested in theoretical questions of change detection. It is ideal for graduate students and researchers of engineering, statistics, economics, and finance.
Author(s): H. Vincent Poor, Olympia Hadjiliadis
Edition: 1
Publisher: Cambridge University Press
Year: 2008
Language: English
Commentary: 43457
Pages: 245
Tags: Математика;Теория вероятностей и математическая статистика;Математическая статистика;Прикладная математическая статистика;
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Figures......Page 12
Preface......Page 13
Frequently used notation......Page 15
1 Introduction......Page 17
2.2.1 Probability spaces......Page 22
2.2.2 Random variables......Page 23
2.2.3 Expectation......Page 24
2.2.4 Radon-Nikodym derivatives......Page 26
2.2.5 Conditional expectation and independence......Page 27
2.2.6 Random sequences......Page 31
2.3 Martingales and stopping times......Page 34
2.3.1 Martingales......Page 35
2.3.2 Stopping times......Page 40
2.3.3 Continuous-time analogs......Page 42
2.4 Brownian motion and Poisson processes......Page 43
2.4.1 Brownian motion......Page 44
2.4.2 Poisson processes......Page 46
2.5 Continuous-time semimartingales......Page 48
2.6 Stochastic integration......Page 50
3.2 Markov optimal stopping problems......Page 56
3.3.1 The general case......Page 57
3.3.2 The Markov case......Page 62
3.4 The infinite-horizon case......Page 66
3.4.1 A martingale interpretation of the finite-horizon results......Page 67
3.4.2 The infinite-horizon case for bounded reward......Page 68
3.4.3 The general infinite-horizon case......Page 71
3.4.4 The infinite-horizon case with Markov rewards......Page 75
3.5 Markov optimal stopping in continuous time......Page 76
3.6 Appendix: a proof of Lemma 3.8......Page 77
4.2 Optimal detection......Page 81
4.3 Performance analysis......Page 90
4.4.1 The Brownian case......Page 97
4.4.2 The Brownian case - an alternative proof......Page 102
4.4.3 An interesting extension of Wald-Wolfowitz......Page 106
4.4.4 The case of Itô processes......Page 107
4.4.5 The Poisson case......Page 109
4.4.6 The compound Poisson case......Page 116
4.5 Discussion......Page 117
5.1 Introduction......Page 118
5.2 Shiryaev's problem......Page 119
5.3.1 Brownian observations......Page 125
5.3.2 Poisson observations......Page 131
5.4 A probability maximizing approach......Page 138
5.5 Other penalty functions......Page 140
5.6 A game theoretic formulation......Page 141
5.7 Discussion......Page 144
6.2 Lorden's problem......Page 146
6.3 Performance of Page's test......Page 158
6.4.1 Brownian observations......Page 160
6.4.2 Itô processes......Page 166
6.4.3 Brownian motion with an unknown drift parameter......Page 168
6.4.4 Poisson observations......Page 170
6.5 Asymptotic results......Page 173
6.5.1 Lorden's approach......Page 174
6.5.2 Brownian motion with two-sided alternatives......Page 183
6.6 Comments on the false-alarm constraint......Page 187
6.7 Discussion......Page 188
7.1 Introduction......Page 190
7.2 Decentralized sequential and quickest detection......Page 191
7.2.1 Decentralized sequential detection with a fusion center......Page 192
7.2.2 Decentralized quickest detection with a fusion center......Page 200
7.2.3 Decentralized sequential detection without fusion......Page 205
7.3.1 Robust quickest detection......Page 210
7.3.2 Adaptive quickest detection......Page 216
7.4.1 Quickest detection with independent likelihood ratio sequences......Page 217
7.4.2 Locally asymptotically normal distributions......Page 219
7.4.3 Sequential detection (local hypothesis approach)......Page 221
7.4.4 Quickest detection (local hypothesis approach)......Page 226
Bibliography......Page 229
Index......Page 241
