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Summary
Summary
The statistical bootstrap is one of the methods that can be used to calculate estimates of a certain number of unknown parameters of a random process or a signal observed in noise, based on a random sample. Such situations are common in signal processing and the bootstrap is especially useful when only a small sample is available or an analytical analysis is too cumbersome or even impossible. This book covers the foundations of the bootstrap, its properties, its strengths and its limitations. The authors focus on bootstrap signal detection in Gaussian and non-Gaussian interference as well as bootstrap model selection. The theory developed in the book is supported by a number of useful practical examples written in MATLAB. The book is aimed at graduate students and engineers, and includes applications to real-world problems in areas such as radar and sonar, biomedical engineering and automotive engineering.
Table of Contents
| Preface | p. ix |
| Notations | p. xii |
| 1 Introduction | p. 1 |
| 2 The bootstrap principle | p. 11 |
| 2.1 The principle of resampling | p. 11 |
| 2.1.1 Some theoretical results for the mean | p. 17 |
| 2.1.2 Examples of non-parametric bootstrap estimation | p. 19 |
| 2.1.3 The parametric bootstrap | p. 26 |
| 2.1.4 Bootstrap resampling for dependent data | p. 28 |
| 2.1.5 Examples of dependent data bootstrap estimation | p. 33 |
| 2.2 The principle of pivoting and variance stabilisation | p. 49 |
| 2.2.1 Some examples | p. 51 |
| 2.3 Limitations of the bootstrap | p. 57 |
| 2.4 Trends in bootstrap resampling | p. 59 |
| 2.5 Summary | p. 60 |
| 3 Signal detection with the bootstrap | p. 62 |
| 3.1 Principles of hypothesis testing | p. 62 |
| 3.1.1 Sub-optimal detection | p. 72 |
| 3.2 Hypothesis testing with the bootstrap | p. 73 |
| 3.3 The role of pivoting | p. 74 |
| 3.4 Variance estimation | p. 78 |
| 3.5 Detection through regression | p. 83 |
| 3.6 The bootstrap matched filter | p. 93 |
| 3.6.1 Tolerance interval bootstrap matched filter | p. 99 |
| 3.7 Summary | p. 101 |
| 4 Bootstrap model selection | p. 103 |
| 4.1 Preliminaries | p. 103 |
| 4.2 Model selection | p. 105 |
| 4.3 Model selection in linear models | p. 106 |
| 4.3.1 Model selection based on prediction | p. 107 |
| 4.3.2 Bootstrap based model selection | p. 108 |
| 4.3.3 A consistent bootstrap method | p. 109 |
| 4.3.4 Dependent data in linear models | p. 114 |
| 4.4 Model selection in nonlinear models | p. 114 |
| 4.4.1 Data model | p. 114 |
| 4.4.2 Use of bootstrap in model selection | p. 115 |
| 4.5 Order selection in autoregressions | p. 117 |
| 4.6 Detection of sources using bootstrap techniques | p. 119 |
| 4.6.1 Bootstrap based detection | p. 121 |
| 4.6.2 Null distribution estimation | p. 124 |
| 4.6.3 Bias correction | p. 126 |
| 4.6.4 Simulations | p. 127 |
| 4.7 Summary | p. 127 |
| 5 Real data bootstrap applications | p. 130 |
| 5.1 Optimal sensor placement for knock detection | p. 130 |
| 5.1.1 Motivation | p. 131 |
| 5.1.2 Data model | p. 131 |
| 5.1.3 Bootstrap tests | p. 134 |
| 5.1.4 The experiment | p. 135 |
| 5.2 Confidence intervals for aircraft parameters | p. 136 |
| 5.2.1 Introduction | p. 136 |
| 5.2.2 Results with real passive acoustic data | p. 139 |
| 5.3 Landmine detection | p. 143 |
| 5.4 Noise floor estimation in over-the-horizon radar | p. 147 |
| 5.4.1 Principle of the trimmed mean | p. 148 |
| 5.4.2 Optimal trimming | p. 150 |
| 5.4.3 Noise floor estimation | p. 151 |
| 5.5 Model order selection for corneal elevation | p. 154 |
| 5.6 Summary | p. 158 |
| Appendix 1 Matlab codes for the examples | p. 159 |
| A1.1 Basic non-parametric bootstrap estimation | p. 159 |
| A1.2 The parametric bootstrap | p. 160 |
| A1.3 Bootstrap resampling for dependent data | p. 160 |
| A1.4 The principle of pivoting and variance stabilisation | p. 161 |
| A1.5 Limitations of bootstrap procedure | p. 163 |
| A1.6 Hypothesis testing | p. 163 |
| A1.7 The bootstrap matched filter | p. 167 |
| A1.8 Bootstrap model selection | p. 167 |
| A1.9 Noise floor estimation | p. 170 |
| Appendix 2 Bootstrap Matlab Toolbox | p. 174 |
| References | p. 201 |
| Index | p. 215 |
