## Wavelet Methods in Statistics with RWavelet methods have recently undergone a rapid period of development with important implications for a number of disciplines including statistics. This book has three main objectives: (i) providing an introduction to wavelets and their uses in statistics; (ii) acting as a quick and broad reference to many developments in the area; (iii) interspersing R code that enables the reader to learn the methods, to carry out their own analyses, and further develop their own ideas. The book code is designed to work with the freeware R package WaveThresh4, but the book can be read independently of R. The book introduces the wavelet transform by starting with the simple Haar wavelet transform, and then builds to consider more general wavelets, complex-valued wavelets, non-decimated transforms, multidimensional wavelets, multiple wavelets, wavelet packets, boundary handling, and initialization. Later chapters consider a variety of wavelet-based nonparametric regression methods for different noise models and designs including density estimation, hazard rate estimation, and inverse problems; the use of wavelets for stationary and non-stationary time series analysis; and how wavelets might be used for variance estimation and intensity estimation for non-Gaussian sequences. The book is aimed both at Masters/Ph.D. students in a numerate discipline (such as statistics, mathematics, economics, engineering, computer science, and physics) and postdoctoral researchers/users interested in statistical wavelet methods. Guy Nason is Professor of Statistics at the University of Bristol. He has been actively involved in the development of various wavelet methods in statistics since 1993. He was awarded the Royal Statistical Society’s 2001 Guy Medal in Bronze for work on wavelets in statistics. He was the author of the first, free, generally available wavelet package for statistical purposes in S and R (WaveThresh2). |

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### Contents

Introduction | 1 |

12 Why Use Wavelets? | 2 |

13 Why Wavelets in Statistics? | 11 |

14 Software and This Book | 13 |

Wavelets | 15 |

22 Haar Wavelets on Functions | 28 |

23 Multiresolution Analysis | 37 |

24 Vanishing Moments | 40 |

315 Block Thresholding | 128 |

316 Miscellanea and Discussion | 130 |

Related Wavelet Smoothing Techniques | 133 |

43 NonGaussian Noise | 138 |

44 Multidimensional Data | 140 |

45 Irregularly Spaced Data | 143 |

46 Confidence Bands | 150 |

47 Density Estimation | 155 |

25 WaveThresh Wavelets and What Some Look Like | 41 |

26 Other Wavelets | 45 |

27 The General Fast Discrete Wavelet Transform | 50 |

28 Boundary Conditions | 55 |

29 Nondecimated Wavelets | 57 |

210 Multiple Wavelets | 66 |

211 Wavelet Packet Transforms | 68 |

212 Nondecimated Wavelet Packet Transforms | 75 |

213 Multivariate Wavelet Transforms | 76 |

214 Other Topics | 78 |

Wavelet Shrinkage | 83 |

32 Wavelet Shrinkage | 84 |

33 The Oracle | 85 |

34 Test Functions | 88 |

36 Primary Resolution | 96 |

38 Crossvalidation | 98 |

39 False Discovery Rate | 100 |

310 Bayesian Wavelet Shrinkage | 101 |

312 NonDecimated Wavelet Shrinkage | 110 |

313 Multiple Wavelet Shrinkage Multiwavelets | 118 |

314 Complexvalued Wavelet Shrinkage | 120 |

48 Survival Function Estimation | 158 |

49 Inverse Problems | 163 |

Multiscale Time Series Analysis | 167 |

52 Stationary Time Series | 169 |

53 Locally Stationary Time Series | 174 |

54 Forecasting with Locally Stationary Wavelet Models | 192 |

55 Time Series with Wavelet Packets | 197 |

56 Related Topics and Discussion | 198 |

Multiscale Variance Stabilization | 201 |

61 Why the Square Root for Poisson? | 202 |

62 The Fisz Transform | 203 |

63 Poisson Intensity Function Estimation | 206 |

64 The HaarFisz Transform for Poisson Data | 207 |

65 Datadriven HaarFisz | 217 |

66 Discussion | 227 |

R Software for Wavelets and Statistics | 229 |

Notation and Some Mathematical Concepts | 231 |

Survival Function Code | 235 |

237 | |

253 | |