Kernel Smoothing E-Book

Sucharita Ghosh

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 288

Veröffentlichungsjahr: 2017

Beschreibung

Comprehensive theoretical overview of kernel smoothing methods with motivating examples

Kernel smoothing is a flexible nonparametric curve estimation method that is applicable when parametric descriptions of the data are not sufficiently adequate. This book explores theory and methods of kernel smoothing in a variety of contexts, considering independent and correlated data e.g. with short-memory and long-memory correlations, as well as non-Gaussian data that are transformations of latent Gaussian processes. These types of data occur in many fields of research, e.g. the natural and the environmental sciences, and others. Nonparametric density estimation, nonparametric and semiparametric regression, trend and surface estimation in particular for time series and spatial data and other topics such as rapid change points, robustness etc. are introduced alongside a study of their theoretical properties and optimality issues, such as consistency and bandwidth selection.

Addressing a variety of topics, Kernel Smoothing: Principles, Methods and Applications offers a user-friendly presentation of the mathematical content so that the reader can directly implement the formulas using any appropriate software. The overall aim of the book is to describe the methods and their theoretical backgrounds, while maintaining an analytically simple approach and including motivating examples—making it extremely useful in many sciences such as geophysics, climate research, forestry, ecology, and other natural and life sciences, as well as in finance, sociology, and engineering.

• A simple and analytical description of kernel smoothing methods in various contexts
• Presents the basics as well as new developments
• Includes simulated and real data examples

Kernel Smoothing: Principles, Methods and Applications is a textbook for senior undergraduate and graduate students in statistics, as well as a reference book for applied statisticians and advanced researchers. 

Leseprobe

Kernel Smoothing

Principles, Methods and Applications

Sucharita Ghosh

Swiss Federal Research Institute WSLBirmensdorf, Switzerland

This edition first published 2018 © 2018 by John Wiley & Sons Ltd.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this titleis available at http://www.wiley.com/go/permissions.

The right of Sucharita Ghosh to be identified as the author of this work has been asserted in accordance with law.

Registered Office(s)John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK

Editorial OfficeThe Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK

For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com.

Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats.

Limit of Liability/Disclaimer of WarrantyWhile the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

Library of Congress Cataloging-in-Publication Data

Names: Ghosh, S. (Sucharita), author. Title: Kernel smoothing : principles, methods and applications / by Sucharita Ghosh. Description: First edition. | Hoboken, NJ : John Wiley & Sons, 2018. | Includes bibliographical references and index. | Identifiers: LCCN 2017039516 (print) | LCCN 2017046749 (ebook) | ISBN 9781118890509 (pdf) | ISBN 9781118890516 (epub) | ISBN 9781118456057 Subjects: LCSH: Smoothing (Statistics) | Kernel functions. Classification: LCC QA278 (ebook) | LCC QA278 .G534 2018 (print) | DDC 511/.42--dc23 LC record available at https://lccn.loc.gov/2017039516

Cover Design: Wiley

Cover Image: © PASIEKA/SPL/Gettyimages

Preface

Typically, patterns in real data, which we may call curves or surfaces, will not follow simple rules. However, there may be a sufficiently good description in terms of a finite number of interpretable parameters. When this is not the case, or if the parametric description is too complex, a nonparametric approach is an option. In developing nonparametric curve estimation methods, however, sometimes we may take advantage of the vast array of available parametric statistical methods and adapt these to the nonparametric setting. While assessing properties of the nonparametric curve estimators, we will use asymptotic arguments.

This book grew out of a set of lecture notes for a course on smoothing given to the graduate students of Seminar für Statistik (Department of Mathematics, ETH, Zürich). To understand the material presented here, knowledge of linear algebra, calculus, and a background in statistical inference, in particular the theory of estimation, testing, and linear models should suffice. The textbooks Statistical Inference (Chapman & Hall) by Samuel David Silvey, Regression Analysis, Theory, Methods and Applications (Springer-Verlag) by Ashis Sen and Muni Srivastava, Linear Statistical Inference, second edition (John Wiley) by Calyampudi Radhakrishna Rao, and Robert Serfling’s book Approximation Theorems of Mathematical Statistics (John Wiley) are excellent sources for background material. For nonparametric curve estimation, there are several good books and in particular the classic Density Estimation (Chapman & Hall) by Bernard Silverman is a must-have for anyone venturing into this topic. The present text also includes some discussions on nonparametric curve estimation with time series and spatial data, in particular with different correlation types such as long-memory. A nice monograph on long-range dependence is Statistics for Long-Memory Processes (Chapman & Hall) by Jan Beran. Additional references to this topic as well as an incomplete list of textbooks on smoothing methods are included in the list of references.

Our discussion on nonparametric curve estimation starts with density estimation (Chapter 1) for continuous random variables, followed by a chapter on nonparametric regression (Chapter 2). Inspired by applications of nonparametric curve estimation techniques to dependent data, several chapters are dedicated to a selection of problems in nonparametric regression, specifically trend estimation (Chapter 3) and semiparametric regression (Chapter 4), with time series data and surface estimation with spatial observations (Chapter 5). While, for such data sets, types of dependence structures can be vast, we mainly focus on (slow) hyperbolic decays (long memory), as these types of data occur often in many important fields of applications in science as well as in business. Results for short-memory and anti-persistence are also presented in some cases. Of additional interest are spatial or temporal observations that are not necessarily Gaussian, but are unknown transformations of latent Gaussian processes. Moreover, their marginal probability distributions may be time (or spatial location) dependent and assume arbitrary (non-Gaussian) shapes. These types of model assumptions provide flexible yet parsimonious alternatives to stronger distributional assumptions such as Gaussianity or stationarity. An overview of the relevant literature on this topic is in Long Memory Processes – Probabilistic Properties and Statistical Models (Springer-Verlag) by Beran et al. (2013). This is advantageous for analyzing large-scale and long-term spatial and temporal data sets occurring, for instance, in the geosciences, forestry, climate research, medicine, finance, and others. The literature on nonparametric curve estimation is vast. There are other important methods that have not been covered here, such as wavelets – see Percival and Walden (2000), splines (a very brief discussion is included here in Chapter 2 of this book); see in particular Wahba (1990) and Eubank (1988), as well as other approaches. This book looks at kernel smoothing methods and even for kernel based approaches, admittedly, not all topics are presented here, and the focus is merely on a selection. The book also includes a few data examples, outlines of proofs are included in several cases, and otherwise references to relevant sources are provided. The data examples are based on calculations done using the S-plus statistical package (TIBCO Software, TIBCO Spotfire) and the R-package for statistical computing (The R Foundation for Statistical Computing).

Various people have been instrumental in seeing through this project. First and foremost, I am very grateful to my students at ETH, Zürich, for giving me the motivation to write this book and for pointing out many typos in earlier versions of the lecture notes. A big thank you goes to Debbie Jupe, Heather Kay, Richard Davies, and Liz Wingett, at John Wiley & Sons in Chichester, West Sussex, Alison Oliver at Oxford and to the editors at Wiley, India, for their support from the start of the project and for making it possible. I am grateful to the Swiss National Science Foundation for funding PhD students, the IT unit of the WSL for infallible support and for maintaining an extremely comfortable and state-of-the-art computing infrastructure, and the Forest Resources and Management Unit, WSL for generous funding and collaboration. Special thanks go to Jan Beran (Konstanz, Germany) for many helpful remarks on earlier versions of the manuscript and long-term collaboration on several papers on this and related topics. I also wish to thank Yuanhua Feng (Paderborn, Germany), Philipp Sibbertsen (Hannover, Germany), Rafal Kulik (Ottawa, Canada), Hans Künsch (Zurich, Switzerland), and my graduate students Dana Draghicescu, Patricia Menéndez, Hesam Montazeri, Gabrielle Moser, Carlos Ricardo Ochoa Pereira, and Fan Wu, for close collaboration, as well as Bimal Roy and various other colleagues at the Indian Statistical Institute, Kolkata and Liudas Giraitis at Queen Mary, University of London, for fruitful discussions and warm hospitality during recent academic trips. I want to thank the following for sharing data and subject specific knowledge, which have been used in related research elsewhere or in this book: Christoph Frei at MeteoSwiss and ETH, Zürich, various colleagues at the University of Bern, in particular, Willy Tinner at the Oeschger Centre for Climate Change Research, Brigitta Ammann at the Institute of Plant Sciences and Jakob Schwander at the Department of Physics, as well as Matthias Plattner at Hintermann & Weber, AG, Switzerland and various colleagues from the Swiss Federal Research Institute WSL, Birmensdorf, in particulear Urs-Beat Brändli, Fabrizio Cioldi and Andreas Schwyzer, all at the Forest Resources and Management unit. Data obtained from the MeteoSwiss, the Swiss National Forest Inventory, the Federal Office of the Environment (FOEN) in Switzerland, and various public domain data sets made available through the web platforms of the National Aeronautics and Space Administration (NASA), the National Oceanic and Atmospheric Administration (NOAA), and the Meteorological Office, UK (Met Office) used in related research elsewhere or used in this book for methodological illustrations are gratefully acknowledged.

My deepest gratitude goes to my family and friends. I want to thank my family Céline and Jan for being with me every step of the way, making sure that I finish this book at last, my family in India for their unfailing support, our colleagues Suju and Yuanhua for their hospitality on many occasions, Maria, Gunnar, Shila, and Goutam for holding the fort during conferences and other long trips, Wolfgang for his sense of humor, and last but not the least, Sir Hastings, our lovely Coton de Tuléar, for keeping us all on track with his incredible wit and judgment.

Sucharita Ghosh

Birmensdorf

CONTENTS

Cover Page

Title Page

Copyright Page

Preface

1 Density Estimation

1.1 Introduction

1.2 Histograms

1.3 Kernel density estimation

1.4 Multivariate density estimation

2 Nonparametric Regression

2.1 Introduction

2.2 Priestley–Chao regression estimator

2.3 Local polynomials

2.4 Nadaraya–Watson regression estimator

2.5 Bandwidth selection

2.6 Further remarks

3 Trend Estimation

3.1 Time series replicates

3.2 Irregularly spaced observations

3.3 Rapid change points

3.4 Nonparametric M-estimation of a trend function

4 Semiparametric Regression

4.1 Partial linear models with constant slope

4.2 Partial linear models with time-varying slope

5 Surface Estimation

5.1 Introduction

5.2 Gaussian subordination

5.3 Spatial correlations

5.4 Estimation of the mean and consistency

5.5 Variance estimation

5.6 Distribution function and spatial Gini index

References

Author Index

Subject Index

WILEY END USER LICENSE AGREEMENT

Guide

Cover

Table of Contents

Preface

Pages

Page_ix

Page_x

Page_xi

Page_xii

Page_1

Page_2

Page_3

Page_4

Page_7

Page_8

Page_9

Page_11

Page_12

Page_13

Page_14

Page_15

Page_16

Page_17

Page_18

Page_19

Page_20

Page_21

Page_22

Page_23

Page_25

Page_26

Page_27

Page_29

Page_30

Page_31

Page_32

Page_33

Page_34

Page_35

Page_36

Page_37

Page_38

Page_39

Page_40

Page_41

Page_42

Page_43

Page_44

Page_45

Page_46

Page_47

Page_48

Page_49

Page_50

Page_51

Page_52

Page_53

Page_54

Page_55

Page_56

Page_57

Page_59

Page_60

Page_61

Page_62

Page_63

Page_64

Page_65

Page_66

Page_67

Page_68

Page_69

Page_70

Page_71

Page_72

Page_73

Page_74

Page_75

Page_76

Page_77

Page_78

Page_79

Page_80

Page_81

Page_82

Page_83

Page_84

Page_85

Page_86

Page_87

Page_88

Page_89

Page_90

Page_91

Page_92

Page_93

Page_94

Page_95

Page_96

Page_97

Page_98

Page_99

Page_100

Page_101

Page_102

Page_103

Page_104

Page_105

Page_108

Page_109

Page_110

Page_111

Page_113

Page_114

Page_115

Page_116

Page_117

Page_118

Page_119

Page_120

Page_121

Page_122

Page_123

Page_124

Page_125

Page_126

Page_127

Page_128

Page_129

Page_130

Page_131

Page_132

Page_133

Page_134

Page_135

Page_136

Page_137

Page_138

Page_139

Page_140

Page_141

Page_142

Page_143

Page_144

Page_145

Page_146

Page_147

Page_148

Page_149

Page_150

Page_151

Page_152

Page_153

Page_154

Page_155

Page_156

Page_157

Page_158

Page_159

Page_160

Page_161

Page_163

Page_164

Page_165

Page_166

Page_167

Page_168

Page_169

Page_170

Page_171

Page_172

Page_173

Page_174

Page_175

Page_176

Page_177

Page_178

Page_179

Page_181

Page_182

Page_184

Page_185

Page_187

Page_188

Page_189

Page_190

Page_191

Page_192

Page_193

Page_194

Page_195

Page_196

Page_197

Page_198

Page_199

Page_200

Page_201

Page_201

Page_202

Page_203

Page_204

Page_205

Page_206

Page_207

Page_208

Page_209

Page_210

Page_211

Page_212

Page_213

Page_214

Page_215

Page_217

Page_218

Page_219

Page_220

Page_221

Page_222

Page_223

Page_224

Page_225

Page_226

Page_227

Page_228

Page_229

Page_230

Page_231

Page_232

Page_233

Page_234

Page_235

Page_236

Page_237

Page_238

Page_239

Page_240

Page_241

Page_242

Page_243

Page_244

Page_245

Page_246

Page_247

Page_248

Page_249

Page_251

Page_252

Page_253

Page_254

Page_255

Page_256

Page_257

Page_258

Page_259

Page_260