Spatial Statistics and Spatio-Temporal Data E-Book

Michael Sherman

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

Seitenzahl: 424

Veröffentlichungsjahr: 2011

Beschreibung

In the spatial or spatio-temporal context, specifying the correct covariance function is fundamental to obtain efficient predictions, and to understand the underlying physical process of interest. This book focuses on covariance and variogram functions, their role in prediction, and appropriate choice of these functions in applications. Both recent and more established methods are illustrated to assess many common assumptions on these functions, such as, isotropy, separability, symmetry, and intrinsic correlation.

After an extensive introduction to spatial methodology, the book details the effects of common covariance assumptions and addresses methods to assess the appropriateness of such assumptions for various data structures.

Key features:

• An extensive introduction to spatial methodology including a survey of spatial covariance functions and their use in spatial prediction (kriging) is given.
• Explores methodology for assessing the appropriateness of assumptions on covariance functions in the spatial, spatio-temporal, multivariate spatial, and point pattern settings.
• Provides illustrations of all methods based on data and simulation experiments to demonstrate all methodology and guide to proper usage of all methods.
• Presents a brief survey of spatial and spatio-temporal models, highlighting the Gaussian case and the binary data setting, along with the different methodologies for estimation and model fitting for these two data structures.
• Discusses models that allow for anisotropic and nonseparable behaviour in covariance functions in the spatial, spatio-temporal and multivariate settings.
• Gives an introduction to point pattern models, including testing for randomness, and fitting regular and clustered point patterns. The importance and assessment of isotropy of point patterns is detailed.

Statisticians, researchers, and data analysts working with spatial and space-time data will benefit from this book as well as will graduate students with a background in basic statistics following courses in engineering, quantitative ecology or atmospheric science.

Leseprobe

Contents

Preface

1 Introduction

1.1 Stationarity

1.2 The effect of correlation in estimation and prediction

2 Geostatistics

2.1 A model for optimal prediction and error assessment

2.2 Optimal prediction (kriging)

2.3 Prediction intervals

2.4 Universal kriging

2.5 The intuition behind kriging

3 Variogram and covariance models and estimation

3.1 Empirical estimation of the variogram or covariance function

3.2 On the necessity of parametric variogram and covariance models

3.3 Covariance and variogram models

3.4 Convolution methods and extensions

3.5 Parameter estimation for variogram and covariance models

3.6 Prediction for the phosphorus data

3.7 Nonstationary covariance models

4 Spatial models and statistical inference

4.1 Estimationinthe Gaussian case

4.2 Estimation for binary spatial observations

5 Isotropy

5.1 Geometric anisotropy

5.2 Other typesofanisotropy

5.3 Covariance modeling under anisotropy

5.4 Detectionofanisotropy: the rose plot

5.5 Parametric methodstoassess isotropy

5.6 Nonparametric methods of assessing anisotropy

5.7 Assessment of isotropy for general sampling designs

5.8 An assessment of isotropy for the longleaf pine sizes

6 Space–time data

6.1 Space–time observations

6.2 Spatio-temporal stationarity and spatio-temporal prediction

6.3 Empirical estimation of the variogram, covariance models, and estimation

6.4 Spatio-temporal covariance models

6.5 Space–time models

6.6 Parametric methods of assessing full symmetry and space–time separability

6.7 Nonparametric methods of assessing full symmetry and space–time separability

6.8 Nonstationary space–time covariance models

7 Spatial point patterns

7.1 The Poisson process and spatial randomness

7.2 Inhibition models

7.3 Clustered models

8 Isotropy for spatial point patterns

8.1 Some large sample results

8.2 Atest for isotropy

8.3 Practical issues

8.4 Numerical results

8.5 An application to leukemia data

9 Multivariate spatial and spatio-temporal models

9.1 Cokriging

9.2 Analternativetocokriging

9.3 Multivariate covariance functions

9.4 Testing and assessing intrinsic correlation

9.5 Numerical experiments

9.6 Adata applicationtopollutants

9.7 Discussion

10 Resampling for correlated observations

10.1 Independent observations

10.2 Other data structures

10.3 Model-based bootstrap

10.4 Model-free resampling methods

10.5 Spatial resampling

10.6 Model-free spatial resampling

10.7 Unequally spaced observations

Bibliography

Index

Spatial Statistics and Spatio-Temporal Data

WILEY SERIES IN PROBABILITY AND STATISTICS

Established by WALTER A. SHEWHART and SAMUEL S. WILKS

Editors

David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Harvey Goldstein,

Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F. M. Smith,

Ruey S. Tsay, Sanford Weisberg

Editors Emeriti

Vic Barnett, Ralph A. Bradley, J. Stuart Hunter, J.B. Kadane, David G. Kendall,

Jozef L. Teugels

A complete list of the titles in this series appears at the end of this volume

This edition first published 2011

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Library of Congress Cataloging-in-Publication Data

Sherman, Michael, 1963–

Spatial statistics and spatio-temporal data : covariance functions and directional properties / Michael Sherman.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-69958-4 (cloth)

1. Spatial analysis (Statistics) 2. Analysis of covariance. I. Title.

QA278.2.S497 2010

519.5–dc22

2010029551

A catalogue record for this book is available from the British Library.

Print ISBN: 978-0-470-69958-4

ePDF ISBN: 978-0-470-97440-7

eBook ISBN: 978-0-470-97439-1 ePub

ISBN: 978-0-470-97492-6

Preface

The fields of spatial data and spatio-temporal data have expanded greatly over the previous 20 years. This has occurred as the amount of spatial and spatio-temporal data has increased. One main tool in spatial prediction is the covariance function or the variogram. Given these functions, we know how to make optimal predictions of quantities of interest at unsampled locations. In practice, these covariance functions are unknown and need to be estimated from sample data. Covariance functions and their estimation is a subset of the field of geostatistics. Several previous texts in geostatistics consider these topics in great detail, specifically those of Cressie (1993), Chiles and Delfiner (1999), Diggle and Ribeiro (2007), Isaaks and Srivastava (1989), and Stein (1999).

A common assumption on variogram or covariance functions is that they are isotropic, that is, not direction dependent. For spatio-temporal covariance functions, a common assumption is that the spatial and temporal covariances are separable. For multivariate spatial observations, a common assumption is intrinsic correlation; that is, that the variable correlations and spatial correlations are separable. All these types of assumptions make models simpler, and thus aid in effective parameter estimation in these covariance models. Much of this book details the effects of these assumptions, and addresses methods to assess the appropriateness of such assumptions for these various data structures.

Chapters 1–3 are an introduction to the topics of stationarity, spatial prediction, variogram and covariance models, and estimation for these models. Chapter 4 gives a brief survey of spatial models, highlighting the Gaussian case and the binary data setting and the different methodologies for these two data structures. Chapter 5 discusses the assumption of isotropy for spatial covariances, and methods to assess and correct for anisotropies; while Chapter 6 discusses models for spatio-temporal covariances and assessment of symmetry and separability assumptions. Chapter 7 serves as an introduction to spatial point patterns. In this chapter we discuss testing for spatial randomness and models for both regular and clustered point patterns. These and further topics in the analysis of point patterns can be found in, for example, Diggle (2003) or Illian et al. (2008). The isotropy assumption for point pattern models has not been as often addressed as in the geostatistical setting. Chapter 8 details methods for testing for isotropy based on spatial point pattern observations. Chapter 9 considers models for multivariate spatial and spatio-temporal observations and covariance functions for these data. Due to spatial correlations and unwieldy likelihoods in the spatial setting, many statistics are complicated. In particular, this means that variances and other distributional properties are difficult to derive analytically. Resampling methodology can greatly aid in estimating these quantities. For this reason, Chapter 10 gives some background and details on resampling methodology for independent, time series, and spatial observations.

The first four chapters and Chapters 7 and 10 of this book are relatively non-technical, and any necessary technical items should be accessible on the way. Chapters 5, 6, 8, and 9 often make reference to large sample theory, but the basic methodology can be followed without reference to these large sample results. The chapters that address the testing of various assumptions of covariance functions, Chapters 5, 6, 8, and 9, often rely on a common testing approach. This approach is repeated separately, to some extent, within each of these chapters. Hopefully, this will aid the data analyst who may be interested in only one or two of the data structures addressed in these chapters. There are no exercises given at the end of chapters. It is hoped that some of the details within the chapters will lend themselves to further exploration, if desired, for an instructor. All data analyses have been carried out using the R language, and various R packages. I have not listed any specific packages, as the continual growth and improvement of these packages would make this inappropriate. Furthermore, as R is freeware, users can experiment, and find the software they are most comfortable with.

This book introduces spatial covariance models and discusses their importance in making predictions. Whenever building models based on data, a key component is to assess the validity of any model assumptions. It is hoped that this book shows how this can be done, and hopefully suggests further methodology to expand the applicability of such assessments.

The content of this book could never have come into being without the benefits of associations with mentors, colleagues, collaborators, and students. Specifically, I greatly appreciate Ed Carlstein and Martin Tanner for their imparting of wisdom and experience to me when I was a student.I greatly thank colleagues, with whom many of the results in this book have been obtained. Specifically, I thank Tanya Apanosovich, Jim Calvin, Ray Carroll, Marc Genton, Yongtao Guan, Bo Li, Johan Lim, Arnab Maity, Dimitris Politis, Gad Ritvo, and Michael Speed for their collaborative efforts over the years. I also thank Professor Christopher K. Wikle for the use of the Pacific Ocean wind-speed data in Chapters 5 and 6, and Professor Sue Carrozza for use of the leukemia data in Chapter 8. Lastly, I sincerely appreciate the loving aid of my wife, Aviva Sherman, who provided the utmost emotional and technical support in the writing of this book.