66,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Probability and Statistics
- Sprache: Englisch
SPATIAL ANALYSIS
Explore the foundations and latest developments in spatial statistical analysis
In Spatial Analysis, two distinguished authors deliver a practical and insightful exploration of the statistical investigation of the interdependence of random variables as a function of their spatial proximity. The book expertly blends theory and application, offering numerous worked examples and exercises at the end of each chapter.
Increasingly relevant to fields as diverse as epidemiology, geography, geology, image analysis, and machine learning, spatial statistics is becoming more important to a wide range of specialists and professionals. The book includes:
- Thorough introduction to stationary random fields, intrinsic and generalized random fields, and stochastic models
- Comprehensive exploration of the estimation of spatial structure
- Practical discussion of kriging and the spatial linear model
Spatial Analysis is an invaluable resource for advanced undergraduate and postgraduate students in statistics, data science, digital imaging, geostatistics, and agriculture. It’s also an accessible reference for professionals who are required to use spatial models in their work.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 585
Veröffentlichungsjahr: 2022
Ähnliche
Table of Contents
Cover
Title Page
Copyright
Dedication
Dedication
List of Figures
List of Tables
Preface
List of Notation and Terminology
1 Introduction
1.1 Spatial Analysis
1.2 Presentation of the Data
1.3 Objectives
1.4 The Covariance Function and Semivariogram
1.5 Behavior of the Sample Semivariogram
1.6 Some Special Features of Spatial Analysis
Exercises
2 Stationary Random Fields
2.1 Introduction
2.2 Second Moment Properties
2.3 Positive Definiteness and the Spectral Representation
2.4 Isotropic Stationary Random Fields
2.5 Construction of Stationary Covariance Functions
2.6 Matérn Scheme
2.7 Other Examples of Isotropic Stationary Covariance Functions
2.8 Construction of Nonstationary Random Fields
2.9 Smoothness
2.10 Regularization
2.11 Lattice Random Fields
2.12 Torus Models
2.13 Long‐range Correlation
2.14 Simulation
Exercises
3 Intrinsic and Generalized Random Fields
3.1 Introduction
3.2 Intrinsic Random Fields of Order
3.3 Characterizations of Semivariograms
3.4 Higher Order Intrinsic Random Fields
3.5 Registration of Higher Order Intrinsic Random Fields
3.6 Generalized Random Fields
3.7 Generalized Intrinsic Random Fields of Intrinsic Order
3.8 Spectral Theory for Intrinsic and Generalized Processes
3.9 Regularization for Intrinsic and Generalized Processes
3.10 Self‐Similarity
3.11 Simulation
3.12 Dispersion Variance
Exercises
4 Autoregression and Related Models
4.1 Introduction
4.2 Background
4.3 Moving Averages
4.4 Finite Symmetric Neighborhoods of the Origin in
4.5 Simultaneous Autoregressions (SARs)
4.6 Conditional Autoregressions (CARs)
4.7 Limits of CAR Models Under Fine Lattice Spacing
4.8 Unilateral Autoregressions for Lattice Random Fields
4.9 Markov Random Fields (MRFs)
4.10 Markov Mesh Models
Exercises
5 Estimation of Spatial Structure
5.1 Introduction
5.2 Patterns of Behavior
5.3 Preliminaries
5.4 Exploratory and Graphical Methods
5.5 Maximum Likelihood for Stationary Models
5.6 Parameterization Issues for the Matérn Scheme
5.7 Maximum Likelihood Examples for Stationary Models
5.8 Restricted Maximum Likelihood (REML)
5.9 Vecchia's Composite Likelihood
5.10 REML Revisited with Composite Likelihood
5.11 Spatial Linear Model
5.12 REML for the Spatial Linear Model
5.13 Intrinsic Random Fields
5.14 Infill Asymptotics and Fractal Dimension
Exercises
6 Estimation for Lattice Models
6.1 Introduction
6.2 Sample Moments
6.3 The AR(1) Process on
6.4 Moment Methods for Lattice Data
6.5 Approximate Likelihoods for Lattice Data
6.6 Accuracy of the Maximum Likelihood Estimator
6.7 The Moment Estimator for a CAR Model
Exercises
7 Kriging
7.1 Introduction
7.2 The Prediction Problem
7.3 Simple Kriging
7.4 Ordinary Kriging
7.5 Universal Kriging
7.6 Further Details for the Universal Kriging Predictor
7.7 Stationary Examples
7.8 Intrinsic Random Fields
7.9 Intrinsic Examples
7.10 Square Example
7.11 Kriging with Derivative Information
7.12 Bayesian Kriging
7.13 Kriging and Machine Learning
7.14 The Link Between Kriging and Splines
7.15 Reproducing Kernel Hilbert Spaces
7.16 Deformations
Exercises
8 Additional Topics
8.1 Introduction
8.2 Log‐normal Random Fields
8.3 Generalized Linear Spatial Mixed Models (GLSMMs)
8.4 Bayesian Hierarchical Modeling and Inference
8.5 Co‐kriging
8.6 Spatial–temporal Models
8.7 Clamped Plate Splines
8.8 Gaussian Markov Random Field Approximations
8.9 Designing a Monitoring Network
Exercises
Appendix A Mathematical Background
A.1 Domains for Sequences and Functions
A.2 Classes of Sequences and Functions
A.3 Matrix Algebra
A.4 Fourier Transforms
A.5 Properties of the Fourier Transform
A.6 Generalizations of the Fourier Transform
A.7 Discrete Fourier Transform and Matrix Algebra
A.8 Discrete Cosine Transform (DCT)
A.9 Periodic Approximations to Sequences
A.10 Structured Matrices in
Dimension
A.11 Matrix Approximations for an Inverse Covariance Matrix
A.12 Maximum Likelihood Estimation
A.13 Bias in Maximum Likelihood Estimation
Appendix B A Brief History of the Spatial Linear Model and the Gaussian Process Approach
B.1 Introduction
B.2 Matheron and Watson
B.3 Geostatistics at Leeds 1977–1987
B.4 Frequentist vs. Bayesian Inference
References and Author Index
Index
Wiley End User License Agreement
List of Tables
Chapter 1
Table 1.1 Illustrative data
, on a
regular grid, represented in various ...
Table 1.2 Elevation data: elevation
in feet above the sea level, where
,...
Table 1.3 Bauxite data: percentage ore grade for bauxite at
locations.
Table 1.4 Gravimetric data: local gravity measurements in Quebec, Canada.
Table 1.5 Semivariograms in each direction for the gravimetric data.
Table 1.6 Soil data: surface pH in
on an
grid.
Table 1.7 Mercer–Hall wheat yield (in lbs.) for 20 (rows)
25 (columns) ag...
Table 1.8 Aggregated Mercer–Hall wheat data for plots aggregated into block...
Chapter 2
Table 2.1 Some radial covariance functions.
Table 2.2 Special cases of the Matérn covariance function in 2.34 for half‐...
Table 2.3 Some examples of stationary covariance functions
on the circle,...
Chapter 3
Table 3.1 Self‐similar random fields with spectral density
: some particul...
Chapter 5
Table 5.1 Parameter estimates (and standard errors) for the bauxite data us...
Table 5.2 Parameter estimates (and standard errors) for the elevation data ...
Table 5.3 Parameter estimates (with standard errors in parentheses) for Vec...
Chapter 7
Table 7.1 Notation used for kriging at the data sites
, and at the predict...
Table 7.2 Various methods of determining the kriging predictor
, where
c...
Table 7.3 Comparison between the terminology and notation of this book for ...
Appendix A
Table A.1 Types of domain.
Table A.2 Domains for Fourier transforms and inverse Fourier transforms in ...
Table A.3 Three types of boundary condition for a one‐dimensional set of da...
Table A.4 Some examples of Toeplitz, circulant, and folded circulant matric...
Table A.5 First and second derivatives of
and
with respect to
and
....
List of Illustrations
Chapter 1
Figure 1.1 Fingerprint of R A Fisher, taken from Mardia's personal collectio...
Figure 1.2 Elevation data: (a) raw plot giving the elevation at each site an...
Figure 1.3 Panels (a), (b), and (c) show interpolated plots for the elevatio...
Figure 1.4 Bauxite data: (a) raw plot giving the ore grade at each site and ...
Figure 1.5 Landsat data (
pixels): image plot.
Figure 1.6 Synthetic Landsat data: image plot.
Figure 1.7 Typical semivariogram, showing the range, nugget variance, and si...
Figure 1.8 Angle convention for polar coordinates. Angles are measured clock...
Figure 1.9 Fingerprint section data (218 pixels wide by 356 pixels high): (a...
Figure 1.10 Elevation data: (a) directional semivariograms and (b) omnidirec...
Figure 1.11 Bauxite data: (a) directional semivariograms and (b) omnidirecti...
Figure 1.12 Directional semivariograms for (a) the Landsat data and (b) the ...
Figure 1.13 Gravimetric data: (a) bubble plot and (b) directional semivariog...
Figure 1.14 Soil data: (a) bubble plot and (b) directional semivariograms.
Figure 1.15 Mercer–Hall wheat data: log–log plot of variance vs. block size....
Chapter 2
Figure 2.1 Matérn covariance functions for varying index parameters. The ran...
Chapter 3
Figure 3.1 Examples of radial semivariograms: the power schemes
for
and ...
Figure 3.2 A linear semivariogram with a nugget effect:
.
Chapter 4
Figure 4.1 Panels (a) and (b) illustrate the first‐order basic and full neig...
Figure 4.2 Three notions of “past” of the origin in
: (a) quadrant past (
)...
Figure 4.3 (a) First‐order basic neighborhood (nbhd) of the origin ○ in
di...
Figure 4.4 (a) First‐order full neighborhood (nbhd) of the origin ○ in
dim...
Chapter 5
Figure 5.1 Bauxite data: Bubble plot and directional semivariograms.
Figure 5.2 Elevation data: Bubble plot and directional semivariograms.
Figure 5.3 Bauxite data: Profile log‐likelihoods together with 95% confidenc...
Figure 5.4 Bauxite data: sample isotropic semivariogram values and fitted Ma...
Figure 5.5 Elevation data: Profile log‐likelihoods together with 95% confide...
Figure 5.6 Unilateral lexicographic neighborhood of full size
for lattice ...
Figure 5.7 Profile log‐likelihoods for self‐similar models of intrinsic orde...
Chapter 6
Figure 6.1 Mercer–Hall data: bubble plot. See Example 6.1 for an interpretat...
Figure 6.2 A plot of the sample and two fitted covariance functions (“biased...
Figure 6.3 A plot of the sample and two fitted covariance functions (“fold‐m...
Figure 6.4 Relative efficiency of the composite likelihood estimator in AR(1...
Chapter 7
Figure 7.1 Kriging predictor for
data points assumed to come from a statio...
Figure 7.2 Kriging predictor for
data points assumed to come from a statio...
Figure 7.3 Panel (a) shows the interpolated kriging surface for the elevatio...
Figure 7.4 Panel (a) shows a contour plot for the kriged surface fitted to t...
Figure 7.5 Kriging predictor and kriging standard errors for
data points a...
Figure 7.6 Panel (a) shows the interpolated kriging surface for the gravimet...
Figure 7.7 Kriging predictors for Example 7.6. For Panel (a), the kriging pr...
Figure 7.8 Deformation of a square (a) into a kite (b) using a thin‐plate sp...
Appendix B
Figure B.1 Creators of Kriging: Danie Krige and Georges Matheron.
Figure B.2 Letter from Matheron to Mardia, dated 1990.
Figure B.3 Translation of the letter from Matheron to Mardia, dated 1990.
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
Dedication
List of Figures
List of Tables
Preface
List of Notation and Terminology
Begin Reading
Appendix A Mathematical Background
Appendix B A Brief History of the Spatial Linear Model and the Gaussian Process Approach
References and Author Index
Index
Wiley End User License Agreement
Pages
ii
iii
iv
v
vi
xiii
xiv
xv
xvi
xvii
xviii
xix
xx
xxi
xxii
xxiii
xxv
xxvi
xxvii
xxviii
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
347
348
349
350
351
352
353
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
WILEY SERIES IN PROBABILITY AND STATISTICS
Established by Walter A. Shewhart and Samuel S. Wilks
The Wiley Series in Probability and Statistics is well established and authoritative. It covers many topics of current research interest in both pure and applied statistics and probability theory. Written by leading statisticians and institutions, the titles span both state‐of‐the‐art developments in the field and classical methods.
Reflecting the wide range of current research in statistics, the series encompasses applied, methodological and theoretical statistics, ranging from applications and new techniques made possible by advances in computerized practice to rigorous treatment of theoretical approaches.
This series provides essential and invaluable reading for all statisticians, whether in academia, industry, government, or research.
Spatial Analysis
John T. KentUniversity of Leeds, UK
Kanti V. MardiaUniversity of Leeds, UKUniversity of Oxford, UK
This edition first published 2022© 2022 John Wiley and 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 title is available at http://www.wiley.com/go/permissions.
The right of John T. Kent and Kanti V. Mardia to be identified as the authors of this work has been asserted in accordance with law.
Registered OfficesJohn Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USAJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK
Editorial Office9600 Garsington Road, Oxford, OX4 2DQ, 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 WarrantyThe contents of this work are intended to further general scientific research, understanding, and discussion only and are not intended and should not be relied upon as recommending or promoting scientific method, diagnosis, or treatment by physicians for any particular patient. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of medicines, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each medicine, equipment, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While 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 applied for
ISBN 9780471632054 (hardback); ISBN 9781118763568 (adobe pdf);ISBN 9781118763575 (epub); 9781118763551 (obook)
Cover Design: WileyCover Images: Courtesy of John Kent and Kanti Mardia
To my wife Sue for all her patience during the many years it has taken to complete the book
(John Kent)
To my son Hemant and daughter‐in‐law Preeti
— with Jainness
(Kanti Mardia)
“Whatever there is in all the three worlds, which are possessed of moving and non‐moving beings, cannot exist as apart from the ‘Ganita’ (mathematics/statistics).”
Acharya Mahavira (Jain monk, AD 850)
List of Figures
Figure 1.1 Fingerprint of R A Fisher, taken from Mardia's personal collection. A blowup of the marked rectangular section is given in Figure 1.9.
Figure 1.2 Elevation data: (a) raw plot giving the elevation at each site and (b) bubble plot where larger elevations are indicated by bigger circles.
Figure 1.3 Panels (a), (b), and (c) show interpolated plots for the elevation data, as a contour map, a perspective plot (viewed from the top of the region), and an image plot, respectively. Panel (d) shows a contour map of the corresponding standard errors.
Figure 1.4 Bauxite data: (a) raw plot giving the ore grade at each site and (b) bubble plot where larger ore grades are indicated by bigger circles.
Figure 1.5 Landsat data ( pixels): image plot.
Figure 1.6 Synthetic Landsat data: image plot.
Figure 1.7 Typical semivariogram, showing the range, nugget variance, and sill.
Figure 1.8 Angle convention for polar coordinates. Angles are measured clockwise from vertical.
Figure 1.9 Fingerprint section data (218 pixels wide by 356 pixels high): (a) image plot and (b) directional semivariograms.
Figure 1.10 Elevation data: (a) directional semivariograms and (b) omnidirectional semivariogram.
Figure 1.11 Bauxite data: (a) directional semivariograms and (b) omnidirectional semivariogram.
Figure 1.12Directional semivariograms for (a) the Landsat data and (b) the synthetic Landsat data.
Figure 1.13 Gravimetric data: (a) bubble plot and (b) directional semivariograms.
Figure 1.14 Soil data: (a) bubble plot and (b) directional semivariograms.
Figure 1.15 Mercer–Hall wheat data: log–log plot of variance vs. block size.
Figure 2.1 Matérn covariance functions for varying index parameters. The range and scale parameters have been chosen so that the covariance functions match at lags and .
Figure 3.1 Examples of radial semivariograms: the power schemes for and the exponential scheme . All the semivariograms have been scaled to take the same value for .
Figure 3.2 A linear semivariogram with a nugget effect: .
Figure 4.1 Panels (a) and (b) illustrate the first‐order basic and full neighborhoods of the origin in the plane. Panel (c) illustrates the second‐order basic neighborhood.
Figure 4.2 Three notions of “past” of the origin in : (a) quadrant past (), (b) lexicographic past (), and (c) weak past (). In each plot, ○ denotes the origin, denotes a site in the past, and denotes a site in the future.
Figure 4.3 (a) First‐order basic neighborhood (nbhd) of the origin ○ in dimensions. Neighbors of the origin are indicated by . (b) Two types of clique in addition to singleton cliques: horizontal and vertical edges.
Figure 4.4 (a) First‐order full neighborhood (nbhd) of the origin ○ in dimensions. Neighbors of the origin are indicated by . (b) Seven types of clique in addition to singleton cliques: horizontal and vertical edges, four shapes of triangle and a square.
Figure 5.1 Bauxite data: Bubble plot and directional semivariograms.
Figure 5.2 Elevation data: Bubble plot and directional semivariograms.
Figure 5.3 Bauxite data: Profile log‐likelihoods together with 95% confidence intervals. Exponential model, no nugget effect.
Figure 5.4 Bauxite data: sample isotropic semivariogram values and fitted Matérn semivariograms with a nugget effect, for (solid), (dashed), and (dotted).
Figure 5.5Elevation data: Profile log‐likelihoods together with 95% confidence intervals. Exponential model, no nugget effect.
Figure 5.6 Unilateral lexicographic neighborhood of full size for lattice data; current site marked by ; neighborhood sites in the lexicographic past marked by . Other sites are marked by a dot.
Figure 5.7 Profile log‐likelihoods for self‐similar models of intrinsic order , as a function of the index , both without a nugget effect (dashed line) and with a nugget effect (solid line). In addition, the log‐likelihood for each , with the parameters estimated by MINQUE (described In Section 5.13), is shown (dotted line).
Figure 6.1 Mercer–Hall data: bubble plot. See Example 6.1 for an interpretation.
Figure 6.2 A plot of the sample and two fitted covariance functions (“biased‐mom‐ML”) for a CAR model fitted to the leftmost 13 columns of the Mercer–Hall data (Example 6.1). The data have been summarized by the biased sample covariance function. The four panels show the covariance function in the four principal directions with the sample covariances (open circles) together with the fitted covariances using moment estimation (solid lines) and maximum likelihood estimation (dashed lines).
Figure 6.3 A plot of the sample and two fitted covariance functions (“fold‐mom‐ML”) for a CAR model fitted to the leftmost 13 columns of the Mercer–Hall data (Example 6.2). The data have been summarized by the folded sample covariance function. The four panels show the covariance function in the four principal directions with the sample covariances (open circles) together with the fitted covariances using moment estimation (solid lines) and maximum likelihood estimation (dashed lines).
Figure 6.4 Relative efficiency of the composite likelihood estimator in AR(1) model relative to the ML estimator.
Figure 7.1 Kriging predictor for data points assumed to come from a stationary random field with a squared exponential covariance function 7.55, without a nugget effect, with mean 0. Panels (a)–(c) show the kriging predictor for three choices of the range parameter, , respectively. Each panel shows the true unknown shifted sine function (solid), together with the fitted kriging curve (dashed), plus/minus twice the kriging standard errors (dotted).
Figure 7.2Kriging predictor for data points assumed to come from a stationary random field with a squared exponential covariance function 7.55, plus a nugget effect, with mean 0. The size of the relative nugget effect in Panels (a)–(c) is given by , respectively. Each panel shows the true unknown shifted sine function (solid), together with the fitted kriging curve (dashed), plus/minus twice the kriging standard errors (dotted).
Figure 7.3 Panel (a) shows the interpolated kriging surface for the elevation data, as a contour map. Panel (b) shows a contour map of the corresponding kriging standard errors. This figure is also included in Figure 1.3.
Figure 7.4 Panel (a) shows a contour plot for the kriged surface fitted to the bauxite data assuming a constant mean and an exponential covariance function for the error terms. Panel (b) shows the same plot assuming a quadratic trend and independent errors. Panels (c) and (d) show the kriging standard errors for the models in (a) and (b), respectively.
Figure 7.5 Kriging predictor and kriging standard errors for data points assumed to come from an intrinsic random field, , no nugget effect. The intrinsic drift is constant. Panel (a): no extrinsic drift; Panel (b): linear extrinsic drift. Each panel shows the fitted kriging curve (solid), plus/minus twice the kriging standard errors (dashed).
Figure 7.6 Panel (a) shows the interpolated kriging surface for the gravimetric data, as a contour map. Panel (b) shows a contour map of the corresponding kriging standard errors.
Figure 7.7
List of Tables
Table 1.1 Illustrative data , on a regular grid, represented in various ways, .
Table 1.2 Elevation data: elevation in feet above the sea level, where , .
Table 1.3 Bauxite data: percentage ore grade for bauxite at locations.
Table 1.4 Gravimetric data: local gravity measurements in Quebec, Canada.
Table 1.5 Semivariograms in each direction for the gravimetric data.
Table 1.6 Soil data: surface pH in on an grid.
Table 1.7 Mercer–Hall wheat yield (in lbs.) for 20 (rows) 25 (columns) agricultural plots (Mercer and Hall, 1911), where top–bottom corresponds to West–East, and left–right corresponds to North–South.
Table 1.8 Aggregated Mercer–Hall wheat data for plots aggregated into blocks, giving the array layout for the blocks, the block dimensions, the block size (number of plots in each block), the number of blocks , and sample variance .
Table 2.1 Some radial covariance functions.
Table 2.2 Special cases of the Matérn covariance function in (2.34) for half‐integer with scale parameter .
Table 2.3 Some examples of stationary covariance functions on the circle, together with the terms in their Fourier series.
Table 3.1 Self‐similar random fields with spectral density : some particular cases
Table 5.1Parameter estimates (and standard errors) for the bauxite data using the Matérn model with a constant mean and with various choices for the index .
Table 5.2 Parameter estimates (and standard errors) for the elevation data using the Matérn model with a constant mean and with various choices for the index .
Table 5.3 Parameter estimates (with standard errors in parentheses) for Vecchia's composite likelihood in Example 5.7 for the synthetic Landsat data, using different sizes of neighborhood
Table 7.1 Notation used for kriging at the data sites , and at the prediction and data sites .
Table 7.2 Various methods of determining the kriging predictor , where can be defined in terms of transfer matrices by or in terms of by .
Table 7.3
Preface
Spatial statistics is concerned with data collected at various spatial locations or sites, typically in a Euclidean space . The important cases in practice are , corresponding to the data on the line, in the plane, or in 3‐space, respectively. A common property of spatial data is “spatial continuity,” which means that measurements at nearby locations will tend to be more similar than measurements at distant locations. Spatial continuity can be modeled statistically using a covariance function of a stochastic process for which observations at nearby sites are more highly correlated than at distant sites. A stochastic process in space is also known as a random field.
One distinctive feature of spatial statistics, and related areas such as time series, is that there is typically just one realization of the stochastic process to analyze. Other branches of statistics often involve the analysis of independent replications of data.
The purpose of this book is to develop the statistical tools to analyze spatial data. The main emphasis in the book is on Gaussian processes. Here is a brief summary of the contents. A list of Notation and Terminology is given at the start for ease of reference. An introduction to the overall objectives of spatial analysis, together with some exploratory methods, is given in Chapter 1. Next is the specification of possible covariance functions (Chapter 2 for the stationary case and Chapter 3 for the intrinsic case). It is helpful to distinguish discretely indexed, or lattice, processes from continuously indexed processes. In particular, for lattice processes, it is possible to specify a covariance function through an autoregressive model (the SAR and CAR models of Chapter 4), with specialized estimation procedures (Chapter 6). Model fitting through maximum likelihood and related ideas for continuously indexed processes is covered in Chapter 5. An important use of spatial models is kriging, i.e. the prediction of the process at a collection of new sites, given the values of the process at a collection of training sites (Chapter 7), and in particular the links to machine learning are explained. Some additional topics, for which there was not space for in the book, are summarized in Chapter 8. The technical mathematical tools have been collected in Appendix A for ease of reference. Appendix B contains a short historical review of the spatial linear model.
The development of statistical methodology for spatial data arose somewhat separately in several academic disciplines over the past century.
Agricultural field trials
. An area of land is divided into long, thin plots, and different crop is grown on each plot. Spatial correlation in the soil fertility can cause spatial correlation in the crop yields (Webster and Oliver,
2001
).
Geostatistics
. In mining applications, the concentration of a mineral of interest will often show spatial continuity in a body of ore. Two giants in the field of spatial analysis came out of this field. Krige (
1951
) set out the methodology for spatial prediction (now known as kriging) and Matheron (
1963
) developed a comprehensive theory for stationary and intrinsic random fields; see Appendix B.
Social and medical science
. Spatial continuity is an important property when describing characteristics that vary across a region of space. One application is in geography and environmetrics and key names include Cliff and Ord (
1981
), Anselin (
1988
), Upton and Fingleton (
1985
,
1989
), Wilson (
2000
), Lawson and Denison (
2002
), Kanevski and Maignan (
2004
), and Schabenberger and Gotway (
2005
). Another application is in public health and epidemiology, see, e.g., Diggle and Giorgi (2019).
Splines
. A very different approach to spatial continuity has been pursued in the field of nonparametric statistics. Spatial continuity of an underlying smooth function is ensured by imposing a
roughness penalty
when fitting the function to data by least squares. It turns out that fitted spline is identical to the kriging predictor under suitable assumptions on the underlying covariance function. Key names here include Wahba (
1990
) and Watson (
1984
). A modern treatment is given in Berlinet and Thomas‐Agnan (
2004
).
Mainstream statistics
. From at least the 1950s, mainstream statisticians have been closely involved in the development of suitable spatial models and suitable fitting procedures. Highlights include the work by Whittle (
1954
), Matérn (
1960
,
1986
), Besag (
1974
), Cressie (
1993
), and Diggle and Ribeiro (
2007
).
Probability theory and fractals
. For the most part, statisticians interested in asymptotics have focused on “outfill” asymptotics – the data sites cover an increasing domain as the sample size increases. The other extreme is “infill asymptotics” in which the interest is on the local smoothness of realizations from the spatial process. This infill topic has long been of interest to probabilists (e.g. Adler,
1981
). The smoothness properties of spatial processes underlie much of the theory of fractals (Mandelbrot,
1982
).
Machine learning
. Gaussian processes and splines have become a fundamental tool in machine learning. Key texts include Rasmussen and Williams (
2006
) and Hastie et al. (
2009
).
Morphometrics
. Starting with Bookstein (
1989
), a pair of thin‐plate splines have been used for the construction of deformations of two‐dimensional images. The thin‐plate spline is just a special case of kriging.
Image analysis
. Stationary random fields form a fundamental model for randomness in images, though typically the interest is in more substantive structures. Some books include Grenander and Miller (
2007
), Sonka et al. (
2013
), and Dryden and Mardia (
2016
). The two edited volumes Mardia and Kanji (
1993
) and Mardia (
1994
) are still relevant for the underlying statistical theory in image analysis; in particular, Mardia and Kanji (
1993
) contains a reproduction of some seminal papers in the area.
The book is designed to be used in teaching. The statistical models and methods are carefully explained, and there is an extensive set of exercises. At the same time the book is a research monograph, pulling together and unifying a wide variety of different ideas.
A key strength of the book is a careful description of the foundations of the subject for stationary and related random fields. Our view is that a clear understanding of the basics of the subject is needed before the methods can be used in more complicated situations. Subtleties are sometimes skimmed over in more applied texts (e.g. how to interpret the “covariance function” for an intrinsic process, especially of higher order, or a generalized process, and how to specify their spectral representations). The unity of the subject, ranging from continuously indexed to lattice processes, has been emphasized. The important special case of self‐similar intrinsic covariance functions is carefully explained. There are now a wide variety of estimation methods, mainly variants and approximations to maximum likelihood, and these are explored in detail.
There is a careful treatment of kriging, especially for intrinsic covariance functions where the importance of drift terms is emphasized. The link to splines is explained in detail. Examples based on real data, especially from geostatistics, are used to illustrate the key ideas.
The book aims at a balance between theory and illustrative applications, while remaining accessible to a wide audience. Although there is now a wide variety of books available on the subject of spatial analysis, none of them has quite the same perspective. There have been many books published on spatial analysis, and here we just highlight a few. Ripley (1988) was one of the first monographs in the mainstream Statistics literature. Some key books that complement the material in this book, especially for applications, include Cressie (1993), Diggle and Ribeiro (2007), Diggle and Giorgi (2019), Gelfand et al. (2010), Chilés and Delfiner (2012), Banerjee et al. (2015), van Lieshout (2019), and Rasmussen and Williams (2006).
What background does a reader need? The book assumes a knowledge of the ideas covered by intermediate courses in mathematical statistics and linear algebra. In addition, some familiarity with multivariate statistics will be helpful. Otherwise, the book is largely self‐contained. In particular, no prior knowledge of stochastic processes is assumed. All the necessary matrix algebra is included in Appendix A. Some knowledge of time series is not necessary, but will help to set some of the ideas into context.
There is now a wide selection of software packages to carry out spatial analysis, especially in R, and it is not the purpose in this book to compare them. We have largely used the package geoR (Ribeiro Jr and Diggle, 2001) and the program of Pardo‐Igúzquiza et al. (2008), with additional routines written where necessary. The data sets are available from a public repository at https://github.com/jtkent1/spatial-analysis-datasets.
Several themes receive little or no coverage in the book. These include point processes, discretely valued processes (e.g. binary processes), and spatial–temporal processes. There is little emphasis on a full Bayesian analysis when the covariance parameters needed to be estimated. The main focus is on methods related to maximum likelihood.
The book has had a long gestation period. When we started writing the book the 1980s, the literature was much sparser. As the writing of the book progressed, the subject has evolved at an increasing rate, and more sections and chapters have been added. As a result the coverage of the subject feels more complete. At last, this first edition is finished (though the subject continues to advance).
A series of workshops at Leeds University (the Leeds Annual Statistics Research [LASR] workshops), starting from 1979, helped to develop the cross‐disciplinary fertilization of ideas between Statistics and other disciplines. Some leading researchers who presented their work at these meetings include Julian Besag, Fred Bookstein, David Cox, Xavier Guyon, John Haslett, Chris Jennison, Hans Künsch, Alain Marechal, Richard Martin, Brian Ripley, and Tata Subba‐Rao.
We are extremely grateful to Wiley for their patience and help during the writing of the book, especially Helen Ramsey, Sharon Clutton, Rob Calver, Richard Davies, Kathryn Sharples, Liz Wingett, Kelvin Matthews, Alison Oliver, Viktoria Hartl‐Vida, Ashley Alliano, Kimberly Monroe‐Hill, and Paul Sayer. Secretarial help at Leeds during the initial development was given by Margaret Richardson, Christine Rutherford, and Catherine Dobson.
We have had helpful discussions with many participants at the LASR workshops and with colleagues and students about the material in the book. These include Robert Adler, Francisco Alonso, Jose Angulo, Robert Aykroyd, Andrew Baczkowski, Noel Cressie, Sourish Das, Pierre Delfiner, Peter Diggle, Peter Dowd, Ian Dryden, Alan Gelfand, Christine Gill, Chris Glasbey, Arnaldo Goitía, Colin Goodall, Peter Green, Ulf Grenander, Luigi Ippoliti, Anil Jain, Giovanna Jona Lasinio, André Journel, Freddie Kalaitzis, David Kendall, Danie Krige, Neil Lawrence, Toby Lewis, John Little, Roger Marshall, Georges Matheron, Lutz Mattner, Charles Meyer, Michael Miller, Mohsen Mohammadzadeh, Debashis Mondal, Richard Morris, Ali Mosammam, Nitis Mukhopadhyay, Keith Ord, E Pardo‐Igúzquiza, Anna Persson, Sophia Rabe, Ed Redfern, Allen Royale, Sujit Sahu, Paul Sampson, Bernard Silverman, Nozer Singpurwalla, Paul Switzer, Charles Taylor, D. Vere‐Jones, Alan Watkins, Geof Watson, Chris Wikle, Alan Wilson, and Jim Zidek.
John is grateful to his wife Sue for her support in the writing of this book, especially with the challenges of the Covid pandemic. Kanti would like to thank the Leverhulme Trust for an Emeritus Fellowship and Anna Grundy of the Trust for simplifying the administration process. Finally, he would like to express his sincere gratitude to his wife and his family for continuous love, support and compassion during his research writings such as this monograph.
We would be pleased to hear about any typographical or other errors in the text.
30 June 2021
John T. Kent
Kanti V. Mardia
List of Notation and Terminology
Here is a list of some of the key notations and terminology used in the book.
and
denote the real numbers and integers.
For a dimension
, a
site
is a location
or
. The elements or components of a site
are written using square brackets
Note is not in bold face.
A
random field
is synonymous with a
stochastic process
. A random field on
is written as
, using function notation. A random field on the
lattice
is written as
, using subscript notation. A random field is often assumed to be a Gaussian process (GP).
The mean function and covariance function are written as
and covariance function
. In the stationary case,
is constant and
depends only on the
lag
. In the lattice case, use subscripts, e.g.
.
A stationary covariance function
can be written as a product of a
marginal variance σ
2
and an
autocorrelation function
ρ
(
h
).
An
intrinsic
random field extends the idea of a stationary random field. Write
for an intrinsic random field of order
(IRF‐
) with
intrinsic
covariance function
. For an intrinsic random field of order 0 (IRF‐0), the
semivariogram
is given by
. A
registered
version of an intrinsic random field is denoted
.
For a stationary model, a
scheme
is a parameterized family of covariance functions. For an intrinsic model, a scheme is a parameterized family of intrinsic covariance functions (or equivalently for an IRF‐0 model, a parameterized family of semivariograms).
A
nugget effect
refers to observations from a random field subject to measurement error, with variance typically denoted
.
The vector of covariance parameters for a stationary or intrinsic model, possibly including a nugget effect, is denoted
and can be partitioned as
in terms of an overall scale parameter and the remaining parameters.
Spaces of polynomials in
(
Section
3.4):
–
: Space of homogeneous polynomials of degree
, with dimension denoted
–
: Space of all polynomials of degree
in
, with dimension denoted
.
denotes the isotropic
self‐similar
intrinsic random field of index
and with drift space
(
Section
3.10). The intrinsic covariance function is denoted
and spectral density is denoted
.
Most of the book is concerned with
ordinary
random fields. There are also
generalized
random fields indexed by functions rather than sites and written as
with covariance functional
.
The surface area of the unit sphere in
is denoted
.
denotes a domain of sites in
or
. The notation encompasses several possibilities, including the following:
– An open subset
, e.g.
– A finite collection of sites
in
or
– The infinite lattice
– A finite rectangular lattice in
,
with dimension vector
and of size
. In the lattice case, sites in
can be denoted using letters such as
to emphasize the link to the continuous case, or using letters such as
to emphasize the fact that the components are integers.
For a finite domain, the notation stands for the number of sites in .
Frequencies in the Fourier domain are denoted
.
Vectors indexing data are treated as column vectors and are written as
in bold lowercase letters, with the components indicated by subscripts. The transpose of
is denoted
. This subscript convention is typical in multivariate analysis. Note the difference from the convention for sites
and frequencies
.
Random vectors, e.g.,
or
are written in bold letters, with the components indicated by subscripts. In particular, upper case is used when the distinction between a random quantity and its possible values needs emphasis.
Matrices are written using nonbold uppercase letters, e.g.
and
, with the elements of
written as
or as
. The two notations are synonymous. The columns of
are written using bracketed subscripts,
. For a square matrix, the determinant is denoted by either
or
; the notation
should not be confused with
, the size of a domain
described above.
If
and
are sites, then
is the
inner product
and
is the
squared Euclidean norm
.
Modulo notation
(for numbers) and
(for vectors) (
Section
A.1)
Check
and
convolution
notation. If
is a function of
, let
. Then
and the latter is symmetric in
.
The
Kronecker delta
and
Dirac delta
functions are denoted
and
, respectively.
. A finite symmetric neighborhood of the origin in
. The augmented neighborhood
includes the origin. Half of the neighborhood
is denoted
(
Section 4.4
).
. A half‐space in
, especially the lexicographic half‐space
(
Section 4.8
). Related ideas are the weak past
and quadrant past
(
Section 4.8
), and the partial past (
Section 5.9
).
Kriging
is essentially prediction for random fields. It comes in various forms including
simple kriging
,
ordinary kriging
,
universal kriging
, and
1Introduction
1.1 Spatial Analysis
Spatial analysis involves the analysis of data collected in a spatial region. A key aspect of such data is that observations at nearby sites tend to be highly correlated with one another. Any adequate statistical analysis should take these correlations into account.
The region in which the data lie is a subset of ‐dimensional space, , for some . The important cases in practice are , corresponding to the data on the line, in the plane, or in 3‐space, respectively.
The one‐dimensional case, , is already well known from the analysis of time series. Therefore, it will come as no surprise that many of the techniques introduced in this book represent generalizations of standard methodology from time‐series analysis. However, just as multivariate analysis contains techniques with no counterpart in univariate statistical analysis, spatial analysis includes techniques with no counterpart in time‐series analysis.
Spatial data arise in many applications. In mining we may have measurements of ore grade at a set of boreholes. If all the observations along each borehole are averaged together, we obtain data in dimensions, whereas if we retain the depth information at which each observation in the borehole is made, we obtain three‐dimensional data. In agriculture, experiments are usually performed on experimental plots, which are regularly spaced in a field. For environmental monitoring, data are collected at an array of monitoring sites, possibly irregularly located. There may also be a temporal component to this monitoring application as data are collected through time.
Digital images can also be viewed as spatial data sets. Examples include Landsat satellite images of areas of the earth's surface, medical images of the interior of the human body, and fingerprint images.
1.2 Presentation of the Data
The points in at which the data are collected are known as sites. A data set consists of a collection of sites and real‐valued observations or values. Note that each represents a vector in .
If the sites are located arbitrarily in , the data are known as irregularly spaced data. However, if the components of the sites are restricted to have integer values, and the sites cover a rectangular region in , then the data are known as regular lattice or regularly spaced data. (There is also the case of irregular lattice data for which the data do not fill a rectangular region.) For convenience, we shall often write lattice data using subscripts rather than with parentheses to emphasize the link with sequences of data in dimension. For example, in dimensions stands for , though we shall usually avoid the need to expand the suffix in full.
There are two ways to represent spatial data.
Regularly spaced data can be represented as a two‐way table of numbers. The other representation, which can be used both for regularly spaced and irregularly spaced data, is a list of spatial sites and data values.
Example 1.1 Illustrative data
Table 1.1 gives a simple illustrative regularly spaced data set in . In Panel (a), the data are presented as a two‐way array of numbers. Panel (b) shows a matrix coordinate system in which the origin is at the upper left of the table, with increasing down the rows of the table and increasing across the rows. Although the matrix coordinate system is conventional for multivariate analysis, we do not use matrix coordinates in this book. Instead, we use graphical coordinates, as in Panel (c), for which the ‐axis increases horizontally to the right, and the ‐axis increases vertically upward. Finally, in Panel (d) the data are presented as a list of spatial sites and data values.
A digital image can be regarded as a spatial data set on a large regular grid; typically, and or . In this context, the sites are known as pixels (picture elements).
Example 1.2 Fingerprint data
Figure 1.1 shows the gray level image of a fingerprint of R A Fisher. The sites of the data lie on a rectangular grid 3003 pixels wide by 3339 pixels high. The values of have been scaled to lie between 0 and 1. The marked rectangular section is investigated in more detail in Example 1.8.
Table 1.1 Illustrative data , on a regular grid, represented in various ways, .
(a) Table of values on a
grid
6
7
3
10
13
2
4
3
22
9
2
5
(b) Matrix coordinates (not generally used in this book)
1
2
3
4
1
6
7
3
10
2
13
2
4
3
3
22
9
2
5
(c) Graphical coordinates
(used for all spatial data sets in this book)
. The asterisks are explained in
Example 1.7
3
6
7
3
10
2
13
2
4
3
1
22
9
2
5
1
2
3
4
(d) List of graphical coordinates and values
t
t
t
t
1
1
22
3
2
4
2
1
9
4
2
3
3
1
2
1
3
6
4
1
5
2
3
7
1
2
13
3
3
3
2
2
2
4
3
10
Example 1.3 Elevation data
The topographic elevation data of Davis (1973) are given in Table 1.2 and consist of irregularly spaced observations. The data contain geographic coordinates and elevations of control points for a surveying problem. The elevation is measured in feet above the sea level. The coordinates are expressed in 50‐feet units measured from an arbitrary origin located in the southwest corner; is the East–West coordinate and is the North–South coordinate. Figure 1.2 gives two plots of the data. The raw plot in Panel (a) shows the elevation values printed at each site. The bubble plot in Panel (b) shows a circle plotted at each site, where the size of the circle encodes graphically the elevation information; larger elevations are indicated by bigger circles. Patterns in the data are often easier to pick out using the bubble plot. Note that the elevations are high near the edges of the region with a basin in the middle. There are extra features associated with the data such as river locations, but we will limit ourselves here to just the elevation information for illustrative purposes.
Figure 1.1 Fingerprint of R A Fisher, taken from Mardia's personal collection. A blowup of the marked rectangular section is given in Figure 1.9.
One of the objectives for this sort of data is to predict the elevation throughout the region and to represent the result graphically. Using a statistical method called kriging (see Chapter 7 for details), the elevation was predicted or smoothed on a fine grid of points throughout the region. The result for this data set can be summarized visually in different ways including:
Table 1.2 Elevation data: elevation in feet above the sea level, where , .
E‐W
N‐S
Elevation
E‐W
N‐S
Elevation
0.3
6.1
870
5.2
3.2
805
1.4
6.2
793
6.3
3.4
840
2.4
6.1
755
0.3
2.4
890
3.6
6.2
690
2.0
2.7
820
5.7
6.2
800
3.8
2.3
873
1.6
5.2
800
6.3
2.2
875
2.9
5.1
730
0.6
1.7
873
3.4
5.3
728
1.5
1.8
865
3.4
5.7
710
2.1
1.8
841
4.8
5.6
780
2.1
1.1
862
5.3
5.0
804
3.1
1.1
908
6.2
5.2
855
4.5
1.8
855
0.2
4.3
830
5.5
1.7
850
0.9
4.2
813
5.7
1.0
882
2.3
4.8
762
6.2
1.0
910
2.5
4.5
765
0.4
0.5
940
3.0
4.5
740
1.4
0.6
915
3.5
4.5
765
1.4
0.1
890
4.1
4.6
760
2.1
0.7
880
4.9
4.2
790
2.3
0.3
870
6.3
4.3
820
3.1
0.0
880
0.9
3.2
855
4.1
0.8
960
1.7
3.8
812
5.4
0.4
890
2.4
3.8
773
6.0
0.1
860
3.7
3.5
812
5.7
3.0
830
4.5
3.2
827
3.6
6.0
705
Coordinates are expressed in 50‐feet units measured from an arbitrary origin located in the southwest corner, with being the East–West coordinate and being the South–North coordinate.
Source: Davis (1973).
Figure 1.2 Elevation data: (a) raw plot giving the elevation at each site and (b) bubble plot where larger elevations are indicated by bigger circles.
A contour map (
Figure 1.3
a)
A perspective plot (
Figure 1.3
b), viewed from the top of the region
A digital image using gray level (or color) to indicate ore grade, where white denotes the lower values and black denotes higher values (
Figure 1.3
c)
These images all show that the data have a valley in the top middle of the the image and a peak in the bottom middle.
In addition, the contour plot in Figure 1.3d shows the standard error of the predictor. Notice that the predictor is perfect with zero standard error at the data sites, and it has a larger standard error in places where the data sites are sparse. See Example 7.2 for more details.
Example 1.4
