Spatial Analysis

John T. Kent is a Professor in the Department of Statistics at the University of Leeds, UK. He began his career as a research fellow at Sidney Sussex College, Cambridge before moving to the University of Leeds. He has published extensively on various aspects of statistics, including infinite divisibility, directional data analysis, multivariate analysis, inference, robustness, shape analysis, image analysis, spatial statistics, and spatial-temporal modelling. Kanti V. Mardia is a Senior Research Professor and Leverhulme Emeritus Fellow in the Department of Statistics at the University of Leeds, and a Visiting Professor at the University of Oxford. During his career he has received many prestigious honours, including in 2003 the Guy Medal in Silver from the Royal Statistical Society, and in 2013 the Wilks memorial medal from the American Statistical Society. His research interests include bioinformatics, directional statistics, geosciences, image analysis, multivariate analysis, shape analysis, spatial statistics, and spatial-temporal modelling. Kent and Mardia are also joint authors of a well-established monograph on Multivariate Analysis.

List of Figures xiii

List of Tables xvii

Preface xix

List of Notation and Terminology xxv

1 Introduction 1

1.1 Spatial Analysis 1

1.2 Presentation of the Data 2

1.3 Objectives 9

1.4 The Covariance Function and Semivariogram 11

1.4.1 General Properties 11

1.4.2 Regularly Spaced Data 13

1.4.3 Irregularly Spaced Data 14

1.5 Behavior of the Sample Semivariogram 16

1.6 Some Special Features of Spatial Analysis 22

Exercises 27

2 Stationary Random Fields 31

2.1 Introduction 31

2.2 Second Moment Properties 32

2.3 Positive Definiteness and the Spectral Representation 34

2.4 Isotropic Stationary Random Fields 36

2.5 Construction of Stationary Covariance Functions 41

2.6 Matérn Scheme 43

2.7 Other Examples of Isotropic Stationary Covariance Functions 45

2.8 Construction of Nonstationary Random Fields 48

2.8.1 Random Drift 48

2.8.2 Conditioning 49

2.9 Smoothness 49

2.10 Regularization 51

2.11 Lattice Random Fields 53

2.12 Torus Models 56

2.12.1 Models on the Continuous Torus 56

2.12.2 Models on the Lattice Torus 57

2.13 Long-range Correlation 58

2.14 Simulation 61

2.14.1 General Points 61

2.14.2 The Direct Approach 61

2.14.3 Spectral Methods 62

2.14.4 Circulant Methods 66

Exercises 67

3 Intrinsic and Generalized Random Fields 73

3.1 Introduction 73

3.2 Intrinsic Random Fields of Order k = 0 74

3.3 Characterizations of Semivariograms 80

3.4 Higher Order Intrinsic Random Fields 83

3.5 Registration of Higher Order Intrinsic Random Fields 86

3.6 Generalized Random Fields 87

3.7 Generalized Intrinsic Random Fields of Intrinsic Order k ≥ 0 91

3.8 Spectral Theory for Intrinsic and Generalized Processes 91

3.9 Regularization for Intrinsic and Generalized Processes 95

3.10 Self-Similarity 96

3.11 Simulation 100

3.11.1 General Points 100

3.11.2 The Direct Method 101

3.11.3 Spectral Methods 101

3.12 Dispersion Variance 102

Exercises 104

4 Autoregression and Related Models 115

4.1 Introduction 115

4.2 Background 118

4.3 Moving Averages 120

4.3.1 Lattice Case 120

4.3.2 Continuously Indexed Case 121

4.4 Finite Symmetric Neighborhoods of the Origin in Z d 122

4.5 Simultaneous Autoregressions (SARs) 124

4.5.1 Lattice Case 124

4.5.2 Continuously Indexed Random Fields 125

4.6 Conditional Autoregressions (CARs) 127

4.6.1 Stationary CARs 128

4.6.2 Iterated SARs and CARs 130

4.6.3 Intrinsic CARs 131

4.6.4 CARs on a Lattice Torus 132

4.6.5 Finite Regions 132

4.7 Limits of CAR Models Under Fine Lattice Spacing 134

4.8 Unilateral Autoregressions for Lattice Random Fields 135

4.8.1 Half-spaces in Z d 135

4.8.2 Unilateral Models 136

4.8.3 Quadrant Autoregressions 139

4.9 Markov Random Fields (MRFs) 140

4.9.1 The Spatial Markov Property 140

4.9.2 The Subset Expansion of the Negative Potential Function 143

4.9.3 Characterization of Markov Random Fields in Terms of Cliques 145

4.9.4 Auto-models 147

4.10 Markov Mesh Models 149

4.10.1 Validity 149

4.10.2 Marginalization 150

4.10.3 Markov Random Fields 150

4.10.4 Usefulness 151

Exercises 151

5 Estimation of Spatial Structure 159

5.1 Introduction 159

5.2 Patterns of Behavior 160

5.2.1 One-dimensional Case 160

5.2.2 Two-dimensional Case 161

5.2.3 Nugget Effect 162

5.3 Preliminaries 164

5.3.1 Domain of the Spatial Process 164

5.3.2 Model Specification 164

5.3.3 Spacing of Data 165

5.4 Exploratory and Graphical Methods 166

5.5 Maximum Likelihood for Stationary Models 168

5.5.1 Maximum Likelihood Estimates – Known Mean 169

5.5.2 Maximum Likelihood Estimates – Unknown Mean 171

5.5.3 Fisher Information Matrix and Outfill Asymptotics 172

5.6 Parameterization Issues for the Matérn Scheme 173

5.7 Maximum Likelihood Examples for Stationary Models 174

5.8 Restricted Maximum Likelihood (REML) 179

5.9 Vecchia’s Composite Likelihood 180

5.10 REML Revisited with Composite Likelihood 182

5.11 Spatial Linear Model 185

5.11.1 MLEs 186

5.11.2 Outfill Asymptotics for the Spatial Linear Model 188

5.12 REML for the Spatial Linear Model 188

5.13 Intrinsic Random Fields 189

5.14 Infill Asymptotics and Fractal Dimension 192

Exercises 195

6 Estimation for Lattice Models 201

6.1 Introduction 201

6.2 Sample Moments 203

6.3 The AR(1) Process on Z 205

6.4 Moment Methods for Lattice Data 208

6.4.1 Moment Methods for Unilateral Autoregressions (UARs) 209

6.4.2 Moment Estimators for Conditional Autoregression (CAR) Models 210

6.5 Approximate Likelihoods for Lattice Data 212

6.6 Accuracy of the Maximum Likelihood Estimator 215

6.7 The Moment Estimator for a CAR Model 218

Exercises 219

7 Kriging 231

7.1 Introduction 231

7.2 The Prediction Problem 233

7.3 Simple Kriging 236

7.4 Ordinary Kriging 238

7.5 Universal Kriging 240

7.6 Further Details for the Universal Kriging Predictor 241

7.6.1 Transfer Matrices 241

7.6.2 Projection Representation of the Transfer Matrices 242

7.6.3 Second Derivation of the Universal Kriging Predictor 244

7.6.4 A Bordered Matrix Equation for the Transfer Matrices 245

7.6.5 The Augmented Matrix Representation of the Universal Kriging Predictor 245

7.6.6 Summary 247

7.7 Stationary Examples 248

7.8 Intrinsic Random Fields 253

7.8.1 Formulas for the Kriging Predictor and Kriging Variance 253

7.8.2 Conditionally Positive Definite Matrices 254

7.9 Intrinsic Examples 256

7.10 Square Example 258

7.11 Kriging with Derivative Information 259

7.12 Bayesian Kriging 262

7.12.1 Overview 262

7.12.2 Details for Simple Bayesian Kriging 264

7.12.3 Details for Bayesian Kriging with Drift 264

7.13 Kriging and Machine Learning 266

7.14 The Link Between Kriging and Splines 269

7.14.1 Nonparametric Regression 269

7.14.2 Interpolating Splines 271

7.14.3 Comments on Interpolating Splines 273

7.14.4 Smoothing Splines 274

7.15 Reproducing Kernel Hilbert Spaces 274

7.16 Deformations 275

Exercises 277

8 Additional Topics 283

8.1 Introduction 283

8.2 Log-normal Random Fields 284

8.3 Generalized Linear Spatial Mixed Models (GLSMMs) 285

8.4 Bayesian Hierarchical Modeling and Inference 286

8.5 Co-kriging 287

8.6 Spatial–temporal Models 291

8.6.1 General Considerations 291

8.6.2 Examples 292

8.7 Clamped Plate Splines 294

8.8 Gaussian Markov Random Field Approximations 295

8.9 Designing a Monitoring Network 296

Exercises 298

Appendix A Mathematical Background 303

A. 1 Domains for Sequences and Functions 303

A. 2 Classes of Sequences and Functions 305

A.2. 1 Functions on the Domain Rd 305

A.2. 2 Sequences on the Domain Zd 305

A.2. 3 Classes of Functions on the Domain S d1 306

A.2 4 Classes of Sequences on the Domain ZNd, Where N = (n[1], .,n[d]) 306

A. 3 Matrix Algebra 306

A.3. 1 The Spectral Decomposition Theorem 306

A.3. 2 Moore–Penrose Generalized Inverse 307

A.3. 3 Orthogonal Projection Matrices 308

A.3. 4 Partitioned Matrices 308

A.3. 5 Schur Product 309

A.3. 6 Woodbury Formula for a Matrix Inverse 310

A.3. 7 Quadratic Forms 311

A.3. 8 Toeplitz and Circulant Matrices 311

A.3. 9 Tensor Product Matrices 312

A.3. 10 The Spectral Decomposition and Tensor Products 313

A.3. 11 Matrix Derivatives 313

A. 4 Fourier Transforms 313

A. 5 Properties of the Fourier Transform 315

A. 6 Generalizations of the Fourier Transform 318

A. 7 Discrete Fourier Transform and Matrix Algebra 318

A.7. 1 DFT in d = 1Dimension 318

A.7. 2 Properties of the Unitary Matrix G, d = 1 319

A.7. 3 Circulant Matrices and the DFT, d = 1 320

A.7. 4 The Case d > 1 321

A.7. 5 The Periodogram 322

A. 8 Discrete Cosine Transform (DCT) 322

A.8. 1 One-dimensional Case 322

A.8. 2 The Case d > 1 323

A.8. 3 Indexing for the Discrete Fourier and Cosine Transforms 323

A. 9 Periodic Approximations to Sequences 324

A. 10 Structured Matrices in d = 1Dimension 325

A. 11 Matrix Approximations for an Inverse Covariance Matrix 327

A.1. 1 The Inverse Covariance Function 328

A.11. 2 The Toeplitz Approximation to Σ − 1 330

A.11. 3 The Circulant Approximation to Σ − 1 330

A.11. 4 The Folded Circulant Approximation to Σ − 1 330

A.11. 5 Comments on the Approximations 331

A.11. 6 Sparsity 332

A. 12 Maximum Likelihood Estimation 332

A.2. 1 General Considerations 332

A.1. 2 The Multivariate Normal Distribution and the Spatial Linear Model 333

A.12. 3 Change of Variables 335

A.12. 4 Profile Log-likelihood 335

A.12. 5 Confidence Intervals 336

A.12. 6 Linked Parameterization 337

A.12. 7 Model Choice 338

A. 13 Bias in Maximum Likelihood Estimation 338

A.3. 1 A General Result 338

A.13. 2 The Spatial Linear Model 340

Appendix B A Brief History of the Spatial Linear Model and the Gaussian Process Approach 347

B.1 Introduction 347

B.2 Matheron and Watson 348

B.3 Geostatistics at Leeds 1977–1987 349

B.3.1 Courses, Publications, Early Dissemination 349

B.3.2 Numerical Problems with Maximum Likelihood 351

B.4 Frequentist vs. Bayesian Inference 352

References and Author Index 355

Index 367