Spatial Analysis (Hardcover)

· 제목 : Spatial Analysis (Hardcover) 
· 분류 : 외국도서 > 과학/수학/생태 > 수학 > 확률과 통계 > 일반
· ISBN : 9780471632054
· 쪽수 : 400쪽
· 출판일 : 2022-05-31

1 Introduction 37

1.1 Spatial analysis 37

1.2 Presentation of the data 38

1.3 Objectives 44

1.4 The covariance function and semivariogram 46

1.4.1 General properties 46

1.4.2 Regularly spaced data 49

1.4.3 Irregularly spaced data 51

1.5 Behavior of the sample semivariogram 52

1.6 Some special features of spatial analysis 58

2 Stationary Random Fields 69

2.1 Introduction 69

2.2 Second moment properties 70

2.3 Positive de_niteness and the spectral representation 72

2.4 Isotropic stationary random _elds 75

2.5 Construction of stationary covariance functions 79

2.6 Mat_ern scheme 82

2.7 Other examples of isotropic stationary covariance functions 84

2.8 Construction of nonstationary random _elds 87

7

8 CONTENTS

2.9 Smoothness 89

2.10 Regularization 91

2.11 Lattice random _elds 93

2.12 Torus models 96

2.12.1 Models on the continuous torus 96

2.12.2 Models on the lattice torus 97

2.13 Long-range correlation 98

2.14 Simulation 101

2.14.1 General points 101

2.14.2 The direct method 101

2.14.3 Spectral methods 102

2.14.4 Circulant methods 106

3 Intrinsic and generalized random _elds 115

3.1 Introduction 115

3.2 Intrinsic random _elds of order k = 0 117

3.3 Characterizations of semivariograms 122

3.4 Higher order intrinsic random _elds 126

3.5 Registration of higher order intrinsic random _elds 128

3.6 Generalized random _elds 130

3.7 Generalized intrinsic random _elds of intrinsic order k _ 0 134

3.8 Spectral theory for intrinsic and generalized processes 134

3.9 Regularization for intrinsic and generalized processes 138

3.10 Self-Similarity 139

3.11 Simulation 144

3.11.1 General points 144

3.11.2 The direct method 145

CONTENTS 9

3.11.3 Spectral methods 145

3.12 Dispersion variance 146

4 Autoregressive and related Models 159

4.1 Introduction 159

4.2 Background 162

4.3 Moving averages 165

4.3.1 Lattice case 165

4.3.2 Continuously-indexed case 165

4.4 Finite symmetric neighborhoods of the origin in Zd 167

4.5 Simultaneous autoregressions (SARs) 168

4.5.1 Lattice case 168

4.5.2 Continuously-indexed random _elds 170

4.6 Conditional autoregressions (CARs) 172

4.6.1 Stationary CARs 172

4.6.2 Iterated SARs and CARs 175

4.6.3 Intrinsic CARs 176

4.6.4 CARs on a lattice torus 177

4.6.5 Finite regions 177

4.7 Limits of CAR models under _ne lattice spacing 179

4.8 Unilateral autoregressions for lattice random _elds 181

4.8.1 Half-spaces in Zd 181

4.8.2 Unilateral models 182

4.8.3 Quadrant unilateral autoregressions 184

4.9 Markov random _elds (MRFs) 186

4.9.1 The spatial Markov property 186

4.9.2 The subset expansion of the negative potential function 188

10 CONTENTS

4.9.3 Characterization of Markov random _elds in terms of cliques 191

4.9.4 Auto-models 193

4.10 Markov mesh models 194

5 Estimation of Spatial Structure 203

5.1 Introduction 203

5.2 Patterns of behavior 204

5.2.1 One-dimensional case 205

5.2.2 Two-dimensional case 206

5.2.3 Nugget e_ect 207

5.3 Preliminaries 208

5.3.1 Domain of the spatial process 208

5.3.2 Model speci_cation 209

5.3.3 Spacing of data 210

5.4 Exploratory and graphical methods 211

5.5 Maximum likelihood for stationary models 213

5.5.1 Maximum likelihood estimates | known mean 215

5.5.2 Maximum likelihood estimates| unknown mean 216

5.5.3 Fisher information matrix and out_ll asymptotics 218

5.6 Parameterization issues for the Mat_ern scheme 219

5.7 ML Examples for stationary models 220

5.8 Restricted maximum likelihood (REML) 225

5.9 Vecchia's composite likelihood 226

5.10 REML revisited with composite likelihood 228

5.11 Spatial linear model 232

5.11.1 MLEs 233

5.11.2 Out_ll asymptotics for the spatial linear model 234

CONTENTS 11

5.12 REML for the spatial linear model 235

5.13 Intrinsic random _elds 236

5.14 In_ll asymptotics and fractal dimension 240

6 Estimation for lattice models 249

6.1 Introduction 249

6.2 Sample moments 252

6.3 The AR(1) process on Z 253

6.4 Moment methods for lattice data 257

6.4.1 Moment methods for unilateral autoregressions (UARs) 257

6.4.2 Moment estimators for conditional autoregression (CAR) models 258

6.5 Approximate likelihoods for lattice data 261

6.6 Accuracy of the maximum likelihood estimator 265

6.7 The moment estimator for a CAR model 268

7 Kriging 281

7.1 Introduction 281

7.2 The prediction problem 283

7.3 Simple kriging 286

7.4 Ordinary Kriging 288

7.5 Universal kriging 290

7.6 Further details for the universal kriging predictor 292

7.6.1 Transfer matrices 292

7.6.2 Projection representation of the transfer matrices 293

7.6.3 Second derivation of the universal kriging predictor 295

7.6.4 A bordered matrix equation for the transfer matrices 296

7.6.5 The augmented matrix representation of the universal kriging predictor

 296

12 CONTENTS

7.6.6 Summary 298

7.7 Stationary examples 299

7.8 Intrinsic random _elds 305

7.8.1 Formulas for the kriging predictor and kriging variance 305

7.8.2 Conditionally positive de_nite matrices 306

7.9 Intrinsic examples 307

7.10 Square example 310

7.11 Kriging with derivative information 311

7.12 Bayesian kriging 315

7.12.1 Overview 315

7.12.2 Details for simple Bayesian kriging 316

7.12.3 Details for Bayesian kriging with drift 317

7.13 Kriging and machine learning 319

7.14 The link between kriging and splines 322

7.14.1 Nonparametric regression 322

7.14.2 Interpolating splines 324

7.14.3 Comments on interpolating splines 326

7.14.4 Smoothing splines 327

7.15 Reproducing kernel Hilbert spaces 328

7.16 Deformations 328

7.17 Exercises 330

8 Additional topics 337

8.1 Introduction 337

8.2 Log-normal random _elds 338

8.3 Generalized linear spatial mixed models (GLSMMs) 339

8.4 Bayesian hierarchical modeling and inference 340

CONTENTS 13

8.5 Co-kriging 342

8.6 Spatial-temporal models 345

8.6.1 General considerations 345

8.6.2 Examples 346

8.7 Clamped plate splines 348

8.8 Gaussian Markov random _eld approximations 350

8.9 Designing a monitoring network 351

Appendix A: Mathematical background 357

A.1 Domains for sequences and functions 357

A.2 Classes of sequences and functions 358

A.2.1 Functions on the domain Rd 359

A.2.2 Sequences on the domain Zd 359

A.2.3 Classes of functions on the domain Sd

1 360

A.2.4 Classes of sequences on the domain Zd

N, where N = (n[1]; : : : ; n[d]) 360

A.3 Matrix algebra 360

A.3.1 The spectral decomposition theorem 360

A.3.2 Moore-Penrose generalized inverse 361

A.3.3 Orthogonal projection matrices 362

A.3.4 Partitioned matrices 363

A.3.5 Schur product 364

A.3.6 Woodbury formula for a matrix inverse 364

A.3.7 Quadratic forms 365

A.3.8 Toeplitz and circulant matrices 366

A.3.9 Tensor product matrices 367

A.3.10 The spectral decomposition and tensor products 367

A.3.11 Matrix derivatives 367

14 CONTENTS

A.4 Fourier transforms 368

A.5 Properties of the Fourier transform 370

A.6 Generalizations of the Fourier transform 372

A.7 Discrete Fourier transform and matrix algebra 373

A.7.1 DFT in d = 1 dimension 373

A.7.2 Properties of the unitary matrix G, d = 1 374

A.7.3 Circulant matrices and the DFT, d = 1 375

A.7.4 The case d > 1 376

A.7.5 The periodogram 377

A.8 Discrete cosine transform (DCT) 377

A.8.1 One-dimensional case 378

A.8.2 The case d > 1 379

A.8.3 Indexing for the discrete Fourier and cosine transforms 379

A.9 Periodic approximations to sequences 379

A.10 Structured matrices in d = 1 dimension 380

A.11 Matrix approximations for an inverse covariance matrix 383

A.11.1 The inverse covariance function 383

A.11.2 The Toeplitz approximation to _􀀀1 386

A.11.3 The circulant approximation to _􀀀1 386

A.11.4 The folded circulant approximation to _􀀀1 386

A.11.5 Comments on the approximations 387

A.11.6 Sparsity 388

A.12 Maximum likelihood estimation 389

A.12.1 General considerations 389

A.12.2 The multivariate normal distribution and the spatial linear model 390

A.12.3 Change of variables 391

A.12.4 Pro_le log-likelihood 392

CONTENTS 15

A.12.5 Con_dence intervals 392

A.12.6 Linked parameterization 393

A.12.7 Model choice 394

A.13 Bias in maximum likelihood estimation 395

A.13.1 A general result 395

A.13.2 The Spatial Linear Model 396

Appendix B: A brief history of the spatial linear model and the Gaussian

process approach 403

B.1 Introduction 403

B.2 Matheron and Watson 404

B.3 Geostatistics at Leeds 1977-1987 405

B.3.1 Courses, Publications, Early Dissemination 405

B.3.2 Numerical problems with maximum likelihood 408

B.4 Frequentist vs Bayesian inference 409

References and Author Index 411

Index 426