Texts in Statistícal Science

Statistícal Methodsfor Spatial Data Analysis

V-

Oliver SchabenbergerCarol A. Gotway

PCTCHAPMAN & K

Contents

Preface xv

1 Introduction 11.1 The Need for Spatial Analysis 11.2 Types of Spatial Data 6

1.2.1 Geostatistical Data 71.2.2 Lattice Data, Regional Data 81.2.3 Point Patterns 11

1.3 Autocorrelation—Concept and Elementary Measures 141.3.1 Mantel's Tests for Clustering 141.3.2 Measures on Lattices 181.3.3 Localized Indicators of Spatial Autocorrelation 23

1.4 Autocorrelation Functions 251.4.1 The Autocorrelation Function of a Time Series 251.4.2 Autocorrelation Functions in Space—Covariance and

Semivariogram 261.4.3 From Mantel's Statistic to the Semivariogram 29

1.5 The Effects of Autocorrelation on Statistical Inference 311.5.1 Effects on Prediction 321.5.2 Effects on Precisión of Estimators 34

1.6 Chapter Problems 37

2 Some Theory on Random Fields 412.1 Stochastic Processes and Samples of Size One 412.2 Stationarity, Isotropy, and Heterogeneity 422.3 Spatial Continuity and Differentiability 482.4 Random Fields in the Spatial Domain 52

2.4.1 Model Representation 532.4.2 Convolution Representation 57

2.5 Random Fields in the Frequency Domain 622.5.1 Spectral Representation of Deterministic Functions 622.5.2 Spectral Representation of Random Processes 652.5.3 Covariance and Spectral Density Function 662.5.4 Properties of Spectral Distribution Functions 702.5.5 Continuous and Discrete Spectra 722.5.6 Linear Location-Invariant Filters 74

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2.5.7 Importance of Spectral Analysis 772.6 Chapter Problems 78

3 Mapped Point Patterns 813.1 Random, Aggregated, and Regular Patterns 813.2 Binornial and Poisson Processes 83

3.2.1 Bernoulli and Binomial Processes 833.2.2 Poisson Processes 843.2.3 Process Equivalence 85

3.3 Testing for Complete Spatial Randomness 863.3.1 Monte Cario Tests 873.3.2 Simulation Envelopes 883.3.3 Tests Based on Quadrat Counts 903.3.4 Tests Based on Distances 97

3.4 Second-Order Properties of Point Patterns 993.4.1 The Reduced Second Moment Measure—

The K-Function 1013.4.2 Estimation of K- and L-Functions 1023.4.3 Assessing the Relationship between Two Patterns 103

3.5 The Inhomogeneous Poisson Process 1073.5.1 Estimation of the Intensity Function 1103.5.2 Estimating the Ratio of Intensity Functions 1123.5.3 Clustering and Cluster Detection 114

3.6 Marked and Multivariate Point Patterns 1183.6.1 Extensions 1183.6.2 Intensities and Moment Measures for Multivariate

Point Patterns 1203.7 Point Process Models 122

3.7.1 Thinning and Clustering 1233.7.2 Clustered Processes 1253.7.3 Regular Processes 128

3.8 Chapter Problems 129

Semivariogram and Covariance Function Analysisand Estimation 1334.1 Introduction 1334.2 Semivariogram and Covariogram 135

4.2.1 Defmition and Empirical Counterparts 1354.2.2 Interpretation as Structural Tools 138

4.3 Covariance and Semivariogram Models 1414.3.1 Model Validity 1414.3.2 The Matérn Class of Covariance Functions 1434.3.3 The Spherical Family of Covariance Functions 1454.3.4 Isotropic Models Allowing Negative Correlations 1464.3.5 Basic Models Not Second-Order Stationary 1494.3.6 Models with Nugget Effects and Nested Models 150

4.3.7 Accommodating Anisotropy 1514.4 Estimating the Semivariogram 153

4.4.1 Matheron's Estimator 1534.4.2 The Cressie-Hawkins Robust Estimator 1594.4.3 Estiraators Based on Order Statistics and Quantiles 161

4.5 Parametric Modeling 1634.5.1 Least Squares and the Semivariogram 1644.5.2 Máximum and Restricted Máximum Likelihood 1664.5.3 Composite Likelihood and Generalized Estimating

Equations 1694.5.4 Comparisons 172

4.6 Nonparametric Estimation and Modeling 1784.6.1 The Spectral Approach 1794.6.2 The Moving-Average Approach 1834.6.3 Incorporating a Nugget Effect 186

4.7 Estimation and Inference in the Frequency Domain 1884.7.1 The Periodogram on a Rectangular Lattice 1904.7.2 Spectral Density Functions 1984.7.3 Analysis of Point Patterns 200

4.8 On the Use of Non-Euclidean Distances in Geostatistics 2044.8.1 Distance Metrics and Isotropic Covariance Functions 2054.8.2 Multidimensional Scaling 206

4.9 Supplement: Bessel Functions 2104.9.1 Bessel Function of the First Kind 2104.9.2 Modified Bessel Functions of the First and Second Kind 210

4.10 Chapter Problems 211

Spatial Prediction and Kriging 2155.1 Optimal Prediction in Random Fields 2155.2 Linear Prediction—Simple and Ordinary Kriging 221

5.2.1 The Mean Is Known—Simple Kriging 2235.2.2 The Mean Is Unknown and Constant—Ordinary Kriging 2265.2.3 Effects of Nugget, Sill, and Range 228

5.3 Linear Prediction with a Spatially Varying Mean 2325.3.1 Trend Surface Models 2345.3.2 Localized Estimation 2385.3.3 Universal Kriging 241

5.4 Kriging in Practice 2435.4.1 On the Uniqueness of the Decomposition 2435.4.2 Local Versus Global Kriging 2445.4.3 Filtering and Smoothing 248

5.5 Estimating Covariance Parameters 2545.5.1 Least Squares Estimation 2565.5.2 Máximum Likelihood 2595.5.3 Restricted Máximum Likelihood 261

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5.5.4 Prediction Errors When Covariance Parameters AreEstimated 263

5.6 Nonlinear Prediction 2675.6.1 Lognormal Kriging 2675.6.2 Trans-Gaussian Kriging 2705.6.3 Indicator Kriging 2785.6.4 Disjunctive Kriging 279

5.7 Change of Support 2845.7.1 Block Kriging 2855.7.2 The Multi-Gaussian Approach 2895.7.3 The Use of Indicator Data 2905.7.4 Disjunctive Kriging and Isofactorial Models 2905.7.5 Constrained Kriging 291

5.8 On the Popularity of the Multivariate Gaussian Distribution 2925.9 Chapter Problems 295

6 Spatial Regression Models 2996.1 Linear Models with Uncorrelated Errors 301

6.1.1 Ordinary Least Squares—Inference and Diagnostics 3036.1.2 Working with OLS Residuals 3076.1.3 Spatially Explicit Models 316

6.2 Linear Models with Correlated Errors 3216.2.1 Mixed Models 3256.2.2 Spatial Autoregressive Models 3356.2.3 Generalizad Least Squares—Inference and Diagnostics 341

6.3 Generalized Linear Models 3526.3.1 Background 3526.3.2 Fixed Effects and the Marginal Specification 3546.3.3 A Caveat 3556.3.4 Mixed Models and the Conditional Specification 3566.3.5 Estimation in Spatial GLMs and GLMMs 3596.3.6 Spatial Prediction in GLMs 369

6.4 Bayesian Hierarchical Models 3836.4.1 Prior Distributions 3856.4.2 Fitting Bayesian Models 3866.4.3 Selected Spatial Models 390

6.5 Chapter Problems 400

7 Simulation of Random Fields 4057.1 Unconditional Simulation of Gaussian Random Fields 406

7.1.1 Cholesky (LU) Decomposition 4077.1.2 Spectral Decomposition 407

7.2 Conditional Simulation of Gaussian Random Fields 4077.2.1 Sequential Simulation 4087.2.2 Conditioning a Simulation by Kriging 409

7.3 Simulated Annealing 409

7.4 Simulating from Convolutions 4137.5 Simulating Point Processes 418

7.5.1 Homogeneous Poisson Process on the Rectangle (0,0) x(a, 6) with Intensity A 418

7.5.2 Inhomogeneous Poisson Process with Intensity A(s) 4197.6 Chapter Problems 419

8 Non-Stationary Covariance 4218.1 Types of Non-Stationarity 4218.2 Global Modeling Approaches 422

8.2.1 Parametric Models 4228.2.2 Space Deformation 423

8.3 Local Stationarity 4258.3.1 Moving Windows 4258.3.2 Convolution Methods 4268.3.3 Weighted Stationary Processes 428

9 Spatio-Temporal Processes 4319.1 A New Dimensión 4319.2 Separable Covariance Functions 4349.3 Non-Separable Covariance Functions 435

9.3.1 Monotone Function Approach 4369.3.2 Spectral Approach 4369.3.3 Mixture Approach 4389.3.4 Differential Equation Approach 439

9.4 The Spatio-Temporal Semivariogram 4409.5 Spatio-Temporal Point Processes 442

9.5.1 Types of Processes 4429.5.2 Intensity Measures 4439.5.3 Stationarity and Complete Randomness 444

References 447

Author Index 463

i,Subject Index 467