Preface v 1 Introduction 1 1.1 Motivating examples 1 Example 1.1. Surface elevations 1 Example 1.2. Residual contamination from nuclear weapons testing 2 Example 1.3. Childhood malaria in The Gambia 4 Example 1.4. Soil data 5 1.2 Terminology and notation 9 1.2.1 Support 9 1.2.2 Multivariate responses and explanatory variables 10 1.2.3 Sampling design 12 1.3 Scientific objectives 12 1.4 Generalised linear geostatistical models 13 1.5 What is in this book? 15 1.5.1 Organisation of the book 16 1.5.2 Statistical pre-requisites 17 1.6 Computation 17 1.6.1 Elevation data 17 1.6.2 More on the geodata object 20 1.6.3 Rongelap data 22 1.6.4 The Gambia malaria data 24 1.6.5 The soil data 24 1.7 Exercises 26 2 An overview of model-based geostatistics 27 2.1 Design 27 2.2 Model formulation 28 2.3 Exploratory data analysis 30 2.3.1 Non-spatial exploratory analysis 30 2.3.2 Spatial exploratory analysis 31 2.4 The distinction between parameter estimation and spatial prediction 35 2.5 Parameter estimation 36 2.6 Spatial prediction 37 2.7 Definitions of distance 39 2.8 Computation 40 2.9 Exercises 45 3 Gaussian models for geostatistical data 46 3.1 Covariance functions and the variogram 46 3.2 Regularisation 48 3.3 Continuity and differentiability of stochastic processes 49 3.4 Families of covariance functions and their properties 51 3.4.1 The Mat e rn family 51 3.4.2 The powered exponential family 52 3.4.3 Other families 55 3.5 The nugget effect 56 3.6 Spatial trends 57 3.7 Directional effects 57 3.8 Transformed Gaussian models 60 3.9 Intrinsic models 62 3.10 Unconditional and conditional simulation 66 3.11 Low-rank models 68 3.12 Multivariate models 69 3.12.1 Cross-covariance, cross-correlation and cross-variogram 70 3.12.2 Bivariate signal and noise 71 3.12.3 Some simple constructions 72 3.13 Computation 74 3.14 Exercises 76 4 Generalized linear models for geostatistical data 78 4.1 General formulation 78 4.2 The approximate covariance function and variogram 80 4.3 Examples of generalised linear geostatistical models 81 4.3.1 The Poisson log-linear model 81 4.3.2 The binomial logistic-linear model 82 4.3.3 Spatial survival analysis 83 4.4 Point process models and geostatistics 85 4.4.1 Cox processes 86 4.4.2 Preferential sampling 88 4.5 Some examples of other model constructions 92 4.5.1 Scan processes 92 4.5.2 Random sets 93 4.6 Computation 93 4.6.1 Simulating from the generalised linear model 93 Poisson model 93 Bernoulli model 94 Binomial model 95 4.6.2 Preferential sampling 95 4.7 Exercises 96 5 Classical parameter estimation 98 5.1 Trend estimation 99 5.2 Variograms 99 5.2.1 The theoretical variogram 99 5.2.2 The empirical variogram 101 5.2.3 Smoothing the empirical variogram 101 5.2.4 Exploring directional effects 103 5.2.5 The interplay between trend and covariance structure 104 5.3 Curve-fitting methods for estimating covariance structure 106 5.3.1 Ordinary least squares 107 5.3.2 Weighted least squares 107 5.3.3 Comments on curve-fitting methods 109 5.4 Maximum likelihood estimation 111 5.4.1 General ideas 111 5.4.2 Gaussian models 111 5.4.3 Profile likelihood 113 5.4.4 Application to the surface elevation data 113 5.4.5 Restricted maximum likelihood estimation for the Gaussian linear model 115 5.4.6 Trans-Gaussian models 116 5.4.7 Analysis of Swiss rainfall data 117 5.4.8 Analysis of soil calcium data 120 5.5 Parameter estimation for generalized linear geostatistical models 122 5.5.1 Monte Carlo maximum likelihood 123 5.5.2 Hierarchical likelihood 124 5.5.3 Generalized estimating equations 124 5.6 Computation 125 5.6.1 Variogram calculations 125 5.6.2 Parameter estimation 129 5.7 Exercises 131 6 Spatial prediction 133 6.1 Minimum mean square error prediction 133 6.2 Minimum mean square error prediction for the stationary Gaussian model 135 6.2.1 Prediction of the signal at a point 135 6.2.2 Simple and ordinary kriging 136 6.2.3 Prediction of linear targets 137 6.2.4 Prediction of non-linear targets 137 6.3 Prediction with a nugget effect 138 6.4 What does kriging actually do to the data? 139 6.4.1 The prediction weights 140 6.4.2 Varying the correlation parameter 143 6.4.3 Varying the noise-to-signal ratio 145 6.5 Trans-Gaussian kriging 146 6.5.1 Analysis of Swiss rainfall data (continued) 148 6.6 Kriging with non-constant mean 150 6.6.1 Analysis of soil calcium data (continued) 150 6.7 Computation 150 6.8 Exercises 154 7 Bayesian inference 156 7.1 The Bayesian paradigm: a unified treatment of estimation and prediction 156 7.1.1 Prediction using plug-in estimates 156 7.1.2 Bayesian prediction 157 7.1.3 Obstacles to practical Bayesian prediction 159 7.2 Bayesian estimation and prediction for the Gaussian linear model 159 7.2.1 Estimation 160 7.2.2 Prediction when correlation parameters are known 162 7.2.3 Uncertainty in the correlation parameters 163 7.2.4 Prediction of targets which depend on both the signal and the spatial trend 164 7.3 Trans-Gaussian models 165 7.4 Case studies 166 7.4.1 Surface elevations 166 7.4.2 Analysis of Swiss rainfall data (continued) 168 7.5 Bayesian estimation and prediction for generalized linear geostatistical models 171 7.5.1 Markov chain Monte Carlo 171 7.5.2 Estimation 172 7.5.3 Prediction 175 7.5.4 Some possible improvements to the MCMC algorithm 176 Bayesian inference 177 7.6 Case studies in generalized linear geostatistical modelling 178 7.6.1 Simulated data 178 7.6.2 Rongelap island 180 7.6.3 Childhood malaria in The Gambia 184 7.6.4 Loa loa prevalence in equatorial Africa 186 7.7 Computation 192 7.7.1 Gaussian models 192 7.7.2 Non-Gaussian models 195 7.8 Exercises 195 8 Geostatistical design 198 8.1 Choosing the study region 200 8.2 Choosing the sample locations: uniform designs 201 8.3 Designing for efficient prediction 202 8.4 Designing for efficient parameter estimation 203 8.5 A Bayesian design criterion 205 8.5.1 Retrospective design 205 8.5.2 Prospective design 208 8.6 Exercises 210 A Statistical background 212 A.1 Statistical models 212 A.2 Classical inference 212 A.3 Bayesian inference 214 A.4 Prediction 215 References 217 Index 227