Marta Blangiardo, MRC-PHE Centre for Environment and Health, Department of Epidemiology and Biostatistics, Imperial College London, UK

Michela Cameletti, Department of Management, Economics and Quantitative Methods, University of Bergamo, Italy

Dedication iii

Preface ix

1 Introduction 1

1.1 Why spatial and spatio-temporal statistics? 1

1.2 Why do we use Bayesian methods for modelling spatial and spatio-temporal structures? 2

1.3 Why INLA? 3

1.4 Datasets 3

2 Introduction to 21

2.1 The language 21

2.2 objects 22

2.3 Data and session management 34

2.4 Packages 35

2.5 Programming in 36

2.6 Basic statistical analysis with 39

3 Introduction to Bayesian Methods 53

3.1 Bayesian Philosophy 53

3.2 Basic Probability Elements 57

3.3 Bayes Theorem 62

3.4 Prior and Posterior Distributions 64

3.5 Working with the Posterior Distribution 66

3.6 Choosing the Prior Distribution 68

4 Bayesian computing 83

4.1 Monte Carlo integration 83

4.2 Monte Carlo method for Bayesian inference 85

4.3 Probability distributions and random number generation in 86

4.4 Examples of Monte Carlo simulation 89

4.5 Markov chain Monte Carlo methods 97

4.6 The Integrated Nested Laplace Approximations algorithm 113

4.7 Laplace approximation 113

4.8 The package 123

4.9 How INLA works: step by step example 127

5 Bayesian regression and hierarchical models 139

5.1 Linear Regression 139

5.2 Nonlinear regression: random walk 145

5.3 Generalized Linear Models 150

5.4 Hierarchical Models 159

5.5 Prediction 176

5.6 Model Checking and Selection 179

6 Spatial Modeling 189

6.1 Areal data -GMRF 192

6.2 Ecological Regression 203

6.3 Zero inflated models 204

6.4 Geostatistical data 210

6.5 The Stochastic Partial Diferential Equation approach 211

6.6 SPDE within 215

6.7 SPDE toy example with simulated data 217

6.8 More advanced operations through the function 226

6.9 Prior specification for the stationary case 233

6.10 SPDE for Gaussian response: Swiss rainfall data 237

6.11 SPDE with nonnormal outcome: Malaria in the Gambia 245

6.12 Prior specification for the nonstationary case 249

7 Spatio-Temporal Models 257

7.1 Spatio-temporal Disease mapping 258

7.2 Spatio-temporal Modeling particulate matter concentration 268

8 Advanced modeling 283

8.1 Bivariate model for spatially misaligned data 283

8.2 Semicontinuous model to daily rainfall 295

8.3 Spatio-temporal dynamic models 308

8.4 Space-time model lowering the time resolution 321