Bayesian Spatial Modelling with Conjugate Prior Models

This book offers a comprehensive overview of statistical methodology for modelling and evaluating spatial variables useful in a variety of applications. These spatial variables fall into three categories: continuous, like terrain elevation; events, like tree locations; and mosaics, like medical images. Definitions and discussions of random field models are included for each of these three previously mentioned spatial variable types. Moreover, the readers will have access to algorithms suitable for applying this methodology in practical problem solving, and the computational efficiency of these algorithms are discussed. The presentation is made in a consistent predictive Bayesian framework, which allows separate modelling of the observation acquisition procedure, as a likelihood model, and of the spatial variable characteristics, as a prior spatial model. The likelihood and prior models uniquely define the posterior spatial model, which provides the basis for spatial simulations, spatial predictions with associated precisions, and model parameter inference. The emphasis is on Bayesian spatial modelling with conjugate pairs of likelihood and prior models that are analytically tractable and hence suitable for data abundant spatial studies. Alternative methods frequently used in spatial statistics are presented using a unified notation. The book is suitable as a textbook for a ‘Spatial Statistics’ course at the MSc or PhD level, as it also includes algorithm descriptions, project texts, and exercises. Preface Reader's Guide Acknowledgement Contents List of Abbreviations 1 Introduction 2 Bayesian Spatial Modelling 2.1 Motivating Examples 2.2 Posterior Model: Simulation-Based Assessment 2.3 Notation 3 Conjugate Bayesian Models 4 Random Field Models 4.1 Continuous Random Fields 4.1.1 Gaussian RF Models 4.1.2 Hierarchical Gaussian RF Models 4.2 Event Random Fields 4.2.1 Poisson RF Models 4.2.2 Hierarchical Poisson RF Models 4.3 Mosaic Random Fields 4.3.1 Markov RF Models 4.3.2 Hierarchical Markov RF Models 5 Likelihood Models 5.1 Continuous Spatial Variables 5.1.1 Example: Continuous Spatial Variable 5.2 Event Spatial Variables 5.2.1 Example: Event Spatial Variable 5.3 Mosaic Spatial Variables 5.3.1 Example: Mosaic Spatial Variable 6 Prior Models 6.1 Continuous Spatial Variables: Gaussian RF Models 6.2 More on Gaussian RF Prior Models 6.2.1 Model Parameters and Characteristics 6.2.2 Model Validation 6.2.3 Explanatory Spatial Variables 6.2.4 Related Topics 6.2.5 Example: Gaussian RF 6.3 Hierarchical Gaussian RF Models 6.4 Event Spatial Variables: Poisson RF Models 6.5 More on Poisson RF Prior Models 6.5.1 Model Parameters and Characteristics 6.5.2 Model Validation 6.5.3 Explanatory Spatial Variables 6.5.4 Related Topics 6.5.5 Example: Poisson RF 6.6 Hierarchical Poisson RF Models 6.7 Mosaic Spatial Variables: Markov RF Models 6.8 More on Markov RF Prior Models 6.8.1 Model Parameters and Characteristics 6.8.2 Model Validation 6.8.3 Explanatory Spatial Variables 6.8.4 Related Topics 6.8.5 Example: Markov RF 7 Posterior Models 7.1 Continuous Spatial Variables 7.1.1 Gaussian RF Models 7.1.2 Example: Gaussian RF 7.1.3 Hierarchical Gaussian RF Models 7.2 Event Spatial Variables 7.2.1 Poisson RF Models 7.2.2 Example: Poisson RF 7.2.3 Hierarchical Poisson RF Models 7.3 Mosaic Spatial Variables 7.3.1 Markov RF Models 7.3.2 Example: Markov RF 8 Model Parameter Inference 8.1 Gaussian RF Models 8.1.1 Example: Gaussian RF 8.2 Poisson RF Models 8.2.1 Example: Poisson RF 8.3 Markov RF Models 8.3.1 Example: Markov RF 9 Computational Challenges 9.1 Gaussian RF Models 9.1.1 Sparse Matrix Representations 9.1.2 Localisation Approximations 9.2 Poisson RF Models 9.3 Markov RF Models 10 Special Topics 10.1 Classes of Simulation Algorithms 10.1.1 Transformation Algorithms 10.1.2 Sequential Algorithms 10.1.3 Rejection Algorithms 10.1.4 Iterative Algorithms 10.1.5 Approximate Algorithms 10.2 Geostatistics: Kriging Prediction Models 10.3 Gaussian Markov RF Models 10.4 Gaussian Basis Function RF Models 10.5 Kernel Predictors for Gaussian RF Models 10.6 Clustered/Repulsive Event RF 10.7 INLA Framework for Poisson RF 10.8 Markov Random Profile Versus Markov Random Chain 10.9 Model Parameter Inference in Hierarchical RF Models 11 Selected Applications 11.1 Continuous Spatial Variables 11.2 Event Spatial Variables 11.3 Mosaic Spatial Variables 12 Projects and Exercises 12.1 Projects in Spatial Modelling 12.1.1 Project Text: Continuous Spatial Variables 12.1.2 Project Text: Event Spatial Variables 12.1.3 Project Text: Mosaic Spatial Variables 12.2 Exercises in Spatial Modelling 12.2.1 Exercise Sets: Continuous Spatial Variables 12.2.2 Exercise Sets: Event Spatial Variables 12.2.3 Exercise Sets: Mosaic Spatial Variables References