Modelling Spatial and Spatial-Temporal Data: A Bayesian Approach

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
What Are the Aims of the Book?
What Are the Key Features of the Book?
The Structure of the Book
Acknowledgements
Part I Fundamentals for Modelling Spatial and Spatial-Temporal Data
1 Challenges and Opportunities Analysing Spatial and Spatial-Temporal Data
1.1 Introduction
1.2 Four Main Challenges When Analysing Spatial and Spatial-Temporal Data
1.2.1 Dependency
1.2.2 Heterogeneity
1.2.3 Data Sparsity
1.2.4 Uncertainty
1.2.4.1 Data Uncertainty
1.2.4.2 Model (or Process) Uncertainty
1.2.4.3 Parameter Uncertainty
1.3 Opportunities Arising from Modelling Spatial and Spatial-Temporal Data
1.3.1 Improving Statistical Precision
1.3.2 Explaining Variation in Space and Time
1.3.2.1 Example 1: Modelling Exposure-Outcome Relationships
1.3.2.2 Example 2: Testing a Conceptual Model at the Small Area Level
1.3.2.3 Example 3: Testing for Spatial Spillover (Local Competition) Effects
1.3.2.4 Example 4: Assessing the Effects of an Intervention
1.3.3 Investigating Space-Time Dynamics
1.4 Spatial and Spatial-Temporal Models: Bridging between Challenges and Opportunities
1.4.1 Statistical Thinking in Analysing Spatial and Spatial-Temporal Data: The Big Picture
1.4.2 Bayesian Thinking in a Statistical Analysis
1.4.3 Bayesian Hierarchical Models
1.4.3.1 Thinking Hierarchically
1.4.3.2 Incorporating Spatial and Spatial-Temporal Dependence Structures in a Bayesian Hierarchical Model Using Random Effects
1.4.3.3 Information Sharing in a Bayesian Hierarchical Model through Random Effects
1.4.4 Bayesian Spatial Econometrics
1.5 Concluding Remarks
1.6 The Datasets Used in the Book
1.7 Exercises
2 Concepts for Modelling Spatial and Spatial-Temporal Data: An Introduction to “Spatial Thinking”
2.1 Introduction
2.2 Mapping Data and Why It Matters
2.3 Thinking Spatially
2.3.1 Explaining Spatial Variation
2.3.2 Spatial Interpolation and Small Area Estimation
2.4 Thinking Spatially and Temporally
2.4.1 Explaining Space-Time Variation
2.4.2 Estimating Parameters for Spatial-Temporal Units
2.5 Concluding Remarks
2.6 Exercises
Appendix: Geographic Information Systems
3 The Nature of Spatial and Spatial-Temporal Attribute Data
3.1 Introduction
3.2 Data Collection Processes in the Social Sciences
3.2.1 Natural Experiments
3.2.2 Quasi-Experiments
3.2.3 Non-Experimental Observational Studies
3.3 Spatial and Spatial-Temporal Data: Properties
3.3.1 From Geographical Reality to the Spatial Database
3.3.2 Fundamental Properties of Spatial and Spatial-Temporal Data
3.3.2.1 Spatial and Temporal Dependence
3.3.2.2 Spatial and Temporal Heterogeneity
3.3.3 Properties Induced by Representational Choices
3.3.4 Properties Induced by Measurement Processes
3.4 Concluding Remarks
3.5 Exercises
4 Specifying Spatial Relationships on the Map: The Weights Matrix
4.1 Introduction
4.2 Specifying Weights Based on Contiguity
4.3 Specifying Weights Based on Geographical Distance
4.4 Specifying Weights Based on the Graph Structure Associated with a Set of Points
4.5 Specifying Weights Based on Attribute Values
4.6 Specifying Weights Based on Evidence about Interactions
4.7 Row Standardisation
4.8 Higher Order Weights Matrices
4.9 Choice of W and Statistical Implications
4.9.1 Implications for Small Area Estimation
4.9.2 Implications for Spatial Econometric Modelling
4.9.3 Implications for Estimating the Effects of Observable Covariates on the Outcome
4.10 Estimating the W Matrix
4.11 Concluding Remarks
4.12 Exercises
4.13 Appendices
Appendix 4.13.1 Building a Geodatabase in R
Appendix 4.13.2 Constructing the W Matrix and Accessing Data Stored in a Shapefile
5 Introduction to the Bayesian Approach to Regression Modelling with Spatial and Spatial-Temporal Data
5.1 Introduction
5.2 Introducing Bayesian Analysis
5.2.1 Prior, Likelihood and Posterior: What Do These Terms Refer To?
5.2.2 Example: Modelling High-Intensity Crime Areas
5.3 Bayesian Computation
5.3.1 Summarising the Posterior Distribution
5.3.2 Integration and Monte Carlo Integration
5.3.3 Markov Chain Monte Carlo with Gibbs Sampling
5.3.4 Introduction to WinBUGS
5.3.5 Practical Considerations when Fitting Models in WinBUGS
5.3.5.1 Setting the Initial Values
5.3.5.2 Checking Convergence
5.3.5.3 Checking Efficiency
5.4 Bayesian Regression Models
5.4.1 Example I: Modelling Household-Level Income
5.4.2 Example II: Modelling Annual Burglary Rates in Small Areas
5.5 Bayesian Model Comparison and Model Evaluation
5.6 Prior Specifications
5.6.1 When We Have Little Prior Information
5.6.2 Towards More Informative Priors for Modelling Spatial and Spatial-Temporal Data
5.7 Concluding Remarks
5.8 Exercises
Part II Modelling Spatial Data
6 Exploratory Analysis of Spatial Data
6.1 Introduction
6.2 Techniques for the Exploratory Analysis of Univariate Spatial Data
6.2.1 Mapping
6.2.2 Checking for Spatial Trend
6.2.3 Checking for Spatial Heterogeneity in the Mean
6.2.3.1 Count Data
6.2.3.2 A Monte Carlo Test
6.2.3.3 Continuous-Valued Data
6.2.4 Checking for Global Spatial Dependence (Spatial Autocorrelation)
6.2.4.1 The Moran Scatterplot
6.2.4.2 The Global Moran’s I Statistic
6.2.4.3 Other Tests for Assessing Global Spatial Autocorrelation
6.2.4.4 The Global Moran’s I Applied to Regression Residuals
6.2.4.5 The Join-Count Test for Categorical Data
6.2.5 Checking for Spatial Heterogeneity in the Spatial Dependence Structure: Detecting Local Spatial Clusters
6.2.5.1 The Local Moran’s I
6.2.5.2 The Multiple Testing Problem When Using Local Moran’s I
6.2.5.3 Kulldorff’s Spatial Scan Statistic
6.3 Exploring Relationships between Variables
6.3.1 Scatterplots and the Bivariate Moran Scatterplot
6.3.2 Quantifying Bivariate Association
6.3.2.1 The Clifford-Richardson Test of Bivariate Correlation in the Presence of Spatial Autocorrelation
6.3.2.2 Testing for Association “At a Distance” and the Global Bivariate Moran’s I
6.3.3 Checking for Spatial Heterogeneity in the Outcome-Covariate Relationship: Geographically Weighted Regression (GWR)
6.4 Overdispersion and Zero-Inflation in Spatial Count Data
6.4.1 Testing for Overdispersion
6.4.2 Testing for Zero-Inflation
6.5 Concluding Remarks
6.6 Exercises
Appendix. An R Function to Perform the Zero-Inflation Test by van Den Broek (1995). See more detail in Section 6.4.2.
7 Bayesian Models for Spatial Data I: Non-Hierarchical and Exchangeable Hierarchical Models
7.1 Introduction
7.2 Estimating Small Area Income: A Motivating Example and Different Modelling Strategies
7.2.1 Modelling the 109 Parameters Non-Hierarchically
7.2.2 Modelling the 109 Parameters Hierarchically
7.3 Modelling the Newcastle Income Data Using Non-Hierarchical Models
7.3.1 An Identical Parameter Model Based on Strategy 1
7.3.2 An Independent Parameters Model Based on Strategy 2
7.4 An Exchangeable Hierarchical Model Based on Strategy 3
7.4.1 The Logic of Information Borrowing and Shrinkage
7.4.2 Explaining the Nature of Global Smoothing Due to Exchangeability
7.4.3 The Variance Partition Coefficient (VPC)
7.4.4 Applying an Exchangeable Hierarchical Model to the Newcastle Income Data
7.5 Concluding Remarks
7.6 Exercises
7.7 Appendix: Obtaining the Simulated Household Income Data
8 Bayesian Models for Spatial Data II: Hierarchical Models with Spatial Dependence
8.1 Introduction
8.2 The Intrinsic Conditional Autoregressive (ICAR) Model
8.2.1 The ICAR Model Using a Spatial Weights Matrix with Binary Entries
8.2.1.1 The WinBUGS Implementation of the ICAR Model
8.2.1.2 Applying the ICAR Model Using Spatial Contiguity to the Newcastle Income Data
8.2.1.3 Results
8.2.1.4 A Summary of the Properties of the ICAR Model Using a Binary Spatial Weights Matrix
8.2.2 The ICAR Model with a General Weights Matrix
8.2.2.1 Expressing the ICAR Model as a Joint Distribution and the Implied Restriction on W
8.2.2.2 The Sum-to-Zero Constraint
8.2.2.3 Applying the ICAR Model Using General Weights to the Newcastle Income Data
8.2.2.4 Results
8.3 The Proper CAR (pCAR) Model
8.3.1 Prior Choice for ρ
8.3.2 ICAR or pCAR?
8.3.3 Applying the pCAR Model to the Newcastle Income Data
8.3.4 Results
8.4 Locally Adaptive Models
8.4.1 Choosing an Optimal W Matrix from All Possible Specifications
8.4.2 Modelling the Elements in the W Matrix
8.4.3 Applying Some of the Locally Adaptive Spatial Models to a Subset of the Newcastle Income Data
8.5 The Besag, York and Mollié (BYM) Model
8.5.1 Two Remarks on Applying the BYM Model in Practice
8.5.2 Applying the BYM Model to the Newcastle Income Data
8.6 Comparing the Fits of Different Bayesian Spatial Models
8.6.1 DIC Comparison
8.6.2 Model Comparison Based on the Quality of the MSOA-Level Average Income Estimates
8.7 Concluding Remarks
8.8 Exercises
9 Bayesian Hierarchical Models for Spatial Data: Applications
9.1 Introduction
9.2 Application 1: Modelling the Distribution of High Intensity Crime Areas in a City
9.2.1 Background
9.2.2 Data and Exploratory Analysis
9.2.3 Methods Discussed in Haining and Law (2007) to Combine the PHIA and EHIA Maps
9.2.4 A Joint Analysis of the PHIA and EHIA Data Using the MVCAR Model
9.2.5 Results
9.2.6 Another Specification of the MVCAR Model and a Limitation of the MVCAR Approach
9.2.7 Conclusion and Discussion
9.3 Application 2: Modelling the Association Between Air Pollution and Stroke Mortality
9.3.1 Background and Data
9.3.2 Modelling
9.3.3 Interpreting the Statistical Results
9.3.4 Conclusion and Discussion
9.4 Application 3: Modelling the Village-Level Incidence of Malaria in a Small Region of India
9.4.1 Background
9.4.2 Data and Exploratory Analysis
9.4.3 Model I: A Poisson Regression Model with Random Effects
9.4.4 Model II: A Two-Component Poisson Mixture Model
9.4.5 Model III: A Two-Component Poisson Mixture Model with Zero-Inflation
9.4.6 Results
9.4.7 Conclusion and Model Extensions
9.5 Application 4: Modelling the Small Area Count of Cases of Rape in Stockholm, Sweden
9.5.1 Background and Data
9.5.2 Modelling
9.5.2.1 A “Whole-Map” Analysis Using Poisson Regression
9.5.2.2 A “Localised” Analysis Using Bayesian Profile Regression
9.5.3 Results
9.5.3.1 “Whole Map” Associations for the Risk Factors
9.5.3.2 “Local” Associations for the Risk Factors
9.5.4 Conclusions
9.6 Exercises
10 Spatial Econometric Models
10.1 Introduction
10.2 Spatial Econometric Models
10.2.1 Three Forms of Spatial Spillover
10.2.2 The Spatial Lag Model (SLM)
10.2.2.1 Formulating the Model
10.2.2.2 An Example of the SLM
10.2.2.3 The Reduced Form of the SLM and the Constraint on δ
10.2.2.4 Specification of the Spatial Weights Matrix
10.2.2.5 Issues with Model Fitting and Interpreting Coefficients
10.2.3 The Spatially-Lagged Covariates Model (SLX)
10.2.3.1 Formulating the Model
10.2.3.2 An Example of the SLX Model
10.2.4 The Spatial Error Model (SEM)
10.2.5 The Spatial Durbin Model (SDM)
10.2.5.1 Formulating the Model
10.2.5.2 Relating the SDM Model to the Other Three Spatial Econometric Models
10.2.6 Prior Specifications
10.2.7 An Example: Modelling Cigarette Sales in 46 US States
10.2.7.1 Data Description, Exploratory Analysis and Model Specifications
10.2.7.2 Results
10.3 Interpreting Covariate Effects
10.3.1 Definitions of the Direct, Indirect and Total Effects of a Covariate
10.3.2 Measuring Direct and Indirect Effects without the SAR Structure on the Outcome Variables
10.3.2.1 For the LM and SEM Models
10.3.2.2 For the SLX Model
10.3.3 Measuring Direct and Indirect Effects When the Outcome Variables are Modelled by the SAR Structure
10.3.3.1 Understanding Direct and Indirect Effects in the Presence of Spatial Feedback
10.3.3.2 Calculating the Direct and Indirect Effects in the Presence of Spatial Feedback
10.3.3.3 Some Properties of Direct and Indirect Effects
10.3.3.4 A Property (Limitation) of the Average Direct and Average Indirect Effects Under the SLM Model
10.3.3.5 Summary
10.3.4 The Estimated Effects from the Cigarette Sales Data
10.4 Model Fitting in WinBUGS
10.4.1 Derivation of the Likelihood Function
10.4.2 Simplifications to the Likelihood Computation
10.4.3 The Zeros-Trick in WinBUGS
10.4.4 Calculating the Covariate Effects in WinBUGS
10.5 Concluding Remarks
10.5.1 Other Spatial Econometric Models and the Two Problems of Identifiability
10.5.2 Comparing the Hierarchical Modelling Approach and the Spatial Econometric Approach: A Summary
10.6 Exercises
11 Spatial Econometric Modelling: Applications
11.1 Application 1: Modelling the Voting Outcomes at the Local Authority District Level in England from the 2016 EU Referendum
11.1.1 Introduction
11.1.2 Data
11.1.3 Exploratory Data Analysis
11.1.4 Modelling Using Spatial Econometric Models
11.1.5 Results
11.1.6 Conclusion and Discussion
11.2 Application 2: Modelling Price Competition Between Petrol Retail Outlets in a Large City
11.2.1 Introduction
11.2.2 Data
11.2.3 Exploratory Data Analysis
11.2.4 Spatial Econometric Modelling and Results
11.2.5 A Spatial Hierarchical Model with t4 Likelihood
11.2.6 Conclusion and Discussion
11.3 Final Remarks on Spatial Econometric Modelling of Spatial Data
11.4 Exercises
Appendix: Petrol Retail Price Data
Part III Modelling Spatial-Temporal Data
12 Modelling Spatial-Temporal Data: An Introduction
12.1 Introduction
12.2 Modelling Annual Counts of Burglary Cases at the Small Area Level: A Motivating Example and Frameworks for Modelling Spatial-Temporal Data
12.3 Modelling Small Area Temporal Data
12.3.1 Issues to Consider When Modelling Temporal Patterns in the Small Area Setting
12.3.1.1 Issues Relating to Temporal Dependence
12.3.1.2 Issues Relating to Temporal Heterogeneity and Spatial Heterogeneity in Modelling Small Area Temporal Patterns
12.3.1.3 Issues Relating to Flexibility of a Temporal Model
12.3.2 Modelling Small Area Temporal Patterns: Setting the Scene
12.3.3 A Linear Time Trend Model
12.3.3.1 Model Formulations
12.3.3.2 Modelling Trends in the Peterborough Burglary Data
12.3.4 Random Walk Models
12.3.4.1 Model Formulations
12.3.4.2 The RW1 Model: Its Formulation Via the Full Conditionals and Its Properties
12.3.4.3 WinBUGS Implementation of the RW1 Model
12.3.4.4 Example: Modelling Burglary Trends Using the Peterborough Data
12.3.4.5 The Random Walk Model of Order 2
12.3.5 Interrupted Time Series (ITS) Models
12.3.5.1 Quasi-Experimental Designs and the Purpose of ITS Modelling
12.3.5.2 Model Formulations
12.3.5.3 WinBUGS Implementation
12.3.5.4 Results
12.4 Concluding Remarks
12.5 Exercises
Appendix: Three Different Forms for Specifying the Impact Function f.
13 Exploratory Analysis of Spatial-Temporal Data
13.1 Introduction
13.2 Patterns of Spatial-Temporal Data
13.3 Visualising Spatial-Temporal Data
13.4 Tests of Space-Time Interaction
13.4.1 The Knox Test
13.4.1.1 An Instructive Example of the Knox Test and Different Methods to Derive a p-Value
13.4.1.2 Applying the Knox Test to the Malaria Data
13.4.2 Kulldorff’s Space-Time Scan Statistic
13.4.2.1 Application: The Simulated Small Area COPD Mortality Data
13.4.3 Assessing Space-Time Interaction in the Form of Varying Local Time Trend Patterns
13.4.3.1 Exploratory Analysis of the Local Trends in the Peterborough Burglary Data
13.4.3.2 Exploratory Analysis of the Local Time Trends in the England COPD Mortality Data
13.5 Concluding Remarks
13.6 Exercises
14 Bayesian Hierarchical Models for Spatial-Temporal Data I: Space-Time Separable Models
14.1 Introduction
14.2 Estimating Small Area Burglary Rates Over Time: Setting the Scene
14.3 The Space-Time Separable Modelling Framework
14.3.1 Model Formulations
14.3.2 Do We Combine the Space and Time Components Additively or Multiplicatively?
14.3.3 Analysing the Peterborough Burglary Data Using a Space-Time Separable Model
14.3.4 Results
14.4 Concluding Remarks
14.5 Exercises
15 Bayesian Hierarchical Models for Spatial-Temporal Data II: Space-Time Inseparable Models
15.1 Introduction
15.2 From Space-Time Separability to Space-Time Inseparability: The Big Picture
15.3 Type I Space-Time Interaction
15.3.1 Example: A Space-Time Model with Type I Space-Time Interaction
15.3.2 WinBUGS Implementation
15.4 Type II Space-Time Interaction
15.4.1 Example: Two Space-Time Models with Type II Space-Time Interaction
15.4.2 WinBUGS Implementation
15.5 Type III Space-Time Interaction
15.5.1 Example: A Space-Time Model with Type III Space-Time Interaction
15.5.2 WinBUGS Implementation
15.6 Results from Analysing the Peterborough Burglary Data
15.7 Type IV Space-Time Interaction
15.7.1 Strategy 1: Extending Type II to Type IV
15.7.2 Strategy 2: Extending Type III to Type IV
15.7.2.1 Examples of Strategy 2
15.7.3 Strategy 3: Clayton’s Rule
15.7.3.1 Structure Matrices and Gaussian Markov Random Fields
15.7.3.2 Taking the Kronecker Product
15.7.3.3 Exploring the Induced Space-Time Dependence Structure via the Full Conditionals
15.7.4 Summary on Type IV Space-Time Interaction
15.8 Concluding Remarks
15.9 Exercises
16 Applications in Modelling Spatial-Temporal Data
16.1 Introduction
16.2 Application 1: Evaluating a Targeted Crime Reduction Intervention
16.2.1 Background and Data
16.2.2 Constructing Different Control Groups
16.2.3 Evaluation Using ITS
16.2.4 WinBUGS Implementation
16.2.5 Results
16.2.6 Some Remarks
16.3 Application 2: Assessing the Stability of Risk in Space and Time
16.3.1 Studying the Temporal Dynamics of Crime Hotspots and Coldspots: Background, Data and the Modelling Idea
16.3.2 Model Formulations
16.3.3 Classification of Areas
16.3.4 Model Implementation and Area Classification
16.3.5 Interpreting the Statistical Results
16.4 Application 3: Detecting Unusual Local Time Patterns in Small Area Data
16.4.1 Small Area Disease Surveillance: Background and Modelling Idea
16.4.2 Model Formulation
16.4.3 Detecting Unusual Areas with a Control of the False Discovery Rate
16.4.4 Fitting BaySTDetect in WinBUGS
16.4.5 A Simulated Dataset to Illustrate the Use of BaySTDetect
16.4.6 Results from the Simulated Dataset
16.4.7 General Results from Li et al. (2012) and an Extension of BaySTDetect
16.5 Application 4: Investigating the Presence of Spatial-Temporal Spillover Effects on Village-Level Malaria Risk in Kalaburagi, Karnataka, India
16.5.1 Background and Study Objective
16.5.2 Data
16.5.3 Modelling
16.5.4 Results
16.5.5 Concluding Remarks
16.6 Conclusions
16.7 Exercises
Part IV Addendum
17 Modelling Spatial and Spatial-Temporal Data: Future Agendas?
17.1 Topic 1: Modelling Multiple Related Outcomes Over Space and Time
17.2 Topic 2: Joint Modelling of Georeferenced Longitudinal and Time-to-Event Data
17.3 Topic 3: Multiscale Modelling
17.4 Topic 4: Using Survey Data for Small Area Estimation
17.5 Topic 5: Combining Data at Both Aggregate and Individual Levels to Improve Ecological Inference
17.6 Topic 6: Geostatistical Modelling
17.6.1 Spatial Dependence
17.6.2 Mapping to Reduce Visual Bias
17.6.3 Modelling Scale Effects
17.7 Topic 7: Modelling Count Data in Spatial Econometrics
17.8 Topic 8: Computation
References
Index