1st Edition
Modelling Spatial and Spatial-Temporal Data A Bayesian Approach
Modelling Spatial and Spatial-Temporal Data: A Bayesian Approach is aimed at statisticians and quantitative social, economic and public health students and researchers who work with small-area spatial and spatial-temporal data. It assumes a grounding in statistical theory up to the standard linear regression model. The book compares both hierarchical and spatial econometric modelling, providing both a reference and a teaching text with exercises in each chapter. The book provides a fully Bayesian, self-contained, treatment of the underlying statistical theory, with chapters dedicated to substantive applications. The book includes WinBUGS code and R code and all datasets are available online.
Part I covers fundamental issues arising when modelling spatial and spatial-temporal data. Part II focuses on modelling cross-sectional spatial data and begins by describing exploratory methods that help guide the modelling process. There are then two theoretical chapters on Bayesian models and a chapter of applications. Two chapters follow on spatial econometric modelling, one describing different models, the other substantive applications. Part III discusses modelling spatial-temporal data, first introducing models for time series data. Exploratory methods for detecting different types of space-time interaction are presented, followed by two chapters on the theory of space-time separable (without space-time interaction) and inseparable (with space-time interaction) models. An applications chapter includes: the evaluation of a policy intervention; analysing the temporal dynamics of crime hotspots; chronic disease surveillance; and testing for evidence of spatial spillovers in the spread of an infectious disease. A final chapter suggests some future directions and challenges.
Robert Haining is Emeritus Professor in Human Geography, University of Cambridge, England. He is the author of Spatial Data Analysis in the Social and Environmental Sciences (1990) and Spatial Data Analysis: Theory and Practice (2003). He is a Fellow of the RGS-IBG and of the Academy of Social Sciences.
Guangquan Li is Senior Lecturer in Statistics in the Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle, England. His research includes the development and application of Bayesian methods in the social and health sciences. He is a Fellow of the Royal Statistical Society.
Preface
Section I. Fundamentals for modelling spatial and spatial-temporal data
1. Challenges and opportunities analysing spatial and spatial-temporal data
Introduction
Four main challenges when analysing spatial and spatial-temporal data
Dependency
Heterogeneity
Data sparsity
Uncertainty
Data uncertainty
Model (or process) uncertainty
Parameter uncertainty
Opportunities arising from modelling spatial and spatial-temporal data
Improving statistical precision
Explaining variation in space and time
Example 1: Modelling exposure-outcome relationships
Example 2: Testing a conceptual model at the small area level
Example 3: Testing for spatial spillover (local competition) effects
Example 4: Assessing the effects of an intervention
Investigating space-time dynamics
Spatial and spatial-temporal models: bridging between challenges and opportunities
Statistical thinking in analysing spatial and spatial-temporal data: the big picture
Bayesian thinking in a statistical analysis
Bayesian hierarchical models
Thinking hierarchically
The data model
The process model
The parameter model
Incorporating spatial and spatial-temporal dependence structures in a Bayesian hierarchical model using random effects
Information sharing in a Bayesian hierarchical model through random effects
Bayesian spatial econometrics
Concluding remarks
The datasets used in the book
Exercises
2. Concepts for modelling spatial and spatial-temporal data: an introduction to "spatial thinking"
Introduction
Mapping data and why it matters
Thinking spatially
Explaining spatial variation
Spatial interpolation and small area estimation
Thinking spatially and temporally
Explaining space-time variation
Estimating parameters for spatial-temporal units
Concluding remarks
Exercises
Appendix: Geographic Information Systems
3. The nature of spatial and spatial-temporal attribute data
Introduction
Data collection processes in the social sciences
Natural experiments
Quasi-experiments
Non-experimental observational studies
Spatial and spatial-temporal data: properties
From geographical reality to the spatial database
Fundamental properties of spatial and spatial-temporal data
Spatial and temporal dependence.
Spatial and temporal heterogeneity
Properties induced by representational choices
Properties induced by measurement processes
Concluding remarks
Exercises
4. Specifying spatial relationships on the map: the weights matrix
Introduction
Specifying weights based on contiguity
Specifying weights based on geographical distance
Specifying weights based on the graph structure associated with a set of points
Specifying weights based on attribute values
Specifying weights based on evidence about interactions
Row standardisation
Higher order weights matrices
Choice of W and statistical implications
Implications for small area estimation
Implications for spatial econometric modelling
Implications for estimating the effects of observable covariates on the outcome
Estimating the W matrix
Concluding remarks
Exercises
Appendices
Appendix: Building a geodatabase in R
Appendix: 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
Introduction
Introducing Bayesian analysis
Prior, likelihood and posterior: what do these terms refer to?
Example: modelling high-intensity crime areas
Bayesian computation
Summarizing the posterior distribution
Integration and Monte Carlo integration
Markov chain Monte Carlo with Gibbs sampling
Introduction to WinBUGS
Practical considerations when fitting models in WinBUGS
Setting the initial values
Checking convergence
Checking efficiency
Bayesian regression models
Example I: modelling household-level income
Example II: modelling annual burglary rates in small areas
Bayesian model comparison and model evaluation
Prior specifications
When we have little prior information
Towards more informative priors for spatial and spatial-temporal data
Concluding remarks
Exercises
Section II Modelling spatial data
6. Exploratory analysis of spatial data
Introduction
Techniques for the exploratory analysis of univariate spatial data
Mapping
Checking for spatial trend
Checking for spatial heterogeneity in the mean
Count data
A Monte Carlo test
Continuous-valued data
Checking for global spatial dependence (spatial autocorrelation)
The Moran scatterplot
The global Moran’s I statistic
Other test statistics for assessing global spatial autocorrelation
The join-count test for categorical data
The global Moran’s I applied to regression residuals
Checking for spatial heterogeneity in the spatial dependence structure: detecting local spatial clusters
The Local Moran’s I
The multiple testing problem when using local Moran’s I
Kulldorff’s spatial scan statistic
Exploring relationships between variables:
Scatterplots and the bivariate Moran scatterplot
Quantifying bivariate association
The Clifford-Richardson test of bivariate correlation in the presence of spatial autocorrelation
Testing for association "at a distance" and the global bivariate Moran’s I
Checking for spatial heterogeneity in the outcome-covariate relationship: Geographically weighted regression (GWR)
Overdispersion and zero-inflation in spatial count data
Testing for overdispersion
Testing for zero-inflation
Concluding remarks
Exercises
Appendix: An R function to perform the zero-inflation test by van den Broek (1995)
7. Bayesian models for spatial data I: Non-hierarchical and exchangeable hierarchical models
Introduction
Estimating small area income: a motivating example and different modelling strategies
Modelling the 109 parameters non-hierarchically
Modelling the 109 parameters hierarchically
Modelling the Newcastle income data using non-hierarchical models
An identical parameter model based on Strategy 1
An independent parameters model based on Strategy 2
An exchangeable hierarchical model based on Strategy 3
The logic of information borrowing and shrinkage
Explaining the nature of global smoothing due to exchangeability
The variance partition coefficient (VPC)
Applying an exchangeable hierarchical model to the Newcastle income data
Concluding remarks
Exercises
Appendix: Obtaining the simulated household income data
8. Bayesian models for spatial data II: hierarchical models with spatial dependence
Introduction
The intrinsic conditional autoregressive (ICAR) model
The ICAR model using a spatial weights matrix with binary entries
The WinBUGS implementation of the ICAR model
Applying the ICAR model using spatial contiguity to the Newcastle income data
Results
A summary of the properties of the ICAR model using a binary spatial weights matrix
The ICAR model with a general weights matrix
Expressing the ICAR model as a joint distribution and the implied restriction on W
The sum-to-zero constraint
Applying the ICAR model using general weights to the Newcastle income data
Results
The proper CAR (pCAR) model
Prior choice for ?
ICAR or pCAR?
Applying the pCAR model to the Newcastle income data
Results
Locally adaptive models
Choosing an optimal W matrix from all possible specifications
Modelling the elements of the W matrix
Applying some of the locally adaptive spatial models to a subset of the Newcastle income data
The Besag, York and Mollié (BYM) model
Two remarks on applying the BYM model in practice
Applying the BYM model to the Newcastle income data
Comparing the fits of different Bayesian spatial models
DIC comparison
Model comparison based on the quality of the MSOA-level average income estimates
Concluding remarks
Exercises
9. Bayesian hierarchical models for spatial data: applications
Introduction
Application 1: Modelling the distribution of high intensity crime areas in a city
Background
Data and exploratory analysis
Methods discussed in Haining and Law (2007) to combine the PHIA and EHIA maps
A joint analysis of the PHIA and EHIA data using the MVCAR model
Results
Another specification of the MVCAR model and a limitation of the MVCAR approach
Conclusion and discussion
Application 2: Modelling the association between air pollution and stroke mortality
Background and data
Modelling
Interpreting the statistical results
Conclusion and discussion
Application 3: Modelling the village-level incidence of malaria in a small region of India
Background
Data and exploratory analysis
Model I: A Poisson regression model with random effects
Model II: A two-component Poisson mixture model
Model III: A two-component Poisson mixture model with zero-inflation
Results
Conclusion and model extensions
Application 4: Modelling the small area count of cases of rape in Stockholm, Sweden
Background and data
Modelling
"whole map" analysis using Poisson regression
"localised" analysis using Bayesian profile regression
Results
"Whole map" associations for the risk factors
"Local" associations for the risk factors
Conclusions
Exercises
10. Spatial econometric models
Introduction
Spatial econometric models
Three forms of spatial spillover
The spatial lag model (SLM)
Formulating the model
An example of the SLM
The reduced form of the SLM and the constraint on?
Specification of the spatial weights matrix
Issues with model fitting and interpreting coefficients
The spatially lagged covariates model (SLX)
Formulating the model
An example of the SLX model
The spatial error model (SEM)
The spatial Durbin model (SDM)
Formulating the model
Relating the SDM model to the other three spatial econometric models
Prior specifications
An example: modelling cigarette sales in 46 US states
Data description, exploratory analysis and model specifications
Results
Interpreting covariate effects
Definitions of the direct, indirect and total effects of a covariate
Measuring direct and indirect effects without the SAR structure on the outcome variables
For the LM and SEM models
For the SLX model
Measuring direct and indirect effects when the outcome variables are modelled by the SAR structure
Understanding direct and indirect effects in the presence of spatial feedback
Calculating the direct and indirect effects in the presence of spatial feedback
Some properties of direct and indirect effects
A property (limitation) of the average direct and average indirect effects under the SLM model
Summary
The estimated effects from the cigarette sales data
Model fitting in WinBUGS
Derivation of the likelihood function
Simplifications to the likelihood function
The zeros-trick in WinBUGS
Calculating the covariate effects in WinBUGS
Concluding remarks
Other spatial econometric models and two problems of identifiability
Comparing the hierarchical modelling approach and the spatial econometric modelling approach: a summary
Exercises
11. Spatial Econometric Modelling: applications
Application 1: Modelling the voting outcomes at the local authority district level in England from the 2016 EU referendum
Introduction
Data
Exploratory data analysis
Modelling using spatial econometric models
Results
Conclusion and discussion
Application 2: Modelling price competition between petrol retail outlets in a large city
Introduction
Data
Exploratory data analysis
Spatial econometric modelling and results
A spatial hierarchical model with t4 likelihood
Conclusion and discussion
Final remarks on spatial econometric modelling of spatial data
Exercises
Appendix: Petrol price data
Section III Modelling spatial-temporal data
12. Modelling spatial-temporal data: an introduction
Introduction
Modelling annual counts of burglary cases at the small area level: a motivating example and frameworks for modelling spatial-temporal data
Modelling small area temporal data
Issues to consider when modelling temporal patterns in the small area setting
Issues relating to temporal dependence
Issues relating to temporal heterogeneity and spatial heterogeneity in modelling small area temporal patterns
Issues relating to flexibility of a temporal model
Modelling small area temporal patterns: setting the scene
A linear time trend model
Model formulations
Modelling trends in the Peterborough burglary data
Results from fitting the linear trend model without temporal noise
Results from fitting the linear trend model with temporal no
Random walk models
Model formulations
The RW(1) model: its formulation via the full conditionals and its properties
WinBUGS implementation of the RW(1) model
Example: modelling burglary trends using the Peterborough data
The random walk model of order 2
Interrupted time series (ITS) models
Quasi-experimental designs and the purpose of ITS modelling
Model formulations
WinBUGS implementation
Results
Concluding remarks
Exercises
Appendix Three different forms for specifying the impact function, f
13. Exploratory analysis of spatial-temporal data
Introduction
Patterns of spatial-temporal data
Visualizing spatial-temporal datayou
Tests of space-time interaction
The Knox test
An instructive example of the Knox test and different methods to derive a p-value
Applying the Knox test to the malaria data
Kulldorff’s space-time scan statistic
Application: the simulated small area COPD mortality data
Assessing space-time interaction in the form of varying local time trend patterns
Exploratory analysis of the local trends in the Peterborough burglary data
Exploratory analysis of the local time trends in the England COPD mortality data
Concluding remarks
Exercises
14. Bayesian hierarchical models for spatial-temporal data I: space-time separable models
Introduction
Estimating small area burglary rates over time: setting the scene
The space-time separable modelling framework
Model formulations
Do we combine the space and time components additively or multiplicatively?
Analysing the Peterborough burglary data using a space-time separable model
Results
Concluding remarks
Exercises
15. Bayesian hierarchical models for spatial-temporal data II: space-time inseparable models
Introduction
From space-time separability to space-time inseparability: the big picture
Type I space-time interaction
Example: a space-time model with Type I space-time interaction
WinBUGS implementation
Type II space-time interaction
Example: two space-time models with Type II space-time interaction
WinBUGS implementation
Type III space-time interaction
Example: a space-time model with Type III space-time interaction
WinBUGS implementation
Results from analysing the Peterborough burglary data: Part I
Type IV space-time interaction
Strategy 1: extending Type II to Type IV
Strategy 2: extending Type III to Type IV
Examples of strategy 2
Strategy 3: Clayton’s rule
Structure matrices and Gaussian Markov random fields
Taking the Kronecker product
Exploring the induced space-time dependence structure via the full conditionals
Summary on Type IV space-time interaction
Concluding remarks
Exercises
16. Modelling spatial-temporal data: applications
Introduction
Application 1: evaluating a targeted crime reduction intervention
Background and data
Constructing different control groups
Evaluation using ITS
WinBUGS implementation
Results
Some remarks
Application 2: assessing the stability of risk in space and time
Studying the temporal dynamics of crime hotspots and coldspots: background, data and the modelling idea
Model formulations
Classification of areas
Model implementation and area classification
Interpreting the statistical results
Application 3: detecting unusual local time patterns in small area data
Small area disease surveillance: background and modelling idea
Model formulation
Detecting unusual areas with a control of the false discovery rate
Fitting BaySTDetect in WinBUGS
A simulated dataset to illustrate the use of BaySTDetect
Results from the simulated dataset
General results from Li et al. (2012) and an extension of BaySTDetect
Application 4: Investigating the presence of spatial-temporal spillover effects on village-level malaria risk in Kalaburagi, Karnataka, India
Background and study objective
Data
Modelling
Results
Concluding remarks
Conclusions
Section IV Directions in spatial and spatial-temporal data analysis
17. Modelling spatial and spatial-temporal data: Future agendas?
Topic 1: Modelling multiple related outcomes over space and time
Topic 2: Joint modelling of georeferenced longitudinal and time-to-event data
Topic 3: Multiscale modelling
Topic 4: Using survey data for small area estimation
Topic 5: Combining data at both aggregate and individual levels to improve ecological inference
Topic 6: Geostatistical modelling
Spatial dependence
Mapping to reduce visual bias
Modelling scale effects
Topic 7: Modelling count data in spatial econometrics
Topic 8: Computation
Biography
Robert Haining is Emeritus Professor in Human Geography, University of Cambridge, England. He is the author of Spatial Data Analysis in the Social and Environmental Sciences (1990) and Spatial Data Analysis: Theory and Practice (2003). He is a Fellow of the RGS-IBG and of the Academy of Social Sciences.
Guangquan Li is Senior Lecturer in Statistics in the Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle, England. His research includes the development and application of Bayesian methods in the social and health sciences. He is a Fellow of the Royal Statistical Society.
"Knowledge on statistical theory and regression concepts are essential to read, comprehend, appreciate, and use the rich contents of this fascinating book. This well-written book is a good source for the Bayesian concepts and methods to practice the spatial-temporal analysis using R and WinBugs codes . . . I recommend this book to economics, health, statistics and computing professionals and researchers."
~ Ramalingam Shanmugam, Texas State University