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Modelling Spatial and Spatial-Temporal Data

A Bayesian Approach

; Guangquan Li

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. Les mer
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Om boka

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.

Fakta

Innholdsfortegnelse

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 Mollie (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

Om forfatteren

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.