Modelling Spatial and Spatial-Temporal Data : A Bayesian Approach (Paperback)

PrefaceSectionI. Fundamentals for modelling spatial and spatial-temporal data1. Challenges and opportunities analysing spatial and spatial-temporal dataIntroductionFour main challenges when analysing spatial and spatial-temporal dataDependencyHeterogeneity Data sparsityUncertaintyData uncertainty Model (or process) uncertainty Parameter uncertaintyOpportunities arising from modelling spatial and spatial-temporal dataImproving statistical precision Explaining variation in space and time Example 1: Modelling exposure-outcome relationshipsExample 2: Testing a conceptual model at the small area levelExample 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 pictureBayesian thinking in a statistical analysisBayesian hierarchical modelsThinking hierarchicallyThe data modelThe process modelThe parameter modelIncorporating spatial and spatial-temporal dependence structures in a Bayesian hierarchical model using random effectsInformation sharing in a Bayesian hierarchical model through random effects Bayesian spatial econometricsConcluding remarksThe datasets used in the bookExercises2. Concepts for modelling spatial and spatial-temporal data: an introduction to "spatial thinking"IntroductionMapping data and why it mattersThinking spatiallyExplaining spatial variationSpatial interpolation and small area estimationThinking spatially and temporallyExplaining space-time variationEstimating parameters for spatial-temporal unitsConcluding remarksExercisesAppendix: Geographic Information Systems3. The nature of spatial and spatial-temporal attribute dataIntroductionData collection processes in the social sciencesNatural experimentsQuasi-experimentsNon-experimental observational studies Spatial and spatial-temporal data: propertiesFrom geographical reality to the spatial databaseFundamental properties of spatial and spatial-temporal dataSpatial and temporal dependence.Spatial and temporal heterogeneityProperties induced by representational choicesProperties induced by measurement processesConcluding remarksExercises4. Specifying spatial relationships on the map: the weights matrixIntroductionSpecifying weights based on contiguitySpecifying weights based on geographical distanceSpecifying weights based on the graph structure associated with a set of pointsSpecifying weights based on attribute valuesSpecifying weights based on evidence about interactionsRow standardisationHigher order weights matricesChoice of W and statistical implicationsImplications for small area estimationImplications for spatial econometric modellingImplications for estimating the effects of observable covariates on the outcomeEstimating the W matrixConcluding remarksExercisesAppendicesAppendix: Building a geodatabase in RAppendix: Constructing the W matrix and accessing data stored in a shapefile5. Introduction to the Bayesian approach to regression modelling with spatial and spatial-temporal dataIntroductionIntroducing Bayesian analysisPrior, likelihood and posterior: what do these terms refer to?Example: modelling high-intensity crime areasBayesian computationSummarizing the posterior distributionIntegration and Monte Carlo integrationMarkov chain Monte Carlo with Gibbs samplingIntroduction to WinBUGSPractical considerations when fitting models in WinBUGSSetting the initial valuesChecking convergenceChecking efficiencyBayesian regression modelsExample I: modelling household-level incomeExample II: modelling annual burglary rates in small areasBayesian model comparison and model evaluationPrior specificationsWhen we have little prior informationTowards more informative priors for spatial and spatial-temporal dataConcluding remarksExercisesSection II Modelling spatial data6. Exploratory analysis of spatial dataIntroductionTechniques for the exploratory analysis of univariate spatial dataMappingChecking for spatial trendChecking for spatial heterogeneity in the meanCount dataA Monte Carlo testContinuous-valued dataChecking for global spatial dependence (spatial autocorrelation)The Moran scatterplotThe global Moran’s I statisticOther test statistics for assessing global spatial autocorrelationThe join-count test for categorical dataThe global Moran’s I applied to regression residualsChecking for spatial heterogeneity in the spatial dependence structure: detecting local spatial clustersThe Local Moran’s I The multiple testing problem when using local Moran’s IKulldorff’s spatial scan statisticExploring relationships between variables:Scatterplots and the bivariate Moran scatterplotQuantifying bivariate associationThe Clifford-Richardson test of bivariate correlation in the presence of spatial autocorrelationTesting for association "at a distance" and the global bivariate Moran’s IChecking for spatial heterogeneity in the outcome-covariate relationship: Geographically weighted regression (GWR)Overdispersion and zero-inflation in spatial count dataTesting for overdispersionTesting for zero-inflationConcluding remarksExercisesAppendix: 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 modelsIntroductionEstimating small area income: a motivating example and different modelling strategiesModelling the 109 parameters non-hierarchicallyModelling the 109 parameters hierarchicallyModelling the Newcastle income data using non-hierarchical modelsAn identical parameter model based on Strategy 1An independent parameters model based on Strategy 2An exchangeable hierarchical model based on Strategy 3The logic of information borrowing and shrinkageExplaining the nature of global smoothing due to exchangeabilityThe variance partition coefficient (VPC)Applying an exchangeable hierarchical model to the Newcastle income dataConcluding remarksExercisesAppendix: Obtaining the simulated household income data8. Bayesian models for spatial data II: hierarchical models with spatial dependenceIntroductionThe intrinsic conditional autoregressive (ICAR) modelThe ICAR model using a spatial weights matrix with binary entriesThe WinBUGS implementation of the ICAR modelApplying the ICAR model using spatial contiguity to the Newcastle income dataResultsA summary of the properties of the ICAR model using a binary spatial weights matrixThe ICAR model with a general weights matrixExpressing the ICAR model as a joint distribution and the implied restriction on WThe sum-to-zero constraintApplying the ICAR model using general weights to the Newcastle income dataResultsThe proper CAR (pCAR) modelPrior choice for ?ICAR or pCAR?Applying the pCAR model to the Newcastle income dataResultsLocally adaptive modelsChoosing an optimal W matrix from all possible specificationsModelling the elements of the W matrixApplying some of the locally adaptive spatial models to a subset of the Newcastle income dataThe Besag, York and Mollie (BYM) modelTwo remarks on applying the BYM model in practiceApplying the BYM model to the Newcastle income dataComparing the fits of different Bayesian spatial modelsDIC comparisonModel comparison based on the quality of the MSOA-level average income estimatesConcluding remarksExercises9. Bayesian hierarchical models for spatial data: applicationsIntroductionApplication 1: Modelling the distribution of high intensity crime areas in a cityBackgroundData and exploratory analysisMethods discussed in Haining and Law (2007) to combine the PHIA and EHIA mapsA joint analysis of the PHIA and EHIA data using the MVCAR modelResultsAnother specification of the MVCAR model and a limitation of the MVCAR approachConclusion and discussionApplication 2: Modelling the association between air pollution and stroke mortalityBackground and dataModellingInterpreting the statistical resultsConclusion and discussionApplication 3: Modelling the village-level incidence of malaria in a small region of IndiaBackgroundData and exploratory analysisModel I: A Poisson regression model with random effectsModel II: A two-component Poisson mixture modelModel III: A two-component Poisson mixture model with zero-inflationResultsConclusion and model extensionsApplication 4: Modelling the small area count of cases of rape in Stockholm, SwedenBackground and dataModelling"whole map" analysis using Poisson regression"localised" analysis using Bayesian profile regressionResults"Whole map" associations for the risk factors"Local" associations for the risk factorsConclusionsExercises 10. Spatial econometric modelsIntroductionSpatial econometric modelsThree forms of spatial spilloverThe spatial lag model (SLM)Formulating the modelAn example of the SLMThe reduced form of the SLM and the constraint on?Specification of the spatial weights matrixIssues with model fitting and interpreting coefficientsThe spatially lagged covariates model (SLX)Formulating the modelAn example of the SLX modelThe spatial error model (SEM)The spatial Durbin model (SDM)Formulating the modelRelating the SDM model to the other three spatial econometric modelsPrior specificationsAn example: modelling cigarette sales in 46 US states Data description, exploratory analysis and model specificationsResultsInterpreting covariate effectsDefinitions of the direct, indirect and total effects of a covariateMeasuring direct and indirect effects without the SAR structure on the outcome variablesFor the LM and SEM modelsFor the SLX modelMeasuring direct and indirect effects when the outcome variables are modelled by the SAR structureUnderstanding direct and indirect effects in the presence of spatial feedbackCalculating the direct and indirect effects in the presence of spatial feedbackSome properties of direct and indirect effectsA property (limitation) of the average direct and average indirect effects under the SLM modelSummaryThe estimated effects from the cigarette sales dataModel fitting in WinBUGSDerivation of the likelihood functionSimplifications to the likelihood functionThe zeros-trick in WinBUGSCalculating the covariate effects in WinBUGSConcluding remarksOther spatial econometric models and two problems of identifiabilityComparing the hierarchical modelling approach and the spatial econometric modelling approach: a summaryExercises11. Spatial Econometric Modelling: applicationsApplication 1: Modelling the voting outcomes at the local authority district level in England from the 2016 EU referendumIntroductionDataExploratory data analysisModelling using spatial econometric modelsResultsConclusion and discussionApplication 2: Modelling price competition between petrol retail outlets in a large cityIntroductionDataExploratory data analysisSpatial econometric modelling and resultsA spatial hierarchical model with t4 likelihoodConclusion and discussionFinal remarks on spatial econometric modelling of spatial dataExercisesAppendix: Petrol price data Section III Modelling spatial-temporal data12. Modelling spatial-temporal data: an introductionIntroductionModelling annual counts of burglary cases at the small area level: a motivating example and frameworks for modelling spatial-temporal dataModelling small area temporal dataIssues to consider when modelling temporal patterns in the small area settingIssues relating to temporal dependenceIssues relating to temporal heterogeneity and spatial heterogeneity in modelling small area temporal patternsIssues relating to flexibility of a temporal modelModelling small area temporal patterns: setting the sceneA linear time trend modelModel formulationsModelling trends in the Peterborough burglary dataResults from fitting the linear trend model without temporal noiseResults from fitting the linear trend model with temporal no Random walk modelsModel formulationsThe RW(1) model: its formulation via the full conditionals and its propertiesWinBUGS implementation of the RW(1) modelExample: modelling burglary trends using the Peterborough dataThe random walk model of order 2Interrupted time series (ITS) modelsQuasi-experimental designs and the purpose of ITS modellingModel formulationsWinBUGS implementationResults Concluding remarksExercisesAppendix Three different forms for specifying the impact function, f13. Exploratory analysis of spatial-temporal dataIntroductionPatterns of spatial-temporal dataVisualizing spatial-temporal datayouTests of space-time interactionThe Knox testAn instructive example of the Knox test and different methods to derive a p-valueApplying the Knox test to the malaria dataKulldorff’s space-time scan statisticApplication: the simulated small area COPD mortality dataAssessing space-time interaction in the form of varying local time trend patternsExploratory analysis of the local trends in the Peterborough burglary dataExploratory analysis of the local time trends in the England COPD mortality dataConcluding remarksExercises14. Bayesian hierarchical models for spatial-temporal data I: space-time separable modelsIntroductionEstimating small area burglary rates over time: setting the sceneThe space-time separable modelling frameworkModel formulationsDo we combine the space and time components additively or multiplicatively?Analysing the Peterborough burglary data using a space-time separable modelResultsConcluding remarksExercises15. Bayesian hierarchical models for spatial-temporal data II: space-time inseparable modelsIntroductionFrom space-time separability to space-time inseparability: the big pictureType I space-time interactionExample: a space-time model with Type I space-time interactionWinBUGS implementationType II space-time interactionExample: two space-time models with Type II space-time interactionWinBUGS implementationType III space-time interactionExample: a space-time model with Type III space-time interactionWinBUGS implementationResults from analysing the Peterborough burglary data: Part IType IV space-time interactionStrategy 1: extending Type II to Type IVStrategy 2: extending Type III to Type IVExamples of strategy 2Strategy 3: Clayton’s ruleStructure matrices and Gaussian Markov random fieldsTaking the Kronecker productExploring the induced space-time dependence structure via the full conditionalsSummary on Type IV space-time interactionConcluding remarksExercises16. Modelling spatial-temporal data: applicationsIntroductionApplication 1: evaluating a targeted crime reduction interventionBackground and dataConstructing different control groupsEvaluation using ITSWinBUGS implementationResultsSome remarksApplication 2: assessing the stability of risk in space and timeStudying the temporal dynamics of crime hotspots and coldspots: background, data and the modelling ideaModel formulationsClassification of areasModel implementation and area classificationInterpreting the statistical resultsApplication 3: detecting unusual local time patterns in small area dataSmall area disease surveillance: background and modelling ideaModel formulationDetecting unusual areas with a control of the false discovery rateFitting BaySTDetect in WinBUGSA simulated dataset to illustrate the use of BaySTDetectResults from the simulated datasetGeneral results from Li et al. (2012) and an extension of BaySTDetectApplication 4: Investigating the presence of spatial-temporal spillover effects on village-level malaria risk in Kalaburagi, Karnataka, IndiaBackground and study objectiveDataModellingResultsConcluding remarksConclusionsSection IV Directions in spatial and spatial-temporal data analysis17. Modelling spatial and spatial-temporal data: Future agendas?Topic 1: Modelling multiple related outcomes over space and timeTopic 2: Joint modelling of georeferenced longitudinal and time-to-event dataTopic 3: Multiscale modellingTopic 4: Using survey data for small area estimationTopic 5: Combining data at both aggregate and individual levels to improve ecological inferenceTopic 6: Geostatistical modellingSpatial dependenceMapping to reduce visual biasModelling scale effectsTopic 7: Modelling count data in spatial econometricsTopic 8: Computation