Modelling Spatial and Spatial-Temporal Data A Bayesian Approach

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

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

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

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

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

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

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

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

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

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

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