Bayesian Modelling of Spatio-Temporal Data with R

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Introduction
Preface
1. Examples of spatio-temporal data
	1.1. Introduction
	1.2. Spatio-temporal data types
	1.3. Point referenced data sets used in the book
		1.3.1. New York air pollution data set
		1.3.2. Air pollution data from England and Wales
		1.3.3. Air pollution in the eastern US
		1.3.4. Hubbard Brook precipitation data
		1.3.5. Ocean chlorophyll data
		1.3.6. Atlantic ocean temperature and salinity data set
	1.4. Areal unit data sets used in the book
		1.4.1. Covid-19 mortality data from England
		1.4.2. Childhood vaccination coverage in Kenya
		1.4.3. Cancer rates in the United States
		1.4.4. Hospitalization data from England
		1.4.5. Child poverty in London
	1.5. Conclusion
	1.6. Exercises
2. Jargon of spatial and spatio-temporal modeling
	2.1. Introduction
	2.2. Stochastic processes
	2.3. Stationarity
	2.4. Variogram and covariogram
	2.5. Isotropy
	2.6. Matèrn covariance function
	2.7. Gaussian processes (GP) GP(0;C(
j ))
	2.8. Space-time covariance functions
	2.9. Kriging or optimal spatial prediction
	2.10. Autocorrelation and partial autocorrelation
	2.11. Measures of spatial association for areal data
	2.12. Internal and external standardization for areal data
	2.13. Spatial smoothers
	2.14. CAR models
	2.15. Point processes
	2.16. Conclusion
	2.17. Exercises
3. Exploratory data analysis methods
	3.1. Introduction
	3.2. Exploring point reference data
		3.2.1. Non-spatial graphical exploration
		3.2.2. Exploring spatial variation
	3.3. Exploring spatio-temporal point reference data
	3.4. Exploring areal Covid-19 case and death data
		3.4.1. Calculating the expected numbers of cases and deaths
		3.4.2. Graphical displays and covariate information
	3.5. Conclusion
	3.6. Exercises
4. Bayesian inference methods
	4.1. Introduction
	4.2. Prior and posterior distributions
	4.3. The Bayes theorem for probability
	4.4. Bayes theorem for random variables
	4.5. Posterior / Likelihood x Prior
	4.6. Sequential updating of the posterior distribution
	4.7. Normal-Normal example
	4.8. Bayes estimators
		4.8.1. Posterior mean
		4.8.2. Posterior median
		4.8.3. Posterior mode
	4.9. Credible interval
	4.10. Prior Distributions
		4.10.1. Conjugate prior distribution
		4.10.2. Locally uniform prior distribution
		4.10.3. Non-informative prior distribution
	4.11. Posterior predictive distribution
		4.11.1. Normal-Normal example
	4.12. Prior predictive distribution
	4.13. Inference for more than one parameter
	4.14. Normal example with both parameters unknown
	4.15. Model choice
		4.15.1. The Bayes factor
		4.15.2. Posterior probability of a model
		4.15.3. Hypothesis testing
	4.16. Criteria-based Bayesian model selection
		4.16.1. The DIC
		4.16.2. The WAIC
		4.16.3. Posterior predictive model choice criteria (PMCC)
	4.17. Bayesian model checking
		4.17.1. Nuisance parameters
	4.18. The pressing need for Bayesian computation
	4.19. Conclusion
	4.20. Exercises
5. Bayesian computation methods
	5.1. Introduction
	5.2. Two motivating examples for Bayesian computation
	5.3. Monte Carlo integration
	5.4. Importance sampling
	5.5. Rejection sampling
	5.6. Notions of Markov chains for understanding MCMC
	5.7. Metropolis-Hastings algorithm
	5.8. The Gibbs sampler
	5.9. Hamiltonian Monte Carlo
	5.10. Integrated nested Laplace approximation (INLA)
	5.11. MCMC implementation issues and MCMC output processing
		5.11.1. Diagnostics based on visual plots and autocorrelation
		5.11.2. How many chains?
		5.11.3. Method of batching
	5.12. Computing Bayesian model choice criteria
		5.12.1. Computing DIC
		5.12.2. Computing WAIC
		5.12.3. Computing PMCC
		5.12.4. Computing the model choice criteria for the New York air pollution data
	5.13. Conclusion
	5.14. Exercises
6. Bayesian modeling for point referenced spatial data
	6.1. Introduction
	6.2. Model versus procedure based methods
	6.3. Formulating linear models
		6.3.1. Data set preparation
		6.3.2. Writing down the model formula
		6.3.3. Predictive distributions
	6.4. Linear model for spatial data
		6.4.1. Spatial model fitting using bmstdr
	6.5. A spatial model with nugget effect
		6.5.1. Marginal model implementation
	6.6. Model fitting using software packages
		6.6.1. spBayes
		6.6.2. R-Stan
		6.6.3. R-inla
	6.7. Model choice
	6.8. Model validation methods
		6.8.1. Four most important model validation criteria
		6.8.2. K-fold cross-validation
		6.8.3. Illustrating the model validation statistics
	6.9. Posterior predictive checks
	6.10. Conclusion
	6.11. Exercises
7. Bayesian modeling for point referenced spatio-temporal data
	7.1. Introduction
	7.2. Models with spatio-temporal error distribution
		7.2.1. Posterior distributions
		7.2.2. Predictive distributions
		7.2.3. Simplifying the expressions: 12H?1 and 12H?121
		7.2.4. Estimation of v
		7.2.5. Illustration of a spatio-temporal model fitting
	7.3. Independent GP model with nugget effect
		7.3.1. Full model implementation using spTimer
		7.3.2. Marginal model implementation using Stan
	7.4. Auto regressive (AR) models
		7.4.1. Hierarchical AR Models using spTimer
		7.4.2. AR modeling using INLA
	7.5. Spatio-temporal dynamic models
		7.5.1. A spatially varying dynamic model spTDyn
		7.5.2. A dynamic spatio-temporal model using spBayes
	7.6. Spatio-temporal models based on Gaussian predictive processes (GPP)
	7.7. Performance assessment of all the models
	7.8. Conclusion
	7.9. Exercises
8. Practical examples of point referenced data modeling
	8.1. Introduction
	8.2. Estimating annual average air pollution in England and Wales
	8.3. Assessing probability of non-compliance in air pollution
	8.4. Analyzing precipitation data from the Hubbard Experimental Forest
		8.4.1. Exploratory data analysis
		8.4.2. Modeling and validation
		8.4.3. Predictive inference from model fitting
			8.4.3.1. Selecting gauges for possible downsizing
			8.4.3.2. Spatial patterns in 3-year rolling average annual precipitation
			8.4.3.3. Catchment specific trends in annual precipitation
			8.4.3.4. A note on model fitting
	8.5. Assessing annual trends in ocean chlorophyll levels
	8.6. Modeling temperature data from roaming ocean Argo floats
		8.6.1. Predicting an annual average temperature map
	8.7. Conclusion
	8.8. Exercises
9. Bayesian forecasting for point referenced data
	9.1. Introduction
	9.2. Exact forecasting method for GP
		9.2.1. Example: Hourly ozone levels in the Eastern US
	9.3. Forecasting using the models implemented in spTimer
		9.3.1. Forecasting using GP models
		9.3.2. Forecasting using AR models
		9.3.3. Forecasting using the GPP models
	9.4. Forecast calibration methods
		9.4.1. Theory
		9.4.2. Illustrating the calibration plots
	9.5. Example comparing GP, AR and GPP models
	9.6. Example: Forecasting ozone levels in the Eastern US
	9.7. Conclusion
	9.8. Exercises
10. Bayesian modeling for areal unit data
	10.1. Introduction
	10.2. Generalized linear models
		10.2.1. Exponential family of distributions
		10.2.2. The link function
		10.2.3. Offset
		10.2.4. The implied likelihood function
		10.2.5. Model specification using a GLM
	10.3. Example: Bayesian generalized linear model
		10.3.1. GLM fitting with binomial distribution
		10.3.2. GLM fitting with Poisson distribution
		10.3.3. GLM fitting with normal distribution
	10.4. Spatial random effects for areal unit data
	10.5. Revisited example: Bayesian spatial generalized linear model
		10.5.1. Spatial GLM fitting with binomial distribution
		10.5.2. Spatial GLM fitting with Poisson distribution
		10.5.3. Spatial GLM fitting with normal distribution
	10.6. Spatio-temporal random effects for areal unit data
		10.6.1. Linear model of trend
		10.6.2. Anova model
		10.6.3. Separable model
		10.6.4. Temporal autoregressive model
	10.7. Example: Bayesian spatio-temporal generalized linear model
		10.7.1. Spatio-temporal GLM fitting with binomial distribution
		10.7.2. Spatio-temporal GLM fitting with Poisson distribution
		10.7.3. Examining the model fit
		10.7.4. Spatio-temporal GLM fitting with normal distribution
	10.8. Using INLA for model fitting and validation
	10.9. Conclusion
	10.10. Exercises
11. Further examples of areal data modeling
	11.1. Introduction
	11.2. Assessing childhood vaccination coverage in Kenya
	11.3. Assessing trend in cancer rates in the US
	11.4. Localized modeling of hospitalization data from England
		11.4.1. A localized model
		11.4.2. Model fitting results
	11.5. Assessing trend in child poverty in London
		11.5.1. Adaptive CAR-AR model
		11.5.2. Model fitting results
	11.6. Conclusion
	11.7. Exercises
12. Gaussian processes for data science and other applications
	12.1. Introduction
	12.2. Learning methods and their Bayesian interpretations
		12.2.1. Learning with empirical risk minimization
		12.2.2. Learning by complexity penalization
		12.2.3. Supervised learning and generalized linear models
		12.2.4. Ridge regression, LASSO and elastic net
		12.2.5. Regression trees and random forests
	12.3. Gaussian Process (GP) prior-based machine learning
		12.3.1. Example: predicting house prices
	12.4. Use of GP in Bayesian calibration of computer codes
	12.5. Conclusion
	12.6. Exercises
Appendix A: Statistical densities used in the book
	A.1. Continuous
	A.2. Discrete
Appendix B: Answers to selected exercises
	B.1. Solutions to Exercises in Chapter 4
	B.2. Solutions to Exercises in Chapter 5
Bibliography
Glossary
Index