Beginner's Guide to Spatial, Temporal and Spatial-Temporal Ecological Data Analysis with R-INLA

In Chapter 2 we discuss an important topic: dependency. Ignoring this means that we have pseudoreplication. We present a series of examples and discuss how dependency can manifest itself. We briefly discuss frequentist tools that are available for the analysis of temporal and spatial data in Chapters 3 and 4, and we will conclude that their application is rather limited, especially if non-Gaussian distributions are required. We will therefore consider alternative models, but these require Bayesian techniques. In Chapter 5 we discuss linear mixed-effects models to analyse hierarchical (i.e. clustered or nested) data, and in Chapter 6 we outline how we add spatial and spatial-temporal dependency to regression models via spatial (and/or temporal) correlated random effects. In Chapter 7 we introduce Bayesian analysis, Markov chain Monte Carlo techniques (MCMC), and Integrated Nested Laplace Approximation (INLA). INLA allows us to apply models to spatial, temporal, or spatial-temporal data. In Chapters 8 through 16 we present a series of INLA examples. We start by applying linear regression and mixed-effects models in INLA (Chapters 8 and 9), followed by GLM examples in Chapter 10. In Chapters 11 through 13 we show how to apply GLM models on spatial data. In Chapter 14 we discuss time-series techniques and how to implement them in INLA. Finally, in Chapters 15 and 16 we analyse spatial-temporal models in INLA.

Table of Contents:
1 OVERVIEW OF THIS BOOK 1 1.1 VOLUMES I AND II 1 1.1.1 Volume I 1 1.1.2 Volume II 1 1.2 WHAT TYPE OF SPATIAL DATA DO WE ANALYSE IN THIS BOOK? 1 1.2.1 Areal and lattice data 1 1.2.2 Geostatistical data 2 1.2.3 Spatial point pattern data 3 1.3 OUTLINE OF THIS BOOK 3 1.4 PREREQUISITES 4 1.5 AVAILABILITY OF R CODE AND DATA 4 2 RECOGNISING STATISTICAL DEPENDENCY 5 2.1 PSEUDOREPLICATION 5 2.2 LINEAR REGRESSION APPLIED TO SPATIAL DATA 7 2.2.1 Irish pH data 7 2.2.2 Protocol from Zuur et al. (2016) 8 2.2.3 Visualisation of the experimental design 9 2.2.4 Data exploration 9 2.2.5 Dependency 12 2.2.6 Statistical model 15 2.2.7 Fit the model 16 2.2.8 Model validation 17 2.3 GAM APPLIED TO TEMPORAL DATA 21 2.3.1 Subnivium temperature data 21 2.3.2 Sources of dependency 22 2.3.3 The model 23 2.3.4 Model validation 24 2.4 GLMM APPLIED ON HIERARCHICAL AND SPATIAL DATA 26 2.5 TECHNICALITIES 28 2.5.1 Matrix notation 28 2.5.2 How is dependency causing problems? 31 2.6 DISCUSSION 32 3 TIME SERIES AND GLS 33 3.1 OSPREYS 33 3.2 COVARIANCE AND CORRELATION COEFFICIENTS 33 3.3 LINEAR REGRESSION MODEL 35 3.4 FOCUSSING ON THE RESIDUAL COVARIANCE MATRIX 35 3.5 DEPENDENCY AND THE COVARIANCE MATRIX 36 3.6 GLS: DEALING WITH TEMPORAL DEPENDENCY 39 3.6.1 Adelie penguins 39 3.6.2 Do we have dependency? 40 3.6.3 Formulation of the linear regression model 40 3.6.4 Application of the linear regression model 41 3.6.5 R code for acf and variogram 45 3.6.6 Formulation of the GLS model 46 3.6.7 Implementation using the gls function 50 3.7 MULTIPLE TIME SERIES 51 3.8 DISCUSSION 53 4 SPATIAL DATA AND GLS 55 4.1 VARIOGRAM MODELS FOR SPATIAL DEPENDENCY 55 4.2 APPLICATION ON THE IRISH PH DATA 57 4.3 MATERN CORRELATION FUNCTION 59 5 LINEAR MIXED EFFECTS MODELS AND DEPENDENCY 61 5.1 WHITE STORKS 61 5.2 CONSIDERING THE DATA (WRONGLY) AS ONE-WAY NESTED 62 5.3 FITTING THE ONE-WAY NESTED MODEL USING LMER 65 5.4 MODEL VALIDATION 67 5.5 SKETCHING THE FITTED VALUES 68 5.6 CONSIDERING THE DATA (CORRECTLY) AS TWO-WAY NESTED 69 5.7 APPLICATIONS TO SPATIAL AND TEMPORAL DATA 72 5.8 DIFFERENCE WITH THE AR1 PROCESS APPROACH 72 6 MODELLING SPACE EXPLICITLY 73 6.1 MODEL FORMULATION 73 6.2 COVARIANCE MATRIX OF THE SPATIAL RANDOM EFFECT 75 6.3 SPATIAL-TEMPORAL CORRELATION* 79 7 INTRODUCTION TO BAYESIAN STATISTICS 83 7.1 WHY GO BAYESIAN? 83 7.2 GENERAL PROBABILITY RULES 84 7.3 THE MEAN OF A DISTRIBUTION* 85 7.4 BAYES' THEOREM AGAIN 87 7.5 CONJUGATE PRIORS 88 7.6 MARKOV CHAIN MONTE CARLO SIMULATION 93 7.6.1 Underlying idea 93 7.6.2 Installing JAGS and R2jags 94 7.6.3 Flowchart for running a model in JAGS 94 7.6.4 Preparing the data for JAGS 95 7.6.5 JAGS code 96 7.6.6 Initial values and parameters to save 98 7.6.7 Running JAGS 99 7.6.8 Accessing numerical output from JAGS 100 7.6.9 Assess mixing 100 7.6.10 Posterior information 101 7.7 INTEGRATED NESTED LAPLACE APPROXIMATION* 103 7.7.1 Joint posterior distribution 103 7.7.2 Marginal distributions 105 7.7.3 Back to high school 107 7.7.4 INLA 109 7.8 EXAMPLE USING R-INLA 110 7.9 DISCUSSION 114 8 MULTIPLE LINEAR REGRESSION IN R-INLA 115 8.1 INTRODUCTION 115 8.2 DATA EXPLORATION 116 8.3 MODEL FORMULATION 117 8.4 LINEAR REGRESSION RESULTS 117 8.4.1 Executing the model in R-INLA 117 8.4.2 Output for the betas 117 8.4.3 Output for the hyper-parameters 119 8.4.4 Fitted model 123 8.5 MODEL VALIDATION 123 8.6 MODEL SELECTION 126 8.6.1 Should we do it? 126 8.6.2 Using the DIC 126 8.6.3 Out of sample prediction 131 8.6.4 Posterior predictive check 133 8.7 VISUALISING THE MODEL 135 9 MIXED EFFECTS MODELLING IN R-INLA TO ANALYSE OTOLITH DATA 139 9.1 OTOLITHS IN PLAICE 139 9.2 MODEL FORMULATION 140 9.3 DEPENDENCY 140 9.4 DATA EXPLORATION 141 9.5 RUNNING THE MODEL IN R-INLA 143 9.6 MODEL VALIDATION 146 9.7 MODEL SELECTION 149 9.8 MODEL INTERPRETATION 149 9.8.1 Option 1 for prediction: Adding extra data 150 9.8.2 Option 2 for prediction: Using the inla.make.lincombs 153 9.8.3 Adding extra data or inla.make.lincombs? 155 9.9 MULTIPLE RANDOM EFFECTS 155 9.10 CHANGING PRIORS OF FIXED PARAMETERS 156 9.11 CHANGING PRIORS OF HYPERPARAMETERS 158 9.12 SHOULD WE CHANGE PRIORS? 164 10 POISSON, NEGATIVE BINOMIAL, BINOMIAL AND GAMMA GLMS IN R-INLA 165 10.1 POISSON AND NEGATIVE BINOMIAL GLMS IN R-INLA 165 10.1.1 Introduction 165 10.1.2 Poisson GLM in R-INLA 166 10.1.3 Negative binomial GLM in R-INLA 172 10.1.4 Model selection for the NB GLM 175 10.1.5 Visualisation of the NB GLM 177 10.2 BERNOULLI AND BINOMIAL GLM 180 10.2.1 Bernoulli GLM 181 10.2.2 Model selection with the marginal likelihood 184 10.2.3 Binomial GLM 185 10.3 GAMMA GLM 187 11 MATERN CORRELATION AND SPDE 191 11.1 CONTINUOUS GAUSSIAN FIELD 191 11.2 MODELS THAT WE HAVE IN MIND 191 11.3 MATERN CORRELATION 192 11.4 SPDE APPROACH 197 12 LINEAR REGRESSION MODEL WITH SPATIAL DEPENDENCY FOR THE IRISH PH DATA 205 12.1 INTRODUCTION 205 12.2 MODEL FORMULATION 205 12.3 LINEAR REGRESSION RESULTS 206 12.4 MODEL VALIDATION 207 12.5 ADDING SPATIAL CORRELATION TO THE MODEL 208 12.6 DEFINING THE MESH FOR THE IRISH PH DATA 212 12.7 DEFINE THE WEIGHT FACTORS AIK 216 12.8 DEFINE THE SPDE 218 12.9 DEFINE THE SPATIAL FIELD 218 12.10 DEFINE THE STACK 218 12.11 DEFINE THE FORMULA FOR THE SPATIAL MODEL 221 12.12 EXECUTE THE SPATIAL MODEL IN R 221 12.13 RESULTS 222 12.14 MODEL SELECTION 227 12.15 MODEL VALIDATION 228 12.16 MODEL INTERPRETATION 228 12.17 DETAILED INFORMATION ABOUT THE STACK* 232 12.17.1 Stack for the fitted model again 232 12.17.2 Stack for the new covariate values 234 12.17.3 Combine the two stacks 236 12.17.4 Run the model 236 13 SPATIAL POISSON MODELS APPLIED TO PLANT DIVERSITY 239 13.1 INTRODUCTION 239 13.2 DATA EXPLORATION 239 13.2.1 Sampling locations 239 13.2.2 Outliers 241 13.2.3 Collinearity 242 13.2.4 Relationships 243 13.2.5 Numbers of zeros 244 13.2.6 Conclusions data exploration 244 13.3 MODEL FORMULATION 244 13.4 GLM RESULTS 245 13.5 ADDING SPATIAL CORRELATION TO THE MODEL 248 13.5.1 Model formulation 248 13.5.2 Mesh 248 13.5.3 Projector matrix 253 13.5.4 SPDE 254 13.5.5 Spatial field 254 13.5.6 Stack 254 13.5.7 Formula 255 13.5.8 Run R-INLA 255 13.5.9 Inspect results 256 13.6 SIMULATING FROM THE MODEL 262 13.7 WHAT TO WRITE IN A PAPER 265 14 TIME-SERIES ANALYSIS IN R-INLA 267 14.1 SIMULATION STUDY 267 14.2 TRENDS IN MIGRATION DATES OF SOCKEYE SALMON 269 14.2.1 Applying a random walk trend model 269 14.2.2 Posterior distribution of the sigmas 272 14.2.3 Covariates and trends 273 14.2.4 Making the trend smoother 274 14.3 TRENDS IN POLAR BEAR MOVEMENTS 280 14.4 TRENDS IN WHALE STRANDINGS 283 14.5 MULTIVARIATE TIME SERIES FOR HAWAIIAN BIRDS 285 14.5.1 Importing and preparing the data 285 14.5.2 Data exploration 286 14.5.3 Model formulation 287 14.5.4 Executing the models 288 14.5.5 Mixing Poisson and negative binomial distributions 295 14.6 AR1 TRENDS 297 14.6.1 AR1 trend for regularly spaced time-series data 297 14.6.2 AR1 trend for irregularly spaced time-series data 299 15 SPATIAL-TEMPORAL MODELS FOR ORANGE-CROWNED WARBLERS COUNT DATA 307 15.1 INTRODUCTION 307 15.2 POISSON GLM 308 15.3 MODEL WITH SPATIAL CORRELATION 312 15.4 SPATIAL-TEMPORAL CORRELATION: AR1 318 15.4.1 Why do it? 318 15.4.2 Explanation of the model 318 15.4.4 Simulating a spatial-temporal AR random field 320 15.4.5 Implementation of AR1 model in R-INLA 323 15.4.6 More detailed information on the code 326 15.5 SPATIAL-TEMPORAL CORRELATION: EXCHANGEABLE 328 15.6 SPATIAL-TEMPORAL CORRELATION: REPLICATED 329 15.7 SIMULATION STUDY 330 15.8 DISCUSSION 333 16 SPATIAL-TEMPORAL BERNOULLI MODELS FOR CORAL DISEASE DATA 335 16.1 INTRODUCTION 335 16.2 BERNOULLI MODEL IN R-INLA 336 16.3 SPATIAL CORRELATED BERNOULLI MODEL 338 16.4 SPATIAL-TEMPORAL CORRELATED BERNOULLI MODEL 342 REFERENCES 347 INDEX 353 OTHER BOOKS 357