Spatial and Spatio-temporal Bayesian Models with R - INLA E-Book

Marta Blangiardo

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 437

Veröffentlichungsjahr: 2015

Beschreibung

Spatial and Spatio-Temporal Bayesian Models with R-INLA provides a much needed, practically oriented & innovative presentation of the combination of Bayesian methodology and spatial statistics. The authors combine an introduction to Bayesian theory and methodology with a focus on the spatial and spatio­-temporal models used within the Bayesian framework and a series of practical examples which allow the reader to link the statistical theory presented to real data problems. The numerous examples from the fields of epidemiology, biostatistics and social science all are coded in the R package R-INLA, which has proven to be a valid alternative to the commonly used Markov Chain Monte Carlo simulations

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Dedication

Preface

Chapter 1: Introduction

1.1 Why spatial and spatio-temporal statistics?

1.2 Why do we use Bayesian methods for modeling spatial and spatio-temporal structures?

1.3 Why INLA?

1.4 Datasets

References

Chapter 2: Introduction to R

2.1 The

R

language

2.2

R

objects

2.3 Data and session management

2.4 Packages

2.5 Programming in

R

2.6 Basic statistical analysis with

R

References

Chapter 3: Introduction to Bayesian methods

3.1 Bayesian philosophy

3.2 Basic probability elements

3.3 Bayes theorem

3.4 Prior and posterior distributions

3.5 Working with the posterior distribution

3.6 Choosing the prior distribution

References

Chapter 4: Bayesian computing

4.1 Monte Carlo integration

4.2 Monte Carlo method for Bayesian inference

4.3 Probability distributions and random number generation in

R

4.4 Examples of Monte Carlo simulation

4.5 Markov chain Monte Carlo methods

4.6 The integrated nested Laplace approximations algorithm

4.7 Laplace approximation

4.8 The

R-INLA

package

4.9 How INLA works: step-by-step example

References

Chapter 5: Bayesian regression and hierarchical models

5.1 Linear regression

5.2 Nonlinear regression: random walk

5.3 Generalized linear models

5.4 Hierarchical models

5.5 Prediction

5.6 Model checking and selection

References

Chapter 6: Spatial modeling

6.1 Areal data – GMRF

6.2 Ecological regression

6.3 Zero-inflated models

6.4 Geostatistical data

6.5 The stochastic partial differential equation approach

6.6 SPDE within

R-INLA

6.7 SPDE toy example with simulated data

6.8 More advanced operations through the

inla.stack

function

6.9 Prior specification for the stationary case

6.10 SPDE for Gaussian response: Swiss rainfall data

6.11 SPDE with nonnormal outcome: malaria in the Gambia

6.12 Prior specification for the nonstationary case

References

Chapter 7: Spatio-temporal models

7.1 Spatio-temporal disease mapping

7.2 Spatio-temporal modeling particulate matter concentration

References

Chapter 8: Advanced modeling

8.1 Bivariate model for spatially misaligned data

8.2 Semicontinuous model to daily rainfall

8.3 Spatio-temporal dynamic models

8.4 Space–time model lowering the time resolution

References

Index

End User License Agreement

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Guide

Cover

Table of Contents

Preface

Begin Reading

List of Illustrations

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

Figure 1.6

Figure 1.7

Figure 1.8

Figure 1.9

Figure 1.10

Figure 1.11

Figure 1.12

Figure 1.13

Figure 1.14

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14

Figure 4.15

Figure 4.16

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Figure 5.15

Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 6.9

Figure 6.10

Figure 6.11

Figure 6.12

Figure 6.13

Figure 6.14

Figure 6.15

Figure 6.16

Figure 6.17

Figure 6.18

Figure 6.19

Figure 6.20

Figure 6.21

Figure 6.22

Figure 6.23

Figure 6.24

Figure 6.25

Figure 7.1

Figure 7.2

Figure 7.3

Figure 7.4

Figure 7.5

Figure 7.6

Figure 7.7

Figure 7.8

Figure 7.9

Figure 7.10

Figure 7.11

Figure 7.12

Figure 8.1

Figure 8.2

Figure 8.3

Figure 8.4

Figure 8.5

Figure 8.6

Figure 8.7

Figure 8.8

Figure 8.9

Figure 8.10

Figure 8.11

Figure 8.12

Figure 8.13

Figure 8.14

Figure 8.15

Figure 8.16

Figure 8.17

Figure 8.18

Figure 8.19

Figure 8.20

Figure 8.21

Figure 8.22

Figure 8.23

Figure 8.24

Figure 8.25

Figure 8.26

Figure 8.27

List of Tables

Table 3.1

Table 3.2

Table 4.1

Table 4.2

Table 4.3

Table 5.1

Table 6.1

Table 6.2

Table 6.3

Table 6.4

Table 6.5

Table 6.6

Table 7.1

Table 7.2

Spatial and Spatio-temporal Bayesian Models with R-INLA

Marta Blangiardo

MRC-PHE Centre for Environment and Health, Department of Epidemiology and Biostatistics, Imperial College London, UK

Michela Cameletti

Department of Management, Economics and Quantitative Methods, University of Bergamo, Italy

This edition first published 2015

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Library of Congress Cataloging-in-Publication Data

Blangiardo, Marta.

Spatial and spatio-temporal Bayesian models with R-INLA / by Marta Blangiardo and Michela Cameletti.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-32655-8 (cloth)

1. Bayesian statistical decision theory. 2. Spatial analysis (Statistics) 3. Asymptotic distributiona (Probability theory) 4. R (Computer program language) I. Cameletti, Michela. II. Title.

QA279.5.B63 2015

519.5′42—dc23

2015000696

A catalogue record for this book is available from the British Library.

ISBN: 9781118326558

To MM, Gianluca, Kobi and Clarissa: now we can enjoy life again!

Preface

This book presents the principles of Bayesian theory for spatial and spatio-temporal modeling, combining three aspects: (1) an introduction to Bayesian thinking and theoretical aspects of the Bayesian approach, (2) a focus on the spatial and spatio-temporal models used within the Bayesian framework, (3) a series of practical examples which allow the reader to link the statistical theory presented to real data problems. All the examples are coded in the R package R-INLA, and based on the recently developed integrated nested Laplace approximation (INLA) method, which has proven to be a valid alternative to the commonly used Markov Chain Monte Carlo (MCMC) simulations.

The book starts with an introduction in Chapter 1, providing the reader with the importance of spatial and spatio-temporal modeling in several fields, such as social science, environmental epidemiology, and infectious diseases epidemiology. We then show why Bayesian models are commonly used in these fields and why we focus on the INLA approach. We also describe the datasets which will be used in the rest of the book, providing information on the topics that will be used as illustration.

As all the examples are run in R, in Chapter 2 we introduce the basic concepts of the R language. Chapter 3 describes the Bayesian methods: first we introduce the paradigms of this approach (i.e., the concepts of prior and posterior distributions, Bayes theorem, conjugacy, how to obtain the posterior distribution, the computational issues around Bayesian statistics for conjugated and non conjugated models). We also include a small section on the differences between the frequentist and the Bayesian approach, focusing on the different interpretation of confidence intervals, parameters, and hypothesis testing.

Chapter 4 discusses the computational issues regarding Bayesian inference. After the Monte Carlo method is introduced, we consider MCMC algorithms, providing some examples in R for the case of conjugated and non conjugated distributions. The focus of the chapter is the INLA method, which is a computationally powerful alternative to MCMC algorithms. In particular, the R-INLA library is described by means of a small tutorial and of a step-by-step example.

Then in Chapter 5 we present the Bayesian modeling framework which is used in the fields introduced in Chapter 1 and focuses on regression models (linear and generalized linear models). In this context, we introduce the concept of exchangeability and explain how this is used to predict values from variables of interest, a topic which will be expanded later in the chapters on spatial andspatio-temporal modeling. The last section of this part is devoted to introducing hierarchical models.

Chapter 6 focuses on models for two types of spatial processes: (1) area level—introducing disease mapping models and small area ecological regressions (including risk factors and covariates) and then presenting zero inflated models for Poisson and Binomial data; (2) point level—presenting Bayesian kriging through the stochastic partial differential equations (SPDE) approach and showing how to model observed data and also to predict for new spatial locations. Chapter 7 extends the topics treated in Chapter 6 adding a temporal dimension, where we also include the time dimension in the models.

Finally, Chapter 8 introduces new developments within INLA and focuses on the following advanced applications: when data are modeled using different likelihoods, when missing data are present in covariates, a spatio-temporal model with dynamic evolution for the regression coefficients, and a spatio-temporal model for high-frequency data on time where a temporal resolution reduction is needed.

We would like to thank many people who helped with this book: Håvard Rue for his precious contribution, his endless encouragement and for introducing us to Elias Krainski, who became involved in the book; Finn Lindgren, Aurelie Cosandey Godin and Gianluca Baio for reading drafts of the manuscript and providing useful comments; Philip Li, Ravi Maheswaran, Birgit Schrödle, Virgilio Gómez-Rubio, and Paola Berchialla, who provided some of the datasets; finally, a huge thank to our families who have supported us during all this time.

We hope that this book can be helpful for readers at any level, wanting to familiarize or increase their practice and knowledge of the INLA method. Those who are approaching the Bayesian way of thinking for the first time could follow it from the beginning, while those who are already familiar with R and Bayesian inference can easily skip the first chapters and focus on spatial and spatio-temporal theory and applications.

Marta Blangiardo and Michela Cameletti

Chapter 1Introduction

1.1 Why spatial and spatio-temporal statistics?

In the last few decades, the availability of spatial and spatio-temporal data has increased substantially, mainly due to the advances in computational tools which allow us to collect real-time data coming from GPS, satellites, etc. This means that nowadays in a wide range of fields, from epidemiology to ecology, to climatology and social science, researchers have to deal with geo-referenced data, i.e., including information about space (and possibly also time).

As an example, we consider a typical epidemiological study, where the interest is to evaluate the incidence of a particular disease such as lung cancer across a given country. The data will usually be available as counts of diseases for small areas (e.g., administrative units) for several years. What types of models allow the researchers to take into account all the information available from the data? It is important to consider the potential geographical pattern of the disease: areas close to each others are more likely to share some geographical characteristics which are related to the disease, thus to have similar incidence. Also how is the incidence changing in time? Again it is reasonable to expect that if there is a temporal pattern, this is stronger for subsequent years than for years further apart.

As a different example, let us assume that we are now in the climatology field and observe daily amount of precipitation at particular locations of a sparse network: we want to predict the rain amount at unobserved locations and we need to take into account spatial correlation and temporal dependency.

Spatial and spatio-temporal models are now widely used: typing “statistical models for spatial data” in ™Google Scholar returns more than 3 million hits and “statistical models for spatio-temporal data” gives about 159,000. There are countless scientific papers in peer review journals which use more or less complex and innovative statistical models to deal with the spatial and/or the temporal structure of the data in hand, covering a wide range of applications; the following list only aims at providing a flavor of the main areas where these types of models are used: Haslett and Raftery (1989), Handcock and Wallis (1994) and Jonhansson and Glass (2008) work in the meteorology field; Shoesmith (2013) presents a model for crime rates and burglaries, while Pavia et al. (2008) used spatial models for predicting election results; in epidemiology Knorr-Held and Richardson (2003) worked on infectious disease, while Waller et al. (1997) and Elliott et al. (2001) presented models for chronic diseases. Finally, Szpiro et al. (2010) focused on air pollution estimates and prediction.

1.2 Why do we use Bayesian methods for modeling spatial and spatio-temporal structures?

Several types of models are used with spatial and spatio-temporal data, depending on the aim of the study. If we are interested in summarizing spatial and spatio-temporal variation between areas using risks or probabilities then we could use statistical methods like disease mapping to compare maps and identify clusters. Moran Index is extensively used to check for spatial autocorrelation (Moran, 1950), while the scan statistics, implemented in SaTScan (Killdorf, 1997), has been used for cluster detection and to perform geographical surveillance in a non-Bayesian approach. The same types of models can also be used in studies where there is an aetiological aim to assess the potential effect of risk factors on outcomes.

A different type of study considers the quantification of the risk of experiencing an outcome as the distance from a certain source increases. This is typically framed in an environmental context, so that the source could be a point (e.g., waste site, radio transmitter) or a line (e.g., power line, road). In this case, the methods typically used vary from nonparametric tests proposed by Stone (1988) to the parametric approach introduced by Diggle . (1998).