# Advanced Spatial Modeling with Stochastic Partial Differential Equations Using R and INLA

*Elias T. Krainski, Virgilio Gómez-Rubio, Haakon Bakka, Amanda Lenzi, Daniela Castro-Camilo, Daniel Simpson, Finn Lindgren and Håvard Rue*

*2018-11-13*

# Preface

This book grew out of a tutorial written by Elias T. Krainski, which he started in 2013 together with his PhD-studies at NTNU, Trondheim, Norway. The tutorial has since then been expanded continuously, based on response from the many users and based on new developments.

Lindgren, Rue, and Lindström (2011) describe an approximation to continuous spatial models with a Matérn covariance that is based on the solution to a stochastic partial differential equation (SPDE). This approximation is computed using a sparse representation that can be effectively implemented using the integrated nested Laplace approximation (INLA, Rue, Martino, and Chopin 2009).

This book will show you how to fit models that contain at least one
effect specified with an SPDE using the `INLA`

package for the `R`

software for statistical computing. An SPDE based model will be used
to define random effects over continuous domains in one or two
dimensions. The usual application is data whose geographical location
is explicitly considered in the analysis.

This book explores `INLA`

functionalities through examples, and it is
structured as follows. Chapter 1 provides an introduction
to the integrated nested Laplace approximation and its associate
`INLA`

package for the `R`

programming language. Chapter
2 introduces Gaussian random fields and the SPDE
framework, develops an example on a toy dataset and works through some
examples of building a mesh. Here, an example with non-Gaussian data
is also discussed. Then three examples on the use of models
with several likelihoods are developed in Chapter 3.
These include a measurement error model, a coregionalization model and
considering part or the entire linear predictor from one outcome in a
linear predictor of another one. Point pattern analysis is included in
Chapter 4 using a log-Gaussian Cox process.
Non-stationary spatial models are developed in Chapter
5, which includes inclusion of covariates in
the covariance parameters and barrier models. Chapter 6
focuses on survival analysis, and models for extremes and non-standard
likelihoods are discussed here. Space-time models are described in
detail in Chapter 7. Some applications of space-time
models are developed in Chapter 8. Two appendices are
included at the end with a summary of the notation used in the book
and information about the `R`

packages required to reproduce the
examples in the book.

The introduction in Chapter 1 can be used as a starting
point for the integrated nested Laplace approximation and the `INLA`

package. Chapter 2 tries to explain some of the
theoretical details behind the SPDE approach by developing two
examples. Going through the more theoretical details may require some
background on stochastic processes, but the
applications of the SPDE approach are described in detail in the
examples in this chapter and throughout the book.

This book focuses on SPDE models with INLA but it does not cover the
basics of Bayesian inference or spatial analysis. For this,
Bivand, Pebesma, and Gomez-Rubio (2013) give a thorough description of spatial analysis in
`R`

. Banerjee, Carlin, and Gelfand (2014) cover Bayesian inference for different types
of spatial models in detail. Blangiardo and Cameletti (2015) and Zuur, Ieno, and Saveliev (2017) give
an introduction to INLA and discuss spatial and spatio-temporal
models. Wang, Faraway, and Yue Ryan (2018) and Gómez-Rubio (2019) provide a good
introduction to INLA and modeling with the `INLA`

package, which are a
good resource to learn about INLA.

There are some other resources available on-line or in the `INLA`

package.
Lindgren and Rue (2015) is an excellent tutorial available at
http://www.r-inla.org/examples/tutorials/spde-tutorial-from-jss.
If you are in a rush to fit a simple geostatistical model, please see
the vignette available in the `INLA`

package which
can be loaded by typing `vignette(SPDEhowto)`

or a
one dimensional example by typing `vignette(SPDE1d)`

.
A mesh building demonstration Shiny app can be opened by typing `meshbuilder()`

.

Finally, a Gitbook version of this book is available from the book
website at
http://www.r-inla.org/spde-book.
Here, `R`

code and datasets used in the examples and figures
of this book are also available. We have tried to use color-blind
friendly palettes throughout the book using packages `RColorBrewer`

and `viridisLite`

, but this can be easily changed in the provided
`R`

code.

## Acknowledgement

We would like to thank Sarah Gallup and Helen Sofaer for some
English review in the tutorial that originated this book.
Our thanks to several people who brought nice problems
and questions to the `INLA`

discussion forum at
http://www.r-inla.org/comments-1 and
directly to us. Finally, we are grateful to John Kimmel and CRC
for being supportive about the publication of this book and for his help
throughout the publication process.

Elias T. Krainski was supported by a grant from the Norwegian Research Council, during the years 2013-2016. Virgilio Gómez-Rubio has been partly supported by grant SBPLY/17/180501/000491, awarded by Consejería de Educación, Cultura y Deportes (JCCM, Spain) and FEDER, grant MTM2016-77501-P, awarded by Ministerio de Economía y Competitividad (Spain) and a grant to support research groups from Universidad de Castilla-La Mancha (Spain).

This book has been written using the `bookdown`

package and `R`

markdown. Map
data from Albacete copyrighted OpenStreetMap contributors and is available from
https://www.openstreetmap.org.

### References

Lindgren, F., H. Rue, and J. Lindström. 2011. “An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach (with Discussion).” *J. R. Statist. Soc. B* 73 (4): 423–98.

Rue, H., S. Martino, and N. Chopin. 2009. “Approximate Bayesian Inference for Latent Gaussian Models Using Integrated Nested Laplace Approximations (with Discussion).” *Journal of the Royal Statistical Society: Series B* 71 (2): 319–92.

Bivand, R. S., E. J. Pebesma, and V. Gomez-Rubio. 2013. *Applied Spatial Data Analysis with R*. 2nd ed. Springer, NY. http://www.asdar-book.org/.

Banerjee, S., B. P. Carlin, and A. E. Gelfand. 2014. *Hierarchical Modeling and Analysis for Spatial Data*. 2nd ed. Boca Raton, FL: Chapman & Hall/CRC.

Blangiardo, M., and M. Cameletti. 2015. *Spatial and SpatioTemporal Bayesian Models with R-INLA*. Chichester, UK: John Wiley & Sons, Ltd.

Zuur, A. F., E. N. Ieno, and A. A. Saveliev. 2017. *Beginner’s Guide to Spatial, Temporal and Spatial-Temporal Ecological Data Analysis with R-INLA*. Highland Statistics Ltd.

Wang, X., J. J. Faraway, and Y. Yue Ryan. 2018. *Bayesian Regression Modeling with INLA*. Boca Raton, FL: Chapman & Hall/CRC.

Gómez-Rubio, V. 2019. *Bayesian Inference with INLA*. Chapman & Hall/CRC.

Lindgren, F., and H. Rue. 2015. “Bayesian Spatial and Spatio-Temporal Modelling with R-INLA.” *Journal of Statistical Software* 63 (19).