Spatial and temporal statistics

Preface

These notes support the module MATH4341: Spatio-temporal statistics IV. Term 1 (Michaelmas) will concentrate on temporal modelling and the analysis of time series data. Term 2 (Epiphany) will be primarily concerned with spatial modelling and spatial statistics. We will look briefly at modelling fully spatio-temporal data at the end of the course.

• These notes: https://darrenjw.github.io/spatio-temporal/
• Also available as a PDF document (better for printing and annotating).

Michaelmas term

Part 1: Introduction to time series and linear filters

• Chapter 1 Introduction

Part 2: Linear stochastic systems

• Chapter 2 Linear systems
• Chapter 3 ARMA models
• Chapter 4 Forecasting and estimation
• Chapter 5 Spectral analysis

Part 3: State space modelling

• Chapter 6 HMMs
• Chapter 7 DLMs
• Chapter 8 State space modelling

Part 4: Introduction to spatial data

• Chapter 9 Introduction to spatial data

Epiphany term

Please note that although the lecture notes for Michaelmas term are essentially complete, the notes for Epiphany term are currently being written, and will not be complete until January 2026.

Part 5: Geostatistics

• Chapter 10 Random fields
• Chapter 11 GPs
• Chapter 12 Spectral theory for GPs
• Chapter 13 Kriging

Part 6: Lattice models

• Chapter 14 Lattice random fields
• Chapter 15 Spatial auto-regressive models
• Chapter 16 Inference for lattice models

Part 7: Spatial point processes

• Chapter 17 Spatial point processes

Part 8: Spatio-temporal modelling

• Chapter 18 Spatio-temporal models and data

Each chapter corresponds (very roughly) to one week (two lectures) of material.

Reading list and other resources

The main recommended text for Michaelmas term is Shumway and Stoffer (2017). This is an excellent reference, but has a different style to this course and covers more/different material, in a different order, and typically in more depth. After that, there are many excellent texts on time series analysis, including some classics. Chatfield and Xing (2019) is a good introductory text. Priestley (1989) is the classic text for spectral analysis. West and Harrison (2013) is the classic reference for DLMs, but Petris, Petrone, and Campagnoli (2009) and Särkkä and Svensson (2023) are interesting additions/alternatives.

The main recommended text for Epiphany term is Cressie (2015), but this classic now seems a little old-fashioned and is not to everyone’s taste. Ripley (2005) is more approachable, but also quite dated. Gelfand et al. (2010) is a more modern overview of the field. Rasmussen and Williams (2005) and Rue and Held (2005) are also useful for parts of the course.

Wikle, Zammit-Mangion, and Cressie (2019) will be useful for the final part of the course and Cressie and Wikle (2015) is a more substantial reference for this part of the course.

Of course, in addition to traditional textbooks, there is a lot of good material available freely online (including wikipedia). I’ll attempt to link to some of this from appropriate points in the notes.

We will use R for all of the computational examples. Some CRAN packages will be used. I will attempt to keep a full list of required installs here (but we will also use some of the required dependencies of these packages).

Michaelmas term:

install.packages(c(
  "astsa", "signal", "netcontrol", "dlm", "mvtnorm"
))

Additional packages required for Epiphany term:

install.packages(c(
  "sf", "sp", "spData", "spdep", "imager", "spatstat", 
  "osmdata", "ggplot2", "spTimer"
))

All of the examples, figures and simulations in the notes are fully reproducible in R. The code blocks for each chapter are intended to be run sequentially from the start of the chapter. These blocks are easy to copy-and-paste from the web version of the notes. Illustrative implementations of many important algorithms from time series analysis are provided, including simulation and fitting of ARMA models, the Kalman filter, and forward-backward algorithms for HMMs. Despite this, there is not a single explicit for, while or repeat loop to be found anywhere in the code examples. So the code examples also serve to illustrate a more functional approach to R programming than is typically adopted.

Note that the CRAN task view for time series analysis gives an overview of a large number of R packages relevant to Michaelmas term. The spatial task view is very useful for Epiphany term. Similarly, the task view for spatio-temporal data gives an overview of packages relevant to the end of the course. Also note that the Big book of R lists several e-books relevant to the more practical aspects of the course.

About these notes

This is a Quarto book. To learn more about Quarto books visit https://quarto.org/docs/books.

Copyright (C) 2024-2025 Darren J Wilkinson, all rights reserved.

Chatfield, C., and H. Xing. 2019. The Analysis of Time Series: An Introduction with R. CRC Press.

Cressie, N. 2015. Statistics for Spatial Data. Wiley.

Cressie, N., and C. K. Wikle. 2015. Statistics for Spatio-Temporal Data. Wiley.

Gelfand, A. E., P. Diggle, P. Guttorp, and M. Fuentes. 2010. Handbook of Spatial Statistics. CRC Press.

Petris, G., S. Petrone, and P. Campagnoli. 2009. Dynamic Linear Models with R. Use R! New York: Springer.

Priestley, M. B. 1989. Spectral Analysis and Time Series. Academic Press.

Rasmussen, C. E., and C. K. I. Williams. 2005. Gaussian Processes for Machine Learning. MIT Press.

Ripley, B. D. 2005. Spatial Statistics. Wiley.

Rue, H., and L. Held. 2005. Gaussian Markov Random Fields: Theory and Applications. CRC Press.

Särkkä, S., and L. Svensson. 2023. Bayesian Filtering and Smoothing. Cambridge University Press.

Shumway, R. H., and D. S. Stoffer. 2017. Time Series Analysis and Its Applications, with R Examples, Fourth Edition. Springer.

West, M., and J. Harrison. 2013. Bayesian Forecasting and Dynamic Models. Springer.

Wikle, C. K., A. Zammit-Mangion, and N. Cressie. 2019. Spatio-Temporal Statistics with R. CRC Press.