Temporal modelling and time series analysis

Preface

These notes correspond to the second half of the module MATH4341: Spatio-temporal statistics IV. The first half of the module was primarily concerned with spatial modelling and spatial statistics. This half will concentrate on temporal modelling and the analysis of time series data. We will look briefly at modelling fully spatio-temporal data at the end of the course.

• These notes: https://darrenjw.github.io/time-series/
• Also available as a PDF document (better for printing and annotating). There is also an experimental EPUB version (for e-readers), but the formatting is suboptimal.

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: Spatio-temporal modelling

• Chapter 9 Spatio-temporal models and data

Reading list and other resources

The main recommended text for this part of the course 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. Wikle, Zammit-Mangion, and Cressie (2019) will also be useful for the latter part of the course. 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. Cressie and Wikle (2015) is a more substantial reference for the latter 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):

install.packages(c(
  "astsa", "signal", "netcontrol", "dlm", "mvtnorm", "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 this course. Similarly, the task view for spatio-temporal data gives an overview of packages relevant to the end 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 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., and C. K. Wikle. 2015. Statistics for Spatio-Temporal Data. Wiley.

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.

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.