MATH5835M — Statistical Computing

Syllabus

Statistical computing is the branch of computational mathematics which studies computational techniques for situations which either directly involve randomness, or where randomness is used as part of a mathematical method. This module gives an introduction to statistical computing, with a focus on Monte Carlo methods. The following topics will be covered:

• Monte-Carlo methods
• Random number generation
• Markov Chain Monte Carlo (MCMC) methods
• Resampling methods
• Implementation of different methods in R

We will cover the following sections of the book An Introduction to Statistical Computing: A Simulation-Based Approach (see References, below): 3.1-3.3, 1.1-1.4, 4.1, 4.2 and 5.2.

Handouts

The following links contain pdf copies of the handouts from the lectures.

Software

For the module we will use the statistical computing package R. This program is free software, and I would recommend that you install R on your own laptop. There are different versions of R available:

• R itself, together with a lot of additional information, can be found on the R project homepage.
• A more polished version is RStudio, which can be found at the RStudio homepage. (Choose the open source version, RStudio Desktop, on the download page.)

Alternatively you can use RStudio or plain R on the university computers.

Resources

Useful resources for learning R include to following:

• The stats department provides a short, four-page introduction to R. This covers, amongst other things, how to start R on the university's computers and has some hints on how to install R on your own computer.
• The basics of R are explained in a bit more detail in my Short Introduction to R.
• The official R manual contains a lot of information.
• The R online help, accessed by typing help() or help.start() in R, can be used to remind yourself about indivdual commands.
• An R tutorial can be found in appendix B of my book An Introduction to Statistical Computing: A Simulation-Based Approach (the first reference below).

R Code from Lectures

• 2019-09-30.R — simple Monte Carlo estimate
• 2019-10-03.R — Monte Carlo estimates for means, probabilities and integrals

The module will be self-contained, i.e. you will not be required to read/buy/borrow any books. In case you want to do further reading, a good source is the following book, which was specially written for the module:

• Jochen Voss,
An Introduction to Statistical Computing: A Simulation-Based Approach.
Wiley, 2014 (Library, Amazon)
More in-depth information, beyond what we will be able to cover in the lectures, is for example contained in the following texts.
• Maria L. Rizzo,
Statistical Computing with R.
Chapman & Hall/CRC, 2008 (Library, Amazon)
• Brian D. Ripley,
Stochastic Simulation.
Wiley, 1987 (Library, Amazon)
• Christian P. Robert and George Casella,
Monte Carlo Statistical Methods.
Springer, 2004 (Library, Amazon)
• Wally R. Gilks, Silvia Richardson and David J. Spiegelhalter,
Markov Chain Monte Carlo in Practice.
Chapman & Hall/CRC, 1995 (Library, Amazon)
• Anthony C. Davison and David V. Hinkley,
Bootstrap methods and their application.
Cambridge University Press, 1997 (Library, Amazon)
• Andrew Gelman, et al.,
Bayesian Data Analysis.
Chapman & Hall/CRC, 3rd edition, 2013 (Library, Amazon)