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

In place of lecture notes, we will use the book An Introduction
to Statistical Computing: A Simulation-Based Approach

(see References, below), which was specially
written for this module. The book is available online
via the university library:

The lecture videos which I recorded during the Covid 19 pandemic can be found by clicking on the video number in the following table, or all together in the MATH5835M video playlist.

We will cover sections 3.1-3.3, 1.1, 1.3, 1.4, 2.3, 4.1, 4.2 and 5.2 of this book. The exact page ranges are:

videos | sections | pages | topic |
---|---|---|---|

1, 2, 3, 4 | 3.1 | 69-74 | Monte Carlo methods |

5, 6, 7 | 3.2 | 75-84 | Monte Carlo error |

8, 9 | 3.3.1 | 84-88 | Importance Sampling |

10, 11, 12 | 3.3.2 | 88-93 | Antithetic Variables method |

13, 14 | 3.3.3 | 93-96 | Control Variates method |

15, 16, 17 | 1.1 | 1-8 | Pseudo Random Number Generators |

18, 19 | 1.3 | 11-15 | Inverse Transform method |

20, 21, 22 | 1.4.1 | 15-18 | basic rejection sampling |

23, 24, 25 | 1.4.2 | 18-22 | envelope rejection sampling |

26, 27, 28 | 2.3 | 50-58 | Markov Chains |

29, 30, 31 | 4.1.1-4.1.2 | 110-116 | Metropolis-Hastings (MH) algorithm |

32, 33, 34 | 4.1.3-4.1.4 | 116-120 | special cases of the MH algorithm |

35, 36, 37 | 4.2.2 | 129-137 | Convergence of MCMC estimates |

38,
39,
40, 41, 42, 43 | (intro of 4.3) | 137-139 | Application to Bayesian Inference |

44, 45, 46 | 5.2.1 | 192-197 | Bootstrap sampling |

47, 48 | 5.2.2.1, 5.2.2.2 | 198-203 | Applications to statistical inference |

The practical is an assessed part of the module. It counts 20% towards the final grade. I will publish more information about the practical later in the semester.

The following links contain problem sheets for self-study.

- problem sheet 1, answers
- problem sheet 2, answers
- problem sheet 3, answers
- problem sheet 4, answers
- problem sheet 5, answers

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.

- 2023-10-02.R
- 2023-10-06.R
- 2023-10-16.R
- 2023-10-23.R
- 2023-10-30.R
- 2023-11-03.R
- 2023-11-06.R
- 2023-11-13.R
- 2023-11-27a.R
- 2023-11-27b.R

Useful resources for learning R include to following:

- 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 main reference for the module is the following book:

- Jochen Voss,
*An Introduction to Statistical Computing: A Simulation-Based Approach*.

Wiley, 2014 (Library, Amazon)

- 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)

- MATH5835M module catalog entry
- The university timetable/room plan