Undergraduate

modules

Modules

MAS3910 : Discrete Stochastic Modelling (Inactive)

Semesters
Semester 2 Credit Value: 10
ECTS Credits: 5.0

Aims

To gain an understanding of some of the areas of stochastic modelling in discrete time that underpin quantitative descriptions of population growth, epidemics and the analysis of DNA sequences

Module summary
Many random processes can be thought of as evolving through a sequence of successive generations. For example, population growth depends on which individuals successfully produce offspring in the next generation; the transmission of a disease through a population depends on how individuals interact from day to day. Models which incorporate random variation allow prediction of important quantities such as the size of a population or duration of an epidemic, as well as variability in these estimates. This module presents techniques for modelling such processes with applications drawn in particular from the biological sciences. Branching processes will be introduced as a means of modelling population growth. Stochastic models which describe epidemic growth by considering the size of the infected and immune subpopulations will then be studied. These have been important in recent years for the analysis of influenza and other epidemics.
DNA sequences can be considered as strings of letters from the four-letter alphabet {A,C,G,T}. Markov chains provide a useful stochastic model to describe the probability of the letter at the next
site in the sequence given the letter at the current site. However, the presence of genes and other functional elements within a sequence suggest that more sophisticated models are required, in particular, models which allow these transition probabilities to vary along the length of the sequence. Originally developed for automatic speech recognition, hidden Markov models have proved to be a remarkably flexible and powerful model for automatic segmentation and gene-finding in DNA sequences. As their name suggests, hidden Markov models are based on an unobserved Markov chain. Methods for estimating this "hidden" Markov chain will be considered. Computational algorithms will be developed in R.

Outline Of Syllabus

Review of Markov chains. Probability generating functions, random sums of discrete random variables. Branching processes and extinction probability. Stochastic models of epidemics: the SIS, Greenwood and Reed-Frost models. Comparison with deterministic models. Duration and size of epidemics.
Markov chain models; model choice. Hidden Markov models; simulation; inference via maximum likelihood; forward-backward algorithm; local and global decoding; Baum-Welch algorithm. Application to DNA sequence analysis.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Guided Independent StudyAssessment preparation and completion113:0013:00Revision for unseen exam
Guided Independent StudyAssessment preparation and completion12:002:00Unseen exam
Scheduled Learning And Teaching ActivitiesLecture31:003:00Problem classes
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
Scheduled Learning And Teaching ActivitiesLecture251:0025:00Formal lectures
Guided Independent StudyIndependent study122:0022:00Studying, practising and gaining understanding of course material
Guided Independent StudyIndependent study33:009:00Review of problem-solving exercises and group project
Guided Independent StudyIndependent study112:0012:00Preparation for group project
Guided Independent StudyIndependent study26:0012:00Preparation for problem-solving exercises
Total100:00
Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems. Tutorials are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. In addition, office hours (two per week) will provide an opportunity for more direct contact between individual students and the lecturer.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Exams
Description Length Semester When Set Percentage Comment
Written Examination1202A85N/A
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises2M5Problem solving exercises
Prob solv exercises2M10Group project
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The problem solving exercises are expected to consist of two assignments of equal weight: the exact nature of assessment will be explained at the start of the module. The exercises and the group project allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these are thus formative as well as summative assessments.

Reading Lists

Timetable