In practice, most data sets are multivariate, consisting of observations on several variables for each individual or object. Such data sets arise in many fields, including science, social sciences, and medicine, making techniques for their analysis an important area of statistics. This course introduces a variety of techniques, some of which generalise univariate methods, while others introduce new approaches (e.g. principal component analysis). The course focuses on continuous multivariate data, providing students with the theoretical tools and foundational knowledge necessary for advanced machine learning studies. Students will gain practical experience using R for data analysis throughout the course.  

Students are not permitted to take more than one of MATH38161 or MATH48061 for credit in the same undergraduate year.  Students are not permitted to take MATH48061 and MATH68061 for credit in an undergraduate programme and then a postgraduate programme.

The unit aims to:

Familiarise students with the ideas and methodology of foundational multivariate statistics, preparing them for advanced machine learning concepts. This includes their application in data analysis using the R statistical computing package. 

On successful completion of the course students will be able to:

• Explain the properties of random vectors and matrices, and calculate sample statistics (mean vectors, covariance matrices, correlation matrices) with transformations to simplify analysis.
• Apply the multivariate normal distribution and related distributions (e.g. Wishart, Hotelling’s T-squared) to analyse and model multivariate data.
• Estimate parameters (mean vector, covariance matrix) from multivariate data using maximum likelihood and Bayesian methods.
• Formulate and test hypotheses about mean structures and covariance matrices, and interpret confidence intervals in multivariate contexts.
• Apply dimension reduction techniques such as PCA, CCA and corresponding analysis to summarise and simplify multivariate data.
• Classify multivariate data using clustering and discriminant analysis.
• Apply popular computing software such as R or Python to visualise, analyse, and interpret multivariate data.

Part A – Core concepts and inferential techniques in multivariate analysis

• Random vectors and matrices – introductory ideas and basic concepts, linear transformations, sample statistics and their properties, overall measures of dispersion in p-space, distances in p-space, simple graphical techniques
• Multivariate normal (MVN) distribution and related distribution – definition, properties, conditional distributions, the Wishart and Hotelling T-squared distribution, sampling distributions of the sample mean vector and covariance matrix, maximum likelihood estimation of the mean vector and covariance matrix
• Inferences about mean structures – hypothesis testing and confidence intervals (one sample and two independent sample procedures), generalised likelihood ratio test, CI for the components of the mean vector(s), other topics such as MANOVA and profile analysis
• Inferences about covariance and correlation matrices – testing covariance structure of a single covariance matrix, testing equality of covariance matrices

Part B – Advanced techniques in multivariate analysis

• Bayesian inference for MVN – including conjugate models with potential exploration of approximation techniques and Monte Carlo methods
• Dimension reduction – techniques such as principal component analysis, factor analysis, canonical correlation analysis, and correspondence analysis
• Cluster Analysis – aims, K-means and hierarchical algorithms, the dendrogram
• Discriminant Analysis – LDA, QDA 

Teaching is composed of two hours of lectures and one tutorial class per week. Teaching materials will be made available online for reference and review. 

• Coursework, which will involve applying methods to real data: weighting 20%
• End of semester examination: weighting 80%

Feedback methods

Feedback will be provided throughout the course, including:

• In tutorials you will be able to ask for and receive feedback on your work and under- standing.
• You will receive feedback on your coursework.
• You can receive feedback from the lecturer in person during the o¿ice hour or at other times.
• You can receive feedback via the Forum on BlackBoard.

Chatfield, C. and Collins, A. J., An Introduction to Multivariate Analysis, Chapman & Hall 1983.

Krzanowski, W. J., Principles of Multivariate Analysis: A User's Perspective, Oxford University Press 1990.

Johnson, R. A. and Wichern, D. W., Applied Multivariate Statistical Analysis 3rd edition, Prentice Hall 1992.

Bishop, Y M, Fienberg, S E, and Holland P W (2007) Discrete Multivariate Analysis. Springer 

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 75-120 minutes of video content. Normally you would spend approximately 2.5-4 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 2-2.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes, short-answer formative exercises or directed reading

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 9 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.