## MATH3772 Multivariate Analysis

### 10 creditsClass Size: 100

Module manager: Dr S. Barber
Email: stuart@maths.leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

### Pre-requisite qualifications

MATH2715 or MATH2735.

### This module is mutually exclusive with

 MATH5772M Multivariate and Cluster Analysis

This module is approved as an Elective

### Module summary

This module introduces the statistical methodology used in analysing multivariate observations, and applications to real data sets.

### Objectives

To introduce the statistical methodology used in analysing multivariate observations, and to understand its application to real data sets.

On completion of this module, students should be able to:
(a) relate joint, marginal and conditional distributions and their properties with particular reference to the multivariate normal distribution;
(b) obtain and use Hotelling's T2 statistic for the one sample and two samples problems;
(c) derive, discuss the properties of, and interpret principal components;
(d) use the factor analysis model, and interpret the results of fitting such a model.
(e) derive, discuss the properties of, and interpret decision rules in discriminate analysis;
(f) use a statistical package with real data to facilitate an appropriate analysis and write a report giving and interpreting the results.

### Syllabus

In multivariate analysis several variables are measured on each individual in the sample. The multivariate normal distribution now plays the same modelling role that the normal distribution does in univariate theory. Many of the univariate results have multivariate analogues and the module will look at generalisation of the t-test and confidence intervals. But a range of new techniques become available in the multivariate setting. For example, reducing the effective number of variables as in principal components analysis, describing the structure of dependence between variables as in factor analysis and classifying observation to populations as in descriminant analysis. Using the computer to do these analyses and look at examples will form an integral part of the course.

Topics covered include:
1. Introduction to multivariate analysis and review of matrix algebra.
2. Multivariate distributions; moments; conditional and marginal distributions; linear combinations.
3. Multivariate normal and Wishart distributions; maximum likelihood estimation.
4. Hotelling's T2 test; likelihood vs. union-intersection approach; simultaneous confidence intervals.
5. Principal component analysis; dimension reduction; covariance vs. correlation matrix.
6. Factor analysis; common and specific factors; Heywood cases; interpretation of factor loadings; determination of number of factors.
7. Discriminant analysis; maximum likelihood and Bayesian discriminant rules for normal data; misclassification probabilities and assessment by cross- validation; Fisher's discriminant rule.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Example Class 4 1.00 4.00 Lecture 18 1.00 18.00 Practical 2 1.00 2.00 Private study hours 76.00 Total Contact hours 24.00 Total hours (100hr per 10 credits) 100.00

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Coursework
 Assessment type Notes % of formal assessment In-course Assessment . 20.00 Total percentage (Assessment Coursework) 20.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 80.00 Total percentage (Assessment Exams) 80.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated