# Level 3 Modules

**AMA3011 Applied Mathematics Project (Sem 1 and Sem 2)**

*Pre-requisite*: While there are no specific pre-requisites, this course is intended for students at stage 3 of BSc Mathematics, Applied Mathematics and Physics or Theoretical Physics programmes, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Module Co-ordinators*: Prof H van der Hart (Sem 1), Dr G Gribakin (Sem 2)

**Introduction**

This module (or, alternatively, PMA3013 or PHY3007) is compulsory for all students on a BSc pathway (apart from Mathematics with Finance) and constitutes a self-study project on an advanced mathematical topic under the supervision of a member of staff. The module will allow a student to study a particular topic in more depth than possible in a taught module. It also offers students the opportunity to see how the mathematical skills developed in the programme are used to investigate more extensive problems.

Students will be offered a choice of topic subjects which can span the entire range of applied mathematics and statistics, including some theoretical physics.

To further develop communication skills, the outcomes of the study will be presented through a written report and through a poster presentation. In the poster presentation, students will explain their work in a short presentation to members of staff, with a brief follow-up discussion on points of interest.

**Contents**

Self-study of an advanced mathematical topic under the supervision of a member of staff. Students will be offered a choice of subjects, which can span the entire range of applied mathematics, including theoretical physics. The study concludes with a written report and a poster presentation.

**Assessment**

Report 80%

Presentation 20%

**AMA3020 Investigations (Sem 2)**

*Pre-Requisites*: While there are no specific pre-requisites, this course is intended for MSci students in Mathematics, Mathematics and Statistics & Operational Research, Applied Mathematics and Physics, Mathematics and Computer Science, and Theoretical Physics, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturers*: Prof M Paternostro and Dr C Ballance

**Introduction**

Problem-solving is a key skill in many domains, from finance to academia, from software development to data analysis. Being able to address accurately a previously unseen problem, finding creative manners to solve it, and presenting complex concepts in a simple and effective way are invaluable assets. This module will facilitate the development of problems-solving skills that are expandable in a broad domain of fields and areas.

This module is taken by MSci students intending to take AMA4005 or SOR4001 projects in Level 4.

**Content**

Students conduct a short practice investigation, followed by two short investigations (in small groups and solo) in a range of problems in Applied Mathematics and Theoretical Physics. This is followed by a long investigation, which is a literature study of a Mathematical or Theoretical Physics topic not covered in the offered (or chosen) modules. The two short and the long investigation are typed up in reports and submitted for assessment.

**Assessment**

Presentation 20%

Report 1 35%

Report 2 45%

**AMA3022 Team Project: Mathematics with Finance (Sem 2)**

*Pre-requisite*: Only available to students on the BSc Mathematics with Finance programme.

*Lecturers*: Dr D Dundas and Dr S Moutari

**Introduction**

This is a company-led Level 3 project module for students on the Financial Mathematics Pathway. It is designed to develop commercial, technical and team-working skills. The companies involved will provide problems that they wanted investigated. Students are asked to form small consultancy teams with defined roles and a produce a team brand. Each team then bids for projects by producing proposals based on the company requirements. Based on these proposals, the companies will award their projects. The teams then work with their partner companies to deliver the project.

**Contents**

Business skills workshop. Presentation skills. Negotiation skills. Customer relationship. Project management/team building. Teams required to negotiate, plan, develop and deliver a completed task working as a group, commissioned by the `client' company. The project will require software development skills.

**Assessment**

Business Proposals 30%

Report & Presentation 40%

Business Plan 10%

Peer Evaluation 20%

**MTH3012 Rings and Modules (Sem 1)**

*Pre-requisite*: MTH2011 Linear Algebra and MTH2014 Group Theory, or

MTH2001 Linear Algebra and Complex Variables and PMA2008 Group Theory [for 2021/22]

*Lecturer*: Dr Y-F Lin

**Introduction**

The purpose of this module is to give a general introduction to the theory of rings and modules, which is a subject of central importance in algebra. A module over a ring is a generalisation of the notion of vector space over a field. Historically, some of the major discoveries have helped to shape the course of developments of modern abstract algebra. Today, ring theory and modules are a possible meeting ground for many algebraic sub-disciplines such as representation theory of groups, Lie theory, algebraic geometry, homological algebra and algebraic topology, to name but a few.

**Content**

Rings, subrings, prime and maximal ideals, quotient rings, homomorphisms, isomorphism theorems, integral domains, principal ideal domains, modules, submodules and quotient modules, module maps, isomorphism theorems, chain conditions (Noetherian and Artinian), direct sums and products of modules, simple and semisimple modules.

**Assessment**

Continuous assessment 30%

Written examination 70%

**MTH3021 Dynamical Systems (Sem 2)**

*Pre-requisite: *MTH2011 Linear Algebra, MTH2013 Metric Spaces, or MTH2001 Linear Algebra and Complex Variables and PMA2002 Analysis [for 2021/22]

*Lecturer*: Dr G Kiss

**Introduction**

A dynamical system is a mathematical description of a real-world process evolving in time, such as the number of infectious individuals in a population during an outbreak, the price of a financial instrument, or the positions and velocities of finite point masses. The mathematical representation of a time dependent phenomenon can be given in terms of differential equations or iterated functions, and very often models are non-linear. In this module, by studying the existence and qualitative properties of invariant objects such as equilibria, closed orbits and their invariant manifolds, we will learn how a well-formulated rather abstract mathematical model can be used, with mathematical rigour, to control an epidemic, predict the unpredictable and obtain low-energy interplanetary spacecraft trajectories.

**Contents**

Continuous dynamical systems

- Fundamental theory: existence, uniqueness and parameter dependence of solutions;

- Linear systems: constant coefficient systems and the matrix exponential; nonautonomous linear systems; periodic linear systems.

- Topological dynamics: invariant sets; limit sets; Lyapunov stability.

- Grobman-Hartman theorem.

- Stable, unstable and centre manifolds.

- Periodic orbits: Poincare-Bendixson theorem.

- Bifurcations

- Applications: the Van der Pol oscillator; the SIR compartmental model; the Lorenz system.

Discrete dynamical systems

- One-dimensional dynamics: the discrete logistic model; chaos; the Cantor middle-third set.

**Assessment**

Continuous assessment 30%

Written examination 70%

**MTH3022 Discrete Mathematics (Sem 2)**

*Pre-requisite*: MTH1011 Introduction to Algebra and Analysis, or MTH1001 Analysis and Calculus and MTH1002 Numbers, Vectors and Matrices [for 2021/22]

*Lecturer*: Dr F Pausinger

**Introduction**

This module gives an introduction to basic techniques in discrete mathematics. The main objects of study in discrete mathematics are integers, graphs and logical statements. Dis- create objects can often be enumerated by integers and, thus, discrete mathematics can be characterised as the branch of mathematics dealing with countable sets. Discrete mathematics is particularly important in our modern world since digital computers typically operate in discrete steps and store data in discrete bits. Thus, the main applications of the methods presented in this module can be found in such diverse areas as theory of computation, cryptography or operations research.

**Content**

1. Intro Enumerative Combinatorics: basic counting, pigeonhole principle, inclusion-exclusion, recurrence relations, generating functions.

2. Intro Elementary Number Theory: Divisibility and primes, Euclidean algorithm, linear congruences, Chinese Remainder Theorem.

3. Intro Graph Theory: basic notions, trees, connectivity, matchings, graph colouring, planarity, basic Ramsey theory.

4. Intro Algorithmics: analysis of algorithms, sorting, greedy and divide-and-conquer algorithms, basic graph and NT algorithms.

**Assessment**

Assignment 15%

Project 15%

Exam 70%

**MTH3023 Numerical Analysis (Sem 1)**

*Pre-requisite*: MTH2011 Linear Algebra, or MTH2001 Linear Algebra and Complex Variables [for 2021/22]

*Lecturer*: Dr D Dundas

**Introduction**

In many real-world applications of mathematics, it may not be possible to obtain analytical solutions. Numerical Analysis is concerned with devising methods for finding approximate, numerical solutions to mathematically expressed problems. These methods can be analysed for their accuracy, efficiency and robustness. For example, some methods will guarantee convergence to a solution, but may require much effort, while other methods may converge quickly with less effort, but may also diverge. Faced with such differing behaviour of the methods, we need to be able to determine the ‘best’ strategy to adopt for a given problem. In MTH3023 we cover the basic introductory material of Numerical Analysis. We investigate the solution of equations, interpolation, function approximation, differentiation, integration and the solution of ordinary differential equations.

**Content**

Introduction and basic properties of errors: Introduction; Review of basic calculus; Taylor's theorem and truncation error; Storage of non-integers; Round-off error; Machine accuracy; Absolute and relative errors; Richardson's extrapolation.

• Solution of equations in one variable: Bisection method; False-position method; Secant method; Newton-Raphson method; Fixed point and one-point iteration; Aitken's "delta-squared" process; Roots of polynomials.

• Solution of linear equations: LU decomposition; Pivoting strategies; Calculating the inverse; Norms; Condition number; Ill-conditioned linear equations; Iterative refinement; Iterative methods.

• Interpolation and polynomial approximation: Why use polynomials? Lagrangian interpolation; Neville's algorithm; Other methods.

• Approximation theory: Norms; Least-squares approximation; Linear least-squares; Orthogonal polynomials; Error term; Discrete least-squares; Generating orthogonal polynomials.

• Numerical quadrature: Newton-Cotes formulae; Composite quadrature; Romberg integration; Adaptive quadrature; Gaussian quadrature (Gauss-Legendre, Gauss-Laguerre, Gauss-Hermite, Gauss-Chebyshev).

• Numerical solution of ordinary differential equations: Boundary-value problems; Finite-difference formulae for first and second derivatives; Initial-value problems; Errors; Taylor-series methods; Runge-Kutta methods.

**Assessment**

Project 1 20%

Project 2 30%

Exam 50%

**MTH3024 Modelling and Simulation (Sem 2)**

*Pre-requisite*: MTH2021 Mathematical Methods 2, or

MTH2002 Introduction to Partial Differential Equations [for 2021/22]

*Lecturers*: Dr M Gruening and Prof H van der Hart

**Introduction**

Mathematical modelling consists in translating a phenomenon observed in the real world into a mathematical framework, often involving associated equations. Only rarely can an analytical solution for these equations be found and instead numerical simulations are required. Numerical simulations are calculations run on a computer according to a program that implements the mathematical model. Mathematical models and numerical simulations are used extensively. Examples are numerous: Major building design must enable rapid emergency evacuation in line with legislation. Sustainable city design requires the modelling of traffic, including vehicles, pedestrians and cyclists. Understanding the green impact of policy decisions requires mathematics to estimate the effects of policy decisions and understanding how these effects impact the environment.

Many of the taught modules introduce particular mathematical models. However, these models are often applied to ideal systems, so that analytic results can be obtained. Most real systems are not ideal systems, however. Modelling and simulation then aims to identify suitable approximations to capture the essence of the real system, so that appropriate outcomes can be obtained. This module aims to introduce students to the modelling and simulation process through hands-on experience.

**Content**

In this module, students will analyse real-life situations, build a mathematical model, solve it using analytical and/or numerical techniques, and analyse and interpret the results and the validity of the model by comparing to actual data. The emphasis will be on the construction and analysis of the model rather than on solution methods. Two group projects will fix the key ideas and incorporate the methodology. This will take 6-7 weeks of term and will be supported with seminars and workshops on the modelling process. Then students will focus on a final project, ideally with real-life application, and work on this for the remaining weeks of term. They will present their results through a variety of approaches. A key element of the module is the discussion of ideas. These discussions can take place not only within a group, but also between groups. This highlights that the same type of problem can often be approached by entirely different means.

The starting group project will be focused and offer a limited number of specific modelling problems. The second project will also be focused, but offer more scope for students to identify their own specific follow-on development. A wider range of projects will be available for the final project, spanning a wide range of disciplines in which mathematical modelling plays a critical role. The projects will place a significant and increasing emphasis on students’ own initiative.

**Assessment**

Project work 100%

**MTH3025 Financial Mathematics (Sem 1)**

*Pre-requisite*: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or a BSc Mathematics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturer*: Dr A Ferraro

**Introduction**

Mathematical skills are highly sought after in the financial services industries, and this employment sector remains a favoured destination for graduates. Around 40% of Mathematics graduates entering employment across the UK (see www.prospects.ac.uk for recent data) go into financial services, which includes, accountancy, retail and investment banking, mergers and acquisitions, insurance and actuarial work, capital market trading, and hedge fund employment, and so on.

At the low end of this sector, retail banking for example, a degree in mathematics is certainly not essential. This work is mainly concerned with simple arithmetic operations. However, at the high end of financial services, in a hedge fund for example, employers expect to see PhD-level qualifications in mathematics from their applicants along with excellent software skills. These mathematicians are involved in the business of derivative pricing and trading and earn salaries well over 100k. Derivatives are financial products (instruments as they are called in the trade) derived from assets that have an unpredictable price. The total outstanding notional value of derivatives contracts today has grown beyond a quadrillion dollars (that's 1015 to you and me). It is a perilous and lucrative business!

Derivatives were originally devised to avoid risk by providing an insurance on a risky asset. Nowadays, they are an essential part of risk taking in capital markets. Indeed the speculation in buying and selling these instruments, specifically credit derivatives, precipitated the current credit crunch. Of course, this trade relies upon knowing the fair price of a derivative. Pioneering work by Black, Merton and Scholes, showed that, under certain assumptions for the unpredictability of the asset, the price of the derivative obeys a partial-differential equation. The construction of such equations and their solution is where mathematicians come in!

The objective of the course is to provide an introduction to the mathematical techniques which can be applied to pricing problems for financial derivatives. Specifically, our focus is on stochastic calculus and the theory and practice of pricing simple derivatives such as contracts and options.

**Content**

Introduction to financial derivatives: forwards, futures, swaps and options; Future markets and prices; Option markets; Binomial methods and risk-free portfolio; Stochastic calculus and random walks; Ito's lemma; the Black-Scholes equation; Pricing models for European Options; Greeks; Credit Risk.

**Assessment**

Presentation 10%

Report 20%

Exam 70%

**MTH3031 Classical Fields (Sem 1)**

*Pre-requisite*: MTH2031 Classical Mechanics or PHY2004 Electricity, Magnetism & Optics, or AMA2001 Classical Mechanics or PHY2004 Electricity, Magnetism & Optics [for 2021/22]

*Lecturer*: Dr G Gribakin

**Introduction**

This module develops further the fundamental ideas that enable mathematics to describe the Universe that was started in Classical Mechanics. It introduces the notion of 4-dimensional space-time and Einstein’s relativity and Lorentz covariance as the main principles that the theory must obey. It then uses the Least Action Principle to build the description of the motion of charged particles and electron-magnetic fields and their interaction. Having set up the equations for the electro-magnetic field in covariant, 4-dimensional form, we then “descend” into the more familiar 3+1 (space + time) form to derive Maxwell’s equations and study their solutions in a range of context, from electrostatics and magnetostatics to waves. The ideas and description of classical (electro-magnetic) fields developed in this module can lead to further generalisation in the form of Einstein’s general relativity. When combined with Quantum Theory, this allows the development of the theory of Quantum Fields which underpins all modern theories of elementary particles.

**Content**

• Recapping of the least action principle in Classical Mechanics. Lagrangians for continuous systems (e.g., a string), and derivation of the wave equation from Lagrange’s equation.

• 4-dimensional space time, interval, 4-vectors, tensors, Lorentz covariance.

• Action and Lagrangian for a particle, energy and momentum.

• 4-potential and the Lagrangian for a charged particle in an electromagnetic field, relativistic equation of motion and Lorentz’s force, electric and magnetic fields.

• Lagrangian of the electromagnetic field, Maxwell’s equations in covariant form, charge density and current density, continuity equation, Maxwell’s equations in conventional (3+1) form, and in integral form.

• Electrostatics (general ideas, Coulomb’s law, fields of various charge distributions, electric dipole moment).

• Magnetostatics (general ideas, Biot-Savart-Laplace law, fields of systems of currents, magnetic dipole moment).

• Electromagnetic waves, plane wave, polarisation, monochromatic wave.

• Electromagnetic radiation: retarded potentials, dipole radiation (electric, magnetic), Larmor formula.

**Assessment**

Assignment 1 15%

Assignment 2 15%

Exam 70%

**MTH3032 Quantum Theory (Sem 1)**

*Pre-requisite*: MTH2031 Classical Mechanics or PHY2001 Quantum and Statistical Physics, and MTH2021 Mathematical Methods 2, or None [for 2021/22]

*Lecturer*: Prof M Paternostro, Dr G Zicari

**Introduction **

Quantum mechanics brought about the most fundamental change in our understanding of how the world works. It describes the behaviour of microscopic particles (electrons, photons, atomic nuclei, etc.) and their interactions in a way that is very different from our everyday experience. Key points of the theory are wave-particle duality (e.g., the ability of particles to display typical wave-like properties, such as interference), quantisation (i.e., restriction of possible values of some physical observables, such as energy or angular momentum, to a discrete set of values), statistical nature of its predictions, and the role played by the observer in a measurement process.

In this module we develop the mathematical methods that enable one to describe how nature behaves at small scales, e.g., that of individual atoms. The mathematical setting of Quantum Theory is that of Hilbert spaces and linear operators, and its practical aspects involve dealing with matrices, eigenvalue problems and differential equations.

Students completing this module will understand the basic principles of quantum mechanics and its mathematical tools, learn how to solve a few simple, fundamental problems, and practice approximate methods that greatly widen the range of problems that can be solved.

Quantum Theory builds on Classical Mechanics and open the route to Quantum Fields which is the framework for modern theories of elementary particles.

**Content**

• Overview of classical physics and the need for new theory.

• Basic principles: states and the superposition principle, amplitude and probability, linear operators, observables, commutators, uncertainty principle, time evolution (Schrödinger equation), wavefunctions and coordinate representation.

• Elementary applications: harmonic oscillator, angular momentum, spin.

• Motion in one dimension: free particle, square well, square barrier.

• Approximate methods: semiclassical approximation (Bohr-Sommerfeld quantisation), variational method, time-independent perturbation theory, perturbation theory for degenerate states (example: spin-spin interaction, singlet and triplet states).

• Motion in three dimensions: Schrödinger equation, orbital angular momentum, spherical harmonics, motion in a central field, hydrogen atom.

• Atoms: hydrogen-like systems, Pauli principle, structure of many-electron atoms and the Periodic Table.

**Assessment**

Assignments 30%

Exam 70%

**MTH3099 Placement Year Out (Full year)**

*Pre-requisite*: MTH2010. Placement documentation (including contract, job description and health & safety agreement) that has been approved by the School must be in place.

*Lecturer: *Dr D Dundas

**Introduction**

This is a 120 CAT point module (equivalent to a year of academic study) that is taken by students who are taking an approved placement year as part of their degree. As part of the placement, students keep a diary of work undertaken and use this to produce a portfolio at the end of the placement. This portfolio will allow students to highlight the work undertaken and to reflect on the skills developed.

**Content**

Course contents is as defined by the School-approved student contract and job description. A mid-placement visit by School staff will take place to ensure that the job role and conditions of employment reflect those described in the job description and contract.

**Compulsory Element**

Submission of all required paperwork associated with the placement.

**Assessment**

A placement portfolio to be submitted at the end of the placement. This portfolio is assessed on a Pass/Fail basis. The placement year does not contribute to your degree classification and therefore does not affect your progression onto the next year of study.

**PMA3013 Mathematical Investigations (Sem 2)**

*Pre-Requisites*: None

*Lecturers*: Dr S Shkarin and Dr A Zhigun

**Introduction**

This module is an alternative to AMA3011 as a compulsory project module in Level 3 for BSc students (apart from Mathematics with Finance). It is also taken by the MSci students who intend to take PMA4001 Project in their final year.

**Content **

This module is concerned with the investigation processes of mathematics, including the construction of conjectures based on simple examples and the testing of these with further examples, aided by computers where appropriate. A variety of case studies will be used to illustrate these processes. A series of group and individual investigations will be made by students under supervision, an oral presentation will be made on one of these investigations. While some of the investigations require little more than GCSE as a background, students will be required to undertake at least one investigation which needs knowledge of Mathematics at Level 2 or Level 3 standard and/or some background reading.

**Assessment **

Solo Project 50%

Group Project 40%

Presentation 10%

**PMA3014 Set Theory (Sem 1)**

*Pre-requisite*: MTH2011 Linear Algebra. This module is a compulsory component of the MSci course in Pure Mathematics.

*Lecturer*: Prof M Mathieu

**Introduction**

Set Theory is a rich and beautiful subject of Pure Mathematics whose fundamental concepts permeate every branch of mathematics. No undergraduate mathematics education is complete without a thorough study of this discipline. This module will teach fluency in the language of elementary axiomatic set theory, facility in the use of key techniques such as transfinite induction and maximality principles, and basic arithmetic of cardinal and ordinal numbers (the ‘arithmetic of infinity’).

**Content**

The language of set theory; constructing number systems; cardinals and ordinals; Zermelo-Fraenkel axiomatization of set theory; the axiom of choice.

**Assessment**

Assignment 1 15%

Assignment 2 15%

Exam 70%

**PMA3017 Metric and Normed Spaces (Sem 2)**

In 2021-22 this module will be co-taught with MTH2013 Metric Spaces, but will be examined separately.

*Pre-requisite: *PMA2002

*Lecturers*: Dr Y-F Lin and Dr G Kiss

**Introduction**

Analysis is the study of convergence and continuity, it is fundamentaly linked to the structure of the real numbers. The aim of this course is to move gradually away from real analysis to the more general setting of metric and normed spaces. The fundamental question is: what aspects of real analysis survive to the setting of metric and normed spaces?A metric space is a set with a notion of distance, called a metric. Them most familiar example is the real line, with the distance from *x* to *y* given by | *x - y *| . This notion of distance allows us to define convergence and continuity in a much more abstract setting.Often, we want the set underlying our metric space to have an addition operation. To get interesting structures, we must then add some extra compatibility conditions to our metric. This leads us to the concept of a normed vector space, this is a vector space as we have encountered before, but now it comes with a measure of the size of a vector, called the norm. As every normed vector space is a metric space, we will begin with the study of metric spaces.

**Content**

Definition and examples of metric spaces; open sets, closed sets, closure points, sequential convergence, compactness; completeness; continuous mappings between metric spaces; Banach's fixed-point theorem and applications; Baire category theorem and applications. Normed spaces; Banach spaces; finite dimensional normed spaces; subspaces and quotient spaces.

**Assessment**

Continuous assessment 30%

Written examination 70%

**SOR3001 Linear and Dynamic Programming (Sem 1)**

*Pre-requisites: *None

*Lecturers*: Dr M Gruening, Dr M Carlesso

**Introduction**

Operational Research is the application of quantitative analysis to problems outside of the physical sciences and in particular to the problems of business, industry and administration. However, all the techniques can be used outside of Operational Research and emphasis is given to formulation of problems and recognition of the appropriate technique. Blackboard examples are given for each technique and the homework problems are drawn from a wide field. Most techniques are in the form of algorithms and for hand calculations the problems have to be fairly simple, but indications are given how computers could be applied to bigger problems in the project and presentation components.

The aim of this course is to develop competence in two of the most important mathematical techniques used in Operational Research, in formulating problems and expressing answers clearly.

This course covers the two main general-purpose mathematical techniques used in Operational Research together with some specialized applications. These are all optimisation techniques, although they also give insight into the problems and the interpretation of answers is stressed. Calculus cannot be used because either (i) the objective function and the constraints cannot be expressed simply, or (ii) the constraints dominate the problem and the optimal solution will be at the edge of the feasible region, or (iii) the variables are not continuous (e.g., they are integer).

Some students may have met some of the techniques in courses at other levels, but no previous knowledge is assumed, and more mathematical rigour and understanding is required than at those levels. However, the mathematical knowledge assumed does not extend beyond that in Level 1. The main knowledge assumed is elementary linear algebra (bases, linear dependence of vectors, matrix notation, partitioning of matrices). No knowledge of economics is required but the economic interpretation of some results is explored. Apart from a passing reference to stochastic Dynamic Programming this course is purely deterministic, and no knowledge of statistics is required.

**Contents**

Dynamic programming; stochastic problems; allocation, production-inventory and optimal route problems, including some non-dynamic programming algorithms. Linear programming: primal, dual and revised simplex methods, duality, post-optimal analysis, transportation and assignment problems, network flow problems.

**Assessment**

Group work 20%

Exam 70%

Presentation 10%

**SOR3004 Linear Models (Sem 1)****NOT TO BE TAKEN AFTER SOR2004**

*Pre-requisite:* SOR2002

*Lecturer*: Dr H Mitchell

**Introduction**

The aim of this module is to cover linear models encompassing multiple linear regression and analysis of variance (ANOVA). These models are the workhorses of statistical data analysis and are found in virtually all branches of the sciences as well as in the industrial and financial sectors.

Multiple linear regression is concerned with modelling a measured response as a function of explanatory variables. For example, a pharmaceutical company might use a a regression model to relate the effectiveness of a new cancer drug to the patients age, gender, weight, diet, tumour size, etc. ANOVA is concerned with the analysis of data from designed experiments. A materials manufacturer for example, may wish to analyse the results from an experiment to compare the heat resisting properties of four different polymers.

Regression and ANOVA will be initially developed using a classic least squares approach and later the correspondence between least squares and the method of maximum likelihood will be examined. After a thorough development of linear models the groundwork will have been laid to allow an extension to the broader class of Genealized Linear Models (GLM). These permit regression models to be applied to situations where the recorded response is not normally distributed. One famous example of the use of GLM was the analysis of O-ring failures on the space shuttle Challenger.

An important element of this module will be a weekly practical data analysis class using the SAS software package. SAS is probably the leading statistical package used in industry. These classes, lasting up to three hours, will introduce the student to elementary data entry in SAS, elementary matrix manipulation using the SAS Interactive Matrix Language (IML) and analysis of data using linear and generalized linear models.

**Contents**

Linear regression. Non-singular case: analysis of variance, extra sum of squares principle, generalised least squares, residuals. Singular case: generalised inverse solution, estimable functions. Experimental designs: completely randomised, randomised block, factorial; contrasts, analysis of covariance; Generalised linear model (GLM): maximum likelihood and least squares; exponential family; Poisson and logistic models; model selection for GLM.

**Assessment**

Exam 70%

Coursework 20%

Presentation 10%

**SOR3008 Statistical Data Mining with Machine Learning (Sem 2)**

*Pre-requisite*: SOR2004 or currently enrolled in SOR3004 in semester 1.*Lecturer*: Dr K Cairns

**Introduction**

In the 1990's there was an explosive growth in both the generation and collection of data due mainly to the advancement of computing technology in processing and storage of data and the ease of scientific data collection. As a result, overwhelming mountains of data are being generated and stored. For example, in the business world large supermarket chains such as Wal-Mart and Sainsbury's collect data amounting to millions of transactions per day. In the US all health-care transactions are stored on computers yielding terabyte databases which are constantly being analysed by insurance companies. There are huge scientific databases as well. Examples include the human genome database project and NASA's Earth Observatory System. This has brought about a need for vital techniques for the modelling and analysis of these large quantities of data: data mining.

Data Mining is the process of selection, exploration, and modelling of large quantities of data to discover previously unknown regularities or relations with the aim of obtaining clear and useful results for the owner of the database. The application of data mining includes many different areas, such as market research (customer preferences), medicine, epidemiology, risk analysis, fraud detection and more recently within bioinformatics for modelling DNA.

This module will focus on data mining techniques which have evolved from and are strongly based on statistical theory.

**Content**

Introduction to Data Mining and Machine Learning; Supervised and Unsupervised Learning; Exploratory Data Analysis including Principal Component Analysis, Multiple Imputation; Cluster analysis; Classification including Decision tree analysis, Bayesian Networks, Probabilistic/Regression-Based Modelling; Prediction including Regression trees, Random Forests, Neural nets, Support Vector Machines; Association Rule Mining.

**Assessment**

Coursework 1 20%

Coursework 2 20%

Exam 60%

**SOR3012 Stochastic Processes and Risk (Sem 2)**

*Pre-requisite*: SOR1020*Lecturer*: Dr G Tribello

**Introduction**

The module teaches you about time series of random variables. A very practical approach to the subject is taken so you learn about these random time series by writing computer programs to generate and analyse random data. The final assessment of the module requires you to use what you have learned about modelling random processes to write a report for a real community transport organisation that is based in Fermanagh.

**Contents**

Logic and Boolean algebra, counting and combinatorics, set algebra, inclustion-exclusion theorem, mutually exclusive events, De Morgan Laws.

Axioms of probability, events and probability spaces, sigma-field, random variables, conditional probability, and expectation, Bayes’ theorem, discrete and continuous random variables, moments and moment generating function. Laws of large numbers and central limit theorem.

Pairs of random variables, marginal probabilities, Cauchy Schwartz Inequality in statistics, correlation and covariance.

Discrete time Markov chains, Chapman Kolmogorov relation, limiting behaviour, transient, recurrent states and periodic states, limiting stationary distribution, hitting times and hitting probabilities.

Continuous time Markov chains, Kolmogorov forward equations, stationary distribution for continuous time Markov chains, Poisson process, MM1 Queue, inhomogeneous Poisson process and compound Poisson process.

**Assessment**

Exam 45%

Practical Lab / Log Book 45%

Report 10%