Guide to:

NOTE:
The following courses are no longer taught:
MATH3181, MATH3241, MATH3301
MATH3421, MATH3511
MATH3830, MATH3880, MATH2890, MATH3930, MATH3931, MATH3941

The following courses have been replaced by other courses:
MATH3541, MATH3610, MATH3620, MATH3641, MATH3690, MATH3700, MATH3710
MATH3830, MATH3930

The following courses have moved to graduate level:
MATH3630, MATH3680, MATH3740, MATH3780, MATH3790

The following are descriptions of courses offered by the Department of Applied Mathematics.

One of the greatest mysteries of our universe is that it can be understood using the tools of mathematics. What better prophet than the mathematics of dynamics? What better selector than the mathematics of optimization? What greater guide through uncertainty than the mathematics of statistics? This course uses tools from the mathematics of dynamics, optimization and statistics to model real world problems.

In addition to learning the skills for turning real world problems into mathematical problems you will learn invaluable associated skills in writing reports, carrying out a literature search, working as part of a team, and general problem solving and research. You will also learn basic skills in computational packages such as MAPLE, MATLAB and EXCEL.

This modern mathematics course has been designed to provide a preparation for entry to the work force or for further research related studies. The course is highly recommended to all mathematics students and all science and engineering students who have a basic knowledge of mathematics up to second year undergraduate level.

Handbook entry

Many mathematical models in engineering, finance and science are based on differential equations. In general these equations cannot be readily solved analytically. This course introduces computational methods for solving partial differential equations that exploit the power of modern computer architectures. Overview of scientific computing, iterative methods for linear systems, parabolic partial differential equations (heat equation), elliptic equations (Poisson equation) including multigrid methods, and hyperbolic partial differential equations.

Old MATH3101 description
Differential equations play an extremely important and useful role in applied mathematics, engineering, finance and science. In general mathematical models based on differential equations cannot be readily solved analytically. Much of the scientific computation machinery has been developed for the solution of differential equations.

In this course, we introduce modern computational methods to solve systemsof both initial and boundary value ordinary differential equations. There is a substantial computing component in this course, involving implementation of the methods and simulation of some mathematical models using MATLAB software package on UNIX and Windows based computer systems. We use MATLAB Spline Toolbox and Ode Suite Package to minimise our programming needs.

Good preparation for this course assumes relatively little in computational mathematics per se, except for basic understanding of calculus and linear algebra. Prior knowledge of numerical methods for differential equations will not be assumed and is not necessarily an advantage. Experience with programming in MATLAB and application of computational techniques will obviously assist comprehension but is neither assumed nor expected.

This course begins with introduction to approximation theory based on global interpolation and splines. The second (major) part of the course is to describe in detail computational techniques for solving initial value ordinary differential systems, based on explicit and implicit computer methods for non-stiff and stiff problems. We then introduce direct computer algebra methods to solve matrix equation. In the final part of the course we introduce shooting, finite difference and orthogonal collocation numerical methods solve boundary value problems.

Handbook entry

Properties of and fundamental methods for the solution of partial differential equations (PDEs). Classification of first and second order PDEs. Dirichlet and Neumann problems including the Wave, Heat and Laplace Equations. Method of Characteristics. Fourier and Laplace Transforms, Green’s Functions and Conformal Maps. Selected applications from finance, physics, engineering and the biosciences.

Old MATH3121 description
This course builds on MATH2120 Mathematical Methods for Differential Equations in that it is concerned with ways of solving the (usually partial) differential equations that arise mainly in physical and engineering applications. We are concerned with methods of actually finding solutions rather than with answering theoretical questions as to the conditions under which solutions exist or are unique.

We start with revising one of the basic methods - the method of separation of variables. This leads on to studying Sturm-Liouville problems in a deeper way than was done in second year. As part of this, we revise Bessel functions and introduce some of the standard orthogonal polynomials associated with the names of Legendre, Chebyshev, Laguerre and Hermite.

Next comes a new concept - that of the Laplace transform. The Laplace transform turns differentiation into multiplication by a transform variable. Hence, some ordinary differential equations are turned into algebraic equations by this technique and partial differential equations with n independent variables into partial differential equations with n-1 independent variables. This is one of the standard tools in electrical engineering for studying the behaviour of circuits and control systems. We use it for solving partial differential equations which describe the flow of heat or the diffusion of pollution. We also introduce a related transform - the Fourier transform - which is also widely used in areas such as data anaysis and signal processing for example. We use it to solve partial differential equations.

A completely different approach is provided by the method of characteristics which, amongst other things, enables us to show that the propagation of sound or light in one direction, say x, may be described by only two functions - one representing propagation towards x= -infinity and one progagation towards x= +infinity.

The final method is that of Green's Functions. These are already familiar to us as, for example, the electric potential due to a point charge. The total electric potential due to a continuous distribution of charge is then obtained by integration. Green's functions are basically generalisations of this idea to other situations.

Handbook entry

Optimization problems, in which one wants to find the values of variables to maximize or minimize an objective function subject to constraints on which variables are allowed, are common throughout the physical and biological sciences, economics, financeé and engineering. This course looks at the formulation of optimization problems as mathematical problems, characterizing solutions using necessary and/or sufficient optimality conditions and modern numerical methods and software for solving the problems. Both finite dimensional problems which involve a vector of variables, including linear and nonlinear programming, and infinite dimensional problems where the variables are functions, including optimal control problems, are covered.

Old MATH3161 description
Optimization is the study of problems in which we wish to optimize (either maximize or minimize) a function (usually of several variables) often subject to a collection of restrictions on these variables. The restrictions are known as constraints and the function to be optimized is the objective function. Optimization problems are widespread in the modelling of real world systems, and cover a very broad range of applications. Problems of engineering design (such as the design of electronic circuits subject to a tolerancing and tuning provision and the design of paths for robots in the presence of obstacles), information technology (such as the extraction of meaningful information from large databases and the classification of data), financial decision making and investment planning (such as the selection of optimal investment portfolios), and transportation management and so on arise in the form of an optimization pro blem.

Optimization has its foundation in the development of calculus by Newton and Leibniz in the 17th century. The solution of large multi-variable optimizat ion problems using computers started with the work of Dantzig in the late 1940s and 1950s on the simplex method for linear programming. Now, nonlinear optimization problems with a few hundred variables can be solved routinely. Optimization methods are part of wider areas known as Operations Research, Management Science and Mathematical Programming. This course focuses on nonlinear optimization problems, as distinct from linear programming problems and discrete optimization problems. In this course you will learn how to formulate, solve and analyze multi-variable optimization problems.

Handbook entry

A dynamical system is any system whose state changes as a function of time. This course studies the regular and irregular behaviour of nonlinear dynamical systems, concentrating on ordinary differential equations (ODEs) and their solutions. Topics from the theory of ODEs include: existence and uniqueness theorems; linear ODEs with constant and periodic coefficients and Floquet theory; linearization and stability analysis; perturbation methods; bifurcation theory; phase plane analysis for autonomous systems. The theory will be illustrated with applications to physical, biological and ecological systems. In addition, a selection from the dynamical concepts: Hamiltonian dynamics, resonant oscillations, chaotic systems, Lyapunov exponents, Poincare maps, homoclinic tangles.

Old MATH3201 description
The aim of dynamics is to understand and predict the behaviour of systems as they evolve in time. This discipline was developed initially by Newton in the mid to late 1600s, simultaneous with his development of calculus. Most of the early applications were to physical systems such as the two body Earth-Sun problem and the three body Earth-Sun-Moon problem. The two body problem was solved by Newton but the three body problem remains unsolved to this day; in the sense of obtaining algebraic expressions for their motions.

In the late 1800s Poincar'e developed a geometrical approach for dynamical systems based on an abstract mathematical space called phase space. The solutions of differential equations are viewed as trajectories in this phase space and even in cases where solutions cannot be obtained as algebraic expressions the phase space trajectories can often still qualitatively be inferred. Through this geometric approach Poincar\'e glimpsed what algebraic solutions could never reveal; two trajectories weaving ``a type of trellis, tissue, or grid with infinitely serrated mesh''. Poincare had glimpsed what was later to be known as ``chaos'' and in describing his geometric vision he reported that: ``Nothing is more suitable for providing us with an idea of the complex nature of the three body problem''.

Algebraic progress in dynamics was revitalized in the mid 1900s with the development of asymptotic series expansion methods. These methods have led to a detailed understanding of nonlinear vibrations and stability. Applications of this work featured in the development of radios, lasers and stable communication satellites. Geometric phase space methods and algebraic series expansion methods are now standard tools in the analysis of nonlinear dynamical systems.

In the latter half of the 1900s the advent of high speed computing and visualization greatly expanded the understanding and applications of dynamics. Computer plots revealed other ``trellis'' like structures similar to that which Poincare had glimpsed in his minds eye. These stochastic webs and ``strange attractors'' are signatures of ``chaos''. It wasn't until 1975 that the word ``chaos'' was introduced in the scientific literature -- to mean randomness and unpredictability in a deterministic dynamical system. Coincidentally, in this same year, Mandelbrot invented the word ``fractal'' to describe geometric objects with infinite complexity. It subsequently became revealed that the phase space trajectories of chaotic dynamical systems were fractals or fractally entwined. In the last few decades of the twentieth century much of the progress in dynamics was directed towards attempts to diagnose, quantify, and control chaos. One of the most famous findings in this body of work was Feigenbaum's discovery of universal laws governing the onset of chaos. The onset of chaos in water dripping from a tap, irregular heartbeats, electrical circuits and numerous other physical phenomena were studied by experimentalists and shown to obey the same mathematical laws.

Nonlinear dynamics, chaos and fractals are now widely studied and applied in science, engineering, technology and economics with some of the most interesting applications in areas such as internet traffic flow, pattern formation and neural dynamics.

In this course we do not have the time to investigate all of the important results in this discipline. However you will learn geometric phase space methods and algebraic series expansion methods and you will gain an understanding of nonlinear vibrations, stability, bifurcations, chaos and universality. This course would appeal to any student seeking a mathematical intellectual challenge and it would equally appeal to any student who wants to learn tools for anaylzing nonlinear dynamical problems in physics, biology, medicine, chemistry, meteorology, engineering or economics.

Handbook entry

The mathematical modelling and theory of problems arising in the flow of fluids, the oceans and the global climate. Cartesian tensors, kinematics, mass conservation, vorticity, Navier-Stokes equation. Topics from inviscid and viscous fluid flow, gas dynamics, sound waves, water waves. The dynamics underlying the circulation of the atmosphere and oceans are detailed using key concepts such as geostrophy, the deformation radius and the conservation of potential vorticity. The role of Rossby waves, shelf waves, turbulent boundary layers and stratification is discussed. The atmosphere-ocean system as a global heat engine for climate variablity is examined using models for buoyant forcing, quasi-geostrophy and baroclinic instability.

Old MATH3261 description
Weather, climate and oceanic currents play an enormous role in the Earth's environment, controlling cycles in the biosphere, regulating natural ecosystems, and impacting on human health. We depend on the ocean and atmosphere for resources and transport. Yet we also subject the system to pollution, including increased greenhouse gases, motor car emissions, oil spills and chlorofluorocarbons.

Understanding the dynamics of the atmosphere and oceans is one of the great challenges in science today. In this course you will learn how mathematical equations can be used to describe aspects of flow in both the ocean and atmosphere, and how solutions to these equations are obtained. We will cover circulation dynamics ranging in spatial scales from global down to molecular, spanning time-scales from fractions of a second up to decadal and beyond (e.g., climate change).

The course will include analysis of key concepts such as geostrophy, the conservation of potential vorticity, waves, tides, and buoyancy controlled flow. For example, the ocean-atmosphere system witnesses wave motions from millimetre scale capillary waves, to sea and swell, and beyond to planetary scales where the Earth's rotation affects dispersion. The study of these phenomena combines theory with mathematical techniques to shed insight into how flow in the ocean and atmosphere is forced and maintained.

Other topics include Rossby waves, shelf waves, turbulent boundary layers and stratification. The atmosphere-ocean system as a global heat engine for driving climate variablity and change. This is studied worldwide using models that are built upon the dynamics introduced in this course.

Handbook entry

In the end, finance is concerned with making definite numerical recommendations which frequently can only be made by analysing sophisticated models using high-speed computers. This course studies the design, implementation and use of computer programs to solve practical mathematical problems of relevance to finance, insurance and risk management. A review of MATLAB, floating point numbers, rounding error and computational complexity. A selection of topics from: approximation and parameter estimation, Fourier series and the FFT, finite difference approximations, partial differential equations (heat equation), sparse linear systems, non-linear algebraic equations, trees, Monte Carlo methods and simulation, random numbers and variance reduction, numerical integration. Computing environments for mathematical finance. Practical examples and programming assignments using MATLAB.

The following are descriptions of courses offered by the Department of Pure Mathematics at the third year ordinary (i.e. not higher) level.

In third year, the courses become more challenging, and begin to reveal what modern mathematics is "really about", from the view point both of the theory and of applications to other disciplines. The course descriptions given here are intended to show not only the content of the courses, but also to give you an idea of their history, their peculiarities and special concerns, and their place in the modern scene. We hope that you will find them sufficiently interesting and relevant to give you the desire to discover them for yourselves.

Well known examples of codes include morse, ASCII, ISBN book numbers, as well as the bar code used on grocery items. A code provides a way of converting a message from an arbitrary source alphabet into a form suitable for transmission or storage. This usually entails some binary procedure suitable for an electronic device. Coding theory is largely motivated by two basic problems:

The theoretical framework to answer the first question was established by Claude Shannon in 1948. At that time he introduced the notion of entropy as the measure of the amount of information in a source alphabet. Shannon showed that the average length of a symbol when coded is greater than the entropy and he was able to show that this average length could be made close to the value of the entropy in a certain sense. The code that generally achieves the aims of (1) is the Huffman code.

Almost all modern computers have memories that are built from silicon chips. Alas, a stored 0 or 1 in a memory chip can spontaneously switch to what it should not be! Such errors are usually caused by alpha particles released during radioactive decay. There is no known economically feasible way to shield a computer memory against alpha particles and as a consequence the mean time before failure for a one megabyte memory is only about 40 days. This physical fact highlights the importance of the second problem of coding theory, for error-correcting codes can cope with this failure. If the (64,57) Hamming code is designed into the architecture the mean time before failure becomes about 63 years. The Reed-Solomon codes used on compact discs are so powerful that holes up to 3mm diameter can be drilled through a disc with no loss of sound quality (or at least so it's said!).

A cipher is a secret code. The ancient Romans are said to have communicated secret messages by shaving a slave's head, inscribing the message on the scalp and then sending the slave to deliver the message after the hair had grown back. This is actually irrelevant to the course but at least adds substance to the claim that the Romans made no significant contribution to mathematics. Modern "field operatives" generally used "one-time pads" and such ciphers are essentially unbreakable provided they are used only once. In any case the course will expose the flaws of traditional ciphers and will introduce the RSA public key cryptosystem, the one which is the rage today.

This course follows on from the plane Euclidean geometry which you learn at high school, but it takes a different viewpoint, dealing with those geometric properties which survive projection, as first studied by Renaissance painters studying perspective. This leads to the idea of defining different types geometry in terms of the properties which remain unchanged by different types of transformations.

The purpose of the course is to give you a basic understanding of the geometry of a space via the study of certain natural transformations which act on the space. For example, in studying the geometry of the Euclidean plane, or Euclidean 3-space, we are naturally led to consider the transformations which arise from rigid motions because these are the mappings which preserve the basic notion of congruence.

This principle - that a geometry is defined by certain allowed transformations - was formulated by Felix Klein in his Erlanger programme of 1872. The set of transformations defining a geometry is an example of an algebraic structure called a group, so the study of different forms of geometry (and of symmetry) links naturally with the study of groups in general. In this way, this course provides a friendly and highly visual introduction to proofs and the more abstract mathematics of group theory.

As groups of transformations give expression to the symmetries of objects, the course will cover many fun and interesting applications to such diverse things as wallpaper designs, Escher's art, tesselations of the plane, Rubik's cube and much else.

This course should be particularly useful to students who intend to teach mathematics.

The study of the integers and their properties has intrigued mathematicians and novices since the ancient Greeks discovered the so called "perfect numbers" and looked for integer solutions to the Pythagorean equation x^2+y^2=z^2. The theory of numbers has advanced a long way since then and still contains many problems which are simply stated but as yet unsolved.

This course takes a fascinating journey through the key ideas of basic Number Theory but from a modern viewpoint, using ideas from Abstract Algebra (rings, groups and fields). Ideas and techniques in basic Group Theory are motivated by problems in Number Theory and then used to solve such problems. The constant interplay between these two areas of mathematics is emphasised. The course then concludes by looking at field extensions which are then used in a simple, yet spectacular way to solve the ancient Greek problems on which geometrical constructions are possible using only a rule and compass. This in turn leads back to some Number Theory.

This course would be of great interest and enjoyment to anyone majoring in mathematics and in particular to those planning to teach High School.

Not offered in 2007

This course begins with some differential geometry. We study curves and surfaces in R^3 and illustrate the theory with many geometrically important examples.

To study curves we define a certain frame field (i.e. a set of three orthonormal vectors) at each point on the curve. By expressing the derivatives of these vectors in terms of the vectors themselves, we get the famous Frenet formulae which naturally lead to the concepts of curvature and torsion. By considering congruence of curves, we see that curvature, torsion and speed determine the shape of a curve completely.

The shape operator, S, on a surface M is the rate of change of the unit normal to M. Thus the shape operator describes the shape of the surface. This operator S turns out to be a linear operator on each tangent plane to M; the invariants of S (such as its determinant, trace etc.) correspond to geometrical properties of the surface M. We obtain efficient methods for calculating these invariants and apply these methods to many geometrically interesting examples.

Early in the last century Gauss speculated about how much of the geometry of a surface is independent of its shape. We now consider this `intrinsic' geometry of a surface, that is, those properties of a surface which do not depend on its relationship to surrounding space.

This brings us to the celebrated Gauss-Bonnet theorem which relates curvature to topology and so leads into the second part of the course, which deals with the topology of surfaces. We study such surfaces as the sphere, cylinder, torus and Moebius strip and eventually we show that any surface can be obtained as a combination of these.

The Gauss-Bonnet Theorem implies that for each surface there is a single number, called the Euler characteristic, which determines many properties of the surface. We calculate Euler characteristics by means of triangulations, and apply this to the surfaces mentioned above and others such as the Klein bottle and the projective plane, which we construct by glueing edges of rectangles together.

The course finishes with a look at many of the famous theorems of elementary topology such as the Brouwer Fixed Point Theorem, the Hairy Ball Theorem and the Ham Sandwich Theorem.

This course is intended to be "fun", and is particularly recommended for those considering a teaching career.

Why were the ancient Greeks so obsessed with geometry? The answer is surprisingly deep and philosophical. A look at Greek mathematics is not what you might think - not a tedious tale of of things we learned in early high school but a venture into a view of mathematics which is radically different from our own and dominated mathematics for more than a thousand years - it was not until Descartes in the 17th century that mathematicians escaped from the restriction imposed by the Greek insistence on `homogeneity' in equations. While we may see some aspects of Greek mathematics as hang-ups, we marvel at their invention of the logical and axiomatic methods we use today and at the ingenuity of Archimedes in finding volumes and centres of gravity without the aid of multiple integrals.

Apart from the ancient Greeks, the other period that we study closely is the time when the ideas of calculus gradually emerged, were established on a sound footing by Newton and Leibnitz and quickly applied to a multitude of problems by such great mathematicians as the Bernoulli brothers, Euler, Lagrange and Laplace. We look at actual texts from the time and learn to recognize familiar ideas and techniques when they are expressed in different language or in a rudimentary form.

In a recent survey, our graduates told us how important it is for students to develop their skills in written and spoken communication. To make a contribution outside the university, you need to have communication skills which go beyond the writing of cryptic mathematical symbols. This course helps you to develop these skills in essay writing and making a presentation.

Calculus is one of humanity's great intellectual achievements, one which has led to precise and accurate modelling of many physical processes. But what makes it work? What does it really mean for a function to be continuous or differentiable, and how bizarre (or regular) can (or must) such a function be? Why is it that Taylor and Fourier Series exhibit quite different convergence phenomena?

This course focuses on basic questions such as these, and on the even more fundamental one of ``what is a real number?'' - a ``point on the real number line'' will not do -, and tries to demystify some of the recipes you have been using since high school. The key idea is that of a limit of a sequence of numbers, from which we eventually build differential and integral calculus.

We also generalise this basic idea to the idea of a limit of a sequence of functions, where there are several different possibilities (pointwise or uniform limits, for example).

It is important to understand such things if you are going to teach or seriously use mathematics, or if you are going to appreciate why computers (approximation machines!) are able to produce satisfactory answers to real-world problems.

Limits and continuity are the central concepts of calculus in one and several variables. These concepts can be extended to quite general situations. The simplest of these is when there is some way of measuring the distance between two objects. Some of the most important examples of these `metric spaces' occur as sets of functions, so this course looks at ways in which one might say that a sequence of functions converges. Taking these ideas one step further, we look at convergence which does not come from a generalized distance function. These are the ideas of point set topology. The course will include topics such as countability, continuity, uniform convergence, compactness and connectedness. This is not a `computational' course, but rather one in which you will develop your ability to think and write abstractly, precisely and creatively.

(MATH3611 replaces MATH3610 andMATH3620.)

This course introduces the mathematical areas of differential geometry and topology and how they are interrelated, and in particular studies various aspects of the differential geometry of surfaces. The approach to the latter taken is built around Cartan's approach, which leads more easily to modern differential geometry and also to its applications in theoretical physics.
The course's major theme is how certain natural questions of "sameness" can be systematically approached and answered, and how these answers can be used.

We begin with a review extention of basic topology, multivariable calculus and linear algebra. Then we study curves and how they bend and twist in space.
This will lead us to look at general ideas in the topology of curves, and the fundamental group. Then we look at one of the original themes of topology as developed by Poincare: vector fields.

Turing to differential geometry, we look at manifolds and structures on them, in particular tangent vectors and tensors. This leads to the idea of differential forms and the further topological idea of cohomology.

With these building blocks, we then consider surfaces, studying the classical fundamental forms introduced by Gauss, the various measures of curvature for surfaces and what they mean for the internal and external appearance and properties of surfaces. A closer look at the intrinsic geometry of surfaces leads to Gauss' famous "Remarkable Theorem" on curvature and provides the starting point that would lead to the fundamental uses of differential geometry in, for example, Einstein's general relativity. In relation to surfaces, we consider geodesics, the Gauss-Bonnet theorem and the Euler characteristic. These lead us to our last topic: consideration of non-Euclidean geometries, especially the hyperbolic plane.

(MATH3701 replaces MATH3700 and MATH3690.)

Handbook entry

In Higher Algebra, we will examine some of the basic notions of modern algebra that arose in the late 19th and early 20th century. The most fundamental notion is that of a group, which is how mathematicians study symmetry. These will be studied in detail both from an abstract point of view and also to study symmetry in 3-dimensional space. The other important concept studied is that of a ring. The algebra of adding and multiplying matrices has many similarities with the algebra of numbers. The notion of rings generalise both these two examples.

(MATH3711 replaces MATH3710 and MATH3720.)

The following course descriptions will give you an idea of what is taught in the third year Statistics courses within the School.

In this course we consider the time evolution of random phenomena such as stock prices, the size of animal populations in a rain forest, the movements of a gas particle or records of the water levels of a river. All these phenomena, while very different in nature, evolve randomly in time and they are very often modelled with so-called Markov chains. In a Markov chain model the future random evolution of the phenomenon in question is determined by its present state only so we do not need to know the whole history of the process.

The course will be devoted to the study of the general theory of Markov chains and more detailed analysis of examples which are particularly important in applications like Birth and Death Processes, Branching Processes, Binomial Processes and the Metropolis Algorithm.

Statistical inference is concerned with using data to answer substantive questions. Rigorous statistical analysis is a compulsory element of any information retrival process involving uncertainty. The theory of statistical inference forms the logical foundations of such statistical analysis, starting from the axioms of probability and evolving through statistical arguments. This course invloves a thorough general discussion of the most important and practically relevant statistical procedures: estimation, confidence set construction and hypothesis testing. Particular attention is paid to motivating and defending the optimality of the procedures. This optimality is shown through limiting results for large sample sizes (asymptotic optimality) and sometimes, when analytically possible, also for any fixed sample size.

It is important to note that optimality depends on the assumptions we make fora statistical model. Often in practice some of the common simplifying assumptions, such as the assumption of normality of the data, might be seriously violated. In such cases standard inference procedures must be modified. This is achieved by applying robust or nonparametric inference procedures instead of standard parametric procedures. Discussion of robustness and nonparametric modelling issues will constitute the second part of the course.

The course will help students to not just apply available statistical computer packages for analysing a given data set but also to understand why a particular algorithm amongst many available is most suitable and should be preferred in a particular situation.

The course is offered at two levels. MATH3811 is the ordinary level, while MATH3911 is the higher level, which offers a more detailed and refined treatment.

Modern statistics is increasingly concerned with the analysis of large and complicated data sets from applications in medicine, social and market research, environmental science and many other fields. Flexible statistical models are needed to analyse these large and complex data sets. Fortunately, with modern computing and statistical software packages it is possible to easily fit complicated statistical models and to visualize the resulting fitted models graphically.

This course gives an introduction to some modern, flexible statistical modelling techniques and introduces the use of various statistical software packages such as SAS and S-PLUS. Some of the methods we consider in this course are extensions of methods considered in the linear models course MATH2831. We consider regression models where the mean of a response variable (some variable to be predicted or explained in terms of related predictor variables) is described by some function of the predictors which is nonlinear in unknown parameters (nonlinear regression models). We also consider the class of generalised linear models, which are models which are commonly used in the analysis of non-normal data.

Modern highly flexible regression methods where the goal is estimation or approximation of a relationship between the mean of a response variable and predictor variables under only very general smoothness assumptions about that relationship will also be considered. We consider scatter plot smoothers, additive models and if time permits neural networks.

Additional topics covered include the estimation of density functions from random samples under only very general assumptions about the unknown density (that is, without assuming some rigid parametric form for the density such as normality) and Monte Carlo statistical methods (where we try to simulate randomness via deterministic rules on a computer). Many applications of Monte Carlo statistical methods in statistical inference will also be discussed.

Handbook entry

Development of statistical methods for design and analysis of data for social and market research. Review of research methodology. Sample survey design. Statistical aspects of survey design. Statistical aspects of questionnaire design and analysis. Estimation of means, totals, proportions and ratios. Estimation using auxiliary information. Methods for analysing cross-classified data, binary and ordinal responses, assessment, control and quantification of errors in survey research.

Measurements on different aspects of individual subjects are usually not independent and to successfully describe the relationships between the various measurements models for correlation or dependence are required. Similarly the successive observations on a time series, such as occur in financial application for example, will exhibit serial dependence and models which describe the serial dependence are useful for forecasting future values of the series. Spatially organised data, such as occur in environmental processes for example, similarly exhibit dependence between values observed at different sites.

The aim of this subject is to extend the student's understanding of statistical modelling, predominantly based upon independently distributed data, to these important practical examples where dependence is required in the models. The first half of the subject covers the multivariate normal distribution and the marginal and conditional distributions derived from it as well as various important properties concerning optimal prediction. The multivariate normal distribution is central to the practicising statistician's understanding of dependence between measurements within subjects, across time or space.

The second half of the subject builds on the basic properties of the multivariate normal distribution by applying the results to a series of examples drawn from time series and spatial processes.

Students who complete this course can expect to have obtained a good understanding of the importance of modelling dependence in observed data as well as an understanding of the basic distributions and models useful in a range of practical situations.

Handbook entry - Replaces MATH2810 and MATH3830

This course will focus on the principles of good experimental design and the statistical tools appropriate for discrete valued data. Topics include factorial designs and their analysis, response surface designs for product and process optimization, random effects models and components of variance, exploratory and graphical analysis of data using modern statistical packages, data visualization, analysis of cross-tabulated data, logistic and Poisson regression for analysis of binary and count data and log-linear models for contingency tables.