Table of Contents


General Information

For general information about postgraduate study in the School of Mathematics at Leeds, see the School of Mathematics postgraduate degrees web page. The University Postgraduate Prospectus is also worth looking at; both include some information on funding.

More general information on postgraduate study in statistics in the UK can be found at the Committee of Professors of Statistics Postgraduate Statistics Opportunities web page.


Research Interests of Staff

Dr R.G. Aykroyd: Bayesian image analysis and inverse-data problems, EM/OSL algorithm, MCMC, archaeological geophysics and medical imaging.

Dr A.J. Baczkowski: geostatistics, spatial statistics, statistical ecology.

Dr S. Barber: wavelet methods in statistics, group sequential clinical trials, survival analysis.

Dr L.V. Bogachev: stochastic processes in random media, branching random walks, probabilistic methods in combinatorics.

Dr C.A. Gill: statistical ecology, diversity, estimating population size.

Prof W.R. Gilks: bioinformatics; computational statistics; Markov chain Monte Carlo methods; statistical genomics.

Dr A. Gusnanto: bioinformatics.

Prof. J.T. Kent: multivariate analysis, image analysis, shape analysis, robustness, spatial statistics, tomography, statistical inference, directional statistics.

Prof. K.V. Mardia: bioinformatics, image analysis, shape analysis, machine vision, statistical inference, multivariate analysis, directional statistics, geostatistics, spatial statistics.

Prof. C.C. Taylor: classification, image analysis, statistical pattern recognition, bioinformatics, nonparametric density estimation, spatial statistics.

Prof. A.Yu. Veretennikov: stochastic analysis, partial differential equations, applications to theoretical statistics.

Dr J Voss: MCMC methods, in particular in infinite dimensional spaces; numerical solution of SPDEs; probability distributions on the torus; large deviations for diffusion processes

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Current Research Themes

A list of specific research projects suggested by potential supervisors is available. In order to put these projects in context with the wide range of the research in the Department we have grouped the main themes of our research as follows. Of course, the list of projects is not exhaustive and many other projects are possible. If you have a particular topic or research area in mind, we would be pleased to discuss other possibilities.

Image And Shape Analysis, Spatial Statistics And Spatial-temporal Modelling

This general area of research deals with statistical problems in a geometric or spatial context.

Image Analysis

[See Project 3 ; Project 4; Project 6; Project 8; Project 12 ]

There are many opportunities for research in the statistical aspects of imaging. Many of the applications are in medical imaging, although we also investigate imaging problems in agriculture, genetics, industry, biology, archaeology, etc.

Shape Analysis

[See Project 4; Project 25; Project 27 ]

Shape Analysis is concerned with the geometric information that is invariant under changes in location, rotation and scale of an object. The aim of Shape Analysis is to develop methods to describe shape and to summarise succinctly variability and evolution of shape. Some particular applications include testing for shape differences in parts of the brain between healthy and diseased people and using images of hands to identify different individuals for security purposes.

Shape Analysis involves the development of specialised multivariate statistical techniques. When the information on shape is given in terms of a set of known key landmark positions, the methodology is now fairly well understood. However, when studying the shapes of surfaces, there are still many unresolved issues including the identification of landmarks and the description of other aspects of surfaces such as curvature.

There are numerous interesting theoretical problems still to be examined, often motivated by important applications in image analysis, medicine, biology, and other fields. Further study of shape distributions and the study of shape change through time are all topics of current work.

A laser scan of the face
surface Kriging to fit a face using
landmarks Curvature map of the human
face

(Left) A laser scan of the surface of the face of a patient with a cleft palate. (Centre) The method of kriging is related to spline fitting. It can be used to fit the surface of a face using a small number of landmarks. (Right)

Statistical Image Averaging And Warping

Image warping or deformation involves the distortion of an image with the goal of superimposing two or more images on top of one another. Sometimes the deformations are of interest in their own right to investigate differences in shape between objects. In other cases the deformation is treated as a nuisance feature; once a set of images is deformed to a common registration, the objective might be to study finer scale differences.

For example, Galton, the famous statistician who invented regression, averaged photographs of faces in an early attempt to define an average face. More recently, there has been work on averaging laser scans of faces by psychologists to study differences between males and females. Suitable statistical models to allow the understanding of variability about any proposed average need much further development.

Tagged MR images of the heart
beating Tagged MR images of the heart
beating Tagged MR images of the heart
beating

Tracking of quadrilaterals (quads) on a tagged MR image in the middle of the heartbeat cycle. The middle image shows the potential quads generated by a quad detector on the left image. The darker the potential quad, the higher the probability. The right image shows the result of the tracking, which is part of a sequence of images.

Spatial Analysis

[See Project 3; Project 17 ]

Spatial analysis is concerned with the development of statistical methodology for observations located in two- or three-dimensional space. Applications are widespread, ranging from ore deposits in mining to pattern recognition. Such data typically have two features: the similarity of nearby observations to one another (autocorrelation), and long-range trends across the region of study. The spatial linear model provides a powerful tool to model such behaviour.

An important extension of purely spatial analysis is spatialtemporal modelling, for example to monitor a cloud of contaminated material as it moves across the countryside. Investigation of new models will be useful in monitoring and prediction of pollution.

Image Reconstruction And Inverse Problems

Tomographic techniques and inverse problems in general are concerned with reconstructing an image of the interior of an object from non-invasive measurements taken from the outside of the object. Such problems are mathematically illposed because the solution does not depend smoothly on the data; hence noise in the measurements creates instability in the solution.

There are applications to many areas, including archaeology (to investigate archaeological remains hidden underground), industrial processes (for on-line monitoring and control), and medical investigations (for diagnosis and treatment).

Development of statistical methods for reconstruction is needed to minimise artefacts, such as blurring, masking, shadowing and distortions, leading to improved visualisation and allowing direct estimation of control and diagnostic parameters. Dynamic monitoring requires temporal modelling such as flow-fields for diffusion and heart-lung rhythms.

Related work on solving inverse problems through deterministic methods is being tackled by the Department of Applied Mathematics.

Statistical Image Techniques for
tumor location

A display tool for combining information from an X-ray (far left) and a nuclear medicine image (far right). Thin-plate spline transformations are used to register the images and perhaps the most useful image is the second from the left which displays the highest nuclear medicine grey levels on top of the X-ray image. These bright areas could indicate possible locations of tumours or other abnormalities. The precise location is easier to see in this new image compared to the pure nuclear medicine image.

Modern Data Analysis

Modern computing technology allows us to apply many statistical techniques that would have been impossible ten years ago. However, in tandem with these advances, modern scientific developments have brought forward a host of new challenges for statistics. Often these problems can only be solved by harnessing modern techniques for data analysis.

Robustness

[See Project 10]

Robustness is a key concept in modern statistical modelling. The motivation is that statistical inferences should not be dramatically affected by a few outlying observations in the data.

One of the classic problems of robustness theory involves the simultaneous estimation of location and scatter from a set of multivariate data. More work is needed to better understand questions of influence, uniqueness, and breakdown. Classic estimators used in multivariate robustness are the M-estimators. These have good local robustness properties but have poor breakdown in higher dimensions.

A new class of estimators developed at Leeds called constrained M-estimators overcomes these theoretical problems at the price of being more difficult to compute in practice. More work is needed to understand and fine-tune the behaviour of these estimators. The use of constrained M-estimators in regression analysis is also a topic of current research.

Bioinformatics

[See Project 19; Project 20; Project 21; Project 22; Project 23; Project 24; Project 25; Project 26; Project 27 ]

Bioinformatics deals with use of quantitative methods involving computers to handle biological information, aiming to characterise the molecular components of living organisms. The human genome project is surely the greatest recent achievement of bioinformatics. However, much more work is needed concerned with the technology of genome databases, and there are still many problems of an essentially statistical nature that remain unresolved.

A key problem in bioinformatics is to provide efficient search facilities which, given a query molecule, will find other similar molecules within a database. This is important, for example, to trace the 'family trees' of different proteins sharing aspects of biological function or structure through evolution. The need for accurate statistics will become even more important in coming years as the genomic projects begin to augment the current rapid growth in the database of known protein structures. The research in this exciting area will develop research in this exciting area will develop structures at several levels from overall fold to functional site structure and shape.

Pattern Recognition And Machine Learning

[See Project 6; Project 7 ]

Recent advances in technology, in particular, falling costs of high capacity storage media, have greatly increased the acquisition and archiving of large amounts of data. Consequently, Data Mining is a rapidly growing theme amongst researchers, and is a potential source of useful information for industry and commerce.

Classification or discrimination involves learning a rule whereby a new observation can be classified into a predefined class. Current approaches can be grouped into three historical strands: statistical, machine learning and neural networks. The classical statistical methods are based on distributional assumptions. There are many others which are distribution free, and which require some regularisation so that the rule performs well on unseen data.

Recent methods in machine learning have focused on ways to improve simple rules by judicious resampling (different versions of) the data, and then combining the results in a committee-like voting procedure. Statistical methods indicate why this can be effective, but many important issues remain - for example in the choice of smoothing parameters and selection of appropriate variables.

Statistical Computation

[See Project 11]

In complex statistical models, it is often impossible to write down the probability density in a closed form amenable to parameter estimation. Two important methodologies have been developed over the past 20 years to estimate the parameters. The EM (Expectation-Maximisation) algorithm is a tool for calculating the maximum likelihood estimate or the posterior mode for problems with incomplete data. Another, more recent, approach is the Markov Chain Monte Carlo (MCMC) technique. This method involves the simulation of a suitable Markov chain whose equilibrium distribution is the desired posterior distribution.

For both of these algorithms there is the question of speed of convergence. Rather surprisingly, it turns out that the two methodologies are closely linked from the point of view of speeding the convergence. However, further research is needed to study these questions in more detail.

Wavelets

[See Project 15; Project 16; Project 17; Project 23 ]

Wavelets form the basis of a new modern methodology for the removal of noise from signals in one or more dimensions. To some extent they are analogous to the use of Fourier series, but they are often more sensitive and powerful when the signal consists of sporadic information. Current interests include both theoretical properties (such as the concept of complex wavelets and distributional behaviour) and practical applications (such as modelling non-stationary time series, solving inverse problems, and analysing long range dependence and self-similar behaviour). Applications range from engineering to environmental science to bioinformatics.

Probability And Stochastic Processes

This research group has interests in the areas of theoretical and applied probability and stochastic analysis, such as limit theorems, Markov chains, diffusion processes, branching random walks, random media, and stochastic differential equations.

Parameter Estimation for Markov Processes

Markov random processes, characterised by the 'lack of memory' property, provide an efficient mathematical tool to model dependence and interaction in various real-life systems. Diffusions are the most general class of Markov processes having continuous sample paths. This class is particularly suitable for modelling because it can be described by using two local characteristics, the expected velocity (drift) and the variance of random motion. In statistical work, one needs to estimate these parameters from observations of the process.

The problem of asymptotically efficient estimation for parameters not changing in time in general Markov systems still needs investigation. Even more challenging is the problem of asymptotically optimal tracking of parameters which may change slowly through time.

Stochastic Financial Models

In recent years, it has been realised that financial time series, like option prices or currency exchange rates, possess 'long-range memory'. Hence, modelling long range temporal dependence is a topical subject in modern stochastic finance. All existing models involve complex non- Markovian stochastic processes leading to considerable difficulties in numerical computations. We propose new models based on Markov processes, which not only mimic the long-range dependence but also reproduce some other desirable features of stock market prices.

Stochastic Differential Equations

[See Project 18]

The standard way to describe diffusions is via stochastic differential equations. It is important to investigate the robustness of numerical methods for these equations perturbed by additional 'noise' and/or by discretisation. A key question is whether the long-term behaviour of the approximating model is close to that of the original process. Answering such questions is essential for the successful application of numerical methods to stochastic systems.

Filtering Theory deals with estimation of one component of a process given observations of another component. Equations arising here combine features of deterministic partial differential equations and stochastic differential equations. It is of theoretical and practical interest to explore the possibility of a direct derivation of filtering equations, thus avoiding the use of complex abstract constructions. Recently, a new approach has been introduced, allowing such analysis for certain simple models; however, more general cases await consideration. This approach will also provide a new efficient tool for modelling and numerical analysis of solutions. Other problems in stochastic analysis are being tackled by the Department of Applied Mathematics.

Extinction And Propagation In Catalytic Media

[See Project 13]

Suppose that a particle is randomly moving in space, and at certain points ('catalysts') it may give birth to random offspring which then evolve in a similar manner. Such stochastic processes are known as branching random walks in a catalytic environment and may be used to model various reactions in chemical kinetics. Research in this area aims to study the asymptotic properties of the particle population, in particular its propagation and extinction.

Intermittency In Random Media

[See Project 14]

In many situations, random processes evolve in an environment that is random in its own right. For example, propagation of heat in an ocean will depend on fluctuating characteristics such as water temperature, salinity, direction and strength of wind etc. Such processes are studied in a modern exciting branch of applied probability - the theory of random media.

Classical mathematical physics proceeds from the assumption that random fluctuations can be averaged out to yield effective macroscopic characteristics of the medium, like temperature, diffusion coefficient, electric conductance etc. However, over the past three decades or so, both experimental and theoretical results have revealed many anomalous effects that cannot be explained by the classical approach. The reason is that untypical fluctuations (large deviations) of the environment may appear predominant in the long run. As a result, the spatial-temporal structure of the system will become intermittent, characterised by the development of rare high peaks on a relatively low-profile background. Real-life examples of intermittency are widespread, ranging from distribution of matter in the Universe to demographic statistics. More work is needed to better understand the intermittency phenomenon, in particular in the case of non-stationary media.

Probabilistic Methods In Number Theory And Combinatorics

Recently, a beautiful probabilistic method based on ideas from statistical physics has been proposed to handle certain topics in pure mathematics. Examples include the statistics of partitions of large natural numbers (a topic dating back to the celebrated work by Hardy and Ramanujan) and the limit shape of convex lattice polygons.

Once incorporated into a probability framework, this area presents a variety of challenging questions of statistical nature in its own right. On the other hand, the method also provides an efficient tool to tackle some classical and modern problems in number theory.

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Supervision and Training

A student works closely with his or her supervisor on their topic of current research. The supervisor guides the student through the work and the usual arrangement involves a regular weekly meeting. In some projects there are two supervisors from the Department or possibly one supervisor from another department. The supervisor carefully looks after the student's progress and at the end of each year a review meeting is held with an internal assessor (another member of the Department).

As well as UK students we make overseas students very welcome in the Department. For example recent postgraduates who have studied with us have come from several countries around the world. Students whose native language is not English can attend pre-sessional English courses if they need to improve their language skills.

In their first year of study students may be recommended to take one or more of our large range of undergraduate modules. Students are required to attend the Departmental seminar programme, as well as local Royal Statistical Society meetings and the LASR Workshop.

At a tea break in the Mathematics Common Room

Specialist statistical training:

Students may, in consultation with their supervisors, opt to attend any of our extensive range of final-year modules on a wide range of topics in statistics. The choice is made both on the grounds of relevance to your research project and your personal interests. In addition, many of our research students attend a selection of the four one-week residential courses provided by the Academy for PhD Training in Statistics

Conference Opportunities:

Research students are encouraged to attend and participate in conferences which are relevant to their research area. A new student may attend a conference to gather expertise and ideas, while students who are well into their research may be able to present a talk or a poster describing their findings. Some recent examples of conference visits by research students include trips to San Francisco, Prague, Budapest and Italy. In addition there is a regular series of conferences aimed specifically at postgraduate UK researchers: EPSRC Graduate School, annually; Postgraduate Conference in Probability and Statistics, annually; Young Statisticians Meeting, annually.

The Leeds Annual Statistics Research Workshop (LASR) is a well established international event. It began as an internal meeting in 1975, but now enjoys the support of a number of visitors each year from other universities and research bodies from home and abroad. The workshop is an important departmental event and research students are encouraged to participate and to present a poster about their research, where appropriate. These workshops usually extend over three days and are usually conducted by some invited speakers who are experts in their fields of interest. Over recent years, the theme of the workshop has reflected the growing, but not exclusive, departmental interest in image analysis and shape analysis. Research students, particularly those working in a closely related field, have found these workshops very informative, as well as providing the opportunity to meet other researchers with similar interests.

Recent workshops:

2004: Bioinformatics, Images and Wavelets
2005: Quantitative Biology, Shape Analysis, and Wavelets
2006: Interdisciplinary Statistics and Bioinformatics
2007: Systems Biology & Statistical Bioinformatics
2008: The Art and Science of Statistical Bioinformatics
2009: Statistical Tools for Challenges in Bioinformatics

Statistical Consultancy:

The consultancy unit, Leeds University Statistical Services (LUSS), was formally established in October 1987 and has grown steadily since then. LUSS has been successful in initiating a variety of collaborative research projects with other departments in the University. This kind of joint research provides academic benefits such as new ideas for research and joint research publications. Consultancy work covers a very wide range both in terms of the expertise required and in the amount of work involved. There have been numerous projects in the medical field, involving work at the design stage as well as statistical analysis of data.

It is current practice to encourage our research students to take part in some consultancy; it is an important part of general statistical training and helps to develop other related skills.

Other Academic Skills:

Apart from getting a PhD there are opportunities for research students to gain some teaching experience. This normally involves taking tutorials, marking of coursework and assisting with computing practicals. Although it is not normal for research students to give lectures to our undergraduate students, they do gain experience of presenting their work through the regular Postgraduate Student seminar series. In addition, active participation in the statistical community is encouraged through the Royal Statistical Society -- in particular the local group meetings.

At a Sun workstation in The Ed Redfern Room

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Current and Recent Research Students

Current students:

  • Fatimah Al Ashwali. Supervisor J.T. Kent.
  • Suaad Ben-Farag. Supervisor C.C. Taylor.
  • Philippa Burdett. Statistical models for hydrogen bonding and compactness of proteins. Supervisors S. Barber and K.V. Mardia.
  • Chris Fallaize.Shape analysis in bioinformatics. Supervisors S. Barber, K.V. Mardia and R.M. Jackson.
  • Asaad Ganeiber. Supervisor J.T. Kent.
  • Kersin Hommola.Bioinformatics. Supervisors W.R. Gilks and K.V. Mardia.
  • Monique Inguanez. Supervisor J.T. Kent.
  • Jennifer Klapper. Multiscale analysis of mass spectroscopy data Supervisor S. Barber.
  • Sam Peck. A wavelet-lifting approach to crop monitoring. Supervisors R.G. Aykroyd and S. Barber.
  • Orathai Polsen. Supervisor C.C. Taylor.
  • Janos Szabo. Supervisor L.V. Bogachev.
  • Zhengzheng Zhang.Bioinformatics. Supervisors K.V. Mardia and W.R.Gilks and C.C. Taylor.

Recent Students:

We have a separate list of recently completed PhD students, including in some cases the text of the students' theses.

Destinations: To give a flavour of our past students, here is a list of some recent destinations.

  • Statisticians in Pharmaceutical Industry
  • Statisticians in City Banking
  • Research Fellows (in University and Hospital)
  • Actuarial profession
  • Statistician and software developer
  • Computer programmer
  • University Lecturer

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The City of Leeds and Yorkshire

Leeds is a vibrant and forward-looking European city with some 750,000 residents. It is a very pleasant city to live in, with excellent facilities, plenty to do and very friendly inhabitants. The city is booming - it is England's fastest growing city and the major financial centre of the North of England. The shopping in the centre is superb - major retailers, the markets and designer stores are all very conveniently situated in and around several Victorian arcades.

The city is home to many superb leisure attractions, including the highly respected Opera North, West Yorkshire Playhouse, Art Gallery, and various major musical events. Leeds is also renowned for its sporting connections - Leeds United Football Club, Test and County cricket at Headingley and top class Rugby League are major attractions. The city has many green areas and parks including Roundhay Park, Woodhouse Moor (next to the University) and the 11th century Kirkstall Abbey.

Leeds is a thriving multi-cultural city. Its cultural and ethnic richness is reflected in community groups and varied facilities that exist for social and religious purposes. The array of restaurants and pubs is diverse, with many situated close to the University. The cost of entertainment and leisure in Leeds is low.

Leeds is very close to some of England's most spectacular countryside and National Parks. The Yorkshire Dales, Pennines, Lake District, North York Moors and Yorkshire Coast are all beautiful and well worth visiting. There are plenty of historical attractions nearby, including several Abbeys and stately homes. Medieval York and small market towns such as Otley, Haworth (home of the Brontes), and Wetherby make pleasant day trips and the spa towns of Ilkley and Harrogate offer a pleasant relaxing air.

Leeds is very conveniently situated in West Yorkshire in the geographical centre of Britain, 320 km north of London (about 2 hours by train) and 320 km miles south of Edinburgh (about 3 hours by train). There are excellent bus and coach connections on motorways to the major British cities. Leeds/Bradford International Airport offers convenient direct links to many countries and the whole world is within easy reach when flying from Leeds via Amsterdam or London.

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Who to contact

We are sure that our expertise and facilities will give you the perfect atmosphere for studying for a research degree. The research topics that are offered provide very interesting and stimulating projects in Statistics. Our staff are very happy to discuss any questions that you have about our programme, so please don't hesitate to contact us.

We look forward to hearing from you!

An application form and brochures can be downloaded electronically from here:

Application form

Further PG Study information.

To obtain hard copies of an application form and the brochures, please write to:

Dr Leonid Bogachev
Postgraduate Research Tutor
Department of Statistics
University of Leeds
Leeds LS2 9JT
United Kingdom
Tel: +44 (0)113 343 4972, Fax: +44 (0)113 343 5090
E-mail: bogachev@maths.leeds.ac.uk

Alternatively, please fill in and submit the following form:

Title (Mr, Mrs, Miss, Ms etc.):

Name(s):

Postal address:

E-mail address:

Background/qualification(s)/degree(s):

Preferred area of research (optional):

Special query, request or message (optional):

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