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Table of Contents
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.
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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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.
(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.
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.
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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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.
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.
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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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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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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:
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