Network Optimization

Mathematics & Statistics Available PhD projects

Dr Robin Cook

Evaluation of Maximum Sustainable Yield (MSY) in an ecosystem context

MSY has become the standard basis on which fishery management reference points, such a Fishing mortality rate (F) and Spawning Stock Biomass(SSB), are set.

In almost all stocks these reference points are estimated using single species steady state models that assume invariance of life history parameters such as growth reproduction and natural mortality. There are good reasons to suppose that these life history parameters are not constant and it would be expected that natural mortality would change as a result of species interactions. Thus while it might be possible to fish at a value of F based in single species models, there is not good reason to believe that by so doing, SSB and catch will be achieved. Despite this major difficulty many analyses of the status of global fish stock are based on the assumption that single species MSY is observable and has been used to estimate foregone catch and estimate “safe” limits for SSB.

The purpose of this project will be to use current ecosystem models to assess the implications for both realised SSB and catch when stocks in a multispecies complex are all fished at single species Fmsy. The project would also examine methods that have been used to classify the exploitation status of stocks based on catch data alone when it has been assume that single species MSY is observable. Based on this analysis alternative reference points will be proposed that take into account species interactions and hence provide a more robust basis for fishery management.

Development of Bayesian statistical methods for the assessment of data poor fish stocks

MSY has become the standard basis on which fishery management reference points, such a Fishing mortality rate (F) and Spawning Stock Biomass(SSB), are set.

In almost all stocks these reference points are estimated using single species steady state models that assume invariance of life history parameters such as growth reproduction and natural mortality. There are good reasons to suppose that these life history parameters are not constant and it would be expected that natural mortality would change as a result of species interactions. Thus while it might be possible to fish at a value of F based in single species models, there is not good reason to believe that by so doing, SSB and catch will be achieved.

Despite this major difficulty many analyses of the status of global fish stock are based on the assumption that single species MSY is observable and has been used to estimate foregone catch and estimate “safe” limits for SSB. The purpose of this project will be to use current ecosystem models to assess the implications for both realised SSB and catch when stocks in a multispecies complex are all fished at single species Fmsy.

The project would also examine methods that have been used to classify the exploitation status of stocks based on catch data alone when it has been assume that single species MSY is observable. Based on this analysis alternative reference points will be proposed that take into account species interactions and hence provide a more robust basis for fishery management.

Dr Penny Davies

Tissue modelling and simulation

Modern investigative and surgical techniques (such as ultrasound imaging or keyhole surgery) require increasingly sophisticated models of human tissue. Nonlinear elasticity seems to provide a good framework for modelling, and the aim is to develop models for particular tissue types and add in other physically realistic behaviour, such as viscoelasticity.

Once good mathematical models have been derived and verified, the object is to use them to compute the tissue's behaviour. The aim here is to develop efficient numerical methods since there are no "exact" solutions in general.

Time-dependent scattering problems

Time dependent wave propagation and scattering is important in acoustics, electromagnetics and seismology. These problems involve transient wave fields (often short pulses) and give rise to systems of hyperbolic PDEs, which can be approximated directly or first reformulated as boundary integral equations (BIEs) posed on the surface of the scatterer.

The BIE approach is computationally attractive because it only requires quantities to be approximated on a two-dimensional surface, but standard methods are complicated to implement and/or have numerical stability problems which need to be overcome. Recently progress has been made by using time-stepping approximations based on convolution quadrature or other methods which share its "backwards-in-time" framework, but several interesting open problems remain, such as coupling these time-stepping methods to spatial approximations based on B-splines.

The overall aim is to develop reliable and efficient approximation schemes.

Professor Ernesto Estrada

Repulsion-Attraction Model for Network Transitivity

The study of complex networks represents an important area of multidisciplinary research involving physics, mathematics, chemistry, biology, social sciences, and information sciences, among others

[1]. A characteristic feature of these networks is that the number of transitive relationships is significantly larger than the expected one from a random evolution of the system. The transitivity is measured by the ration of the number of triangles to the maximum possible number of triangles in the graph [1]. In this Thesis we are interested in developing a mathematical model to predict the creation of triagles in complex networks.

The model is based on the concept of network communicability [2] and communicability distances [3, 4]. The communicability can be seen as a Green’s function for a network of coupled harmonic oscillators [2], which represents the attraction between pairs of nodes in a network. We will extend this idea to create a new communicability function which represents the repulsion between pairs of nodes separated by two links in a network.

By combining the attractive and repulsive communicability we are going to develop a model that account for the probability that two nodes separated by two links can form a triangle in an evolutionary process. The aims of this Thesis are: (i) to create a mathematical model to predict triangle closure in complex networks; (ii) by using a modified Schrödinger equation to study the dynamics of the propagation of repulsive effects in networks; (iii) to implement computationally the model in order to analyse some real-world networks in different scenarios, e.g. biological, social, ecological and technological networks.

[1] Estrada, E., The Structure of Complex Networks. Oxford University Press. 480 pages 305 b/w line illustrations and halftones | 246x189mm 978-0-19-959175-6 | Hardback | October 2011.

[2] Estrada, E., Hatano, N., Benzi, M., The physics of communicability in complex networks. Physics Reports, 514 2012, 89-119

[3] Estrada, E., The communicability distance in graphs. Linear Algebra and its Applications, 436 2012, 4317-4328.

[4] Estrada, E., Complex networks in the Euclidean space of communicability distances. Physical Review E, 85 2012, 066122.

Dr Alison Gray

Digital image analysis for the classification of tea

Classification of teas (types, quality grades, region of origin etc) has been examined by many researchers, using both chemical composition and more recently digital image analysis techniques to extract features from the image that are useful for classification.

Factors affecting the success of classification include the choice of features, the classifier and the imaging modality. Building on previous work at Strathclyde in collaboration with the EEE department, this project will allow the student to examine the choice of any of these to achieve optimal results. Intending students should have a strong statistical background and excellent computer skills, and be competent/be able to quickly become competent in the use of both R and Matlab. 

Honey bee colony losses and associated risk factors

Second supervisors: George Gettinby, Magnus Peterson

Worldwide losses of honey bee colonies have attracted considerable media attention in recent years and a huge amount of research. Researchers at Strathclyde have experience since 2006 of carrying out a series of surveys of beekeepers in Scotland (http://personal.strath.ac.uk/a.j.gray/) and now have 5 years of data arising from these surveys. These data have been used to estimate colony loss rates in Scotland and to provide a picture of beekeepers’ experience and management practices.

This project will examine the data in more detail than has been done so far, and is likely to involve data modelling and multivariate methods to identify risk factors. Part of the project will involve establishing the spatial distribution of various bee diseases.

This project will build on links with the Scottish Beekeepers’ Association and membership of COLOSS, a network linking honey bee researchers in Europe and beyond. Intending students should have a strong statistical background and excellent computer skills, and be competent/be able to quickly become competent in the use of R.

Dr David Greenhalgh

Stochastic Models in Epidemiology and Population Dynamics

Introduction
In this project we shall look at how stochastic models can be used to describe how infectious diseases spread. We will start off by looking at one of the simplest epidemic models, the SIS (susceptible-infected-susceptible) model. In this model a typical individual starts off susceptible, at some stage catches the disease and after a short infectious period becomes susceptible again.

These models are used for diseases such as pneumococcus amongst children and sexually transmitted diseases such as gonorrhea amongst adults (Bailey, 1975). Previous work has already looked at introducing stochastic noise into this model via the disease transmission term (Gray et al. 2011). This is called environmental stochasticity which means introducing the random effects of the environment into how the disease spreads. This results in a stochastic differential equation (SDE) model which we have analysed. We have derived an expression for a key epidemiological parameter, the basic reproduction number.

In the deterministic model this is defined as the expected number of secondary cases caused by a single newly-infected individual entering the disease-free population at equilibrium. The basic reproduction number is different in the stochastic model than the deterministic one, but in both cases it determines whether the disease dies out or persists. In the stochastic SIS SDE model we have shown the existence of a stationary distribution and that the disease will persist if the basic reproduction number exceeds one and die out if it is less than one.

SDE Models with Environmental Stochasticity
We have also looked at other SDE models for environmental stochasticity. One of these took a simple deterministic model for the effect of condom use on the spread of HIV amongst a homosexual population and introduced environmental stochasticity into the disease transmission term. Again we found that a key parameter was the basic reproduction number which determined the behaviour of the system.

As before this was different in the deterministic model than the stochastic one. Indeed it was possible for stochastic noise to stabilise the system and cause an epidemic which would have taken off in the deterministic model to die out in the stochastic model (Dalal et al., 2007). Similar effects were observed in a model for the internal viral dynamics of HIV within an HIV-infected individual (Dalal et al., 2008).

Demographic Stochasticity
The real world is stochastic, not deterministic, and it is difficult to predict with certainty what will happen. Another way to introduce stochasticity into epidemic models is demographic stochasticity. If we take the simple homogeneously mixing SIS epidemic model with births and deaths in the population we can derive a stochastic model to describe this situation by defining p(i, j, t) to be the probability that at time t there are exactly i susceptible and j infected individuals and deriving the differential equations satisfied by these probabilities.

Then we shall look at how stochastic differential equations can be used to approximate the above set of equations for p(i, j, t). This is called demographic stochasticity and arises from the fact that we are trying to approximate a deterministic process by a stochastic one (Allen, 2007). Although the reasons for demographic and environmental stochasticity are quite different the SDEs which describe the progress of the disease are similar. The first project which we shall look at is analysis of the SIS epidemic model with demographic stochasticity along the lines of our analysis of the SIS epidemic model with environmental stochasticity.

Further Work
After this we intend to look at other classical epidemiological models, in particular the SIR (susceptible-infected-removed) model in which an individual starts off susceptible, at some stage he or she catches the disease and after a short infectious period he or she becomes permanently immune. These models are used for common childhood diseases such as measles, mumps and rubella (Anderson and May, 1991). We would look at introducing both environmental and demographic stochasticity into this model.

Other epidemiological models which could be analysed include the SIRS (susceptible-infected-removed-susceptible) epidemic model, which is similar to the SIR epidemic model, except that immunity is not permanent, the SEIS (susceptible-exposed-infected-susceptible) model which is similar to the SIS model, but includes an exposed or latent class, and the SEIR (susceptible-exposedinfected- removed) model, which similarly extends the SIR model.

We would also aim to look at other population dynamic models such as the Lotka-Volterra predator-prey model. There is also the possibility of developing methods for parameter estimation in all of these epidemiological and population dynamic models, and we have started work on this with another Ph. D. student (J. Pan).

References
1. E. Allen, Modelling with Itˆo Stochastic Differential Equations, Springer-Verlag, 2007.
2. R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.
3. N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Second Edition, Griffin, 1975.
4. A.J. Gray, D. Greenhalgh, L. Hu, X.
5. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for AIDS and condom-use. J. Math. Anal. Appl. 325, 36-53, 2007.
6. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics. J. Math. Anal. Appl. 341, 1084-1101, 2008.

Mathematical Modelling of Disease Awareness Programs

Differential equation models are commonly used to model infectious diseases. The population is divided up into compartments and the flow of individuals through the various classes such as susceptible, infected and removed is modelled using a set of ordinary differential equations (Anderson and May, 1991, Bailey, 1975). A basic epidemiological parameter is the basic reproduction number. This is defined as the expected number of secondary cases produced by a single newly infected case entering a disease-free population at equilibrium (Diekman and Heesterbeek). Typically the disease takes off if R0 > 1 and dies out if R0≤1.

However media awareness campaigns are often used to influence behaviour and if successful can alter the behaviour of the population. This is an area which has not been studied much until recently. The student would survey the existing literature on media awareness models in the literature and with the supervisor formulate mathematical models using differential equations for the effect of behavioural change on disease incidence. These would be examined using both analytical methods and computer simulation with parameters drawn from real data where appropriate. The mathematical techniques used would be differential equations, equilibrium and stability analyses and computer simulation.

References
1. R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.
2. N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Second Edition, Griffin, 1975.
3. O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and computation of the basic reproduction number R0 for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365-382.
4. A. K. Misra, A. Sharma and J. B. Shukla, Modelling and analysis of effects of awareness programs by media on the spread of infectious diseases. Math. Comp. Modelling 53, 1221- 1228.
5. A. K. Misra, A. Sharma, V. Singh, Effect of awareness programs in controlling the prevalence of an epidemic with time delay, J. Biol. Systems, 19(2), 389-402,

Nonintrusive management of complex systems

This studentship will be of 3 years duration with stipend and fees for a UK/EU student, funded by the University (funding application submitted outcome pending). This project is part of a collaboration with the Department of Mechanical and Aerospace Engineering.

Our immune system, the movement of people in congested environments, and the economy, are all examples of complex systems that are key to our well-being. These systems normally operate in a predictable way, that is, when the immune system is healthy, crowds move homogeneously, and economic business cycles oscillate steadily about an increasing mean.

However, catastrophe in these systems can occur, in these cases, through disease, crowd panic and economic shocks such as acts of terrorism, that lead to undesirable and unpredictable behaviour. This unpredictable behaviour is seemingly impossible to manage effectively.

For example government bodies may introduce constrictive legislation to regulate the decision maker but this is often at the expense of efficiency and increased bureaucracy. This studentship will investigate a mathematically informed method to introduce a robustness in these systems that will not require aggressive short term fixes but uses continuous non-intrusive management instead.

Moreover, current strategies and control methods fail to cope with the complexities of these systems and there is an opportunity here to test the possibility of a new management philosophy through fundamental research.

Our methodology will be informed by a technique used in physics to control chaos in low-dimensional systems of ordinary differential equations (ODEs). This technique is coined time-delayed auto-synchronization (T-DAS) [1,2]. Analytical and numerical techniques will be used to investigate models in the areas of disease management, economic management and crowd control.

References
1. Pyragas, K., Continuous control of chaos by self-controlling feedback, Physics Letters A, vol. 170, pp. 421-428, 1992.
2. Scholl, E. and Schuster, H.G. (Editors): Handbook of Chaos Control, Wiley-VCH, Weinheim, 2008.

Dr Michael Grinfeld

Dynamics of reaction-diffusion systems with non-Fickian diffusion

In recent years there has been much work on reaction-diffusion equations in which the diffusion mechanism is not the usual Fickian one. Examples are integro-differential equations, porous media type equations, pseudodifferential equations, p-Laplacian type equations and prescribed curvature type (saturating flux) equations.

The motivation for this work comes from material science and mathematical ecology. However, there are applied contexts where these diffusion mechanisms have never been considered. One is in the area of combustion and the other is in the area of regularised conservation laws and shock propagation. This project, which would build on the work I did through the years on integrodifferential models and recently with M. Burns on the prescribed curvature equations, will use PDE, asymptotic, and topological methods to explore the dynamics of blowup and of shock propagation in canonical examples of reaction equations and nonlinear scalar conservation laws regularised by non-Fickian diffusion terms.

References:

[1] M. Burns and M. Grinfeld, Steady state solutions of a bistable quasilinear equation with saturating flux, European J. Appl. Math. 22 (2011), 317-331.

[2] M. Burns and M. Grinfeld, Steady state solutions of a bistable quasilinear equation with saturating flux, European J. Appl. Math. 22 (2011), 317-331.

Coarsening in integro-differential models of materials science

Recently, a new class of model has been developed to describe, for example, phase separation in materials such as binary alloys. These take the form of integrodifferential equations. Coarsening, that is, creation of large scale patterns in such models is poorly understood.

There are partial results [1, 2] that use the maximum principle, while for most interesting problems such a tool is not available. This will be a mixture of analytic and numerical work and will need tools of functional analysis and semigroup theory.

References:

[1] D. B. Duncan, M. Grinfeld, and I. Stoleriu, Coarsening in an integro-differential model of phase transitions, Euro. J. Appl. Math. 11 (2000), 561-572.

[2]  V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math. 17 (2006), 221-232.

Professor Des Higham

Centrality Measures for Dynamic Networks

The digital revolution is generating novel large scale examples of connectivity patterns that change over time. This scenario may be formalized as a graph with a fixed set of nodes whose edges switch on and off. For example, we may have networks of interacting mobile phone users, emailers, Facebookers or Tweeters.

To understand and quantify the key properties of such evolving networks, we can extend classical graph theoretical notions like degree, pathlength and centrality. This project will focus on the development of linear algebra-based algorithms that can capture various aspects of information flow around an evolving network, with the aim of identifying important players. Real data sets from the world of on-line social interaction will be used to test ideas.

Suggested reading:

Collaboration Blooms from SIAM News Article, SIAM News, 2012

Peter Grindrod, Desmond J. Higham, and Peter Laflin http://www.siam.org/pdf/news/2039.pdf

Stochastic Models for Online Interaction

Social scientists have developed empirical rules, including homophily and triadic closure, that aim to describe how human social interactions evolve over time. The emergence of large scale digital data sets opens up the possibility of testing these hypotheses.

This project will involve the development of mathematical models, in the language of stochastic processes, and the validation of these models via computer simulation and Bayesian inference.

Suggested reading:

Bistability through triadic closure, P. Grindrod, D. J. Higham and M. C. Parsons, Internet Mathematics, Internet Math. Volume 8, Number 4 (2012), 402-423.

http://personal.strath.ac.uk/d.j.higham/Plist/P111.pdf

Dynamic Communities

This project will look at the development and validation of computatonal algorithms that aim to summarize the community structure in a dynamically evolving network. The work will involve linear algebra, matrix computation, applied statistics and high performance computing. Applications include the analysis of on-line behaviour, e.g., through Twitter (who tweeted who) and email (who emailed who). The project will also look at methods for real-time summaries of other network properties.

Suggested reading:

A matrix iteration for dynamic network summaries, P. Grindrod and D. J. Higham, SIAM Review, 55(1), 118–128.

http://personal.strath.ac.uk/d.j.higham/Plist/P109.pdf

Dr Philip Knight

How and Why of Matrix Balancing

Matrix balancing aims to transform a nonnegative matrix A by a diagonal scaling by matrices D and E so that P = DAE has prescribed row and column sums. Historical motivation for achieving the balance has included interpreting economic data, preconditioning sparse matrices and understanding traffic circulation. More recently, it has been acknowledged that there is a role for matrix balancing in network analysis, inferring phylogenies (via hierarchical clustering) and increasing fairness in elections.

Application of balancing has been limited by a lack of analysis of existing algorithms in order to assess their efficacy for large scale data sets. We have recently filled in some of the holes in this analysis and have had some success in implementing existing and new algorithms on very large problems. The new algorithms we have developed for balancing have been shown to work an order of magnitude faster than the standard algorithm. We would like to develop these algorithms to find implementations suitable for parallel computations on large data sets. And we would like to develop more full the theory of balancing by drawing from the theory of linear programming as well as linear algebra.

In tandem with algorithmic and theoretical work, applications in disciplines such as electoral policy and phylogenic reconstruction are a central feature of the project. The student will be able to influence the direction of the project based on her/his interests.

The successful candidate will have an undergraduate or postgraduate degree in mathematics or a similar numerate discipline. Experience of numerical analysis (particularly numerical linear algebra) and programming in a high level language such as C++ or FORTRAN would be an advantage.

The studentship is available for UK/EU candidates subject to specific eligibility criteria. In addition to the payment of tuition fees for the 3.5 year duration of the project, the award includes provision for a student maintenance grant at the standard EPSRC rate.

Professor Nigel Mottram

Mathematical modelling of active fluids

The area of active fluids is currently a “hot topic” in biological, physical and mathematical research circles. Such fluids contain active organisms which can be influenced by the flow of fluid around them but, crucially, also influence the flow themselves, i.e. by swimming. When the organisms are anisotropic (as is often the case) a model of such a system must include these inherent symmetries. Models of bacteria and even larger organisms such as fish have started to be developed over the last few years in order to examine the order, self-organisation and pattern formation within these systems, although direct correlation and comparison to real-world situations has been limited.

This project will use the theories and modelling techniques of liquid crystal systems and apply such modelling techniques to the area of anisotropy and self-organisation derived from active agents. The research will involve a continuum description of the fluid, using equations similar to the classical Navier-Staokes equations, as well as both the analytical and numerical solution of ordinary and partial differential equations.

Interaction and collaboration with a number of UK and international groups will be possible: particularly groups in Bath and North Carolina. Although co-funding is not available at present this might be possible through contacts in the fish-farm industry.

Flow of groundwater in soils with vegetation and variable surface influx

Groundwater is the water underneath the surface of the earth which fills the small spaces in the soil and rock and is extremely important as a water supply in many areas of the world. In the UK, groundwater sources, or aquifers, make up over 30% of the water used, and a single borehole can provide up to 10 million litres of water every day (enough for 70,000 people).

The flow of water into and out of these aquifers is clearly an important issue, more so since current extraction rates are using up this groundwater at a faster rate than it is beoing replenished. In any specific location the fluxes of water occur from precipitation infiltrating from the surface, evaporation from the surface, influx from surrounding areas under the surface, the flow of surface water (e.g. rivers) into the area, and the transpiration of water from underground directly into the atmosphere by the action of rooted plants.

This complicated system can be modelled using various models and combined into a single system of differential equations. This project will consider single site depth-only models where, even for systems which include complicated rooting profiles, analytical solutions are possible, but also two- and three-dimensional models in which the relatively shallow depth compared to the plan area of the aquifer can be utilised to make certain "thin-film" approximations to the governing equations.

This project will be undertaken in collaboration with Prof Alessandro Tarantino from the Department of Civil and Environmental Engineering at the University of Strathclyde.

Dr Tony Mulholland

Mathematical Modelling to aid the Detection of Cracks in Safety Critical Structures

Safety critical structures, such as those found in nuclear plants, aircraft engines, oil pipelines, and railway tracks, require to be routinely checked for the presence of cracks. This testing has to be performed non-destructively and often in-situ. To look inside these structures, one of the dominant imaging technologies that is deployed is ultrasound. In the last few years a new non-destructive testing (NDT) technology has emerged that has the potential to transform the capability of ultrasound transducer systems in detecting cracks in steel welds.

This technology consists of a spatial array of individual ultrasound transducers with each being capable of independently transmitting and receiving an ultrasound signal. The challenge is now for mathematicians to develop fast, real-time algorithms that optimise the use of this data in flaw detection and characterisation. This PhD project will use ideas from stochastic calculus (as used in the finance industry) to address this problem.

The project will involve close collaboration with engineers and other disciplines within the Centre for Ultrasonic Engineering (www.cue.ac.uk) and with industry partners.

Mathematical Modelling to Improve Drug Manufacturing

Many high added value products and processes (such as those found in the manufacture of drugs) utilise crystalline solids whose manufacture typically involves several discrete batch processes to achieve the final desired product.

The aim of the Centre for Continuous Manufacturing and Crystallisation (CMAC) is to enable conversion of current batch manufacturing processes to a continuous basis to provide a substantial competitive advantage to UK manufacturing. However, advances are required in the fundamental understanding of the processes and in the use of sensors to measure the crystal shapes as they evolve. This PhD project will look at developing this understanding and in developing mathematical methods for extracting key properties of these crystals from sensor measurements.

The project will involve close collaboration with engineers and other disciplines within the Centre for Ultrasonic Engineering (www.cue.ac.uk) and with industry partners.

Mathematical Analysis of Insect Hearing Systems

The aim of this project is to develop a mathematical understanding of insect tympanal membrane vibrations. Hearing airborne vibrations, or sound, is a sense particularly important in many insects, and plays a key role in both predator and prey detection and mate attraction.

One of the mechanical systems used by insects to detect sound waves is the air pressure receiver. Such a membrane, or tympanum, is a structurally heterogeneous structure and the way it is actuated by incident sound energy has been investigated in only a few systems. Recent work has examined tympanal vibrations in response to sound in the ears of several insects, including the locust, the moth and the cicada. These investigations have unearthed several un-expected phenomena.

The challenge now is to combine the dynamic measurements of the insect tympana vibrations with the sparse structural data in order to explain how the dynamic phenomena occur. This PhD project will involve the mathematical modelling of these systems with the aim of furthering the basic understanding of how these systems operate.

The project will involve close collaboration with engineers and biophysicists within the Centre for Ultrasonic Engineering (www.cue.ac.uk).

Dr Jiazhu Pan

Modelling multivariate time series

We wish to develop innovative methods for modelling high-dimensional time series. Practical time series data, including both continuous-valued and discrete-valued data such as climate record data, medical data, and financial and economic data, are used for empirical analysis.

Models for forecasting multivariate conditional mean and multivariate conditional variance (volatility) are concerned. Techniques for dimension reduction, such as dynamic factor analysis, are used.

The estimation of models for panel data analysis and the option valuation with co-integrated asset prices is discussed.

Functional time series models: estimation and applications

Nowadays people often meet problems in forecasting a functional. A functional may be a curve, a spatial process, or a graph/image. In contrast to conventional time series analysis, in which observations are scalars or vectors, we observe a functional at each time point; for example, daily mean-variance efficient frontiers of portfolios, yield curves, annual production charts and annual weather record charts.

Our goal is to develop new models, methodology and associated theory under a general framework of Functional Time Series Analysis for modelling complex dynamic phenomena. We intend to build functional time series models and to do forecasting.

Modelling large economic systems and high frequency data

When the true economic system consists of many equations, or our economic observations have a very high dimension, one may meet the ``curse of dimensionality" problem. We try to impose a common factor structure to reduce dimension for the parametric and nonparametric stability analysis of a large system. Replacing unobservable common factors by principle components in parametric and nonparametric estimation will be justified.

In contrast to conventional factor models which focuses on reducing dimensions and modeling conditional first moment, the proposed project devotes attention to dimensional reduction and statistical inference for conditional second moments (covariance matrices). The direct motivation lies in the increasing need to model and explain risk and uncertainty of a large economic system.

The other distinctive point is that the proposed project considers factor models for high frequency data. A key application is the analysis of high dimensional and high-frequency financial time series, although the potential uses are much wider.

Dr Dougie Speirs

Surfing the Size Spectrum using Length-Structured Models

Marine ecosystems are generally composed of large numbers of species of widely varying sizes, ranging from unicellular species, through zooplankton and up to large fish and whales. The distribution of the total biomass of all species by size is known as the biomass spectrum.

In addition to the size variation due to differing characteristic sizes of the different species, and unlike the case in terrestrial systems, individuals themselves often undergo increases in body size of several orders of magnitude, from small eggs and larvae at a few millimetres in length up to large adults at the metre length scale.

At different parts of its life cycle an individual will be present at different parts of the biomass spectrum. Despite this apparent complexity it has long been known from field observations that biomass spectra show many regularities. In particular the logarithm of biomass density is approximately linearly related to the logarithm of body length with negative slope. Moreover, the slope of the spectrum is potentially sensitive to environmental and anthropogenic perturbations, for example the removal of large fish due to commercial fishing.

For these reasons biomass spectra have gained currency in recent years as a tool for studying the integrated ecosystem impacts of climate change and human exploitation of the seas.

The relative simplicity of representing the entire ecosystem as a size-structured spectrum has also permitted the development of mathematical representations in terms of partial differential equations, and these can be used to make predictions about ecosystem level responses to fishing and environmental change.

To date most modelling efforts have focused on steady state analyses of biomass spectra, which may be compared to annually averaged size spectra from field observations. In temperate shelf seas such as the North Sea, however, the biomass spectrum is not static but subject to seasonal impulses caused by increased primary production from phytoplankton in the spring bloom, and by seasonal reproduction by zooplankton and fish. This is known from both observational and mathematical models to induce annual ripples in the biomass distribution that propagate up the spectrum, gradually attenuating at larger sizes.

Fish larvae can exploit this by growing in size at a rate that allows them to feed near the peak of the wave, a phenomenon that has been dubbed by John Pope and his co-workers “surf-riding the biomass spectrum”. This project will focus on developing existing models to better represent these processes. In order to test the models the parameters will be estimated using Bayesian inference methods by fitting to a variety of data sets, including a long term (multi-annual) high temporal resolution (weekly) data from the North Sea.

Modelling fisheries-induced evolution

Many fish populations worldwide have been heavily exploited and there is accumulating evidence from both observational and theoretical studies that this harvesting can induce evolutionary changes. Such responses can affect the stock sustainability and catch quality, and so there is a recognized need for new management strategies that minimise these risks. Most results suggest that high mortality on larger fish favours early maturation.

However, recent theoretical work has shown that trade-offs between growth and maturation can lead to more complex evolutionary responses. Surprisingly, harvesting large fish can select for either late or early maturation depending on the effect of maturation on growth rate. To date most theoretical studies have used evolutionary invasion analyses on simple age-based discrete-time models or on continuous-time coupled ODE representations of size structure.

In common with many generic models of fish population dynamics, population control occurs by unspecified density-dependence at settlement. While these simplifications carry the advantage of analytical tractability, the analysis assumes steady state populations. This, together with the stylised life-histories precludes comparing model results with field data on secular changes in size-distributions an sexual maturity.

The work in this project will develop a new generation of testable model for fisheries-induced adaptive changes with the potential to inform future management decisions. This will involve developing a consumer-resource model which a length-structured fish population feeding on a dynamic biomass spectrum.

Differently-sized fish will compete for food by exploiting overlapping parts of the food size spectrum. The population will be partitioned by length at maturity, and this will be the heritable trait under selection. The model will be used to explore how changes in mortality and food abundance affect the evolutionarily stable distribution of maturation lengths.

Comparisons with survey data on North Sea demersal fish will be used to assess whether the historical harvest rates are sufficient to explain growth rate changes as an evolutionary response. Finally the evolutionarily stable optimal harvesting strategies will be identified.

Professor Stephen Wilson

Microfluidics

In recent years there has been an explosive growth of interest in the behaviour and control of fluids at small (typically sub-millimetre) scales motivated by a range of novel applications including ink-jet printing and lab-on-a-chip technologies.

Much of the most exciting current research concerns the interaction between fluids and both rigid and flexible structures at small scales, and so the aim of the present project is to use a judicious combination of asymptotic methods and judiciously chosen numerical calculations to bring new insight into the behaviour of a variety of novel fluid-structure interaction problems in microfluidics.

Dr Alex Wray

Dynamics of Self-Rewetting Droplets

In the last decade or so there has been an explosion of interest in droplet evaporation, driven by new technological applications as diverse as crop spraying, printing, cooling technologies such as heat pipes, and DNA micro-array analysis.

One particularly interesting aspect of this problem which has thus far received relatively little attention is that of fluids whose surface tension exhibits a local minimum with temperature, known as self-rewetting fluids, a property that can have a profound effect on the dynamics of droplets on heated substrates.

The aim of the project is to build on the existing literature on conventional surface-tension-gradient driven spreading and droplet drying (see, for example, the references to some of our recent work on these problems given below) to bring new physical insight into this challenging scientific problem, and hence to harness the novel properties of self-rewetting droplets in a range of applications.

The project will be a collaboration colleagues at the University of Edinburgh who will be undertaking a parallel series of experimental investigations on this problem which will be key to the successful outcome of the project.

Dunn, G.J., Wilson, S.K., Duffy, B.R., David, S., Sefiane, K. “The strong influence of substrate conductivity on droplet evaporation” J. Fluid Mech. 623 329-351 (2009)

Dunn, G.J., Duffy, B.R., Wilson, S.K., Holland, D. “Quasi-steady spreading of a thin ridge of fluid with temperature-dependent surface tension on a heated or cooled substrate” Q. Jl. Mech. appl. Math. 62 (4) 365-402 (2009)

Computer and Information Science

Strathclyde Combinatorics Group

There is an opportunity to do graduate work with the Strathclyde Combinatorics Group in enumerative combinatorics, algebraic combinatorics and graph theory, some of this with connections to physics and theoretical computer science. More information about the group and its research can be found at http://combinatorics.cis.strath.ac.uk.

If you would like to discuss opportunities to do graduate work in combinatorics then please contact Dr. Anders Claesson at anders.claesson@strath.ac.uk

MSP group

The MSP group engages in research at the frontier of Mathematics, Physics and Theoretical Computer Science with particular strengths in category theory, type theory, logic, functional programming and quantum computation. Fundamentally, we want to change the world around us and believe we have the mathematical ideas to do so. If you want to help us achieve this, look us up at http://msp.cis.strath.ac.uk.

If you would like to discuss opportunities to do graduate work with the MSP group then please contact Dr. Anders Claesson at anders.claesson@strath.ac.uk