Statistics and Operational ResearchResearch interests include: Epidemiology and Healthcare Modelling

Health Modelling Centre Cymru (hmc2) is a new initiative of WIMCS which will build out from the current expertise in this field across Wales, fostering collaboration across different research areas of the mathematical and computational sciences, to create a more vibrant and effective interface between the mathematical research community, the medical research community, NHS Wales, the Welsh Assembly Government and the Health industry. Many of the cluster members apply their research expertise to healthcare, covering areas such as mathematical and operational research modelling, statistical analysis and clinical trials, health services research and epidemiology.

Optimisation

Members of the cluster are from the Centre for Optimisation and Its Applications at Cardiff University , which is an interdisciplinary centre encouraging joint research and applied projects among members of the Schools of Mathematics, Computer Science and Business and Manufacturing Engineering Centre. It also encourages increased awareness of the rapidly growing field of optimisation through publications, conferences, joint research and student exchange. The director of the Centre is cluster member and Head of Statistics at Cardiff, Professor Anatoly Zhigljavsky.

Queueing Systems

Queueing studies have focussed on bulk service queues and time-dependent queueing models, including research projects at Gatwick Airport, the Severn Bridge, the Channel Tunnel, and healthcare services. Recent theoretical work has made significant progress with the transient solution of queueing systems with a variety of service mechanisms. Research and application in simulation has involved novel research on the use of simulation models incorporating small-world theory for modelling of disease propagation, modelling consumer choice and incorporating human behaviour.

Meta-heuristics and Scheduling

Research is focussed on meta-heuristics and their application to a variety of combinatorial optimisation problems. These methods include genetic algorithms, simulated annealing and ant colony optimisation. Application areas include examination timetabling, school scheduling, nurse rostering, sports fixture scheduling and vehicle routing problems. Projects have been undertaken for many companies including WH Smiths, John Menzies and The International Rugby Board.

Logistics and Supply Chains

Current research interests involve the application of control theory and statistical techniques to supply chains and e-business scenarios in order to investigate their dynamic and economic performance.

Multivariate Data Analysis and Forecasting

The objective is development of a methodology of exploratory analysis of temporal-spatial data of complex structure with the final aim of construction of suitable parametric models. The applications include various medical, biological, engineering and economical data. Several market research projects where the development of statistical models was a substantial part have taken place. In recent years a powerful technique of time series analysis has been developed and applied to many practical problems. This technique is based on the use of the Singular-value decomposition of the so-called trajectory matrix obtained from the initial time series by the method of delays. See below for further details.

Small World NetworksSmall World Networks

UK-China workshop on SSA and its applicationsUK-China workshop on SSA and its applications

Poster Presenters: Dr D Denisov, Dr I T Vieira

alternative

Mean Geodesic Distance (L)

alternative

Clustering Coefficient (C)

Cv = 1, 1, 1/6, 0, 0. C = 0.433

Degree Distribution (pk)

k = 5 1 2 3 4 ... 270 k

P(k)

1 2 3 4 ... 270 k

P(k)

Properties:Properties:Let G be a ring lattice graph with n vertices and k edge connections per vertex. Now rewire each edge with probability p, (0 < p < 1). Here n = 20 and k = 4.

In this region, the world is small and highly clustered, the network can propagate diseases or information very efficiently both globally and locally.

Surprising Fact: Roughly 5 shortcuts reduce L by factor of ½, regardless of n.

The Small-WorldThe Small-World

Definition:Definition:

Adaptation for Social NetworksAdaptation for Social NetworksLet A be a vector giving the position of an individual on the sphere, then we define the neighbourhood of A as all vector points B such that

Here is the angle between the two vectors and Q the length of the projection of point B on the axis passing through A. The quantity cos() is a convenient quantity for measuring geographic closeness.

Example: People who are at less than 1000Km on the surface of the Earth (radius 6378Km) are at less than 1000/6378 = 0.157 radians. Here cos(0.157 radians) = 0.988, therefore any two individuals for whom A • B ≥ 0.988 are at a distance of less than 1000Km.

HIV Vaccine in a Small World HIV Vaccine in a Small World (Data from Brazil)(Data from Brazil)At the 31.2% efficacy of the RV144 trial, this vaccine can reduce the HIV transmission if coverage is high, while a hypothetical 75% efficacy vaccine could markedly reduce the HIV pandemic with relatively low coverage.

Reduction in condom use due to vaccination

Reduction in condom use due to vaccination

In any case, vaccine intervention must go alongside education and a wide range of effective prevention programmes. Those who receive the vaccine must understand that their risk of contracting HIV infection has lessened but has not vanished. In any case, vaccine intervention must go alongside education and a wide range of effective prevention programmes. Those who receive the vaccine must understand that their risk of contracting HIV infection has lessened but has not vanished.

• Watts, D. J. and Strogatz, S. H. (1998), “Collective dynamics of 'small-world’ networks”, Nature, 393(4), 40-442.• Joint United Nations Programme on HIV/AIDS (2009), AIDS epidemic update. Technical Report UNAIDS/09.36E / JC1700E.• Bastos, F. et al (2008), Sexual behaviour and perceptions of the Brazilian population regarding HIV/AIDS, Saúde Pública, vol.42, suppl 1. • Rerks-Ngarm S, et al (2009), “Vaccination with ALVAC and AIDSVAX to prevent HIV-1 infection in Thailand”. N Engl J Med; 361:2209-20• Vieira, I., Cheng, R., Harper, P. and de Senna, V. (2010), “Small world network models of the dynamics of HIV infection”, Annals of Operations Research, 178, 173-200Re

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1- Date: September 20-22, 20102- Venue: Cardiff School of Mathematics. 3- Number of delegates: 60 from China, USA, UK, Greece, Russia, Czech Republic, Austria, Australia, Switzerland, France, Portugal, Spain. 4- Number of talks: 21.

Topics: Topics: 1- To discuss, promote and challenge recent developments in modern interdisciplinary statistics based around SSA;2- To consider new areas of application of SSA;3- To establish a long-term UK-China collaboration in the area of time series analysis and forecasting.

Aims: Aims: General Information: General Information: 1- Theory and methodology of SSA;2- Applications of SSA in Economics and Finance; 3- Applications of SSA in other areas including: Image processing , Signal Processing, Medicine, Bioinformatics, Climatology, Engineering and Physics.

Future Cluster ActivitiesFuture Cluster Activities

Symposium on Healthcare ModellingSymposium on Healthcare Modelling

1- Date: January 19 , 20092- Venue: Millennium Stadium, Cardiff3- Number of delegates: 30 from across the UK. 4- Number of talks: 25.

Topics: Topics: 1- To encourage and foster collaboration for early career researchers working in the field of OR for healthcare modelling;2- To discuss theoretical challenges and identify areas for joint work;3- To demonstrate the application and benefits of models for the health service.

Aims: Aims: General Information: General Information: 1- Scheduling of healthcare resources;2- Epidemiology; 3- Hospital capacity planning: 4- Emergency services planning and location analysis.

The cluster has various plans for future events and activities, aimed at reinforcing the collaborative spirit within our cluster and links to external academic groups and interfacing with industry. Short-term plans include:Making a significant contribution to the newly established Health Modelling Centre Cymru (hmc2: Prof. Paul Harper, Cluster co-ordinator is the Director of hmc2) including a high quality seminar series and other workshops; Establishment of a young researchers exchange scheme with the University of Toronto and CIMATEC (Integrated Centre of manufacturing and Technology), Salvador, Brazil, whereby each year early career researchers may spend a short period of time in the partner institution (under the guidance of internationally leading figures in Operations Research) and; workshops in conjunction with the Office for National Statistics (ONS) on the topics of survey sampling methodologies and SSA forecasting.

Large deviations and exit times Large deviations and exit times

for Markov processesfor Markov processes

Selected publicationsSelected publications[1] Denisov, Dieker and Shneer. Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Probab. 36 (2008), no. 5, 1946–

1991

[2] Denisov and Shneer. Local asymptotics of the cycle maximum of a heavy-tailed random walk. Adv. in Appl. Probab. 39 (2007), no. 1, 221–244.

[3] Denisov and Shneer. Asymptotics for first-passage times of Lévy processes and random walks.  arXiv:0712.0728, Submitted.

[4] Denisov, Foss and Korshunov. On lower limits and equivalences for distribution tails of randomly stopped sums. Bernoulli 14 (2008), no. 2, 391–404.

[5] Denisov, Foss and Korshunov. Asymptotics of randomly stopped sums in the presence of heavy tails Bernoulli 16 (2010), no. 4, 971–994

[6] Denisov and Leonenko. Exit times for Kolmogorov-Pearson diffusions. In preparation.

[7] Denisov and Wachtel. Conditional limit theorems for ordered random walks. Electron. J. Probab. 15 (2010), no. 11, 292–322. .

[8] Denisov, Foss and Konstantopoulos. Limit theorems for a random directed slab graph. arXiv:1005.4806. Accepted by Ann. Appl. Probab.

[9] Denisov and Leonenko.. Stationary multifractal processes. In preparation

Heavy tails and large deviations in Heavy tails and large deviations in queueing systemsqueueing systemsAs evidenced by many statistical studies, Internet traffic exhibits self-similarity, long-range dependence and the presence of heavy tailed distributions. Also, heavy-tail distributions have been observed in many natural phenomena including stock markets and non-life insurance. When heavy-tailed distributions are present in the system, large deviations from the normal state are caused by a single non-typical event . This is opposed to light tailed distributions, where large deviations are caused by a large sequence of unfavourable events.

Exit times and eigenvalues of the Gaussian Exit times and eigenvalues of the Gaussian Unitary EnsembleUnitary EnsembleInterestingly, the exit times, large deviations and heavy tails have a connection with eigenvalues of Gaussian Unitary Ensemble. To see the connection one can study exit times from a certain region of an n-dimensional random walk. These exit times have heavy tailed distributions and allow to define a Markov chain confined to this region. Such a Markov Chain in the long run is very well approximated by a process of eigenvalues of a GUE. There is also a queueing interpretation of eigenvalues via a construction of a tandem of queues. I have been working on this and similar problems with Wachtel, Foss and Konstantopoulos [7-8].

Multifractality and long-range dependenceMultifractality and long-range dependenceTypically the self similar behaviour gives rise to a fractal (see Cantor set, Brownian motion and the British coast line) . However if self-similarity property is different at different length scales we encounter a new entity – multifractals. This was noted by great Kolmogorov in his studies of turbulence. The multifractals can be found in mathematical finance and some kinds of network traffic. The direction of my research [9] (with Leonenko) in this area is to present rigorous mathematical models which would support the statistical evidence and to give new instrumentals for analysing of such phenomena.

The aim of this research is to establish these heuristics and to understand better the quantitative and qualitative characteristics of these large deviations. This is done on example of a single server queuing systems This system can be analysed using classical stochastic processes: random walks and Levy processes. For example, a busy period of such a system can be described as an exit time of a Levy process . Large deviation of a busy period is a large deviation of the exit time. Besides giving an important understanding of the typical behaviour of the system, the results of this type are of importance in efficient numerical simulation of the rare events. In these studies I collaborated with Foss, Korshunov, Shneer and Dieker [1-5]. My current work (with Leonenko) [6] in this direction is a construction of a direct approach which would allow to model exit times of jump diffusion processes . The aim is to bring together the basic model of finance and insurance mathematics.

WIMCS

Wales Institute of Mathematical and Computational Sciences

Sefydliad Gwyddorau Cyfrifiadurol a Mathemategol Cymru SGCMC