Statistics and Operational Research

1. Sefydliad Gwyddorau Cyfrifiadurol a Mathemategol Cymru SGCMC WIMCS Wales Institute of Mathematical and Computational Sciences Statistics and Operational Research P(k) P(k) k = 5 Research interests include: 1 2 3 4 ... 270 1 2 3 4 ... 270 k k Large deviations and exit times for Markov processes Small World Networks Definition: 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. Mean Geodesic Distance (L) alternative Heavy tails and large deviations in queueing systems As 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. Clustering Coefficient (C) alternative The Small-World Cv = 1, 1, 1/6, 0, 0. C = 0.433 Degree Distribution (pk) 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. 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. Adaptation for Social Networks Multifractality and long-range dependence Typically 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. Let 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. Exit times and eigenvalues of the Gaussian Unitary Ensemble Interestingly, 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]. HIV Vaccine in a Small World (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 Selected 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 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, 173-200 References: UK-China workshop on SSA and its applications General Information: Aims: Topics: 1- Date: September 20-22, 2010 2- 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. 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. 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. Symposium on Healthcare Modelling General Information: Aims: Topics: 1- Date: January 19 , 2009 2- Venue: Millennium Stadium, Cardiff 3- Number of delegates: 30 from across the UK. 4- Number of talks: 25. 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. 1- Scheduling of healthcare resources; 2- Epidemiology; 3- Hospital capacity planning: 4- Emergency services planning and location analysis. Future Cluster Activities 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. Poster Presenters: Dr D Denisov, Dr I T Vieira