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Flexible modelling of extreme events


Department of Mathematical Sciences

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Dr Ilaria Prosdocimi No more applications being accepted Competition Funded PhD Project (European/UK Students Only)

About the Project

In several application the consequences of a variable of interest exceeding a high threshold can be quite severe and costly. For example, excessively high river flows would entail flooding and potentially disruption to structures and transportation systems, while excessively low river flows might cause the deterioration of the river ecosystem and give issues of water provision for the human consumption and irrigation. There is therefore a need for methods to accurately estimate the probability that a process of interest might be exceptionally high (or low): these methods typically rely on Extreme Value Statistic. While much effort in the development of statistical methods aims at characterising some properties of the central tendency of the distribution of a process, Extreme Values statistic aims at characterising the tail of a distribution.

Extreme Value models are typically motivated by some asymptotic laws which give an approximate distribution for the maxima of weakly long-range dependent series, see Coles (2001) for an introduction. It is often the case that rather than simply studying the behaviour of the process of interest Y, the conditional process (Y/X) might be of interest, with X a set of covariates. For example, if Y denotes the process describing high flows at a gauging station on a river, it could be of interest to study the how Y changes with respect to for example the annual rainfall totals and the annual temperature. In particular, it would be of interest to allow the shape of the relationship between the process Y and the covariates X to be determined by the data at hand using non-parametric regression methods in the spirit of Generalised Additive Models (GAM, Wood 2006), rather than imposing a pre-defined parametric shape. Previous works on the use of non-parametric regression for extremes such as Chavez-Demoulin and Davison (2005) or Chavez-Demoulin et al. (2015) use an outdated approach to GAM models, which have been extended and generalised in the recent years noticeably in Wood et al. (2016) and Wood and Fasiolo (2016) which make use of a penalised splines framework for GAMs. In particular, by using penalised splines it is possible to have the complexity of the relationship between the variable of interest and the explanatory variables to be completely determined by properties of the data.

This type of approach for extreme value distributions has been explored Padoan and Wand (2008) and Laurini and Pauli (2009), although these implementation are not very general. Given the potential wide use of non-parametric extreme value models, a more modern implementation of GAMs which would allow for an automatic selection of the model degrees of freedom would be very useful to practitioners of different fields. This project aims at building fast and efficient GAMs for extreme values method, with methods to perform automatic smoothing parameter selection and model checking. The project also aims at applying such methods to existing datasets to assess how external factors impact the risk of exceedance of high threshold in different applications. The lead supervisor has extensive experience in the modelling of flood risk and a possible application of the implemented methods would be the investigation of drivers of change in river flow extremes, which can then be used to ensure the resilience of existing structures to the impacts of climate change. Other applications are also possible, for example in the environmental sciences, finance and engineering. The lead supervisor has connections with colleagues at different departments in the University, so there is a potential for the project to be developed in collaboration with scientists from other disciplines who would have a keen interest in understating how the risk connected to extreme events might be driven by external variables.

Training opportunities:
Access to the course of the Academy for PhD Training in Statistics (APTS); attending both statistical conferences and conferences for the area of application chosen by the student (e.g. for statistical hydrology the international statistical hydrology conference or meetings of the British Hydrological Society); attending the national Research Students’ Conference in Probability and Statistics; discussing with partners in external departments about potential application of the developed statistical methods.

Anticipated start date: 2 October 2017.

Applications may close early if a suitable candidate is found; therefore, early application is recommended.

Funding Notes

UK and EU students applying for this project may be considered for a University Research Studentship which will cover Home/EU tuition fees, a training support fee of £1000 per annum and a tax-free maintenance allowance of £14,553 (2017/18 rate) for 3.5 years.

Note: ONLY UK and EU applicants are eligible for the studentship; unfortunately, applicants who are classed as Overseas for fee paying purposes are NOT eligible for funding.

We welcome all-year round applications from self-funded candidates and candidates who can source their own funding.

References

Chavez-Demoulin, V. and Davison, A. C. (2005). Generalized additive modelling of sample extremes, Journal of the Royal Statistical Society: Series C (Applied Statistics), 54(1), 207-222.

Chavez-Demoulin, V., Embrechts, P. and Hofert, M. (2015). An Extreme Value Approach for Modeling Operational Risk Losses Depending on Covariates, Journal of Risk and Insurance.

Coles, S (2001). An introduction to statistical modeling of extreme values. London: Springer

Laurini, F. and Pauli, F. (2009). Smoothing sample extremes: The mixed model approach. Computational Statistics and Data Analysis, 53, 3842- 3854

Padoan S. A. and Wand, M. P. (2008). Mixed model-based additive models for sample extremes. Statistics and Probability Letters, 78. 2850{285

Wood, S. N. (2006) Generalised Additive Models: An Introduction with R. Boca Raton: Chapman et Hall/CRC

Wood, S. N. Fasiolo, M. (2016). A generalized Fellner-Schall method for smoothing parameter estimation with application to Tweedie location, scale and shape models - arXiv preprint arXiv:1606.04802

Wood, S. N. Pya,N. Säfken, B(2016). Smoothing parameter and model selection for general smooth models. Journal of the American Statistical Association
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