Multivariate Peaks-Over-Threshold (POT) Modelling of Nonstationary Air Pollution Concentration Data

Peaks-over-threshold (POT) method is the preferred modern approach to analyse extreme values in a time series. This is due to a better usage of information as compared to the classic block-maxima method (which utilises only one maximum value in each block, e.g. year). Moreover, in many applications the impact of extremes is often implemented through a few moderately large values rather than due to a single highest maximum.

Threshold exceedances approximately follow a generalised Pareto distribution (GPD) with two parameters (scale, shape), which are constant if the data are stationary (i.e. the observed process is in statistical equilibrium). However, in many practical situations including the air pollution, parameters of the system are likely to significantly change with time. Following Davison & Smith (1990), threshold exceedances in nonstationary data should be modelled by treating the GDP parameters as functions of (time-dependent) covariates (e.g. weather and traffic conditions for air pollutants). However, the Davison-Smith regression model is not threshold stable, which means that the model parameters have to be re-estimated with every new threshold (which may need to vary with time). Recently, Gyarmati-Szabo, Bogachev and Chen (2017) proposed a novel model for nonstationary POT which is threshold stable. This has a strong potential to improve dramatically the computational efficiency of the POT model, making it into a versatile and powerful tool for dynamic estimation and prediction of extremes. In particular, this approach may serve as the basis for a semi- or fully automated computational tool designed for efficient on-line estimation and accurate prediction of future extreme events. Due to the property of threshold stability, such methods will work efficiently with variable threshold selection.

The present project will aim to develop a more general methodology of joint modelling of several observables such as different air pollutants, e.g. NO2, NO, O3 etc., which are highly correlated due to complex photochemical reactions in the atmosphere in the presence of sunlight. The principal innovation to be achieved is to design a suitable multivariate POT model for nonstationary data that will preserve the property of threshold stability. Data analysis based on such a model will involve the MCMC (Markov Chain Monte Carlo) simulations to obtain posterior distributions of the model parameters. Due to an increased computational load, pre-processing of the input data may require a dimension reduction, whereby modern machine learning techniques are expected to be crucial.

Potential for high impact outcome
Improving air quality is one of the key objectives of the NERC. The project is likely to deliver significant results which may be instrumental for dynamic estimation and credible forecasting of future extreme events in air pollution.

This project will be supervised jointly by the Department of Statistics and the School of Computing at Leeds. Also, it has a strong potential to involve collaboration with external organisations such as the Met Office. Full training in the related disciplines and skills will be provided through taught courses and hands-on tuition.

Student profile
The successful PhD candidate should have a solid background in mathematics and statistics, with a strong interest in and a flair for statistical modelling of extreme values. Appreciation of the complexity of modelling air pollution concentrations is desirable, as well as sound grounding in multivariate statistical analysis and Bayesian statistics. Background and experience in machine learning would be an advantage. Key skills required for the project include competent use of R and/or Python and experience with programming and statistical computing in general, including MCMC simulations.

CASE Partner
Negotiations with the Met Office are in progress about their possible involvement as a CASE partner.


  • Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. Statistics of Extremes: Theory and Applications. Wiley, 2004,
  • Davison, A.C. and Smith, R.L. Models for exceedances over high Journal of the Royal Statistical Society, Ser. B 52 (1990), 393–442,
  • Gyarmati-Szabo, J., Bogachev, L.V. and Chen, H. Nonstationary POT modelling of air pollution concentrations: Statistical analysis of the traffic and meteorological impact. Environmetrics 28 (2017), no. 5, Paper e2449, 15 pp,