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

SUMMARY. The project will aim to extend the POT method in statistical modelling of extreme values to incorporate multiple observables (e.g. different air pollutants) when the data is non-stationary due to changing environment. Computation will be based on MCMC simulations to obtain posterior estimates of model parameters. An alternative approach may deploy deep learning techniques, thus surpassing the limitation of regression-like models. Pre-processing of the input data is likely to require a dimension reduction, whereby modern machine learning techniques are expected to be crucial.

<p>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.</p> <p>Threshold exceedances approximately follow a generalised Pareto distribution (GPD) with two parameters (scale, shape), which are constant if the data is 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 non-stationary 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 non-stationary 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. </p> <p>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 non-stationary 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. An alternative approach may deploy deep learning techniques, thus surpassing the limitation of regression-like models but still keeping the valuable interplay between statistical modelling and machine learning. 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.</p> <p> </p> <p><strong>References</strong></p> <ol> <li>Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. <em>Statistics of Extremes: Theory and Applications. </em>Wiley, 2004, <a href="https://doi.org/10.1002/0470012382">https://doi.org/10.1002/0470012382</a></li> <li>Davison, A.C. and Smith, R.L. Models for exceedances over high thresholds.<em> Journal of the Royal Statistical Society, Ser. B </em><strong>52</strong> (1990), 393–442, <a href="http://www.jstor.org/stable/2345667">http://www.jstor.org/stable/2345667</a></li> <li>Gyarmati-Szabo, J., Bogachev, L.V. and Chen, H. Nonstationary POT modelling of air pollution concentrations: Statistical analysis of the traffic and meteorological impact<em>. Environmetrics </em><strong>28</strong> (2017), no. 5, Paper e2449, 15 pp, <a href="https://doi.org/10.1002/env.2449">https://doi.org/10.1002/env.2449</a></li> </ol> <h4>Potential for high impact outcome</h4> <p>Improving air quality is one of the key objectives of the current governmental policies and academic research in environmental science. The project has a strong potential to involve collaboration with external organisations, such as the Leeds City Council, DEFRA, and the Environment Agency. The project is expected to deliver significant results which may be instrumental for dynamic estimation and prediction of future extreme events in air pollution.</p> <h4>Training</h4> <p>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. Supervision will involve weekly meetings between supervisors and the student. Full training in the related disciplines and skills will be provided through taught courses and hands-on tuition. In particular, the student will have access to a broad spectrum of training workshops put on by the Faculty that includes an extensive range of training in theory development, numerical modelling, and data analysis.</p> <h4>Student profile</h4> <p>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 would be an advantage, as well as a sound grounding in multivariate statistical analysis and Bayesian statistics. Key skills required for the project include competent use of R and experience with programming and statistical computing in general, including MCMC simulations.</p>

<p>Formal applications for research degree study should be made online through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University's website</a>. Please state clearly in the Planned Course of Study that you are applying for <em><strong>PHD Statistics FT</strong></em> and in the research information section that the research degree you wish to be considered for is <em><strong>Multivariate Peaks-Over-Threshold (POT) Modelling of Nonstationary Air Pollution Concentration Data</strong></em> as well as <span class="underline_text"><a href="https://eps.leeds.ac.uk/maths/staff/4008/dr-leonid-bogachev">Dr Leonid Bogachev</a></span> as your proposed supervisor and in the finance section, please state clearly <em><strong>the funding that you are applying for, if you are self-funding or externally sponsored</strong></em>.</p> <p>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</p> <p>Applications will be considered on an ongoing basis.  Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application.  We also advise that you apply at the earliest opportunity as the application and selection process may close early, should we receive a sufficient number of applications or that a suitable candidate is appointed.</p> <p><strong>If you are applying for University or School Scholarships for 2026/27 entry, with external sponsorship or you are funding your own study, please ensure you provide your supporting documents at the point you submit your application:</strong></p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date including the grading scheme</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul> <p><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p> <p> </p>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid.

<p><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>Scholarship opportunities open from October 2025</strong></p> <p><strong>UK</strong> – The <a href="https://phd.leeds.ac.uk/funding/138-leeds-doctoral-scholarship-2025-faculty-of-engineering-and-physical-sciences#:~:text=Key%20facts&text=One%20Leeds%20Doctoral%20Scholarship%20is,rata%20for%20part%2Dtime%20study.">Leeds Doctoral Scholarship</a> <strong>(closing date: February 2026)</strong> and <a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a> <strong>(closing date: April 2026)</strong> are available to UK applicants.  <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Non-UK</strong> – The <a href="https://phd.leeds.ac.uk/funding/48-china-scholarship-council-university-of-leeds-scholarships-2021">China Scholarship Council - University of Leeds Scholarship</a> is available to nationals of China <strong>(closing date: January 2026)</strong>. The <a href="https://phd.leeds.ac.uk/funding/73-leeds-marshall-scholarship">Leeds Marshall Scholarship</a> is available to support US citizens. <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p>Please note that if you are successful in securing an academic offer for PhD study, this does not mean that you have been successful in securing an offer of funding.</p> <p>If you are applying for the Leeds Doctoral Scholarship, Leeds Opportunity Research Scholarship, China Scholarship Council-University of Leeds Scholarship or Leeds Marshall Scholarship, you will need to complete a separate application, specific to these scholarships, to be considered for funding.</p> <p>You will be responsible for paying the overtime fee in full in your writing up/overtime year (£340 in Session 2025/26), but the scholarship maintenance allowance will continue to be paid for up to 6 months in the final year of award.</p> <p><strong>Important: </strong>Please note that that the award does not cover the costs associated with moving to the UK.  All such costs (<a href="https://www.leeds.ac.uk/international-visas-immigration/doc/applying-student-visa">visa, Immigration Health Surcharge</a>, flights etc) would have to be met by yourself, or you will need to find an alternative funding source. </p> <p>Please refer to the <a href="https://www.ukcisa.org.uk/">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.</p>

<p>For general enquiries about applications, contact PGR admissions by email to <a href="mailto:phd@engineering.leeds.ac.uk">phd@engineering.leeds.ac.uk</a></p> <p>For questions about the research project, please contact Dr Leonid Bogachev: <a href="mailto:EMAIL@leeds.ac.uk">L.V.Bogachev@leeds.ac.uk</a>, +44 (0)113 343 4972.</p>