[School of Natural Sciences PhD Scholarships] Prediction and Uncertainty Quantification: Multiple-Use Calibration for All Future Observations

About the project

About the Project Regression is one of the fundamental tools in statistics and machine learning. In a standard regression problem, an input (x) is used to predict an output (y). In statistical calibration, however, the problem is reversed: after learning the relationship between (x) and (y) from training data, a new value of (y) is observed and the aim is to infer the unknown (x) that generated it. This inverse-prediction problem arises widely in science, engineering and quantitative measurement. For example, cadmium (Cd) is a toxic heavy metal whose concentration in environmental samples such as drinking water may be measured using Graphite Furnace Atomic Absorption Spectroscopy. If (x) denotes the Cd concentration and (y) the corresponding absorbance signal, observations from samples with known concentrations can be used to fit a regression model. A new absorbance measurement can then be used to estimate the unknown concentration, together with a confidence set quantifying the associated uncertainty. A key challenge arises when the same fitted calibration model is used repeatedly for many future observations. Methods designed for a single future observation generally provide only pointwise guarantees. If the model is to be used for potentially infinitely many future observations, substantially stronger simultaneous uncertainty guarantees are required. The main objective of this PhD project is to develop and investigate statistical methodology for multiple-use calibration, with particular emphasis on simultaneous tolerance bands (STBs) and simultaneous tolerance regions (STRs). The aim is to construct confidence sets for unknown (x)-values associated with future (y)-values while providing rigorous simultaneous coverage guarantees across all future observations. The project will combine statistical theory, methodology and computation. Possible directions include developing calibration procedures for linear, polynomial and multiple regression models; studying theoretical coverage properties and efficiency; comparing pointwise and simultaneous approaches; investigating the effects of sample size, noise level and model complexity; extending the methodology to more flexible regression settings; and exploring connections with modern uncertainty-quantification methods such as conformal prediction. The student will undertake simulation studies and applications to suitable scientific datasets, with implementation in R and/or Python. Potential outcomes include new statistical methodology for reliable multiple-use calibration, theoretical results on simultaneous coverage and efficiency, computational tools for practical implementation, and applications demonstrating the benefits of simultaneous uncertainty quantification. The work is expected to lead to a PhD thesis and potentially peer-reviewed publications. The student will receive regular individual supervision and training in regression, simultaneous inference, tolerance regions, calibration, statistical computing and uncertainty quantification. Further support will include training in R/Python, simulation design, reproducible research, mathematical derivations and scientific writing, together with participation in research seminars, reading groups, specialist courses and relevant conferences. This project is expected to start in September 2027. Before you apply: We strongly recommend that you contact the supervisors for this project before you apply. How to apply: To be considered for this project you must complete a formal application through our online application portal. If you already have an applicant account this link will directly open an application for PhD School of Natural Sciences Scholarships . If you don’t already have an applicant account, please follow the instructions here . When applying, please specify the full title and supervisor/s of the project, details of your previous study, and names and contact details of two referees. You must also upload a Supporting Statement describing your motivation to apply to the project, your CV and transcripts of awarded and in-progress university qualifications . Please note late or incomplete applications will not be considered. Equality, diversity and inclusion are fundamental to the success of The University of Manchester and central to all our activities. A diverse research community strengthens creativity, productivity and quality, while increasing the societal and economic impact of our work. We welcome applicants from all career paths, backgrounds and sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation or transgender status. We welcome applications from candidates returning to study after a career break or experience in other roles. Flexible study arrangements may be available, including part-time study at 50%, 60% or 80%, subject to the requirements of the project and funder. Eligibility : The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree (or international equivalent) in Statistics, Mathematics, Data Science, Biostatistics, or a closely related quantitative discipline OR any upper-second class (2:1) honours degree and a Master’s degree at merit (or international equivalent) in Statistics, Mathematics, Data Science, Biostatistics, or a closely related quantitative discipline. A solid background in probability, statistical inference, regression modelling and mathematical statistics is desirable. Experience with statistical programming, particularly in R, would be advantageous, as would familiarity with simulation studies, bootstrap or resampling methods, and computational statistics. Previous research experience in statistical methodology is desirable but not essential. Candidates should have strong analytical and problem-solving skills, an interest in developing new statistical methodology, and the ability to engage with both theoretical and computational aspects of the project. This project will remain open until filled. If your application is submitted by 1st November 2026, you can expect a decision by 18th December 2026. If your application is submitted by 15th January 2027, you can expect a decision by 30th March 2027. Self or externally funded students can also be considered for this project. FSESoNS

Research areas