[School of Natural Sciences PhD Scholarships] Bootstrap Inference for Rank Based Dependence Measures in Spatio Temporal Data

About the project

About the Project Large spatio temporal datasets in climate science, epidemiology, and environmental monitoring increasingly involve many spatial units but relatively short time series. A key scientific challenge is to assess whether these units exhibit meaningful dependence after accounting for temporal dynamics. Classical tools such as Pearson correlation or spatial autocorrelation indices often miss nonlinear or tail driven relationships and rely on asymptotic approximations that perform poorly when dimensionality is high. Rank based U statistics, including Bergsma’s correlation coefficient ρ, offer a flexible way to quantify dependence because they characterize pairwise independence and capture nonlinear structure. Recent work has applied such statistics to regional time series data to construct global measures of spatial association and derive asymptotic distributions under independence and autoregressive alternatives. These applications—ranging from disease incidence residuals to climate variables—demonstrate the potential of rank based dependence measures for spatio temporal analysis. Yet current inference procedures rely heavily on large sample asymptotics, assume fixed spatial dimension, and provide limited tools for local dependence, pairwise testing, or time varying structure. This PhD project proposes a bootstrap based framework for inference on rank based dependence statistics in spatio temporal settings. The central idea is to treat these statistics as U statistic functionals of dependent data and to design resampling schemes that remain valid when the number of spatial units is large, time series are short or irregular, and dependence structures are complex. The project is primarily methodological and empirical, with scope for moderate theoretical contributions. Objectives Bootstrap methodology for U statistic dependence measures The project will develop and compare resampling procedures—including block bootstrap, wild bootstrap, and multiplier bootstrap—tailored to rank based U statistics. Simulation studies will evaluate type I error, power, robustness, and scalability under autoregressive and moving average dynamics, nonlinear dependence, nonstationarity, and missing data. These results will guide practical recommendations for researchers working with high dimensional spatio temporal datasets. Pairwise and local dependence analysis Beyond global summaries, the project will extend inference to pairwise and local dependence. Bootstrap calibrated critical values will enable testing of individual region pairs while controlling familywise error or false discovery rate. Local dependence measures defined over neighbourhoods or distance bands will support identification of spatial clusters. Significant pairwise links will be represented as edges in a dependence network, enabling analysis of evolving network topology. Change point detection in dependence structure The project will develop rolling window and CUSUM type change point statistics for detecting shifts in spatial association over time. Bootstrap calibration will account for serial dependence and irregular sampling. Applications to climate and epidemiological datasets will illustrate how dependence patterns evolve and how structural breaks relate to external drivers. Expected Contributions The project will deliver: (i) bootstrap inference tools suitable for high dimensional, short time series settings; (ii) methods for pairwise, local, and time varying dependence analysis; (iii) open source software; and (iv) applied insights into climate and health data. Optional theoretical work may establish bootstrap consistency or finite sample guarantees under simplified dependence models. 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, or allied areas OR any upper-second class (2:1) honours degree and a Master’s degree at merit (or international equivalent) in Statistics, Mathematics, or allied areas. Previous research experience / skill in R programming (substantiated by curriculum taken and/or projects), and proficiency in oral & written communication in English are desirable, along with evidence of interest in research, and experience in (statistical) modelling and analysis of spatio-temporal data. This project will remain open until filled. If your application is submitted by 1 st November 2026, you can expect a decision by 18 th December 2026. If your application is submitted by 15 th January 2027, you can expect a decision by 30 th March 2027. Self or externally funded students can also be considered for this project. FSESoNS

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