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Multivariate Regression on High-Dimensional Networks

Project Description

A key feature of many modern datasets is the inherent network structure present in it. Examples include social networks, biological networks, spatial networks among others. Graphical models (Wainwright and Jordan (2008)) are widely used to describe the multivariate relationship among interacting individuals. In this case, each individual corresponds to a node in the graph and the edges represent correlations.

This project is motivated by several applications in biology and in social sciences. Examples include understanding how different
diseases show similar pattern in the number of deaths across spatially nearby states, and mapping crime patterns across neighbouring counties/states (Hӓrdle and Simmer (2003)). In these examples, there are multivariate observations available on each
node of the graph and it is essential to infer about the parameters of a joint model. For example, in biomedical studies, multiple clinical characteristics are measured across several patients and it is essential to understand how those different characteristics are correlated. The objective of the project is to develop a joint optimization framework that can simultaneously estimate the parameters in the model and learn the graph structure as well. Often network structures are sparse in nature and we would like to extend the high dimensional sparse estimation techniques to learn the underlying structure combining across variables over nodes.

There has been growing interest recently to analyse network structures with covariates/meta data (Latouche et al. (2018)). However, most of these methods focus on understanding the interactions through multiple regression based on the covariates. The objective of the project is to extend this framework to multivariate responses and perform multivariate regression utilizing the network structure. Further there are some recent literature (Zhu et al. (2014)) on joint estimation of multiple graph structure (motivated by gene regulatory networks for different subtypes of genes). The regression framework developed in this project will be applied in the case of multiple graph structures as well.

This PhD offers the chance to develop –(i) a joint parameter estimation framework for graphical models with multiple observations on each node when the graph structure is known (spatial network) and (ii) a joint framework for multiple sparse graph estimation (biological network) using techniques available on sparse graphical models, high dimensional estimation and multivariate multiple regression.

The project will be closely linked to the SAMBa CDT and run as an aligned studentship. The student on this project would be exposed
to all of these interactions and encouraged to undertake reading courses and attend workshops and conferences that would give them a
far broader perspective to mathematics than they would gain from a single project.


Applicants should hold, or expect to receive, a First Class or high Upper Second Class UK Honours degree (or the equivalent qualification gained outside the UK) in a relevant subject. A master’s level qualification would also be advantageous. Non-UK applicants must meet our English language entry requirement


Informal enquiries should be directed to Dr Sandipan Roy, email .

Formal applications should be made via the University of Bath’s online application form:

Please ensure that you quote the supervisor’s name and project title in the ‘Your research interests’ section.

More information about applying for a PhD at Bath may be found here:

Anticipated start date: 28 September 2020.

Funding Notes

UK and EU candidates applying for this project will be considered for a University Research Studentship which will cover UK/EU tuition fees, a training support fee of £1,000 per annum and a tax-free maintenance allowance at the UKRI Doctoral Stipend rate (£15,009 in 2019-20) for a period of up to 3.5 years.


1. Hardle, W., & Simar, L. (2003). Applied Multivariate Statistical Analysis, version 29th.
2. Latouche, P., Robin, S., & Ouadah, S. (2018). Goodness of fit of logistic regression models for random graphs. Journal of Computational and Graphical Statistics, 27(1), 98-109.
3. Zhu, Y., Shen, X., & Pan, W. (2014). Structural pursuit over multiple undirected graphs. Journal of the American Statistical Association, 109(508), 1683-1696.

How good is research at University of Bath in Mathematical Sciences?

FTE Category A staff submitted: 44.40

Research output data provided by the Research Excellence Framework (REF)

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