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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. Often, 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.

We would like to integrate the variations across the observations and the individuals in a joint optimization framework that would allow us 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.

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, mapping crime patterns across neighbouring counties/states (Hӓrdle and Simmer (2003)).

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.

The successful candidate will have a solid background in mathematics and a keen interest in statistical modelling.


Informal enquiries are welcomed and should be directed to Dr Sandipan Roy,

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: 30 September 2019.

Funding Notes

Candidates may 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 (£14,777 in 2018-19) for a period of up to 3.5 years.


1. Härdle, W., and Simar, L. (2003). Applied Multivariate Statistical Analysis.
2. Latouche, P., S. Robin, and S. Ouadah (2018). Goodness of fit of logistic regression models for random graphs. Journal of Computational and Graphical Statistics, 27 (1), 98-109
3. Roy S., Atchadѐ Y. and Michailidis G. (2017). Change point estimation in high dimensional Markov random-field models. Journal of Royal Statistical Society, Series B, 79(4):1187-1206
4. Wainwright M. J. and Jordan M. I. (2008). Graphical Models, Exponential Families and Variational Inference. Foundations and Trends in Machine Learning, 1, 1-305
5. Zhu, Y., Shen X. and Pan W. (2014). Structural pursuit over multiple undirected graphs. Journal of American Statistical Association, 109(508), 1683–1696

Related Subjects

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

FTE Category A staff submitted: 44.40

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