## Design of Experiments@Southampton |

We regularly have PhD positions available for students with evidence of strong performance at undergraduate or taught-postgraduate levels. Please contact Dr Antony Overstall if you are interested in studying with us.

Applications for PhD study can be made online.

Current available PhD positions include:

- Hierarchical experiments and likelihood approximations
- Supervisors: Professor Dave Woods and Dr Helen Ogden
- Application deadline: Applications will be considered on merit as they arrive.
- Start date: September 2019
The project will develop new statistical methods for statistical design, modelling and inference using systems and approximations available on at least two hierarchical scales.

Understanding and exploiting hierarchical differences in the accuracy and cost of systems and approximations across different (physical, computational) scales is an important topic in many areas of statistical research. In general, accuracy and cost will be inversely related, with economically or computationally expensive "evaluations" (physical observations or computational approximations) giving highly accurate results. However, project budgets will generally be insufficient to allow statistical modelling and inference to rely solely on results from these evaluations. Instead, use must be made of lower cost, but lower accuracy, evaluations, which will be available in much greater quantities.

This unique project would allow a student to make contributions to both design of experiments and likelihood-based inference by developing generic methods to best combine data and approximations from different scales. Example problems include designing experiments across lab, pilot and manufacturing scales to understand and predict manufactured product performance in the pharmaceutical industry, and constructing likelihood approximations for nonlinear regression models combining analytic (e.g. Laplace) with computational (e.g. Monte Carlo) approximations to obtain statistically valid and efficient inferences.

- Sequential Bayesian design of experiments
- Supervisor: Dr Antony Overstall
- Application deadline: Applications will be considered on merit as they arrive.
- Start date: September 2019
The project will develop new Bayesian statistical methods for the design of sequential experiments.

Often information is gained on a process by conducting a series (or sequence) of experiments. Consider, after each iteration, using the responses collected thus far to design the next iteration of the experiment. A fully sequential approach will be considered where uncertainty is minimised about the quantities of interest after all experimentation is complete. This is in contrast to a myopic approach whereby uncertainty is minimised based on one more iteration of experimentation or to batch design where uncertainty is minimised after all experimentation is complete but only once at the beginning of the experimental process, i.e. not after each iteration.

To be able to apply a fully sequential approach will require the development of new statistical algorithms and approximations. They will have application in diverse fields including automatic experimentation and Bayesian optimisation.

- Mixture experiments with non-normal responses
- Supervisor: Dr Stefanie Biedermann
- Application deadline: Applications will be considered on merit as they arrive.
- Start date: September 2019
A mixture experiment occurs when the response variable, y, is a function of the relative proportions of components in a mixture, rather than a function of the total amount of each component. For example, if the amount of all ingredients in a cake is doubled, respectively, we will get a bigger cake, but the flavour is not affected. Since the proportions of components in a mixture must add to one, this constraint must be incorporated in the model to analyse the data. Several modelling approaches have been developed for continuous response variables in this situation, often assuming a normal distribution, but there is little progress on non-normal responses. For example, a cake may be rated on a scale from 1 - 5. This project will explore a new modelling approach for mixture experiments where the response is binary or ordered. Depending on your interests, there could also be a chapter on how to design these experiments to get the most information from the data.