Dr Jere Koskela
Does high fecundity mask population growth in DNA sequence data?
DNA sequence data is becoming more and more commonplace, and can be used to answer numerous questions about the history of the population from which it was sampled. There are plenty of successful methods for detecting historical population size changes in mammal-like populations, in which family sizes are small and many offspring survive to adulthood. Many organisms, particularly marine ones, reproduce rather differently: each female can generate millions of eggs, but typically most of the eggs do not survive into adult fish. This kind of high fecundity reproduction is less well studied than the mammalian regime, and in particular, methods for detecting historical population size changes are not in widespread use. The purpose of this project is to investigate whether population growth can be detected in the presence of high fecundity, using standard models of mathematical population genetics and simulated data.
Time series inference in genetics
DNA sequence data is now available from multiple generations, and there are standard models (most notably, the Wright-Fisher model) which describe the changes in allele frequencies in a population across time. However, the advent of multi-generation data is recent enough that many practical questions remain unanswered. This project is about finding an optimal balance between how many generations to sequence, and how many individuals to sequence in each generation, when the goal is to minimise the mean square error of estimators of standard genetic quantities of interest, such as the rate with which mutations arise in the population.
Prof. Vassili Kolokoltsov
Probabilistic methods for fractional calculus
Stochastic games with the large number of players including mean-field games
For an introduction to this vast topic see this monograph.
This is a really 21st century very recent development on the cross-road of control, games and quantum technologies. The project will involve the quite new development of dynamic quantum games, an exciting new topic that is just waiting for being properly explored. For an introduction one can see the review "Quantum games: a survey for mathematicians".
Dr Richard Everitt
Approximate Bayesian computation for individual based models
Estimating bowler ability in cricket
Active subspaces using sequential Monte Carlo
Dr Ioannis Kosmidis
High-dimensional logistic regression
Item-response theory models and politics: How liberal are the members of the US House?
Dr Simon Spencer
Simulations for the whole family (of quasi-stationary distributions)
If a Markov process has an absorbing state (reached with probability one in finite time) then the stationary distribution is boring – all the mass falls on the absorbing state. However, if we condition on the process not having reached the absorbing state yet then a so-called quasi-stationary distribution may exist. In fact, there can be infinitely many such quasi-stationary distributions for the same process. The birth-death process is a relatively simple model that has an infinite family of quasi-stationary distributions. One is straightforward: a so-called “low energy” distribution with finite mean, and all others are more exotic, “high energy” distributions with infinite mean. In this project we will look to find ways of simulating from the quasi-stationary distributions of the birth death process, and from the “high energy” distributions in particular. Then, we will look to apply these simulation techniques to more complex models in which the family of quasi-stationary distributions is currently unknown. This project will involve programming in the statistical programming language R.
Key reference: Adam Griffin, Paul A. Jenkins, Gareth O. Roberts and Simon E.F. Spencer (2017). Simulation from quasi-stationary distributions on reducible state spaces. Advances in Applied Probability, 49 (3), 960-980.