Applied probability & stochastic processes
Our research focuses on applied probability and stochastics processes.
Postgraduate research students
Applied Probability and Stochastic Processes - Example Research Projects
Information about postgraduate research opportunities and how to apply can be found on the Postgraduate Research Study page. Below is a selection of projects that could be undertaken with our group.
Evaluating probabilistic forecasts in high-dimensional settings (PhD)
Many decisions are informed by forecasts, and almost all forecasts are uncertain to some degree. Probabilistic forecasts quantify uncertainty to help improve decision-making and are playing an important role in fields including weather forecasting, economics, energy, and public policy. Evaluating the quality of past forecasts is essential to give forecasters and forecast users confidence in their current predictions, and to compare the performance of forecasting systems.
While the principles of probabilistic forecast evaluation have been established over the past 15 years, most notably that of “sharpness subject to calibration/reliability”, we lack a complete toolkit for applying these principles in many situations, especially those that arise in high-dimensional settings. Furthermore, forecast evaluation must be interpretable by forecast users as well as expert forecasts, and assigning value to marginal improvements in forecast quality remains a challenge in many sectors.
This PhD will develop new statistical methods for probabilistic forecast evaluation considering some of the following issues:
- Verifying probabilistic calibration conditional on relevant covariates
- Skill scores for multivariate probabilistic forecasts where “ideal” performance is unknowable
- Assigning value to marginal forecast improvement though the convolution of utility functions and Murphey Diagrams
- Development of the concept of “anticipated verification” and “predicting the of uncertainty of future forecasts”
- Decomposing forecast misspecification (e.g. into spatial and temporal components)
- Evaluation of Conformal Predictions
Good knowledge of multivariate statistics is essential, prior knowledge of probabilistic forecasting and forecast evaluation would be an advantage.
Adaptive probabilistic forecasting (PhD)
Data-driven predictive models depend on the representativeness of data used in model selection and estimation. However, many processes change over time meaning that recent data is more representative than old data. In this situation, predictive models should track these changes, which is the aim of “online” or “adaptive” algorithms. Furthermore, many users of forecasts require probabilistic forecasts, which quantify uncertainty, to inform their decision-making. Existing adaptive methods such as Recursive Least Squares, the Kalman Filter have been very successful for adaptive point forecasting, but adaptive probabilistic forecasting has received little attention. This PhD will develop methods for adaptive probabilistic forecasting from a theoretical perspective and with a view to apply these methods to problems in at least one application area to be determined.
In the context of adaptive probabilistic forecasting, this PhD may consider:
- Online estimation of Generalised Additive Models for Location Scale and Shape
- Online/adaptive (multivariate) time series prediction
- Online aggregation (of experts, or hierarchies)
A good knowledge of methods for time series analysis and regression is essential, familiarity with flexible regression (GAMs) and distributional regression (GAMLSS/quantile regression) would be an advantage.
Modality of mixtures of distributions (PhD)
Supervisors: Surajit Ray
Relevant research groups: Nonparametric and Semi-parametric Statistics, Applied Probability and Stochastic Processes, Statistical Modelling for Biology, Genetics and *omics, Biostatistics, Epidemiology and Health Applications
Finite mixtures provide a flexible and powerful tool for fitting univariate and multivariate distributions that cannot be captured by standard statistical distributions. In particular, multivariate mixtures have been widely used to perform modeling and cluster analysis of high-dimensional data in a wide range of applications. Modes of mixture densities have been used with great success for organizing mixture components into homogenous groups. But the results are limited to normal mixtures. Beyond the clustering application existing research in this area has provided fundamental results regarding the upper bound of the number of modes, but they too are limited to normal mixtures. In this project, we wish to explore the modality of non-normal distributions and their application to real life problems.
Regular seminars relevant to the group are held as part of the Statistics seminar series. The seminars cover various aspects across the AI3 initiative and usually span multiple groups. You can find more information on the Statistics seminar series page, where you can also subscribe to the seminar series calendar.
Stochastic processes are frequently used as mathematical models for phenomena that appear to vary randomly over time or space.
The Applied Probability and Stochastic Processes research group focuses on theory, methodology and applications, developing innovations in the inference and control of stochastic processes across a range of mathematical, scientific and engineering areas.
The group specialises in probabilistic modelling and forecasting for complex systems and processes involving uncertainty, with applications in energy and weather forecasting, population genetics, astronomy, ecology and public policy.