Topic Title: Change-point Problems and their Applications
Supervisor: Dr Georgy Sofronov
Change-point problems (or break point problems, disorder problems) can be considered one of the central problems of mathematical statistics, connecting together asymptotic statistical theory and Monte Carlo methods, frequentist and Bayesian approaches, fixed and sequential procedures. In many real applications, observations are taken sequentially over time, or can be ordered with respect to some other criterion. The basic question, therefore, whether the data obtained are generated by one or by many different probabilistic mechanisms. The project will focus on development of robust and reliable methods for identifying change points in sequences of random variables. The main aim of this project is to develop adequate statistical models fitted to real datasets, investigate new analytical and computational methods to improve accuracy of estimates for parameters of the statistical models and apply the methods to problems in signal processing, bioinformatics and stock exchange modelling.
Topic Title: Optimal Stopping Rules and their Applications
Supervisor: Dr Georgy Sofronov
In many applications data are sequentially collected over time and it is necessary to make decisions based on already obtained information while future observations are not known yet. Examples occur in environmental applications (detecting changes in ecological systems), signal processing (structural analysis of electroencephalographic signals), epidemiology (timely detection and prevention of various types of diseases), and finance (buying or selling an asset). This project aims to develop novel optimal sequential procedures. A significant outcome will be the creation of computational infrastructure for identifying optimal decision rules in real applications.
Topic Title: Statistical Model-based Optimization
Supervisor: A/Prof Jun Ma
The literature concerned with the development of optimization techniques is both large and diverse. Optimization algorithms that construct some kind of statistical model and use this model to influence the search process can be found in areas such as Evolutionary Computation, Machine Learning and Engineering Design, as well as in the fields of stochastic and global optimization. The algorithms considered in this project will be based on a model of the density of promising points from a sample or population evaluated at a given iteration of the algorithm.
Topic Title: Statistical methods for high dimensional inverse problems
Supervisor: Dr Justin Wishart
Inverse problems are scenarios where signals of interest are not directly observable but extracted by inverting data. These are common challenges that pervade both scientific endeavour and everyday life. For example, image blur when using a wobbly camera or the echo heard in long distance phone calls. Inverting the system can be complex, especially when the signal of interest has irregular behaviour in different dimensions. Multiple sources of signals exist either by design or redundancy and should be collated properly when possible for ideal reconstruction. This project aims to develop new statistical methodology to exploit information from multiple noisy sources to produce the best estimates and attach confidence levels to their accuracy.
Topic Title: Theory, implementation and application of mixed conditional transformation models
Supervisor: Dr Maurizio Manuguerra
Transformation models are a wide class of models for the conditional distribution of some output. Subclasses of these models include linear regression, proportional hazard models in survival analysis and ordinal regression models for categorical or continuous scores, to cite a few. Transformation models play a central role in studies in which it is natural to model a latent variable, connected to the observations by some monotonic function. Several opportunities of research exist in the field of generalised additive mixed transformation models, including methodological work and implementation of multidimensional smothers, models with censored data (relevant in several fields, e.g. survival analysis), response-varying effects, constrained optimization. The outputs of the project will include participation in conferences, methodological and applied papers, and contributions to the development of an R package published on the CRAN