This unit will cover advanced numerical and computational methods within an engineering context. The context will include parallel coding using MPI, computational architecture, advanced numerical methods including spectral methods, finite difference schemes and efficient linear solvers including multi-grid solvers and Krylov subspace solvers. Students will develop to skills and confidence to write their own computational software. Applications in fluid and solid mechanics will be covered.

In our interconnected world, networks are an increasingly important representation of datasets and systems. This unit will investigate how this network approach to problems can be pursued through the combination of mathematical models and datasets. You will learn different mathematical models of networks and understand how these models explain non-intuitive phenomena, such as the small world phenomenon (short paths between nodes despite clustering), the friendship paradox (our friends typically have more friends than we have), and the sudden appearance of epidemic-like processes spreading through networks. You will learn computational techniques needed to infer information about the mathematical models from data and, finally, you will learn how to combine mathematical models, computational techniques, and real-world data to draw conclusions about problems. More generally, network data is a paradigm for high-dimensional interdependent data, the typical problem in data science. By doing this unit you will develop computational and mathematical skills of wide applicability in studies of networks, data science, complex systems, and statistical physics.

With explosions in availability of computing power and facilities for gathering data in recent times, a key skill of any graduate is the ability to work with increasingly complex datasets. There may include, for example, data sets with multiple levels of observations gathered from diverse sources using a variety of methods. Being able to apply computational skills to implement appropriate software, as well as bringing to bear statistical expertise in the design of the accompanying algorithms are both vital when facing the challenge of analysing complicated data. This unit is made up of three distinct modules, each focusing on a different aspect of applications of statistical methods to complex data. These include (but are not restricted to) the development of a data product that interrogate large and complicated data structures; using sophisticated statistical methods to improve computational efficiency for large data sets or computationally intensive statistical methods; and the analysis of categorical ordinal data. Across all modules you will develop expertise in areas of statistical methodology, statistical analysis as well as computational statistics. Additional modules may be delivered, depending on the areas of expertise of available staff and distinguished visitors.

Increased computing power has meant that many Bayesian methods can now be easily implemented and provide solutions to problems that have previously been intractable. Bayesian methods allow researchers to incorporate prior knowledge into their statistical models. This unit is made up of three distinct modules, each focusing on a different niche in the application of Bayesian statistical methods to complex data in, for example, geophysics, ecology and hydrology. These include (but are not restricted to) Bayesian methods and models; statistical inversion; approximate Bayesian inference for semiparametric regression. Across all modules you will develop expertise in Bayesian computational statistics. On completion of this unit you will be able to apply appropriate Bayesian methods to a variety of applications in science, and other data-heavy disciplines to develop a better understanding of the information inherent in complex datasets.

There is no doubt of the increasing demand for machine learning models in academia and industry. However, simple linear models and most of online software packages do not work reliably when analysing real world datasets. There is a need to learn more complex probabilistic machine learning models that have the power to generalise and quantify uncertainty to properly inform decision making in complex scenarios. This unit will cover Bayesian machine learning models for the analysis of complex datasets. Starting with linear and non-linear Bayesian parametric regression, the unit will cover generalized linear models, neural networks, Gaussian processes and non-parametric regression models, probabilistic graphical models, finite and infinite mixture models, models for time series and longitudinal data and deep learning models. This unit provides the knowledge to discern between the type of model you should implement for any data you encounter, depending on their advantages and limitations. This unit will also give you insights on the interpretability of models, which is imperative for informed decision making from model predictions. This unit includes practical programming sessions in the tutorials and will also train students in a small research project, which will allow students to put the concepts, knowledge, and experience into practice.

Statistical Machine Learning has become a core skill for solving real world problems with limited data. In this unit you will learn the mathematical foundations and practical implementation of a variety of estimation algorithms. The unit will first introduce advanced statistical inference, followed by a detailed presentation of a variety of methods for achieving inference. These include simulation-based methods: Markov Chain Monte Carlo (MCMC), Hamiltonian Monte Carlo (HMC), Sequential Monte Carlo (SMC), Approximate Bayesian Computation (ABC), Bayesian Optimisation, as well as Variational Inference, Dimension Reduction and Model Selection and Averaging. This course not only provides core and theoretical knowledge, but also covers the practical implementation of these estimation algorithms, including examples in multiple programming languages (R, Python and C++) and with real world datasets. Students who complete this unit will develop critical skills to correctly use advanced statistical machine learning in practice. This unit will not only improve analytical skills but also encourage work in a multidisciplinary team and work on real world problems with datasets provided by the ARC Centre on Data Analytics for Resources and Environment (DARE).

Molecular and systems biology have become data-intensive sciences owing to the fast-growing omics technologies that enable the profiling of genome, epigenome, transcriptome, and proteome at full scale and, increasingly, at the single-cell level. Computational and statistical methodologies are now indispensable for analysing omics data generated from high-throughput technologies. This unit will introduce you to commonly used computational and statistical methods in omics data analysis. You are encouraged to use your own data to construct the models to visualise your research and interpret results. Learning the correct use of computational methods for various omics data analysis applications including your own data, you will develop an essential knowledge of methods and techniques in analysing omics data. This will provide a strong foundation for using computational approaches in omics-based molecular and systems biology research.

The objectives of this unit of study are to develop an understanding of modern computationally intensive methods for statistical learning, inference, exploratory data analysis and data mining. Advanced computational methods for statistical learning will be introduced, including clustering, density estimation, smoothing, predictive models, model selection, combinatorial optimisation methods, sampling methods, the Bootstrap and Monte Carlo approach. In addition, the unit will demonstrate how to apply the above techniques effectively for use on large data sets in practice.

The great power of the discipline of Statistics is the possibility to make inferences concerning a large population based on optimally learning from increasingly large and complex data. Critical to successful inference is a deep understanding of the theory when the number of samples and the number of observed features is large and require complex statistical methods to be analysed correctly. In this unit you will learn how to integrate concepts from a diverse suite of specialities in mathematics and statistics such as optimisation, functional approximations and complex analysis to make inferences for highly complicated data. In particular, this unit explores advanced topics in statistical methodology examining both theoretical foundations and details of implementation to applications. The unit is made up of distinct modules that may include (but are not restricted to) asymptotic theory for statistics and econometrics, theory and algorithms for statistical learning with big data, and introduction to optimal semiparametric optimality.

The great power of the discipline of Statistics is the possibility to make inferences concerning a large population based on only observing a relatively small sample from it. Of course, this "magic" does not come without a price, we must construct statistical models to approximate these populations and samples from them, develop mathematical tools using probability theory, appreciate the limitations of our methods and, most importantly, understand what assumptions need to be made for such inferences to be valid, and develop ways to check these assumptions. Implementing these methods to possibly complex data structures is also a challenge that must be overcome. This unit explores advanced topics in statistical methodology examining both theoretical foundations and details of implementation to applications. The unit is made up of 3 distinct modules. These include (but are not restricted to) Advanced Survival Analysis, Extreme Value Theory and Statistical Methods in Bioinformatics.