MSc Computational Data Science

Semester 1: Mandatory

Delivered by: School of Engineering and Physical Sciences

Course overview

What is the most efficient way to sense, or sample signals that we want to observe? Once data have been acquired, how do we retrieve the sought signal from these acquired data? Such "inference" or "inverse" problems are core to Data Science, particularly when the size of the signal is large and the "inference algorithms" need to be "scalable". This module approaches these questions both from a theoretical perspective (underpinned by the theories of compressive sensing, convex optimisation, deep learning) and in the context of "computational imaging" applications in a variety of domains ranging from astronomy to medicine.

Course syllabus

The first part of the course will be taught. We will review the basic notion of Nyquist sampling of signals and rapidly dive into "computational imaging". In this context, mathematical algorithms need to be designed to solve an "inference" or "inverse" problem for image recovery from incomplete data. The size of the variables of interest in modern imaging application (e.g. in astronomy or medicine) can be very large. In our journey, and concentrating on the data processing (rather than hardware, or application) aspects, we will learn the basics of the theories of compressive sensing (which tells us how to design intelligent data acquisition schemes for sub-Nyquist sampling) and convex optimisation (which provides a whole wealth of algorithms capable to solve inverse problems, and scalable to high-dimensional problems). The second part of the course will take the form of a project which will enable us to explore how machine learning algorithms (more specifically deep neural networks) can provide an alternative framework to solve inverse problems (in particular for imaging applications), or otherwise integrate and enhance optimisation algorithms.

Semester 1: Mandatory

Delivered by: School of Mathematical and Computer Sciences

Course overview

In this course students will study fundamental concepts and techniques used in data mining and machine learning, with a focus on the mathematics underpinning these concepts. The course will discuss the data mining and machine learning techniques, relationships between them, and ways to determine which ones to use in a particular scenario. The course will also explore applications of the various data mining and machine learning techniques.

Course syllabus

The course will start with the introduction of some basic concepts in machine learning (classification, clustering, supervised and unsupervised learning). Students will then work on generative models such as probabilistic graphical models, cluster analysis (including k-means clustering, expectation maximisation and mixture models), regression analysis. We will then focus on discriminative Learning: Instance-based learning and decision tree learning, artificial neural networks (perceptron, multilayer perceptron, back-propagation, deep learning architectures), maximum entropy models, support vector machines, ensemble methods (such as bagging and boosting).

Semester 1: Mandatory

Delivered by: School of Mathematical and Computer Sciences

Course overview

This is a core course where you will learn the methods and tools of classical statistical inference, and the mathematics underpinning them. Students will also be able to apply the results and techniques they learn in this course working on an extended project.

Course syllabus

At the beginning of the course, students will study in detail key concepts in the classical statistics theory: parameter estimation, likelihood, hypothesis testing, credibility theory. An applied project will follow the theory, where students will be able to use the techniques and tools.

Semester 1: Optional

Delivered by: School of Mathematical and Computer Sciences

Course overview

Mathematical Ecology aims to provide students with an advanced knowledge and understanding of the mathematical modelling methods that describe population dynamics, epidemiological processes and evolutionary processes in ecological systems. It will provide training in a wide variety of mathematical techniques which are used to describe ecological systems and provide instruction in the biological interpretation of mathematical results.

Course syllabus

The course starts with a discussion of modelling and analysis of single- and multi-species population models, in both discrete and continuous time. We will then study classical mathematical biology models including models for interacting species, symbiotic, competitive, predator-prey and host-parasite ecological interactions, age-structured models. The second part of the course is devoted to mathematical models of ecological systems, epidemiological models and evolutionary game theory.
 

Semester 1: Optional

Delivered by: School of Engineering and Physical Sciences

Course overview

How can machines learn concepts by just observing examples? Learning theory provides several tools to answer this question and without which machine learning cannot be really understood. Recent developments brought by deep learning have had tremendous impact in several applications, notably in computer vision. Explaining its success, however, requires a revision of learning theory, a topic explored in this module. Applications of the theory and examples of deep learning architectures are illustrated on several computer vision tasks, from image classification to semantic segmentation and scene reconstruction. 

Course syllabus

Part I of the course establishes the foundations of learning theory, introducing concepts like risk, empirical risk, and generalization, and surveying algorithms widely used in machine learning, e.g., stochastic gradient descent. It also introduces the basic components of deep learning architectures and discusses several practical aspects of their deployment. These elements are then put together and illustrated in Part II which, after an introduction to basic concepts in computer vision geometry, surveys different deep learning solutions that have been applied in computer vision tasks.   

Semester 1: Optional

Delivered by: School of Mathematical and Computer Sciences

Course overview

The course introduces students to fundamental stochastic processes which are often used in stochastic modelling and data science. The course introduces key constructs and theoretical results. Students will also work on a project where they can apply the acquired techniques.

Course syllabus

The course starts with the introduction of random walks and introduces some problems one can study using them: rare events and large deviations. We then continue exploring discrete-time processes with a detailed study of Markov chains. The second part of the course looks at continuous-time processes starting with Poisson processes and moving to more general Markov processes. We then study renewal theory and introduce martingales.

Semester 2: Mandatory

Delivered by: School of Engineering and Physical Sciences

Course overview

Iterative optimisation algorithms often break down when the dimensions of its variables and/or the size of its data become very large. Building on the Optimisation and Deep Learning for Imaging and Vision I course, this module peeks into a class of very recent methods to handle very large-scale optimisation problems and showcases their application in computational imaging and computer vision problems. Among the techniques explored in the module are advanced proximal methods, acceleration techniques, development of algorithms via duality, and hybrid methods that combine optimisation algorithms with deep learning. Finally, as data science applications require taking decisions under uncertainty, the course also surveys Bayesian techniques to quantify the uncertainty in the output of an algorithm. 

Course syllabus

The module consists of lectures and discussions, three practical projects, and a reading project. The lectures comprise four chapters: 1) Review of optimisation concepts and motivation for the need of more advanced optimisation algorithms. 2) Advanced proximal algorithms, including acceleration via inertia and scalability using duality. 3) Stochastic algorithms are introduced as solutions to handle large datasets. 4) Basics of Bayesian inference and methods to quantify uncertainty of maximum a posteriori estimates. In the practical projects, students have the opportunity to implement these algorithms and techniques. And the reading project explores connections between optimisation algorithms and deep learning, including unfolded neural networks and plug-and-play algorithms.

Semester 2: Mandatory

Delivered by: School of Mathematical and Computer Sciences

Course overview

Leading successful projects, and in particular research projects, requires methodological tools in addition to technical expertise. This course aims at preparing students for carrying out an extended research or development project in a science or engineering programme by developing their skills in critical thinking, research planning and management, academic writing, experimental design and data handling. Note that this course is delivered to several other MSc programs in Engineering and Physical Sciences. 

Course syllabus

This course is primarily taught and will also involve independent work to prepare your MSc Project. The taught part of the course will discuss how to plan a research/development project, interact efficiently with supervisors and carry out a literature review. This part will also cover how to develop writing skills to prepare the MSc dissertation, as well data analysis tools and how to report results. The tools and skills acquired during the course will then be applied to the preparation of a MSc project portfolio, to be prepared with the project supervisor. 

Semester 2: Mandatory

Delivered by: School of Mathematical and Computer Sciences

Course overview

The course introduces students to modern Bayesian Statistical inference, its theory and applications. The course also studies applications of computational methods in statistics and stochastic simulation methods including Markov Chain Monte Carlo (the famous MCMC). Students will also be able to implement the Bayesian approach in practical situations.

Course syllabus

The course starts with a review of the use of R for probabilistic and statistical calculations. It then discusses the philosophy of Bayesian inference. This includes the comparative treatment of Bayesian and frequentist (classical) approaches. We then proceed to the implementation of the Bayesian approach. This will include the formulation of likelihood for a range of statistical models and sampling designs, the incorporation of prior knowledge through prior density selection, conjugacy, the use of non-informative and non-subjective priors, the interpretation of the posterior distribution as the totality of knowledge, predictive distributions. 

The second part of the course focuses on Markov-chain and other stochastic methods for investigating target distributions. Ideas include simple simulation methods using transformations, distribution function inversion and acceptance-rejection sampling, construction of MCMC methods using standard recipes - Metropolis (and Metropolis-Hastings) algorithm, Gibb's sampler, sequential Monte Carlo, approximate Bayesian computation. Most methods will be implemented using the R computing package.
 

Semester 2: Optional

Delivered by: School of Mathematical and Computer Sciences

Course overview

The course introduces techniques of data assimilation in numerical weather prediction and climate change modelling. These will include basic regression analysis, variational approaches, Kalman filtering, extended and ensemble Kalman filtering and the Bayesian inference approach. The course will teach practical implementation of these data assimilation techniques in the context of computer simulations, which will be illustrated by prototype applications. These methodologies will form the basis for a series of modelling case studies as well as the group-based project component of the course.

Course syllabus

The course starts with some background information including data sets, statistics and elements of analysis. It then introduces and studies various methods of data assimilation including variational approach, Kalman filtering and Bayesian approach. Implementation of these approaches is also discussed. The course then proceeds with an extended simulation project related to biology, climate change or finance, including the modelling and subsequent direct numerical implementation of one or more of the data assimilation approaches above. The project includes a background literature search, development of the underlying model, assessment of the data and appropriate data assimilation techniques that can be applied, hands-on simulation of one or more of these techniques, a group-based presentation and a written report.

Semester 2: Optional 

Delivered by: School of Engineering and Physical Sciences

Course overview

Data is meaningless if it cannot be related to other data. For example, an individual pixel of an image can hardly reveal the object it belongs to unless we look at its surrounding pixels; the social network of a person can only be inferred by looking at the person’s interactions with other people. The appropriate tool to capture such relationships are graphs. They are fundamental and can unveil unknown connections in data analysis, signal processing, and computer vision. In this module, we explore graphs, their properties, and how they can be used to perform inference, including on probabilistic models, and capture dynamic relationships. Their impact and applicability will be illustrated in imaging and vision applications.

Course syllabus

The course consists of three parts. Part I introduces the concept of graphs, describing their properties, and covers spectral graph theory. Part II considers probabilistic models defined on graphs, and introduces several graph-based algorithms, including message-passing, variational methods, and inference with dynamic models. Part III introduces Bayesian neural networks, covering their principles, explores imaging applications of variational auto-encoders, and describes Bayesian deep neural networks. Labs complement the lectures and include implementations of large-scale message-passing algorithms, graph-based methods for dynamic imaging, and training and evaluation of variational auto-encoders using high-performance computing platforms.

Semester 2: Optional

Delivered by: School of Mathematical and Computer Sciences

Course overview

The course is focused on the mathematics underpinning the stochastic modelling of networks of practical interest in the modern world (social networks, computer networks, large server systems) and processes on them (spread of data through a social network, spread of virus through a computer network, data storage). To this end, the course studies branching processes and more general random graphs as well as stochastic processes defined on these objects. The course also provides a formal mathematical interpretation for the exponential growth of epidemics - something that became important to everyone recently.

Course syllabus

The course starts with a review of some probabilistic methods and techniques. We then introduce and study branching processes in discrete and continuous time. These are later generalised to the notion of a random graph. We study important models including Erdos-Renyi random graphs, inhomogeneous random graphs, preferential attachment and configuration models.