In this module, you will be introduced to essential statistical and econometric techniques for data analysis. You will learn how to model relationships between variables and make predictions using the established linear regression model framework. Additionally, you will explore discrete choice models, which are useful for decision-making scenarios with two outcomes, such as yes/no or success/failure.

You will also explore two important estimation methods: Ordinary Least Squares (OLS) and Maximum Likelihood Estimation (MLE) for estimating linear regression and discrete choice models. We will highlight the importance of statistical and diagnostic testing in the context of econometrics applications.

We will illustrate all these techniques empirically using real-word data examples and popular statistical software languages, such as Python and R. This hands-on approach will help you develop practical skills in data analysis and draw meaningful conclusions.

This module will enhance your knowledge in accounting and finance. You will first focus on financial accounting, exploring essential topics like the valuation of inventory and non-current assets such as machinery, buildings and land. This knowledge is critical for understanding financial statements. You will examine financial accounting in limited liability companies and sharpen your skills in analysing financial statements.

In terms of management accounting, you will learn techniques to take control of business costs and performance. You will master the art of costing through absorption and activity-based costing methods, and perfect your approach to budgetary control for planning and effective oversight.

In finance, you will cover key areas of financial markets, including traditional equity, bond and currency markets as well as modern development such as cryptocurrencies. You will also tackle the fascinating subjects of capital investment decision-making, risk and return, and calculating the cost of capital. These are essential tools to help businesses plan for a profitable future.

Building on your knowledge of vectors and matrices, this module explores the elegant framework of linear algebra, a powerful mathematical toolkit with remarkably diverse applications across statistical analysis, advanced algebra, graph theory, and machine learning.

You'll develop a comprehensive understanding of fundamental concepts, including vector spaces and subspaces, linear maps, linear independence, orthogonality, and the spectral decomposition theorem.

Through individual exploration, small-group collaboration, and computational exercises, you'll gain both theoretical insight and practical skills. The module emphasises how these abstract concepts translate into powerful problem-solving techniques across multiple disciplines, preparing you for advanced studies while developing your analytical reasoning abilities.

Statistics allows us to estimate trends and patterns in data and gives a principled way to quantify uncertainty in these estimates. The findings can lead to new insights and support decision-making in fields as diverse as cyber security, human behaviour, finance and economics, medicine, epidemiology, environmental sustainability and many more.

Dive into the behaviour of multivariate random variables and asymptotic probability theory, both of which are central to statistical inference. You will then be equipped to explore one of the most fundamental statistical models, the linear regression model, and learn how to apply general statistical inference techniques to multi-parameter statistical models. Statistical computing is embedded in the module, allowing you to investigate multivariate probability distributions, simulate random data, and implement statistical methods.

Never has the collection of data been more widespread than it is now. The extraction of information from massive, often complex and messy, datasets brings many challenges to fields such as statistics, mathematics and computing.

Develop the skills and understanding to apply modern statistical and data-science tools to gain insight from contemporary data sets. By addressing challenges from a variety of applications, such as social science, public health, industry and environmental science, you will learn how to perform and present an exploratory data analysis and deploy statistical approaches to analyse data and draw conclusions. You will also develop judgement to critically evaluate the appropriateness of different methods for real-world challenges.

Machine learning is at the heart of modern AI systems, and it is a fundamentally mathematical subject. You will learn this mathematics by discovering how techniques are deployed in several AI systems, including the neural networks that have revolutionised the field.

You’ll start by building connections with previously encountered approaches through the unifying concept of a loss function of a parameter vector. For example, with a neural network model the vector input is the set of weights, and the loss function might be the prediction error on a dataset.

The goal is to find a vector input that produces a small loss; in the above example, this is known as training the neural net. You will learn and deploy some of the key mathematical ideas and numerical techniques, such as back propagation and stochastic gradient descent, that enable the automated iterative learning of a good vector input.

Many of the most important real-world challenges, from predicting climate change, to modelling the spread of disease, are described by equations that cannot be solved analytically. To start, you will be introduced to techniques for tackling such problems, beginning with fundamental numerical methods, such as the trapezium rule and Euler’s method, before progressing to more advanced techniques and quantifying the accuracy, stability and limitations of these methods. Alongside numerical approaches, you will also develop heuristic methods to characterise a system's limiting behaviour.?

Other familiar phenomena, such as pulses of light down a fibre optic cable to the shudder of turbulence on a plane, involve multiple variables, such as time and position. Their mathematical description requires differentiation with respect to each of these independent variables, leading to partial differential equations (PDEs). You will learn how to formulate PDEs for complex, real-world problems and practice core techniques for solving them. By the end of the module, you will have the tools to build and analyse mathematical models that underpin emerging challenges across engineering, physics, biology and the environment.?

This advanced module in accounting and finance will give you the tools to navigate the complexities of modern business environments.

In financial accounting, you will explore the critical areas of disclosure and earnings management, unravelling how these practices shape perceptions of a company’s performance. You will also investigate the pivotal role of auditors and the importance of maintaining auditor independence in safeguarding financial integrity.

In management accounting we will focus on strategic management accounting, where you will learn how to align accounting strategies with broader business goals. You will also examine the recent developments in value creation, discovering how organisations generate and sustain competitive advantages.

Finally, in finance, we will delve deeper into various aspects of financial markets, as well as on capital structure and payout policy, key factors that influence how businesses fund their operations and distribute profits to shareholders.

Stochastic processes are fundamental to probability theory and statistics and appear in many places in both theory and practice. For example, they are used in finance to model stock prices and interest rates, in biology to model population dynamics and the spread of disease, and in physics to describe the motion of particles.

During this module, you will focus on the most basic stochastic processes and how they can be analysed, starting with the simple random walk. Based on a model of how a gambler's fortune changes over time, it questions whether there are betting strategies that gamblers can use to guarantee a win. We will focus on Markov processes, which are natural generalisations of the simple random walk, and the most important class of stochastic processes. You will discover how to analyse Markov processes and how they are used to model queues and populations.

This module provides a solid foundation in banking and behavioural finance. You will explore the fundamentals of banking, including the role of financial institutions in the economy and the complexities of their business activities, such as lending and risk management.

The module highlights the role of regulation and supervision in the banking industry, highlighting how these aspects differentiate banking from other industries, and how crises reshape regulation in banking. You will be introduced to the fascinating world of behavioural finance, where you will discover how psychological factors and cognitive biases such as overconfidence, loss aversion and herding behaviour influence decision-making in financial markets.

By combining traditional banking concepts with the latest insights from behavioural finance, this module offers a unique perspective on the interplay between human behaviour and financial systems. With real-world examples, you will gain practical insights into how these concepts apply to both individual and institutional decision-making.

Understanding how data evolves over time is crucial across numerous sectors, from finance and engineering to climate science. Develop the tools to analyse temporal data, detect structural changes and build predictive models.

Using changepoint detection algorithms, you will learn how to identify abrupt changes in the mean or variance of a process, or parameters in a regression model. These methods will be introduced from a foundational perspective, developing both computational and mathematical understanding. You will then learn to handle temporal dependence by studying a popular range of time-series models, using these to generate insights about the data and produce forecasts. Throughout the module, you will learn to critically evaluate and compare models, whilst applying these techniques to real-world challenges.

Models of dynamical systems are fundamental to our understanding of the physical and natural world.

Explore a new class of model, the Markov jump process, for the time evolution of dynamical systems such as the evolution of species populations in the wild and the spread of infectious diseases. You will learn how to simulate from these processes and will study methods for understanding their properties and behaviours. Unlike deterministic differential equation models, Markov jump processes are random, allowing for different behaviour every time they are simulated. You will discover how it is often possible to associate a jump process with a related differential equation approximation and that this can provide important insights into the behaviour of the jump process and the original real-world system.

Statistical techniques are often applied to environmental data, such as air temperatures, rainfall or wildfire locations. You will learn about some of the common features of such datasets and how these features are used to design statistical models. You will first be introduced to the Gaussian process model for continuous spatial processes. You will learn about the properties of the Gaussian process and implement this model for spatial data analysis, before investigating methods for point-reference data, such as earthquake or wildfire locations.

You will also dip into natural hazard risk management, which seeks to mitigate the effects of events, such as flooding or storms, in a manner that is proportionate to the risk. You will learn basic concepts from extreme value theory, including the appropriate distributions for extremes, and how to use these as statistical models for estimating the probability of events more extreme than those in the dataset.

This module provides a solid foundation in Environmental, Social, and Governance (ESG), climate and energy finance by exploring the financial aspects of transitioning to a sustainable future. You will explore how businesses integrate ESG factors into strategy and operations, addressing investor expectations, regulatory demands, and societal pressures.

The module covers:

• ESG data collection
• measurement
• reporting frameworks, including TCFD, GRI, and national regulations

It also examines the challenges of incorporating ESG into risk management, supply chains, and corporate performance assessments. Additionally, you will study how financial markets, investment strategies, and policy decisions align with global climate change efforts, including green bonds, climate risk assessments, and sustainable finance. The module also explores the financial aspects of both traditional and renewable energy markets.

Combining theory with practical insights, this module prepares you to navigate the growing field of sustainable finance and contribute to the transition to a low-carbon economy.

The study of graphs (mathematical objects used to model networks and pairwise relations between objects) is a cornerstone of discrete mathematics. Graphs can represent important real-world situations, and the study of algorithms for graph-theoretical problems has strong practical significance.

You will learn about structural and topological properties of graphs, including graph minors, planarity and colouring. We will introduce several theoretical tools, including matrices relating to graphs and the Tutte polynomial. We will also study fundamental algorithms for network exploration, routing and flows, with applications to the theory of connectivity and trees, considering implementation, proofs of correctness and efficiency of algorithms.

You will gain experience in following and constructing mathematical proofs, correctly and coherently using mathematical notation, and choosing and carrying out appropriate algorithms to solve problems. The module will enable you to develop an appreciation for a range of discrete mathematical techniques.

Linear systems of differential and integral equations provide a mathematical model for a wide range of real-world devices, including communication systems, 5G networks, electrical circuits, heating systems and economic processes. Mathematical analysis of these models gives insight into the behaviour of these devices, with applications in automatic control, signal processing, wireless communications and numerous other areas.

Linear systems are considered in continuous time that reduce to a standard (A,B,C,D) state space representation. Via the Laplace transform, these are reduced further to rational transfer functions. Linear algebra enables us to classify and solve (A,B,C,D) models, while we describe their properties via diagrams in standard computer software. You will consider feedback control for linear systems, describing the rational controllers that stabilise an (A,B,C,D) system. Alongside the development of analytic methods to study linear systems, you will also gain experience in modelling real-world devices by such systems.

Mathematical models are central to financial decision making. You will discover the mathematical foundations necessary to model certain transactions in the world of finance. You will then study stochastic models for financial markets and investigate the pricing of European and American options and other financial products.

You will explore two discrete models, the binomial model and the finite market model, and one continuous model. Following an introduction to some probabilistic terminology, such as sigma algebras and martingales, and some financial terminology such as arbitrage opportunities and self-financing trading strategies, you will deduce the Black Scholes formula. You will also gain a brief overview of Brownian motion.

From denoising diffusion to flow matching, modern generative models are governed by elegant mathematics: stochastic differential equations, PDEs for probability evolution and transport on spaces of measures. This module develops that mathematical toolkit and shows how it underpins today’s state-of-the-art image, audio and scientific generative models.?

We start from how probability distributions evolve over time (continuity and Fokker–Planck equations) and show how this leads to a reverse-time stochastic differential equation and an equivalent probability-flow ODE. We then look at discrete-time diffusion models and explain why their training objective is a practical stand-in for maximum likelihood estimation.

You will be introduced to score matching and denoising score matching, continuous-time formulations and the main numerical solvers (e.g. Euler–Maruyama, predictor–corrector), together with sensible choices of noise/step-size schedules. In parallel, we cover flow-based models: continuous normalising flows and flow matching, which fit a velocity field along a path between distributions, with links to optimal transport and Schrödinger bridges. We finish with fast samplers (e.g., distilled/consistency models) and with how to judge models in practice - negative log-likelihood, bits per dimension and coverage - while balancing against compute cost, stability and common failure modes.?

By the end, you will be able to read and reproduce the derivations that make these models work, implement small-scale prototypes, and reason from first principles about design choices such as noise schedules, guidance and solver accuracy.??

Optimisation is the hidden engine behind the remarkable success of modern AI. Training an AI model, whether a simple regression or a state-of-the-art Transformer architecture, ultimately boils down to minimising a loss function that encodes both the training data and the neural architecture. The optimisation algorithms that make this possible are not only efficient in practice, but also mathematically elegant and broadly applicable across science, engineering and economics.

You will develop the mathematical foundations of optimisation and see how they translate into practice. We will begin with convexity and smoothness of functions before introducing core optimisation schemes and key theoretical notions. Building on this foundation, you will study gradient descent and its many refinements, including momentum, acceleration and adaptive methods that drive modern AI training. Along the way, you will also explore duality and mirror descent, which provides rich algorithmic perspectives, and second-order methods that exploit curvature for faster convergence.

You will gain a solid mathematical understanding of optimisation algorithms and the ability to design algorithms that address real-world constraints such as limited memory, compute and scalability. These skills will prepare you to both analyse algorithms rigorously and adapt them to the practical challenges encountered in AI, data science and beyond.

Building on the statistical techniques explored so far, you deepen your understanding of both the theoretical underpinnings and practical application of frequentist statistical inference. You will then be introduced to an alternative paradigm: Bayesian statistics.

The frequentist perspective views all probabilities in terms of the proportions of outcomes over repeated experimentation and has been the foundation of hypothesis testing and experimental design over years of data-driven science and research. Meanwhile, the increasingly popular Bayesian approach arises directly from Bayes theorem, avoiding hypothetical repeated sampling. As a result, Bayesian statistics is often more intuitive and easier to communicate and naturally takes all forms of uncertainty into account.

With this in mind, you will compare and contrast these two perspectives and their associated tools. You will learn to select and justify an appropriate methodology for inference and model selection, and to reason about the uncertainty in your findings within each paradigm.

Statistics and machine learning share the goal of extracting patterns or trends from very large and complex datasets. These patterns are used to forecast or predict future behaviour or interpolate missing information. Learn about the similarities and differences between statistical inference and machine learning algorithms for supervised learning and how the two approaches can be used for classification and prediction.

You will explore the class of generalised linear models, which is one of the most frequently used classes of supervised learning model. You will learn how to implement these models, how to interpret their output and how to check whether the model is an accurate representation of your dataset. Lastly, you will have the opportunity to see how regularisation and dimension-reduction techniques can be used to apply these models to the case of the ‘large p, small n’ question. This phrase refers to datasets with many more variables than samples.