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Introduction Electricity differs from many financial and commodity assets because large-scale storage remains expensive, requiring continuous balancing of supply and demand. In the UK day-ahead market, hourly electricity prices are determined through auctions that incorporate expected generation and demand, as well as transmission constraints. In Great Britain, this balance is overseen by the National Energy System Operator (NESO), which manages the grid through a combination of forward markets and a real-time Balancing Mechanism. A key feature of the UK energy market is the day-ahead auction, which is facilitated by two exchanges: EPEX SPOT and Nord Pool’s N2EX. ...

Introduction Electricity differs from many financial and commodity assets because large-scale storage remains expensive, requiring supply and demand to be balanced continuously. In the UK day-ahead market, hourly electricity prices are determined through auctions that incorporate expected generation, demand, and transmission constraints. In Great Britain, this balance is overseen by the National Energy System Operator (NESO), which manages the grid through a combination of forward markets and a real-time Balancing Mechanism. A key feature of the UK energy market is the day-ahead auction, which is facilitated by two exchanges: EPEX SPOT and Nord Pool’s N2EX. The auctions operate such that all successful buyers and sellers receive the same clearing price, despite the large difference in cost to produce electricity between renewable and non-renewable methods. The price is determined by the most expensive energy producer to be accepted for that period; this is called the marginal price. The spot prices today were, therefore, set in yesterday’s auctions. Both producers and suppliers commit to their positions a day ahead. ...

Introduction Electricity differs from many financial and commodity assets because large-scale storage remains expensive, requiring supply and demand to be balanced continuously. In the UK day-ahead market, hourly electricity prices are determined through auctions that incorporate expected generation, demand, and transmission constraints. In Great Britain, this balance is overseen by the National Energy System Operator (NESO), which manages the grid through a combination of forward markets and a real-time Balancing Mechanism. A key feature of the UK energy market is the day-ahead auction, which is facilitated by two exchanges: EPEX SPOT and Nord Pool’s N2EX. The auctions operate such that all successful buyers and sellers receive the same clearing price, despite the large difference in cost to produce electricity between renewable and non-renewable methods. The price is determined by the most expensive energy producer to be accepted for that period; this is called the marginal price. The spot prices today were, therefore, set in yesterday’s auctions. Both producers and suppliers commit to their positions a day ahead. ...

Introduction In this article we will discuss how to execute a large number of independent programs in parallel, using a new computer chip capable of executing thousands of independent calculations at once. This may be particularly useful in areas of stochastic modelling and Monte Carlo simulation, where repeated and independent simulations are required to learn about uncertainty in model predictions. The chip in question is called the Intelligence Processing Unit (IPU) and is developed by Graphcore, a UK-based company headquartered in Bristol. The IPU is now (as of February 2023) on its second generation, the Colossus MK2 IPU processor, also known as the GC200. ...

Sampling from a finite set of weighted outcomes is a small task that appears surprisingly often in simulation, stochastic modelling, and Monte Carlo methods. In this post we build a compact C++ function for drawing from a discrete distribution, then compare its behaviour with std::discrete_distribution. We will briefly discuss how to sample a discrete random variable in theory and practice, before constructing our own algorithm and comparing its performance to the std::discrete_distribution in the C++ standard library. Many hand-coded discrete samplers have been proposed, but often these are unnecessarily clunky and, consequently, slow. ...

Before discussing Bayesian classifcation methods we intoduce the concept of mixture models, often used for unsupervised classification tasks. Mixture models are a diverse set of statistical models which can take an endless number of forms, the discussion here is limited on finite mixture models. The most common form of which is likely the Guassian mixture model (GMM), used for a variety of applications including clustering, an unsupervised classification method, and traditional modelling. Mixture models are often used to cluster like data-points into groups. In the Bayesian paradigm this is done by inferring a latent classification variable for each data-point. ...

The t-test is one of the most common tests used in statistics. It compares the means of two normally distributed datasets by calculating a test statistic and comparing it to the t-distribution. Variations of the test exist, including constant and differing variance between groups, paired and unpaired, and non-normality in the data. Here, Bayesian hypothesis testing is discussed, giving an indication of how a two-sample t-test would be conducted within a Bayesian context. Extensions to other situations are available, but the focus of this is to illustrate the how hyptothesis testing is done within the Bayesian paraidgm. ...

Introduction Graphcore, a company based in Bristol, UK, have developed a new kind of processor with the aim of being able to massively parallelise computation. The chip is called an Intelligence Processing Unit (IPU) and is now on its second generation, the Colossus MK2 IPU processor - the GC200. Originally designed to dramatically increase performance in machine learning applications, the IPUs architecture can also be used to provide big performance increases in many other areas. Here, we will show that the IPUs ability for massive parallelisation can be used to decrease the computational time for repeated simulation of dynamic, stochastic models. This family of models are used in a wide variety of applications including biology, where they can be used to perform in silico experiments, and finance, where they can be used to forecast asset prices. ...

Hierarchical models are synonymous with Bayesian methods, which is better equiped to be able to infer the parameters of large and complex models when compared to frequentist methods. They are popular as they allow simple building blocks to be combined to form a large and complex model. The ability of hierarchical models to reflect complex systems means they have been applied to a variety of modelling situations, within all aspects of science. ...

Unlike frequentist statisticians, Bayesian statisticians believe that there is no ’true’ value of the parameters in a statistical model. Instead, Bayesian methods summarise parameter beliefs after observing data (a posteriori) by a probability distribution, giving more weight to more likely values. Parameter beliefs before observing any data (a priori) are called prior beliefs, and are similarly summarised by a probability distribution. Prior beliefs can be as vague or informed as required to reflect the beliefs of relevant experts. A lack of prior can be reflected by vague prior beliefs, with a high variance. In contrast, a large amount of prior knowledge from previous work may result in well-informed priors with high precision. Parameter beliefs are updated by combining prior beliefs and new evidence presented from a dataset. Bayesian methodology is fully probabilistic and considers model parameters, hidden states, as well as missing and observed data in the same vein. The coherent treatment of model parameters, data, and hidden states has allowed Bayesian methods to be used in a wide range of inference problems. ...