TLDR When is a player in form (over performing, or enjoying a hot streak) and how long does this last? If there is such an effect, I suspect it will be a result of some complicated system of personal circumstances. In this post I suggest a popular statistical model (Gaussian process) for approximating the dependencies (how many games back should we look?) and non-linearities (rise and fall of form) that we need. Again, I am suggesting that we should care about uncertainty when trying to model just about anything in football, and using probability is a helpful way of doing so. Ellen White’s dat ..read more

TLDR This is part 2 of an article on fitting a Bayesian partial pooling model to predict expected goals. It has the benefits of (a) quantifying aleatory and epistemic uncertainty, and (b) making both group-level (player-specific) and population-level (team-specific) probabilistic predictions. If you are interested in these ideas but not in statistical language, then you can also check out part 1. Expected Goals Expected Goals (or xG) is a metric that was developed to predict the probability of a football (soccer) player scoring a goal, conditional on some mathematical characterisation of the ..read more

TLDR The Expected Goals (xG) metric is now widely recognised as numerical measure of the quality of a goal scoring opportunity in a football (soccer) match. In this article we consider how to deal with uncertainty in predicting xG, and how each players individual abilities can be accounted for. This is part 1 of the article, which is intended to be free of stats jargon, maths and code. If you are interested in those details, you can also check out part 2. What are Expected Goals? Opta sports tell us that the Expected Goals (or xG) of a shot describe how likely it is to be scored. The cumulati ..read more

TLDR We sometimes have the option to purchase additional information (like paying for an experiment to be performed, or for a report from an expert consultant) to help us in problems of decision making under uncertainty. Deciding whether or not such services are worthwhile is challenging - how do we know how to value some information before we get it? A powerful Bayesian procedure, which unfortunately goes by many names (such as preposterior decision analysis, multi-stage decision analysis, and Bayesian experimental design) allows us to quantify this expected value - in other words, it tells u ..read more

TLDR Maximum Likelihood Estimation (MLE) is one method of inferring model parameters. This post aims to give an intuitive explanation of MLE, discussing why it is so useful (simplicity and availability in software) as well as where it is limited (point estimates are not as informative as Bayesian estimates, which are also shown for comparison). Introduction Distribution parameters describe the shape of a distribution function. A normal (Gaussian) distribution is characterised based on it’s mean, \(\mu\) and standard deviation, \(\sigma\). Increasing the mean shifts the distribution to be cent ..read more

TLDR This post is intended to be a high-level discussion of the merits and challenges of applied Bayesian statistics. It is intended to help the reader answer: Is it worth me learning Bayesian statistics? or Should I look into using Bayesian statistics in my project? Maths, code and technical details have all been left out. Bayes Bayes Introduction Firstly, Bayesian… Statistics Inference Modelling Updating Data Analysis …can be considered the same thing (certainly for the purposes of this post): the application of Bayes theorem to quantify uncertainty. So Bayesian statistics may be of in ..read more

TLDR Logistic regression is a popular machine learning model. One application of it in an engineering context is quantifying the effectiveness of inspection technologies at detecting damage. This post describes the additional information provided by a Bayesian application of logistic regression (and how it can be implemented using the Stan probabilistic programming language). Finally, I’ve also included some recommendations for making sense of priors. Introductions So there are a couple of key topics discussed here: Logistic Regression, and Bayesian Statistics. Before jumping straight into th ..read more

TLDR There are many instances where it may be useful to animate graphical representations of data, perhaps to add an additional dimension to a plot. The below example builds a cumulative map of car accidents in the UK using some of the functionality of the gganimate package. Making Moving Plots with gganimate Graphics made using the ggplot2 package are already extremely customisable. They can be further enhanced using some of the extensions that have been developed. These include providing access to new themes, as well as entirely new functionality. gganimate allows for the animation of exist ..read more