# Research

## PhD: Ranking Systems in Sport

## Motivation:

Pick your favourite sport. Chances are, you have an opinion on who the best in the world is, at this current moment, or of all time, or who would win if A played B. But is it possible to develop a system which returns an objective answer to these questions?

In developing such systems, it is crucial to capture as much information as possible about the dynamic world in which we live. Understand it. Learn from it. Predict it. Athlete’s injuries, the weather, and even economic factors all impact the outcome of these events and the implied ability of the athletes or teams. This project requires a wide range of strategies in order to capture these signals, from graph theory to extreme value theory, and contextual information from news websites, so that the most accurate system of ranking sports teams or athletes is formulated.

Ranking systems in sport are not only interesting to the inquisitive fan, but a fair and accurate system is at the core of all sports organisational bodies and the multi-billion pound industries that they represent.

But these systems are not exclusive to sports.

Methodological advances in the field of sports ranking systems have far-reaching consequences. Ranking systems are used to rank webpages, or to rank schools and hospitals, or even to determine the most essential medical treatments. So, a ranking system based on poor methodology can have much more severe repercussions than incorrectly seeding a tennis tournament… Ultimately, the importance of ranking systems is self-evident, and sport creates a fruitful playground in which ample advancements can be made.

## Details:

The aim of this project is to analyse the statistical properties of world ranking systems for sports, and develop new and improved ranking systems. Defining a ‘good’ ranking system depends on the type of system, and this can be split into three categories:

- Inference: A ranking system used for inference will provide deep insight into the characteristics of the given sport, and the participants.
- Prediction: A ranking system that is used to predict future results must provide accurate predictions.
- Seeding: A system designed to seed players for a tournament needs to be robust to exploitation, fair, and easy to understand.

- Ranking systems must carefully balance long-term performance with recent performance: they must adapt quickly to new information, without over-reacting to the random fluctuations caused by individual results.
- Do the rankings in different sports tend to "evolve" at different speeds, and if so, why?
- Is there any evidence that rankings for team sports evolve more quickly than for individual sports?
- What level of predictive power is provided by the rankings, in terms of forecasting the result of a match or a tournament? How does this predictive power vary across different sports?
- In many ranking systems, different tournaments are assigned different weightings or point values to reflect their perceived level of importance. What statistical evidence is there that these weightings are set appropriately?
- In which situations might it be valuable to assign a multi-dimensional ranking, representing, for instance, a tennis player's ability on different surfaces, or a golfer's ability on different kinds of course? How quickly might we be able to identify players or teams who perform strongly or weakly under particular circumstances?

Listen to this clip from the Royal Statistical Society's audio podcast series, where I speak to Brian Tarran at the 2019 RSS conference in Belfast about some of my current research.

### Publications

- H.Spearing, J.Tawn, D.Irons, T.Paulden, G.Bennett;
*Ranking, and other Properties, of Elite Swimmers using Extreme Value Theory;*submitted [arXiv].

Abstract: The International Swimming Federation (FINA) uses a very simple points system with the aim to rank swimmers across all Olympic events. The points acquired is a function of the ratio of the recorded time and the current world record for that event. With some world records considered better than others however, bias is introduced between events, with some being much harder to attain points where the world record is hard to beat. A model based on extreme value theory will be introduced, where swim-times are modelled through their rate of occurrence, and with the distribution of the best times following a generalised Pareto distribution. Within this framework, the strength of a particular swim is judged based on its position compared to the whole distribution of swim-times, rather than just the world record. This model also accounts for the date of the swim, as training methods improve over the years, as well as changes in technology, such as full body suits. The parameters of the generalised Pareto distribution, for each of the 34 individual Olympic events, will be shown to vary with a covariate, leading to a novel single unified description of swim quality over all events and time. This structure, which allows information to be shared across all strokes, distances, and genders, improves the predictive power as well as the model robustness compared to equivalent independent models. A by-product of the model is that it is possible to estimate other features of interest, such as the ultimate possible time, the distribution of new world records for any event, and to correct swim times for the effect of full body suits. The methods will be illustrated using a dataset of the best 500 swim-times for each event in the period 2001-2018.

### Previous mini-projects

##### Exploring Solutions to the Stochastic Multi-Armed Bandit Problem

This report begins with an introduction to machine learning and its various classifications, and the crossover between machine learning and statistical learning in multi-armed bandit problems. A literature review of solutions to multi-armed bandit problems are presented: the $\epsilon $-greedy algorithm, Thompson sampling algorithms, and Upper-Confidence-Bound (UCB) algorithms. A simulation is then formulated and coded in R and tested on a 10-armed Bernoulli bandit to compare these methods along with an algorithm introduced in this report, the greedy-$\eta (t)$ algorithm. It is shown that the greedy-$\eta (t)$ algorithm outperforms the standard $\epsilon $-greedy algorithm after approximately a 10th of the horizon time, and will always outperform the $\epsilon $-greedy algorithm asymptotically, independent of parameter selection. In a macro test of the code, it is shown that the greedy-η algorithm is more robust and its performance is less dependent on prior information. The simulation is based in a Bayesian setting with a Beta posterior distribution. View the R code for this simulation. Finally a comparison of these methods is presented based on theoretical guarantees on regret bounds, the simulation in this report, and other simulations in the literature.

##### Sequential Monte Carlo Methods in State Space Models.

This report formulates the mathematics behind hidden Markov models, before introducing and simulating a Kalman filter. Later, approximate techniques are presented and applied to a hidden Markov model, with a simulation of one particular example, the Bootstrap filter. Throughout this report it is assumed that the parameters in the models are known, however in practice this is not the case.

##### Markov Chains Applied to Time Dependent Queuing Systems.

Continuous Time Markov chains are applied to an M(t)/M/S queue using a transition rate matrix and the expected number in the queue E(n(t)) is tracked over time. This is tested on two different arrival rate functions and the experimental findings agree with the theoretical values. The upper 10 % tail of the distribution of the number in the queue varies in a similar way to the expected number E(n(t)) in the queue. For 3 servers, this peaks at 97 in our example. We conclude that the number of servers S, has more effect on E(n(t)) than on the upper 10 % tail of the distribution. Discrete Time Markov Decision Chains are applied to and M(t)/M/1 queue and backwards recursion is used to find a closed form for the optimal policy.

##### The Existence, Nature, and Effect of Dark Energy.

The cosmic age problem is approached by comparing observations of globular clusters with the theoretical age of the Universe derived from the Friedmann equation, assuming a cosmological constant and a Universe which is flat and matter dominated. My data concludes that there is indeed a need for dark energy. The possibility of a cyclic Universe is considered, and is shown how this can solve many current problems in the Standard Cosmological Model such as the flatness problem, the coincidence problem, the horizon problem, and the dark energy problem.

##### Further topics

For a light read into a broad range of topics in statistics and operational research, visit my blog.