arXiv subjects

Tim Paulden

Publications and source records attributed to Tim Paulden.

3 recordsLinked to original sources

arXiv2023

A framework for statistical modelling of the extremes of longitudinal data, applied to elite swimming

We develop methods, based on extreme value theory, for analysing observations in the tails of longitudinal data, i.e., a data set consisting of a large number of short time series, which are typically irregularly and non-simultaneously sampled, yet have some commonality in the structure of each series and exhibit independence between time series. Extreme value theory has not been considered previously for the unique features of longitudinal data. Across time series the data are assumed to follow a common generalised Pareto distribution, above a high threshold. To account for temporal dependence of such data we require a model to describe (i) the variation between the different time series properties, (ii) the changes in distribution over time, and (iii) the temporal dependence within each series. Our methodology has the flexibility to capture both asymptotic dependence and asymptotic independence, with this characteristic determined by the data. Bayesian inference is used given the need for inference of parameters that are unique to each time series. Our novel methodology is illustrated through the analysis of data from elite swimmers in the men's 100m breaststroke. Unlike previous analyses of personal-best data in this event, we are able to make inference about the careers of individual swimmers - such as the probability an individual will break the world record or swim the fastest time next year.

stat.ME↗

arXiv2021

Modelling intransitivity in pairwise comparisons with application to baseball data

The seminal Bradley-Terry model exhibits transitivity, i.e., the property that the probabilities of player A beating B and B beating C give the probability of A beating C, with these probabilities determined by a skill parameter for each player. Such transitive models do not account for different strategies of play between each pair of players, which gives rise to {\it intransitivity}. Various intransitive parametric models have been proposed but they lack the flexibility to cover the different strategies across $n$ players, with the $O(n^2)$ values of intransitivity modelled using O(n) parameters, whilst they are not parsimonious when the intransitivity is simple. We overcome their lack of adaptability by allocating each pair of players to one of a random number of $K$ intransitivity levels, each level representing a different strategy. Our novel approach for the skill parameters involves having the $n$ players allocated to a random number of $A<n$ distinct skill levels, to improve efficiency and avoid false rankings. Although we may have to estimate up to $O(n^2)$ unknown parameters for $(A,K)$ we anticipate that in many practical contexts $A+K < n$. Using a Bayesian hierarchical model, $(A,K)$ are treated as unknown, and inference is conducted via a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm. Our semi-parametric model, which gives the Bradley-Terry model when $(A=n-1, K=0)$, is shown to have an improved fit relative to the Bradley-Terry, and the existing intransitivity models, in out-of-sample testing when applied to simulated and American League baseball data. Supplementary materials for the article are available online.

stat.ME↗

arXiv2019

Ranking, and other Properties, of Elite Swimmers using Extreme Value Theory

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.

stat.AP↗