Swimming meets statistics

Harry Spearing applies extreme value theory to personal best swim times to rank swimmers and predict future world record

A Ubiquitous Framework for Statistical Ranking Systems

Ranking systems are everywhere. The thesis will often select sports as its motivating applications, given their accessibility; however, schools and universities, harms of drugs, quality of wines, are all ranked, and all with arguably far greater importance. As such, the methodology is kept necessarily general throughout. In this thesis, a novel conceptual framework for statistical ranking systems is proposed, which separates ranking methodology into two distinct classes: absolute systems, and relative systems. Part I of the thesis deals with absolute systems, with a large portion of the methodology centred on extreme value theory. The methodology is applied to elite swimming, and a statistical ranking system is developed which ranks swimmers, based initially on their personal best times, across different swimming events. A challenge when using extreme value theory in practice is the small number of extreme data, which are by definition rare. By introducing a continuous data-driven covariate, the swim-time can be adjusted for the distance, gender category, or stroke, accordingly, and so allowing all data across all 34 individual events to be pooled into a single model. This results in more efficient inference, and therefore more precise estimates of physical quantities, such as the fastest time possible to swim a particular event. Further increasing inference efficiency, the model is then expanded to include data comprising all the performances of each swimmer, rather than just personal bests. The data therefore have a longitudinal structure, also known as panel data, containing repeated measurements from multiple independent subjects. This work serves as the first attempt at statistical modelling of the extremes of longitudinal data in general and the unique forms of dependence that naturally arise due to the structure of the data. The model can capture a range of extremal dependence structures (asymptotic dependence and asymptotic independence), with this characteristic determined by the data. With this longitudinal model, inference can be made about the careers of individual swimmers - such as the probability an individual will break the world record or swim the fastest time next year. In Part II, the thesis then addresses relative systems. Here, the focus is on incorporating intransitivity into statistical ranking systems. In transitive systems, an object A ranked higher than B implies that A is expected to exhibit preference over B. This is not true in intransitive systems, where pairwise relationships can differ from that which is expected from the underlying rankings alone. In some intransitive systems, a single underlying and unambiguous ranking may not even exist. The seminal Bradley-Terry model is expanded on to allow for intransitivity, and then applied to baseball data as a motivating example. It is found that baseball does indeed contain intransitive elements, and those pairs of teams exhibiting the largest degree of intransitivity are identified. Including intransitivity improves prediction performance for future pairwise comparisons. The thesis ultimately concludes by harmonising the two parts - acknowledging that in reality, there is always some relative element to an absolute system. Forging the armistice between these system types could enflame research into the areas connecting them, which until now remains barren

Modelling intransitivity in pairwise comparisons with application to baseball data

In most commonly used ranking systems, some level of underlying transitivity is assumed. If transitivity exists in a system then information about pairwise comparisons can be translated to other linked pairs. For example, if typically A beats B and B beats C, this could inform us about the expected outcome between A and C. We show that in the seminal Bradley-Terry model knowing the probabilities of A beating B and B beating C completely defines the probability of A beating C, with these probabilities determined by individual skill levels of A, B and C. Users of this model tend not to investigate the validity of this transitive assumption, nor that some skill levels may not be statistically significantly different from each other; the latter leading to false conclusions about rankings. We provide a novel extension to the Bradley-Terry model, which accounts for both of these features: the intransitive relationships between pairs of objects are dealt with through interaction terms that are specific to each pair; and by partitioning the $n$ skills into $A+1\leq n$ distinct clusters, any differences in the objects' skills become significant, given appropriate $A$. With $n$ competitors there are $n(n-1)/2$ interactions, so even in multiple round robin competitions this gives too many parameters to efficiently estimate. Therefore we separately cluster the $n(n-1)/2$ values of intransitivity into $K$ clusters, giving $(A,K)$ estimatable values respectively, typically with $A+K<n$. 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. The model is shown to have an improved fit out of sample in both simulated data and when applied to American League baseball data.Comment: 26 pages, 7 figures, 2 tables in the main text. 17 pages in the supplementary materia

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 unied 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

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 intransitivity. Various intransitive parametric models have been proposed but they lack the flexibility to cover the different strategies across n players, with the O(n2) 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(n2) unknown parameters for (A;K) we anticipate that in many practical contexts A + K < n. 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

Fabrication and structural characterization of self-supporting electrolyte membranes for a micro solid-oxide fuel cell

Micromachined fuel cells are among a class of microscale devices being explored for portable power generation. In this paper, we report processing and geometric design criteria for the fabrication of free-standing electrolyte membranes for microscale solid-oxide fuel cells. Submicron, dense, nanocrystalline yttria-stabilized zirconia (YSZ) and gadolinium-doped ceria (GDC) films were deposited onto silicon nitride membranes using electron-beam evaporation and sputter deposition. Selective silicon nitride removal leads to free-standing, square, electrolyte membranes with side dimensions as large as 1025 µm for YSZ and 525 µm for GDC, with high processing yields for YSZ. Residual stresses are tensile (+85 to +235 MPa) and compressive (–865 to -155 MPa) in as-deposited evaporated and sputtered films, respectively. Tensile evaporated films fail via brittle fracture during annealing at temperatures below 773 K; thermal limitations are dependent on the film thickness to membrane size aspect ratio. Sputtered films with compressive residual stresses show superior mechanical and thermal stability than evaporated films. Sputtered 1025-µm membranes survive annealing at 773 K, which leads to the generation of tensile stresses and brittle fracture at elevated temperatures (923 K)

The revised Approved Instructional Resources score:An improved quality evaluation tool for online educational resources

BACKGROUND: Free Open-Access Medical education (FOAM) use among residents continues to rise. However, it often lacks quality assurance processes and residents receive little guidance on quality assessment. The Academic Life in Emergency Medicine Approved Instructional Resources tool (AAT) was created for FOAM appraisal by and for expert educators and has demonstrated validity in this context. It has yet to be evaluated in other populations.OBJECTIVES: We assessed the AAT's usability in a diverse population of practicing emergency medicine (EM) physicians, residents, and medical students; solicited feedback; and developed a revised tool.METHODS: As part of the Medical Education Translational Resources: Impact and Quality (METRIQ) study, we recruited medical students, EM residents, and EM attendings to evaluate five FOAM posts with the AAT and provide quantitative and qualitative feedback via an online survey. Two independent analysts performed a qualitative thematic analysis with discrepancies resolved through discussion and negotiated consensus. This analysis informed development of an initial revised AAT, which was then further refined after pilot testing among the author group. The final tool was reassessed for reliability.RESULTS: Of 330 recruited international participants, 309 completed all ratings. The Best Evidence in Emergency Medicine (BEEM) score was the component most frequently reported as difficult to use. Several themes emerged from the qualitative analysis: for ease of use-understandable, logically structured, concise, and aligned with educational value. Limitations include deviation from questionnaire best practices, validity concerns, and challenges assessing evidence-based medicine. Themes supporting its use include evaluative utility and usability. The author group pilot tested the initial revised AAT, revealing a total score average measure intraclass correlation coefficient (ICC) of moderate reliability (ICC = 0.68, 95% confidence interval [CI] = 0 to 0.962). The final AAT's average measure ICC was 0.88 (95% CI = 0.77 to 0.95).CONCLUSIONS: We developed the final revised AAT from usability feedback. The new score has significantly increased usability, but will need to be reassessed for reliability in a broad population.</p