Proceedings of the Twelfth Australasian Data Mining Conference (AusDM 2014), Brisbane, Australia Comparison of Athletic Performances across Disciplines Chris Barnes University of Canberra [email protected] Abstract they have some realistic chance of influencing The extreme value (EV) distribution describes the the major placings. asymptotic behaviour of all stationary distributions But these criteria on selection are somewhat in terms of a limiting, three parameter distribution nebulous, and without an obvious clear function. This result can be used to compare elite definition of eligibility; whereas eventually a sport performances for several purposes. choice must be made that an athlete is, or is not, 1. By regressing out most significant fixed and of the requisite performance standard. random effects for a given discipline, gender and Furthermore, since the entrance standard set by class combination, the resulting residuals can be fitted to an EV distribution to determine the the organisers is usually the more stringent, it is optimised parameters describing often the case that the last athlete selected will gender/discipline/class combinations and their be at the expense of an athlete in an entirely uncertainties. This allows the objective ranking of different discipline or event. For a number of athletes across events in terms of their percentiles. reasons it is preferable that these decisions 2. Use of the regression models allows objective should be as transparent as possible, and based estimates of likely future event standards for on objective criteria (rather than purely on the performances in heats, semis and finals placings. selectors subjective experience of relative 3. Determinations of distributional parameters allow merits). comparison between gender, class (including able- In Athletes With Disabilities (AWD) track and bodied or AWD performances) and disciplines via field competition, there is the additional percentiles. 4. Deviations in the regression parameters may difficulty that at anything but World indicate the varying effects of performance Championship (WCh) or Paralympic level enhancement over the time-span of Olympic cycles. competition, there may be too few competitors Keywords: Elite sport; performance analysis; in any one class to yield the requisite closeness Extreme Value theorem; Weibull distribution; of competition or excitement that makes a good performance prediction spectacle; but the entertainment spectacle is 1. Introduction where an increasingly large proportion of In selecting athletes for a track and field team, financial support originates. This is because the generally fairly stringent constraints are placed numbers of AWD athletes in an event is on the number of athletes that can be virtually always less than the potential pool for accommodated. Perhaps the most stringent of able-bodied (AB) athletes in their these, at least for major meets, are those set in corresponding event; and there are often up to place by the organisers because of logistic or greater than 30 AWD classes for each constraints limiting the total number of equivalent AB event. Even at the highest competitors. There are also constraints imposed international level, available numbers are by the team management from their own insufficient to guarantee an appropriate level of logistics and budget, and also through competition in some event/class combinations; consideration of the standard of the meet: there let alone at the far more numerous sub-national is generally little advantage in sending a large AWD competitions that allow us to identify number of athletes likely to be eliminated in athletic talent initially. their first round or heat, or who have no realistic In the past, a number of somewhat ad hoc chance of placing near the front of the race. It solutions to this dilemma have been will almost always be preferable for these experimented with; ranging from a failure to athletes to compete at a lower level, and gain cater for the particular event/class combination experience and confidence from meets where (or equivalently, allowing more severely handicapped classes to compete in a higher 131 CRPIT Volume 158 - Data Mining and Analytics 2014 class, where they will be sorely outgunned), to amounts of data (guaranteed by time) it introducing a class-specific handicap to allow becomes increasingly more precise. For many more even competition, but requiring the AWD event/class combinations, the development of a “fair” handicapping system. methodology is already capable of good This latter choice has been adopted sometimes precision in comparing performances across at the highest levels, but no such handicapping events and classes; possibly including system has been found fully satisfactory, and comparisons between AB and AWD each system tried has eventually lost support. performances. As a side product, it also throws Intuitively, for universal acceptance, a light on variations in elite performances over handicapping system is required to be fair (at time, and also on the probability (prediction) of least in the spirit of “amateur” competition on different levels of performance in the near which both the Olympics and Paralympics are future. still ideologically based); but also sufficiently 2. Methodology understandable that they can be “seen” to be This methodology combines a number of fair. Such a system is yet to be found, and statistical and data-mining processes to arguably cannot reasonably be expected to characterise performances in any athletics event exist, because of the almost diametrically (or event/class combination) in terms of a small opposite features required by each of these number of parameters specific to each event, requirements. but independent of time. In this way, a single Given this last statement, what are the elite performance may be compared accurately alternatives? with other performances achieved under Historically, many handicap schemes for AWD somewhat different conditions at different athletics have attempted to err on the side of times. One price paid for this generality is the simplicity, while attempting to maintain as fair introduction of a degree of uncertainty in a system as possible. So, a number of systems deciding the rankings of superior performances; based on the use of the current world’s best representing the uncertainty due to unknown performance(s) have been proposed and often (ignored but presumed random) effects on used. While experienced proposers were often performance. able to do a very good job in making their In summary, we start with a regression model scheme appear fair to the majority of that includes all measured and identified competitors, unfortunately in all cases significant effects (such as gender, wind speed competitors were found that were now patently and direction (if known), date of performance disadvantaged by the new scheme, leading etc.); also including the identities of each athlete eventually to the demise of that particular as random effects, since the data set will system. An alternative type of handicapping normally contain repeated measures for at least system, while attempting to retain simplicity as some athletes. With all significant effects far as possible, elected to make fairness their identified in this way, it is assumed that the top priority. These systems were necessarily residuals (measured-modelled) are independent more complex than the first type; but all such and identically distributed (iid): in particular systems proposed generally also came up short they should be independent of time if the in terms of fairness. Unsurprisingly, such regression model is adequate. In other words, systems (somewhat complex, and clearly not given the adequacy of the regression model, the totally fair, but in a complicated way) enjoyed residual series can be represented by a even less support than the former type in the stationary iid distribution. long-term. Under these conditions, the Extreme Value In this paper, we take another look at the second (EV) theorem tells us that the tail of this category of solutions to the simplicity vs distribution of residuals asymptotically has a accuracy dilemma: we introduce a relatively fixed form, fully described by only two (or complex type of analysis where fairness is three) parameters. By considering the average more-or-less guaranteed, at least given speed, rather than the total time, for track events sufficient data. While initially the fairness is and similar, we can assume in all athletics only approximately guaranteed, with increasing events that elite performances are characterised 132 Proceedings of the Twelfth Australasian Data Mining Conference (AusDM 2014), Brisbane, Australia by the upper tail of the distribution (bigger is SAS JMP, v. 11.2.0. I have adhered to their better); the relevant form of the distribution is nomenclature wherever possible. then the Weibull (also known, in transformed For free-style (FRS), the data consists of > form, as the Extreme Value) distribution. This 23,000 records (extracted on 07 Aug 2014); distribution is defined on the [0,∞] interval, so each record represents an individual race that it is assumed that there is a non-zero performance (accepted by FINA as an official possibility of any finite performance, although time for all recent performances). the likelihood of really high performances The first part of this analysis uses a linear becomes vanishingly small. regression model to account for effects due to However, the asymptotic