How Good Is Tiger Woods? Baker, RD and Mchale, IG
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How good is Tiger Woods? Baker, RD and McHale, IG Title How good is Tiger Woods? Authors Baker, RD and McHale, IG Type Article URL This version is available at: http://usir.salford.ac.uk/id/eprint/22851/ Published Date 2012 USIR is a digital collection of the research output of the University of Salford. Where copyright permits, full text material held in the repository is made freely available online and can be read, downloaded and copied for non-commercial private study or research purposes. Please check the manuscript for any further copyright restrictions. For more information, including our policy and submission procedure, please contact the Repository Team at: [email protected]. How good is Tiger Woods? Rose D. Baker, C. Math, FIMA and Ian G. McHale Centre for Operational Research and Applied Statistics, University of Salford Abstract tions, or compare the popularity of a set of choices made by consumers. A major objective of professional sport Given the main objective of profes- is to find out which player or team is sional sport is to find the best player or the best. Unfortunately the structure team, ratings models are naturally ap- of some sports means that this is of- plied to sport. Further, data on sport ten a difficult question to answer. For are often widely accessible making it example, there may be too many com- an ideal playground for a statistician. petitors to run a round-robin league, Here we use a ratings model to com- whilst knock-out tournaments do not pare the ability of golfers. The model compare every player with every other adopted is the Plackett-Luce model player. The problem gets worse when which we use to estimate competitor one has to compare players whose `strengths'. performance varies over time. For- The strengths estimated from the tunately mathematical modelling can standard Plackett-Luce model are help and in this article, we use the static in that they do not vary with Plackett-Luce model to estimate time- time. When used to estimate player varying player strengths of golfers. We strength in the context of golf, it seems use the model to investigate how good unreasonable to assume each and ev- golf's current biggest attraction, Tiger ery player has a constant strength Woods, really is. throughout their playing career. It is more likely that a player's strength Introduction will vary during his or her career. As such, we need a model that can al- `Who is the greatest golfer?' is an of- low a player's strength to vary with ten discussed topic at the 19th hole time. Our solution is to estimate of any golf course. In recent times, it the strength parameters at regularly- seems likely that Tiger Woods is. We spaced time points, with a smooth know Tiger is good, in fact we can be interpolation between them. Splines pretty certain he is the best amongst are very often used in statistical mod- his contemporaries, but just how good elling (e.g. Harrell, 2001) but it is is he? How much better than his com- even simpler and arguably better to petitors is he, and given his recent use the newer but little-known method drop in form, can we expect Tiger to of barycentric interpolation. This was get back to his best? It is difficult, mentioned in a recent article in this if not impossible to give definitive an- magazine (Trefethen, 2011), and we swers to these questions. However, as believe we are the first workers to use is often the case in life, mathematical it in a statistical analysis. Adopting modelling and statistical analysis can this methodology means that, unlike shed light on the situation. We use previous work, we model the evolu- a ratings or ranking model, and fit it tion of player strength deterministi- to tournament data by the method of cally rather than stochastically. maximum likelihood. Other authors, when modelling Ratings models have typically been player or team strengths in other used to compare candidates at elec- sports, have most commonly adopted 1 stochastic rather than deterministic player wins is evolution of player strength. This is sometimes justifiable. In team sports, P πi Pi = n ; for example, there is a non-neglible j=1 πj stochastic component of strength evo- lution: as the team buys and sells where the jth player has strength πj. top players, its strength forms part of After a player has been ranked first, a random walk. In individual sports another player ranks second by win- however, there is a strong systematic ning out of the remaining players, and component in strength evolution. For so on down to the last, who `wins' with example, a player may gain experience probability one. in the early part of his career, reach a peak and then decline in strength until he retires. Indeed, as we find, Time-varying player strengths this would appear to describe the ca- reer path of a typical golf professional. In order to allow for time-dependent player strengths, π, we take as model variables for player i the strengths yik tabulated at mi equally-spaced epochs A Plackett-Luce model tik from the first year of tournament with time-varying strengths play to the last. Next, we interpo- late these tabulated strengths, using The Bradley-Terry model (Bradley the barycentric interpolation formula and Terry, 1952) is a ratings model P mi used for contests between two play- wikyik=(t − tik) π (t) = Pk=1 ; i mi − ers. Such a model is appropriate for k=1 wik=(t tik) obtaining strength estimates of ten- k nis players for example, where play- where wik = (−1) , with first and last ers compete in head-to-head matches, weights being halved. This method but many players are observed play- is described in Berrut and Trefethen ing many matches. In golf, the situ- (2004), and is claimed to give smaller ation is more complicated since many errors than spline interpolation. It players compete in the same `match', is trivial to obtain the differential and we observe many of these matches. @πi(t)=@yik which will be needed in Each match is of course a golf tour- computing the likelihood function. nament, and at the end of each tour- To obtain a more precise estimate nament we have a rankings list of the of a player's maximum strength, af- players who competed. But when we ter a preliminary likelihood maximisa- have many such tournaments, with tion, a new set of interpolation epochs different sets of competitors, how can was chosen so that the time of maxi- we combine the results? The answer mum strength fell at a tabulated time. comes from the generalisation of the This necessitated the use of unequally- B-T model, to be used when n players spaced points, when the weight for- are competing in each `match', named mula is the Plackett-Luce model, introduced by Luce (1959), and Plackett (1975). k 1 1 wik = (−1) f + g; There are more complex models (e.g. tik − ti;k−1 ti;k+1 − tik Stern, 1990) which however require computation of multivariate integrals. with terms with out-of-range epochs Here the probability that the ith omitted. 2 The likelihood function playing lifetime. This can be given a simple meaning in terms of ob- Hunter (2004) describes the maximisa- servables. Imagine that a player of tion of Bradley-Terry likelihood func- strength π regularly competes against tions, including the Plackett-Luce ex- 1 a player of very high ability π . The tension. He writes the log-likelihood 2 proportion of games won is π =(π + function (for constant strengths) as 1 1 π2) ' π1/π2. Over the player's play- − ing lifetime, the number of games XN mXj 1 Xmj R won is (ρ/π2) π(t) dt, where ρ is the `(π) = fln πa(j;i)−ln πa(j;s)g; j=1 i=1 s=i rate at whichR these competitions oc- cur. Hence π(t) dt is proportional where there are N tournaments, with to the total number of games that the mj players competing in the jth, player would win against a very strong and a(j; i) denotes the identity of the player. This is a sensible measure of player who came ith in the jth tour- lifetime achievement. nament. The sum is up to mj − 1 only because the probability for the last Estimation player is unity. With ties, however, the formula must be modified to When maximising the likelihood func- tion for the tabulated `strengths', we `(π) = cannot use the slow but sure method of likelihood maximisation recommended XN Xmj Xmj by Hunter (2004).Solving for the tab- fln πa(j;i) − ln πa(j;s)g; ulated `strengths' yik rather than di- j=1 i=1 s=p(j;i) rectly for the π means that we can- where p(j; i) is the position (place- not manipulate the likelihood maximi- ment) of the ith player in the jth sation condition so that yik is the sub- match; several players can have the ject. This is the basis of the ‘fixed same position. This approximation point' method. A further complicat- for ties is that made by Cox and ing factor here is that when fitting the Oakes (1984) in the context of the model to data from golf tournaments proportional hazards model; the anal- we need to maximise a likelihood func- ogy with ranking models was pointed tion with respect to hundreds, or even out in a short paper by Su and Zhou thousands, of parameters.