1 A point-based Bayesian hierarchical model to predict the outcome of tennis matches Martin Ingram, Silverpond September 21, 2017
2 Introduction Predicting tennis matches is of interest for a number of applications: Coaching: Prediction models can provide useful feedback about who players should be able to beat and how they are improving over time Fan engagement: Who is the favourite? By how much? Who is currently the best player?
3 Approaches to tennis prediction Broadly speaking, published tennis prediction models fall into three classes: 1 Regression models 2 Paired comparison models 3 Point based models
4 Regression models Regression models phrase match prediction as a regression task, using a suitable link function (logit/probit) to predict match outcomes. For example, Gilsdorf et al.  predict using a probit model including ranking, prize earnings and demographics
5 Paired comparison models Paired comparison models model match outcomes by assuming that each player has a hidden latent ability The probability of a player winning a match is modelled as a function of the difference of the two latent abilities, θ 1 and θ 2 For example, Elo typically uses the following likelihood: p(p1 wins θ 1, θ 2 ) = (θ 2 θ 1 )/400 (1) A version of Elo (with an optimised k-factor) devised by FiveThirtyEight  has been particularly popular in tennis Other interesting paired comparison models exist but are not as popular for tennis (e.g. TrueSkill , Glicko )
6 Point based models (I) Point based models use a model of a tennis match developed, among others, by Newton & Keller  It assumes that points on serve are independent and identically distributed (i.i.d.) This means that the probabilities p 1 and p 2 of winning a point on serve for players 1 and 2 are assumed constant throughout the match Using recursive equations, it is possible to calculate the probability of holding serve, winning a set, winning a tiebreak, and winning the match as functions of only p 1 and p 2
7 Illustrations of the i.i.d. model For p 1 = 0.63 (ATP average), probability of holding serve is 79.4% For p 1 = 0.65 and p 2 = 0.60, probability of player 1 winning a best-of-three match is 73.7%
8 Point based models (II) Point based prediction models predict p 1 and p 2 and then predict the match winner using the i.i.d. model For example, Barnett and Clarke  propose to calculate the probability as: f ij = f t + (f i f av ) (g j g av ) (2) f t : average serve-winning probability at the tournament f i : the player s average serve-winning probability f av : the tour average serve-winning probability g j : the opponent s average return-winning probability g av : the tour average return-winning probability
9 Comparing model performance In a 2015 paper , Stephanie Kovalchik compares 11 published prediction models, including all three model classes, by predicting the ATP s 2014 season. The best representatives of each model type are: Type Model Accuracy Log loss Regression-based Gilsdorf et al. 68% 0.61 Paired comparison FiveThirtyEight Elo 70% 0.59 Point-based Barnett & Clarke 67% 0.63 Point-based models have the highest log loss and lowest accuracy.
10 Pros and cons of the point-based approach In addition to the worse evaluation, the i.i.d. model is also proven to be wrong, albeit a good approximation (Klaassen & Magnus ). Players do not play i.i.d., although deviations are quite small. Why use it at all? The main attraction is the ability to make a great wealth of predictions beside match outcome: Number of sets (e.g.: two or three sets?) Set scores (e.g.: how likely is a tiebreak?) Many more: number of games, number of points... In addition, in-play win probabilities can be calculated based on the score
11 Example: Set scores At low p 1 and p 2 (average / below average servers), scores like 6-3 or 6-2 are likely. At high p 1 and p 2 (very strong servers), a tiebreak becomes most likely.
12 Rest of the talk Key question of this talk: Can we build a better point-based model?
13 Modelling ideas I wanted the model to account for the following factors: Surface preferences: Tennis is played on a number of surfaces (clay, grass, hard, indoor). Players often do better on one surface than another (e.g. Nadal: 10 French Open titles, 2 Wimbledon titles). Tournament effects: It is easier to win points on serve at some tournaments than at others, raising players averages (and making e.g. tiebreaks more likely). Time dependence: Player skills change over time
14 Likelihood Split each tennis match into two serve-matches and use a binomial likelihood: where: y i Binomial(n i, θ i ) (3) y i : Points won on serve in serve-match i n i : Points played on serve in serve-match i θ i : Serve-winning probability in serve-match i
15 Modelling θ i The model for θ i, the serve-winning probability in serve-match i, is: logit(θ i ) = (α s(i)p(i) β r(i)p(i) )+(γ s(i)m(i) γ r(i)m(i) )+δ t(i) +θ 0 (4) α s(i)p(i) : server s(i) s serving skill in period p(i) β r(i)p(i) : returner r(i) s returning skill in period p(i) γ s(i)m(i) : server s additional skill on surface m(i) γ r(i)m(i) : returner s additional skill on surface m(i) δ t(i) : adjustment to the intercept at tournament t(i) θ 0 : intercept
16 Modelling time dependence Took some inspiration from Glicko . Serve skills α and return skills β follow a Gaussian random walk over time: α.p+1 N(α.p, σ 2 α) (5) β.p+1 N(β.p, σ 2 β ) (6) σ α, σ β N(0, 1) (7) In other words, skills in the next period are a small normal jump away from the previous skills. Note priors on σ are constrained to be positive when fit (unit half-normals).
17 Hierarchical priors Initial skills, tournament intercepts and surface skills all have hierarchical priors: δ N(0, σδ 2 ) (8) γ N(0, σγ) 2 (9) α.1 N(0, σα0) 2 (10) β.1 N(0, σβ0 2 ) (11) All priors for the group σs are unit half-normals.
18 Model Checks & Validation
19 External validation Use data from 2011 onwards to predict 2014 With periods of 3 months, fit model four times, once for each quarter of 2014 Use posterior estimates and i.i.d. model to predict win probabilities Type Model Accuracy Log loss Regression-based Gilsdorf et al. 68% 0.61 Paired comparison FiveThirtyEight Elo 70% 0.59 Point-based Barnett & Clarke 67% 0.63 Point-based Proposed 68% 0.60 Considerable improvement compared to previous best point-based model!
20 Point-level validation For Barnett & Clarke and the model, can also compute metrics on how well the serve-winning probabilities are estimated. Model RMSE R 2 Barnett & Clarke % Proposed % Improved here too (as you would expect).
21 Evaluation: Posterior predictive checks Fit model from 2014 up to 2017 (pre US Open) with three-month periods. Replicate y i using 4,000 simulations of θ i, and compare test quantities computed on data to those on replications. Mean: replications match data exactly (p=0.50). Standard deviation: all replications have greater standard deviation than the data!
22 Evidence of underdispersion? Compare one replication in more detail. Good agreement in general, but at low expected y, the model underpredicts the points won on serve; at high expected y, it overpredicts. Evidence of underdispersion?
23 Results & Examples
24 Results: Serve and return skills Q One advantage of point-based model: can look at serve and return skills. Broadly agree with intuition; some interesting: Nadal very strong on serve, Schwartzman very strong on return.
25 Results: Overall skills Results mostly intuitive: e.g. Nadal and Federer shared all 4 Grand Slams this year Surprises: Kyrgios ranked highly (only number 18 in ATP Rankings); Wawrinka ranked low (number 4 in ATP rankings)
26 Serve compared to return skill
27 Skill evolution Renaissance of Federer (?) and Nadal Nadal and Federer share 4 Grand Slams after droughts of 5 years (Federer) and 3 years (Nadal). Big improvements suggested. But: Federer was better in 2015! Djokovic has declined a great deal. Nadal has improved greatly (particularly on serve). Moya to credit (coach since Dec 16)?
28 Match prediction example I fit the model up to the start of the US Open. How would it have predicted the final: Nadal vs. Anderson? Nadal won
29 Surface effects: Nadal vs. Anderson on grass How would things change in a hypothetical match at Wimbledon?
30 Surface effects: Nadal vs. Anderson on clay How would things change in a hypothetical match at the French Open?
31 Conclusions and future work Introduced a new point-based model with higher prediction accuracy than the previous best Takes into account surface effects and time-varying ability, as well as tournament effects Future work: Accelerating model fit: Currently fit using Stan, takes about 80 minutes. Limited to period lengths of 1 month or over, and data of about 5 years or less. An approximate solution e.g. using variational Bayes or another approach would be of interest. Different skill time evolution: e.g. like Glicko 2, where the jumps are drawn from another distribution , or maybe a Gaussian process. Investigate alternatives to the Binomial link, such as the COM-Poisson model, which could handle underdispersion, and / or investigate causes of underdispersion further
32 Thank you! Thank you for paying attention!
33 Appendix: Comparison to ATP Rankings Broad agreement, but exceptions highlight differences: model rates Kyrgios much higher (injuries), Thiem lower (plays a lot), Murray lower (declining this year). Wawrinka is a very variable player.
34 Appendix: Skill evolution Next generation Kyrgios, Zverev lead among young players Zverev climbing at fastest rate Khachanov improving (slowly), Coric stagnating
35 Appendix: Clay skills Biggest clay boost: Rafael Nadal (unsurprisingly) Players from Latin America and Spain do very well
36 Appendix: The tour is more competitive now than 2015
37 Appendix: Serve compared to return skill in 2017
38 References I K. F. Gilsdorf and V. A. Sukhatme, Testing rosen s sequential elimination tournament model: Incentives and player performance in professional tennis, Journal of Sports Economics, vol. 9, no. 3, pp , B. Morris and C. Bialik, Serena williams and the difference between all-time great and greatest of all time, Sep [Online]. Available: serena-williams-and-the-difference-between-all-time-great-and-greate R. Herbrich, T. Minka, and T. Graepel, Trueskill(tm): A bayesian skill rating system, Advances in Neural Information Processing Systems, pp , 2006.
39 References II M. E. Glickman, Parameter estimation in large dynamic paired comparison experiments, Journal of the Royal Statistical Society: Series C (Applied Statistics), vol. 48, no. 3, pp , P. K. Newton and J. B. Keller, Probability of winning at tennis i. theory and data, Studies in applied Mathematics, vol. 114, no. 3, pp , T. Barnett and S. R. Clarke, Combining player statistics to predict outcomes of tennis matches, IMA Journal of Management Mathematics, vol. 16, no. 2, pp , S. A. Kovalchik, Searching for the goat of tennis win prediction, Journal of Quantitative Analysis in Sports, vol. 12, no. 3, pp , 2016.
40 References III F. J. Klaassen and J. R. Magnus, Are points in tennis independent and identically distributed? evidence from a dynamic binary panel data model, Journal of the American Statistical Association, vol. 96, no. 454, pp , M. E. Glickman, Dynamic paired comparison models with stochastic variances, Journal of Applied Statistics, vol. 28, no. 6, pp , 2001.
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