Professional Tennis: Quantitative Models and Ranking Algorithms
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Imperial College London Department of Computing Professional Tennis: Quantitative Models and Ranking Algorithms Demetris Spanias 30 September 2014 Supervised by William Knottenbelt Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Computing of Imperial College London and the Diploma of Imperial College London 1 Copyright Declaration The copyright of this thesis rests with the author and is made available under a Cre- ative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, trans- form or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 2 Declaration of Originality I declare that the content of this thesis was composed by myself and unless otherwise stated, the work presented is my own. 3 Abstract Professional singles tennis is a popular global sport that attracts spectators and speculators alike. In recent years, financial trading related to sport outcomes has become a reality, thanks to the rise of online betting exchanges and the ever- increasing development and deployment of quantitative models for sports. This thesis investigates the extent to which the outcome of a match between two professional tennis players can be forecast using quantitative models param- eterised by historical data. Three different approaches are explored, each having its own advantages and disadvantages. Firstly, the problem is approached using a Markov chain to model a tennis point, estimating the probability of a player win- ning a point while serving. Such a probability can be used as a parameter to exist- ing hierarchical models to estimate the probability of a player winning the match. We demonstrate how this probability can be estimated using varying subsets of historical player data and investigate their effect on results. Averaged historical data over varying opponents with different skill sets, does not necessarily provide a fair basis of comparison when evaluating the performance of players. The second approach presented is a technique that uses data, which includes only matches played against common opponents, to find the difference between the modelled players’ probability of winning a point on their serve against each common opponent. This difference in probability for each common opponent is a “transitive contribution” towards victory for the match being modelled. By combining these “contributions” the “Common-Opponent” model overcomes the problems of using average historical statistics at the cost of a shrinking data set. Finally, the thesis ventures into the field of player rankings. Rankings provide a fast and simple method for predicting match winners and comparing players. We present a variety of methods to generate such player rankings, either by making use of network analysis or hierarchical models. The generated rankings are then evaluated using their ability to correctly represent the subset of matches that were used to generate them as well as their ability to forecast future matches. 4 I would like to dedicate this thesis first and foremost to my parents, who supported me both financially and mentally throughout my PhD. Secondly, my supervisor, William, who has been my guide and mentor to the world of academia and research. Last but not least, my partner, Sofia, who motivated me through the tough days and kept me going. Had it not been for their understanding and support I would not have reached as far as I did. 5 Contents 1. Introduction 17 1.1. Motivation . 17 1.2. Objectives . 18 1.3. Contributions . 20 1.3.1. Point Model . 20 1.3.2. Common-Opponent Model . 20 1.3.3. Ranking Systems . 21 1.4. Thesis Structure . 22 1.5. Publications . 24 2. Background 25 2.1. The Game of Tennis . 25 2.1.1. Rules . 25 2.1.2. Scoring System and Order of Serve . 28 2.1.3. Tournaments . 31 2.1.4. The Official Ranking Systems . 33 2.2. Theoretical Methods . 36 2.2.1. Probability Theory . 36 2.2.2. Common Distributions . 37 2.2.3. Significance Testing . 42 2.2.4. Stochastic Processes . 44 2.3. Literature Overview . 46 2.3.1. Hierarchical Match Models . 46 2.3.2. Independence of Points . 51 2.3.3. Ranking Models . 51 2.3.4. Using Rankings as Predictive Tools . 52 2.3.5. Other Tennis Model Uses . 53 6 3. Expanding the Hierarchical Tennis Model 60 3.1. Match Markov Chain . 60 3.2. Set Markov Chain . 62 3.3. Game Markov Chain . 64 3.4. Tiebreaker Markov Chain . 65 3.5. Point Markov Chain . 67 3.6. Service and Rally Markov Chains . 68 3.7. Forecasting the Outcome of a Match . 71 3.7.1. Collecting the Data . 71 3.7.2. A Closer Look at the Data . 73 3.7.3. Estimating the Probability of Winning Service Points . 78 3.7.4. Combining Historical Data for Doubles . 82 3.8. Selecting Historical Data . 83 3.8.1. Age of Match Played . 84 3.8.2. Surface of Match Played . 84 3.9. Evaluating the Performance of Tennis Models . 85 3.9.1. A Tennis Model Performance Rating, r .......... 85 3.9.2. The Random Model . 86 3.9.3. Back-testing Using Real Data . 86 3.9.4. Comparing Models Against the Random Model . 91 3.9.5. Uncombined Model vs. Combined Model . 92 3.9.6. Barnett Model vs. Combined Model . 93 3.9.7. Combined Model vs. Bookmaker Models . 93 3.10. Conclusions . 96 4. A Common-Opponent Based Model 98 4.1. Introduction . 98 4.2. The Concept of Transitivity . 99 4.3. Relationship between the probabilistic difference of winning ser- vice points and winning the match . 99 4.4. Match Probabilities Using Common-Opponent Model . 100 4.5. Evaluating Model Performance . 105 4.5.1. Back-testing Results . 105 4.5.2. Common-Opponent Model vs. Random Model . 107 4.5.3. Common-Opponent Model vs. Uncombined Model . 108 4.5.4. Common-Opponent Model vs. Combined Model . 108 7 4.5.5. Common-Opponent Model vs. Bookmaker Models . 109 4.6. Conclusion . 110 5. Ranking Systems for Tennis Players 112 5.1. Introduction . 112 5.2. The PageRank Approach . 112 5.3. Set, Game and Point PageRank Approach . 115 5.4. Comparing PageRank Approaches . 116 5.5. SortRank . 125 5.6. The LadderRank Algorithm . 125 5.7. SortRank and LadderRank Performance . 127 5.8. Forecasting Match Outcomes Using Ranking Systems . 133 5.9. Match Probabilities from Rankings . 136 5.10. PageRank Set rankings vs. Bookmakers . 137 5.11. Conclusion . 138 6. Conclusions and Further Research 139 6.1. Achievements . 139 6.1.1. Tennis Point Markov Model . 139 6.1.2. Common-Opponent Model . 139 6.1.3. Ranking Systems and Forecasting . 140 6.2. Applications . 141 6.3. Further Research . 142 6.3.1. Analysis of Data . 142 6.3.2. Application on Doubles Matches . 142 6.3.3. Analysis of Rallies and Serving . 142 6.3.4. Back-testing with WTA Matches . 143 6.3.5. 2-tier Common-Opponent Model . 143 6.3.6. Expand Data-set to Include Challenger Data . 144 Bibliography 144 Appendices 155 A. Selected Ranking Figures and Tables 156 8 List of Tables 2.1. ATP ranking points structure for larger tournaments (excludes Chal- lenger and Futures tournaments, the Olympics and Tour Finals) . 34 2.2. WTA ranking points structure for larger tournaments (excludes ITF Circuit tournaments, the Olympics and Tour Finals) . 36 2.3. A simple website example for a two-sample Z-test. 44 3.1. Results from a 3-month all surface back-test using the ‘uncom- bined’, ‘combined’ and Barnett’s models to predict 6551 ATP Tour matches played between the 1st of January 2011 and the 31st of December 2013. 88 3.2. Results from a 6-month all surface back-test using the ‘uncom- bined’, ‘combined’ and Barnett’s models to predict 7184 ATP Tour matches played between the 1st of January 2011 and the 31st of December 2013. 88 3.3. Results from a 12-month all surface back-test using the ‘uncom- bined’, ‘combined’ and Barnett’s models to predict 7211 ATP Tour matches played between the 1st of January 2011 and the 31st of December 2013. 89 3.4. Results from a 12-month surface filtered back-test using the ‘un- combined’, ‘combined’ and Barnett’s models to predict 6501 ATP Tour matches played between the 1st of January 2011 and the 31st of December 2013. 90 3.5. Results from a 24-month surface filtered back-test using the ‘un- combined’, ‘combined’ and Barnett’s models to predict 6916 ATP Tour matches played between the 1st of January 2011 and the 31st of December 2013. 90 9 3.6. Results from a 36-month surface filtered back-test using the ‘un- combined’, ‘combined’ and Barnett’s models to predict 7051 ATP Tour matches played between the 1st of January 2011 and the 31st of December 2013. 91 3.7. A two-sample Z-test using results from a 12-month all surface back-test of a random model and the uncombined model for 7211 ATP Tour matches played from 1st of January 2011 to the 31st of December 2013. 91 3.8. A two-sample Z-test using results from 12-month all surface back- test of a random model and the combined model for 7211 ATP Tour matches played from 1st of January 2011 to the 31st of De- cember 2013. 92 3.9. A two-sample Z-test using results from 12-month all surface back- test of a random model and Barnett’s model for 7211 ATP Tour matches played from 1st of January 2011 to the 31st of December 2013.