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| 研究生： |
華軒甫 Hua, Hsuan-Fu |
|---|---|
| 論文名稱： |
Laplace 近似法應用之排名系統探討 An application of Laplace approximation for rating systems |
| 指導教授： |
翁久幸
Weng, Chiu-Hsing |
| 口試委員: |
黃子銘
Huang, Tzee-Ming 陳定立 Chen, Ting-Li |
| 學位類別： |
碩士
Master |
| 系所名稱： |
商學院 - 統計學系 Department of Statistics |
| 論文出版年： | 2022 |
| 畢業學年度： | 110 |
| 語文別： | 英文 |
| 論文頁數： | 33 |
| 中文關鍵詞： | Laplace近似法 、貝氏分析 、Bradley-Terry模型 、GenElo surface 模型 、排名 、網球 |
| 外文關鍵詞： | Laplace approximation, Bayesian estimation, Bradley-Terry model, GenElo surface model, Ranking, Tennis |
| DOI URL： | http://doi.org/10.6814/NCCU202200956 |
| 相關次數： | 點閱：181 下載：31 |
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成對比較模型可以評估運動選手的表現，基於成對比較模型的 Elo 的排名
系統今日被廣泛地用於不同的運動比賽。在網球比賽中，Ingram[5]將網球比賽的場地資訊加入了模型當中，本篇論文基於 Ingram 的貢獻，使用 Laplace Approximation 近似後驗分配，並將球員實力分佈的變異數更新加入討論。因此我們比較了四種不同的演算法，分別為：Ingram 更新平均數、Ingram 更新平均數並加入比賽場地因素、本篇論文更新平均數及變異數、本篇論文更新平均數及變異數同時加入場地因素共四種演算法。本篇論文基於四種演算法給定不同的變異數起始值，發現有更新變異數的演算法可以提升預測率。另外，將比賽場地因素加入考慮之後，也能夠提升模型預測準確率的表現。校準精度圖(Calibration Accuracy plot)也展示了同樣的結果。本篇論文以網球單打比賽作為實驗數據。
Paired comparison models can be used to model the performance of sports players.Elo’s [2] system is a widely used rating system based on paired comparison models.
For tennis games, recently Ingram [5] extended the Elo system by incorporating the surface information into the model. This thesis extends Ingram’s approach by including the uncertainty of skills via Laplace approximation. We compare four different algorithms- Ingram’s mean update, Ingram’s mean update with surface information, our mean-variance update, and our mean-variance update with surface information. By giving different initial variance, we found that updating the variance can increase the prediction accuracy. Furthermore, the surface information can also enhance the prediction performance of the model. The calibration accuracy plot also shows that updating the
variance does improve the model performance. The above comparisons are based on singles tennis games data.
1 Introduction ...... 4
2 Literature Review ...... 6
2.1 Background ...... 6
2.2 Thurstone-Mosteller Model...... 7
2.3 Bradley-Terry Model ...... 7
2.4 Elo...... 8
3 Methodology ...... 9
3.1 GenElo Surface Model ...... 9
3.2 Our method ...... 10
3.2.1 Review Laplace approximation ...... 10
3.2.2 Rating system with Laplace approximation ...... 11
3.2.3 Rating system with Laplace approximation and surface factor ...... 13
4 Dataset ...... 18
5 Results ...... 21
5.1 Setting parameters ...... 21
5.2 Evaluation results ...... 22
5.3 Top 10 Players ...... 29
5.3.1 ATP Ranking ...... 29
5.3.2 Top Lists ...... 30
6 Concluding Remarks ...... 32
REFERENCE ...... 33
Ralph A Bradley and Milton E Terry. Rank analysis of incomplete block designs: I. the method of paired comparisons. Biometrika, 39(3/4):324–345, 1952.
Arpad E Elo. The rating of chessplayers, past and present. BT Batsford Limited, 1978.
Mark E Glickman. Parameter estimation in large dynamic paired comparison experiments. Journal of the Royal Statistical Society: Series C (Applied Statistics), 48(3):377–394, 1999.
John C Handley. Comparative analysis of Bradley-Terry and Thurstone-Mosteller paired comparison models for image quality assessment. In PICS, volume 1, pages 108–112, 2001.
Martin Ingram. How to extend elo: a bayesian perspective. Journal of Quantitative Analysis in Sports, 17(3):203–219, 2021.
Stephanie A Kovalchik. Searching for the goat of tennis win prediction. Journal of Quantitative Analysis in Sports, 12(3):127–138, 2016.
Frederick Mosteller. Remarks on the method of paired comparisons: Ii. the effect of an aberrant standard deviation when equal standard deviations and equal correlations are assumed. Psychometrika, 16(2):203–206, 1951.
Radek Pelánek. Applications of the elo rating system in adaptive educational systems. Computers & Education, 98:169–179, 2016.
Louis L Thurstone. A law of comparative judgment. Psychological review, 34(4): 273, 1927
