Abstract
We propose a novel ranking model that combines the Bradley-Terry-Luce probability model with a nonnegative matrix factorization framework to model and uncover the presence of latent variables that influence the performance of top tennis players. We derive an efficient, provably convergent, and numerically stable majorization-minimization-based algorithm to maximize the likelihood of datasets under the proposed statistical model. The model is tested on datasets involving the outcomes of matches between 20 top male and female tennis players over 14 major tournaments for men (including the Grand Slams and the ATP Masters 1000) and 16 major tournaments for women over the past 10 years. Our model automatically infers that the surface of the court (e.g., clay or hard court) is a key determinant of the performances of male players, but less so for females. Top players on various surfaces over this longitudinal period are also identified in an objective manner.
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Notes
- 1.
The updates for \(\mathbf {W}\) and \(\mathbf {H}\) are not symmetric because the data is in the form of a 3-way tensor \(\{b_{ij}^{(m)}\}\); this is also apparent in (1) and the updates in ().
- 2.
One might be tempted to normalize
. This, however, does not resolve numerical issues as some entries of 
may be zero. - 3.
This may be attributed to its position in the seasonal calendar. The Paris Masters is the last tournament before ATP World Tour Finals. Top players often choose to skip this tournament to prepare for the more prestigious ATP World Tour Finals. This has led to some surprising results, e.g., Ferrer, a strong clay player, won the Paris Masters in 2012 (even though the Paris Masters is a hard court tournament).
- 4.
The solution with the highest likelihood is shown in Trial 2 of Table S-13 but it appears that the solution there is degenerate.
- 5.
Stationary points are not necessarily equivalent up to permutation or rescaling.
. This, however, does not resolve numerical issues as some entries of 
may be zero.References
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Acknowledgements
This work was supported by a Ministry of Education Tier 2 grant (R-263-000-C83-112), an NRF Fellowship (R-263-000-D02-281), and by the European Research Council (ERC FACTORY-CoG-6681839).
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Xia, R., Tan, V.Y.F., Filstroff, L., Févotte, C. (2020). A Ranking Model Motivated by Nonnegative Matrix Factorization with Applications to Tennis Tournaments. In: Brefeld, U., Fromont, E., Hotho, A., Knobbe, A., Maathuis, M., Robardet, C. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2019. Lecture Notes in Computer Science(), vol 11908. Springer, Cham. https://doi.org/10.1007/978-3-030-46133-1_12
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