Hypergraph adjusted plus-minus
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Nathaniel Josephs
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Elizabeth Upton
Abstract
In team sports, traditional ranking statistics do not allow for the simultaneous evaluation of both individuals and combinations of players. Metrics for individual player rankings often fail to include the interaction effects between groups of players, while methods for assessing full lineups cannot be used to identify the value of lower-order combinations of players (pairs, trios, etc.). Given that player and lineup rankings are inherently dependent on each other, these limitations may affect the accuracy of performance evaluations. To address this, we propose a novel adjusted plus-minus (APM) approach that allows for the simultaneous ranking of individual players, lower-order combinations of players, and full lineups within a team. The method adjusts for the complete dependency structure and is motivated by the connection between APM and the hypergraph representation of a team. We discuss the similarities of our approach to other advanced metrics, demonstrate it using NBA data from 2012 to 2022, and suggest potential directions for future work.
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Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Use of Large Language Models, AI and Machine Learning Tools: None declared.
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Conflict of interest: The authors state no conflict of interest.
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Research funding: None declared.
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Data availability: Not applicable.
In the LAPM model specification, we have
To evaluate the uncertainty in our modeling process, one option is to fix a value for σ 2, i.e. view it as a known value. Alternatively, we could extend the model definition by allowing σ 2 to be a random variable with an inverse scaled chi-squared prior:
For the analysis presented in the paper, we chose to use a non-informative prior for σ 2, setting ν = τ 2 = 0. We utilized the Metropolis adjusted Langevin algorithm (Girolami and Calderhead 2011) to sample from the posterior distributions. The posterior distributions displayed throughout the manuscript are based on 1,000 samples after disregarding 10 % for a burn-in period and thinning every 5 samples.
We note that while the center and spread of the posterior distributions (and therefore, the fitted values) are sensitive to prior specification, the resulting rankings of the generalized lineups are not. Also, while we set the hyper-prior parameters to be the same for all teams, it may be worthwhile to investigate the parameters at the team level if that is where one’s interests primarily lie.
In Figure A1, we show the rank correlation of each metric between the first and second halves across each season.
Spearman rank correlation for each method within year to itself for individuals (left) and pairs (right) across the years 2012–2022.
In Figures A2 and A3 we show the full distributions of HAPM and LAPM values for individuals and pairs, respectively, for the Sizers in 2021–22. In Figure 4, we show a network visualization of the Celtics in 2021–22.
HAPM (left) and LAPM (right) for individuals on the Sixers in 2021–22. The boxplots represent the distribution of HAPM and LAPM values obtained via bootstrap and MCMC, respectively.
HAPM (left) and LAPM (right) for the top pairs on the Sixers in 2021–22. The boxplots represent the distribution of HAPM and LAPM values obtained via bootstrap and MCMC, respectively.
Line graph visualization of player/pair (left) and player/trio (right) HAPM rankings for BOS 2021–22. The size of the nodes corresponds to ranking within the individuals/pairs/trios. The green nodes represent players, and the white nodes, labeled with rank, represent the pair/trio.
References
Ahmadalinezhad, M. and Makrehchi, M. (2020). Basketball lineup performance prediction using edge-centric multi-view network analysis. Soc. Netw. Anal. Min. 10: 1–11. https://doi.org/10.1007/s13278-020-00677-0.Suche in Google Scholar
Barrientos, A.F., Sen, D., Page, G.L., and Dunson, D.B. (2023). Bayesian inferences on uncertain ranks and orderings: application to ranking players and lineups. Bayesian Anal. 18: 777–806. https://doi.org/10.1214/22-ba1324.Suche in Google Scholar
Basketball-Reference (2024). Available at: https://www.basketball-reference.com.Suche in Google Scholar
Devlin, S. and Uminsky, D. (2020). Identifying group contributions in NBA lineups with spectral analysis. J. Sports Anal. 6: 215–234. https://doi.org/10.3233/jsa-200407.Suche in Google Scholar
Girolami, M. and Calderhead, B. (2011). Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc., Ser. B: Stat. Methodol. 73: 123–214. https://doi.org/10.1111/j.1467-9868.2010.00765.x.Suche in Google Scholar
Grassetti, L., Bellio, R., Di Gaspero, L., Fonseca, G., and Vidoni, P. (2021). An extended regularized adjusted plus-minus analysis for lineup management in basketball using play-by-play data. IMA J. Manag. Math. 32: 385–409. https://doi.org/10.1093/imaman/dpaa022.Suche in Google Scholar
Hastie, T., Qian, J., and Tay, K. (2021). An introduction to glmnet. CRAN R Repos. 5: 1–35.Suche in Google Scholar
Hvattum, L.M. (2019). A comprehensive review of plus-minus ratings for evaluating individual players in team sports. Int. J. Comp. Sci. Sport 18:1–23. https://sciendo.com/article/10.2478/ijcss-2019-0001.10.2478/ijcss-2019-0001Suche in Google Scholar
Kolaczyk, E.D. and Csárdi, G. (2014). Statistical analysis of network data with R, 65. Springer, Cham, Switzerland.10.1007/978-1-4939-0983-4Suche in Google Scholar
Kubatko, J., Oliver, D., Pelton, K., and Rosenbaum, D.T. (2007). A starting point for analyzing basketball statistics. J. Quant. Anal. Sports 3. https://doi.org/10.2202/1559-0410.1070.Suche in Google Scholar
Lechner, C. and Vidar Gudmundsson, S. (2012). Superior value creation in sports teams: resources and managerial experience. M@ n@ gement: 284–312. https://doi.org/10.3917/mana.153.0284.Suche in Google Scholar
Macdonald, B. (2011). A regression-based adjusted plus-minus statistic for nhl players. J. Quant. Anal. Sports 7. https://doi.org/10.2202/1559-0410.1284.Suche in Google Scholar
Matano, F., Richardson, L., Pospisil, T., Politsch, C.A., and Qin, J. (2023). Augmenting adjusted plus-minus in soccer with fifa ratings. J. Quant. Anal. Sports 19: 43–49. https://doi.org/10.1515/jqas-2021-0005.Suche in Google Scholar
Maymin, A., Maymin, P., and Shen, E. (2013). NBA chemistry: positive and negative synergies in basketball. Int. J. Comput. Sci. Sport, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1935972.Suche in Google Scholar
Pelechrinis, K. (2018). Linnet: probabilistic lineup evaluation through network embedding. In: Joint European conference on machine learning and knowledge discovery in databases. Springer, Cham, Switzerland, pp. 20–36.10.1007/978-3-030-10997-4_2Suche in Google Scholar
Ramsay, J.O. and Silverman, B.W. (2002). Applied functional data analysis: methods and case studies. Springer, New York.10.1007/b98886Suche in Google Scholar
Rosenbaum, D. (2004). Measuring how NBA players help their teams win, Available at: http://www.82games.com/comm30.htm.Suche in Google Scholar
Sabin, R.P. (2021). Estimating player value in American football using plus–minus models. J. Quant. Anal. Sports 17: 313–364. https://doi.org/10.1515/jqas-2020-0033.Suche in Google Scholar
Sill, J. (2010). Improved nba adjusted +/- using regularization and out-of-sample testing. In: Proceedings of the 2010 MIT sloan sports analytics conference.Suche in Google Scholar
Smola, A.J. and Kondor, R. (2003). Kernels and regularization on graphs. In: Learning theory and kernel machines: 16th annual conference on learning theory and 7th kernel workshop, COLT/kernel 2003, Washington, DC, USA, august 24-27, 2003. Proceedings. Springer, pp. 144–158.10.1007/978-3-540-45167-9_12Suche in Google Scholar
Upton, E. and Carvalho, L. (2024). Modeling urban crime occurrences via network regularized regression. Ann. Appl. Stat. 18: 3364–3382. https://doi.org/10.1214/24-aoas1940.Suche in Google Scholar
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