A simple Bayesian procedure for forecasting the outcomes of the UEFA Champions League matches

Foulley, Jean-Louis

Journal de la société française de statistique, Numéro spécial : Sport et Statistique, Tome 156 (2015) no. 2, pp. 38-50.

Cet article présente la mise en œuvre bayésienne d’un modèle de probit cumulatif en vue de la prédiction des résultats de matches de la ligue des champions UEFA. L’argument de la fonction de répartition normale fait intervenir un seuil, un effet de match joué à domicile et la différence entre les valeurs des équipes en compétition. On suppose que la valeur d’une équipe est distribuée normalement avec une espérance qui s’exprime comme la régression sur une évaluation externe de l’équipe provenant du classement UEFA des clubs ou du système de classement mondial des clubs (FCWR). Les a priori sur ces paramètres sont mis à jour au début de chaque saison à partir des a posteriori obtenus en fin de saison précédente. Cela permet de prédire les résultats des matches de chacune des phases de la compétition  : minichampionnat de groupe et tournoi par élimination. On présente une application de cette méthode à la saison 2013-14. L’ajustement par le système FCWR est meilleur que celui obtenu par le coefficient UEFA. L’utilisation du premier conduit à un gain net de 24% sur la précision et de 23% sur le score de Brier par rapport à la situation témoin de non ajustement a priori pour la valeur d’équipe. On discute également d’un classement des équipes sur leurs performances lors de ce championnat et des possibilités d’inclure d’autres sources d’information dans le modèle.

This article presents a Bayesian implementation of a cumulative probit model to forecast the outcomes of the UEFA Champions League matches. The argument of the normal CDF involves a cut-off point, a home vs away playing effect and the difference in strength of the two competing teams. Team strength is assumed to follow a Gaussian distribution the expectation of which is expressed as a linear regression on an external rating of the team from eg. the UEFA Club Ranking (UEFACR) or the Football Club World Ranking (FCWR). Priors on these parameters are updated at the beginning of each season from their posterior distributions obtained at the end of the previous one. This allows making predictions of match results for each phase of the competition: group stage and knock-out. An application is presented for the 2013-2014 season. Adjustment based on the FCWR performs better than on UEFACR. Overall, using the former provides a net improvement of 24% and 23% in accuracy and Brier’s score over the control (zero prior expected difference between teams). A rating and ranking list of teams on their performance at this tournament and possibilities to include extra sources of information (expertise) into the model are also discussed.

[1] Agresti, Alan Analysis of ordinal paired comparison data, Applied Statistics (1992), pp. 287-297 | Zbl

[2] Barnard, John; McCulloch, Robert; Meng, Xiao-Li Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage, Statistica Sinica, Volume 10 (2000) no. 4, pp. 1281-1312 | MR | Zbl

[3] Cattelan, Manuela Models for paired comparison data: A review with emphasis on dependent data, Statistical Science, Volume 27 (2012) no. 3, pp. 412-433 | MR | Zbl

[4] Coulom, Rémi Whole-history rating: A Bayesian rating system for players of time-varying strength, Computers and games, Springer, 2008, pp. 113-124 | Zbl

[5] Cattelan, Manuela; Varin, Cristiano; Firth, David Dynamic Bradley–Terry modelling of sports tournaments, Journal of the Royal Statistical Society: Series C (Applied Statistics), Volume 62 (2013) no. 1, pp. 135-150 | MR

[6] Davidson, Roger R On extending the Bradley–Terry model to accommodate ties in paired comparison experiments, Journal of the American Statistical Association, Volume 65 (1970) no. 329, pp. 317-328

[7] Forrest, David; Goddard, John; Simmons, Robert Odds-setters as forecasters: The case of English football, International Journal of Forecasting, Volume 21 (2005) no. 3, pp. 551-564

[8] Foulley, Jean-Louis; Jaffrézic, Florence Modelling and estimating heterogeneous variances in threshold models for ordinal discrete data via Winbugs/Openbugs, Computer methods and programs in biomedicine, Volume 97 (2010) no. 1, pp. 19-27

[9] Gelman, Andrew; Carlin, John B; Stern, Hal S; Rubin, Donald B Bayesian data analysis, CRC Press/Chapman & Hall, 2004 | MR

[10] Glenn, WA; David, HA Ties in paired-comparison experiments using a modified Thurstone-Mosteller model, Biometrics, Volume 16 (1960) no. 1, pp. 86-109 | Zbl

[11] Gelman Prior distributions for variance parameters in hierarchical models, Bayesian analysis, Volume 3 (2006), pp. 515-534 | Zbl

[12] Goddard, John Regression models for forecasting goals and match results in association football, International Journal of forecasting, Volume 21 (2005) no. 2, pp. 331-340

[13] Glickman, Mark E; Stern, Hal S A state-space model for National Football League scores, Journal of the American Statistical Association, Volume 93 (1998) no. 441, pp. 25-35 | Zbl

[14] Laud, Purushottam W; Ibrahim, Joseph G Predictive model selection, Journal of the Royal Statistical Society. Series B (Methodological), Volume 57 (1995), pp. 247-262 | MR | Zbl

[15] Lunn, David; Jackson, Chris; Best, Nicky; Thomas, Andrew; Spiegelhalter, David The BUGS book: A practical introduction to Bayesian analysis, CRC Press / Chapman & Hall, 2012 | Zbl

[16] Plummer, Martyn Penalized loss functions for Bayesian model comparison, Biostatistics, Volume 9 (2008) no. 3, pp. 523-539 | Zbl

[17] Rao, PV; Kupper, Lawrence L Ties in paired-comparison experiments: A generalization of the Bradley-Terry model, Journal of the American Statistical Association, Volume 62 (1967) no. 317, pp. 194-204 | MR

[18] Shawul, Daniel; Coulom, Rémi Paired Comparisons with Ties: Modeling Game Outcomes in Chess, Web Technical report (2012)