On the Dependency of Soccer Scores - A Sparse Bivariate Poisson Model for the UEFA EURO 2016

Transcription

1 On the Dependency of Soccer Scores - A Sparse Bivariate Poisson Model for the UEFA EURO 2016 A. Groll & A. Mayr & T. Kneib & G. Schauberger Department of Statistics, Georg-August-University Göttingen MathSport International 2017 Conference, Padua, June 28th Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

2 Who will celebrate? Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

3 Who will cry? Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

4 Theoretical Background Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

5 Aims The main aims are to find an explicit model for exact numbers of goals include covariates adjust for possible correlations between numbers of goals of both competing teams. Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

6 Aims The main aims are to find an explicit model for exact numbers of goals include covariates adjust for possible correlations between numbers of goals of both competing teams. Different approaches for EURO 2012 (Groll and Abedieh, 2013) World Cup 2014 (Groll, Schauberger and Tutz, 2015) EURO 2016 Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

7 Univariate Model for International Soccer Tournaments y ijk x ik, x jk Po(λ ijk ) i, j {1,..., n}, i j log(λ ijk ) = β 0 + ξ ik δ jk n: Number of teams y ijk : Number of goals scored by team i against opponent j at tournament k x ik, x jk : Covariate vectors of team i and opponent j varying over tournaments e.g. EURO 2012 (Groll and Abedieh, 2013): ξ ik = x T ikβ ξ + b i δ jk = x T jkβ δ + b j Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

8 Univariate Model for International Soccer Tournaments y ijk x ik, x jk Po(λ ijk ) i, j {1,..., n}, i j log(λ ijk ) = β 0 + ξ ik δ jk n: Number of teams y ijk : Number of goals scored by team i against opponent j at tournament k x ik, x jk : Covariate vectors of team i and opponent j varying over tournaments e.g. World Cup 2014 (Groll, Schauberger and Tutz, 2015): ξ ik = x T ikβ + att i δ jk = x T jkβ + def j Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

9 Univariate Model for International Soccer Tournaments y ijk x ik, x jk Po(λ ijk ) i, j {1,..., n}, i j log(λ ijk ) = β 0 + ξ ik δ jk n: Number of teams y ijk : Number of goals scored by team i against opponent j at tournament k x ik, x jk : Covariate vectors of team i and opponent j varying over tournaments e.g. World Cup 2014 (Groll, Schauberger and Tutz, 2015): ξ ik = x T ikβ + att i δ jk = x T jkβ + def j log(λ ijk ) = β 0 + (x ik x jk ) T β + att i def j Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

10 Correlation between Scores of Both Teams Dixon and Coles (1997) compared marginal distributions of scores with joint distribution correlation! Source: Dixon and Coles (1997) Introduction of additional dependence parameter But: They did not compare conditional distributions! Their linear predictors are not independent! Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

11 The Bivariate Poisson Distribution X k ind. Po(λ k ), k = 1, 2, 3, λ k > 0 Y 1 = X 1 + X 3 and Y 2 = X 2 + X 3 follow a joint bivariate Poisson distribution (Y 1, Y 2 ) Po 2 (λ 1, λ 2, λ 3 ) Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

12 The Bivariate Poisson Distribution X k ind. Po(λ k ), k = 1, 2, 3, λ k > 0 Y 1 = X 1 + X 3 and Y 2 = X 2 + X 3 follow a joint bivariate Poisson distribution Probability function: (Y 1, Y 2 ) Po 2 (λ 1, λ 2, λ 3 ) P Y1,Y 2 (y 1, y 2 ) = P(Y 1 = y 1, Y 2 = y 2 ) = exp( (λ 1 + λ 2 + λ 3 )) λy1 1 y 1! λ y2 min(y 1,y 2) 2 y 2! k=0 ( y 1 k ) ( y 2 k ) k! ( λ k 3 ) λ 1 λ 2 Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

13 The Bivariate Poisson Distribution X k ind. Po(λ k ), k = 1, 2, 3, λ k > 0 Y 1 = X 1 + X 3 and Y 2 = X 2 + X 3 follow a joint bivariate Poisson distribution Probability function: (Y 1, Y 2 ) Po 2 (λ 1, λ 2, λ 3 ) P Y1,Y 2 (y 1, y 2 ) = P(Y 1 = y 1, Y 2 = y 2 ) = exp( (λ 1 + λ 2 + λ 3 )) λy1 1 y 1! λ y2 min(y 1,y 2) 2 y 2! k=0 ( y 1 k ) ( y 2 k ) k! ( λ k 3 ) λ 1 λ 2 E(Y 1 ) = λ 1 + λ 3 E(Y 2 ) = λ 2 + λ 3 cov(y 1, Y 2 ) = λ 3 Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

14 The Bivariate Poisson Distribution λ 1 = 2 λ 2 = 2 λ 3 = 0 λ 1 = 1 λ 2 = 1 λ 3 = 1 Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

15 Re-parametrization of bivariate Poisson distribution Replace λ 1 = γ 1 γ 2 and λ 2 = γ1 γ 2 : P Y1,Y 2 (y 1, y 2) = P(Y 1 = y 1, Y 2 = y 2) = exp( (γ 1(γ 2 + γ 1 2 ) + λ 3)) (γ1γ2)y 1 ( γ 1 γ 2 ) y 2 min(y 1,y 2 ) y 1! y 2! k=0 ( y1 k ) ( y2 k ) k! ( λ3 γ 2 1 k ) γ 1 = exp(β 0 ) γ 2 = exp( x T β) λ 3 = exp(α 0 + x T α) with x = x 1 x 2. Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

16 Re-parametrization of bivariate Poisson distribution Replace λ 1 = γ 1 γ 2 and λ 2 = γ1 γ 2 : P Y1,Y 2 (y 1, y 2) = P(Y 1 = y 1, Y 2 = y 2) = exp( (γ 1(γ 2 + γ 1 2 ) + λ 3)) (γ1γ2)y 1 ( γ 1 γ 2 ) y 2 min(y 1,y 2 ) y 1! y 2! k=0 ( y1 k ) ( y2 k ) k! ( λ3 γ 2 1 k ) γ 1 = exp(β 0 ) λ 1 = exp(β 0 + x T β) γ 2 = exp( x T β) λ 2 = exp(β 0 x T β) λ 3 = exp(α 0 + x T α) with x = x 1 x 2. Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

17 Bivariate Poisson Model for Football Results (y ik, y jk ) x ik, x jk Po 2 (γ 1, γ ijk2, λ ijk3 ) γ 1 = exp(β 0 ) γ ijk2 = exp((x ik x jk ) T β) λ ijk3 = exp(α 0 + x ik x jk T α) Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

18 Bivariate Poisson Model for Football Results (y ik, y jk ) x ik, x jk Po 2 (γ 1, γ ijk2, λ ijk3 ) γ 1 = exp(β 0 ) γ ijk2 = exp((x ik x jk ) T β) λ ijk3 = exp(α 0 + x ik x jk T α) Framework of the so-called Generalized Additive Model for Location, Scale and Shape (GAMLSS; Rigby and Stasinopoulos, 2005) Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

19 Boosting for GAMLSS R-package gamboostlss (Hofner, Mayr and Schmid, 2015) Allows for variable selection within GAMLSS framework Provides a large number of pre-specified distributions Negative binomial distribution Zero-inflated Poisson distribution... Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

20 Boosting for GAMLSS R-package gamboostlss (Hofner, Mayr and Schmid, 2015) Allows for variable selection within GAMLSS framework Provides a large number of pre-specified distributions Negative binomial distribution Zero-inflated Poisson distribution... Mostly restricted to univariate responses, first approach for bivariate normal distribution from Andreas Mayr Users can specify new distributions (also bivariate) by providing loss/risk function neg. log-likelihood neg. gradient of loss function score function possibly suitable offsets for linear predictors Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

21 Boosting for GAMLSS R-package gamboostlss (Hofner, Mayr and Schmid, 2015) Allows for variable selection within GAMLSS framework Provides a large number of pre-specified distributions Negative binomial distribution Zero-inflated Poisson distribution... Mostly restricted to univariate responses, first approach for bivariate normal distribution from Andreas Mayr Users can specify new distributions (also bivariate) by providing loss/risk function neg. log-likelihood neg. gradient of loss function score function possibly suitable offsets for linear predictors We implemented bivariate Poisson distribution Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

22 Application to UEFA Europoean Championship 2016 Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

23 Covariates Economic Factors: GDP per capita population Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

24 Covariates Economic Factors: GDP per capita population Sportive Factors: Home advantage ODDSET odds market value FIFA rank UEFA points Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

25 Covariates Economic Factors: GDP per capita population Sportive Factors: Home advantage ODDSET odds market value FIFA rank UEFA points Factors describing the team s structure (Second) maximum number of teammates Average age Number of CL players Number of Europa League players Age of the national coach Nationality of the national coach Number of players abroad Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

26 Structure of Dataset Data from 2004, 2008 and 2012 All covariates are differences between values of both teams Team 1 Team 2 Goals 1 Goals 2 Year Odds Market Value 1 Portugal Greece Spain Russia Greece Spain Russia Portugal Spain Portugal Russia Greece Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

27 Parameter Estimates γ 1 = exp( ˆβ 0 ): Estimates: ˆβ0 = 0.176, Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

28 Parameter Estimates γ 1 = exp( ˆβ 0 ): Estimates: ˆβ0 = 0.176, γ ijk2 = exp((x ik x jk ) T β): Estimates: ( ˆβ odds, ˆβ marketvalue, ˆβ UEFApoints ) = ( 0.120, 0.143, 0.029) λ ijk3 = exp(α 0 + x ik x jk T α): Estimates: ˆα 0 = 9.21, ˆα = 0 λ ijk3 0 Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

29 Parameter Estimates γ 1 = exp( ˆβ 0 ): Estimates: ˆβ0 = 0.176, γ ijk2 = exp((x ik x jk ) T β): Estimates: ( ˆβ odds, ˆβ marketvalue, ˆβ UEFApoints ) = ( 0.120, 0.143, 0.029) λ ijk3 = exp(α 0 + x ik x jk T α): Estimates: ˆα 0 = 9.21, ˆα = 0 λ ijk3 0 very simple model no (additional) covariance between scores of both teams Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

30 Simulation of the tournament progress Every single match outcome can be simulated by drawing from the respective Poisson distributions Per group, the exact standing after the group stage can be calculated Decision on qualification for round of 16 according to UEFA rules Draws in knockout stage: Simulate extra time with 1/3 of Poisson parameters Simulate penalty shootout by coin flip 1,000,000 simulation runs for the UEFA European Championship 2016 Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

31 Probabilities for UEFA European Football Champion 2016 Round Quarter Semi Final European Oddset of 16 Finals Finals Champion 1 Spain Germany France England Belgium Portugal Italy Croatia Poland Austria Switzerland Turkey Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

32 Most Probable Final Group Standings A B C 1 France England Germany 2 Switzerland Wales Poland 3 Romania Russia Ukraine 4 Albania Slovakia Nor. Ireland 21.2% 15.1% 37.6% D E F 1 Spain Belgium Portugal 2 Croatia Italy Austria 3 Turkey Sweden Iceland 4 Czech Rep. Ireland Hungary 17.7% 17.5% 16.9% Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

33 Most Probable Course of Knockout Stage Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

34 Summary Theoretical Results: Bivariate Poisson model for scores of both teams Implementation into framework of GAMLSS via gamboostlss Very sparse model No additional covariance, reduces to two (conditionally) independent Poisson distributions Prediction Results: Survival rates per team and tournament stage Most probable course of tournament Spain favorite team followed by Germany and France Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

35 References Groll, A. and J. Abedieh (2013). Spain retains its title and sets a new record - generalized linear mixed models on European football championships. Journal of Quantitative Analysis in Sports 9(1), Groll, A., G. Schauberger, and G. Tutz (2015). Prediction of major international soccer tournaments based on team-specific regularized Poisson regression: An application to the FIFA World Cup Journal of Quantitative Analysis in Sports 11(2), Hofner, B., A. Mayr, and M. Schmid (2015). gamboostlss: An R package for model building and variable selection in the GAMLSS framework. Journal of Statistical Software 74(1), Karlis, D. and I. Ntzoufras (2003). Analysis of sports data by using bivariate poisson models. The Statistician 52, Rigby, R. A. and D. M. Stasinopoulos (2005). Generalized additive models for location, scale and shape. Journal of the Royal Statistical Society: Series C (Applied Statistics) 54(3), Stasinopoulos, D. M. and R. A. Rigby (2007). Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software 23(7), Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

36 First Idea for Bivariate Model (y ik, y jk ) x ik, x jk Po 2 (λ ik1, λ jk2, λ ijk3 ) log(λ ik1 ) = β 0 + x T ik β log(λ jk2 ) = β 0 + x T jk β log(λ ijk3 ) = α 0 + x ik x jk T α Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

37 GAMLSS Generalized Additive Models for Location, Scale and Shape g 1 (µ) = η µ = β 0µ + f jµ (x j ) g 2 (σ) = η σ = β 0σ + p 1 j=1 p 2 j=1 f jσ (x j ) "location" "scale" Proposed by Rigby and Stasinopoulos (2005) Extension of generalized additive models (GAMs) The distribution parameters are modeled by specific predictors and associated link functions g k ( ). Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

38 Example for GAMLSS Y N( µ = β 0µ + f µ (x), σ = exp (β 0σ + f σ (x)) x y Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22

39 First Idea for Bivariate Model (y ik, y jk ) x ik, x jk Po 2 (λ ik1, λ jk2, λ ijk3 ) log(λ ik1 ) = β 0 + x T ik β log(λ jk2 ) = β 0 + x T jk β log(λ ijk3 ) = α 0 + x ik x jk T α In general, in GAMLSS effects for predictors differ across different components Restrictions for parameters would become necessary! Solution: Re-parametrize bivariate Poisson distribution Use differences between covariates Groll et al. (MathSport 2017) A Sparse Model for the EURO / 22