After a sombre build up the 2012 UEFA European Football Championship, or Euro 2012, kicked off on Friday with co-hosts Poland drawing 1-1 with 2004 winners Greece followed by a convincing 4-1 win by Russia over the Czech Republic. Saturday saw highly-rated Holland lose to Denmark followed by Germany beating Portugal 1-0, and yesterday defending champions Spain fought back from a goal down to draw 1-1 with 2006 World Cup winners Italy. The Republic of Ireland lost 3-1 to Croatia yesterday and today sees England take on France whilst Sweden take on co-hosts Ukraine. But who will win the tournament? Which teams will make it through their groups and into the quarter-finals? And what about luck? Skill? Expert opinion? This is where statistical analysis comes in.
With Norway not qualifying for Euro 2012 we at the Norwegian Computing Center are statistically modelling the winning chances for all teams involved. The calculations are just for fun (and we’ve not reduced Sweden’s chances!) and have been done to demonstrate that statistical analysis can be fun. And as Norway has not qualified for the tournament we have plenty of time to do this kind of stuff instead!
The probabilities are found by simulating ("playing" on a computer) all remaining matches numerous times. The remaining part of the tournament is "played" 10,000 times and from these simulations we calculate all the teams' chances of winning the Championship, their group, to reach the second round etc. When the remaining matches are "played" we consider details such as:
a) In the group games the order in the table is primarily given by the number of points a team amasses (3 points for a win, 1 for a tie and 0 for a loss). If two or more teams are equal on points their rankings are determined by the number of points, goal difference and number of goals scored in the matches amongst the teams in question. If some teams still have equal ranks, their rankings are then determined by goal difference and number of goals scored in all the group matches, and thereafter by the point coefficients from the qualifying competitions for the two last championships, fair play conduct and finally by drawing lots.
b) Matches in the second round cannot end as a draw. If the result stands as a draw after reaching the end of 90 minutes, extra time of two periods of 15 minutes will be played. If the match is still tied penalties will be taken to determine the winner.
The result of a match will depend on the strength of the two teams but there is also an element of chance. Before the tournament starts expert opinions from twelve Norwegian football enthusiasts give the basis for the strength of each team. As the matches are actually played the strength is updated for every team. Consequently, the strength will increasingly be given by the matches actually played and gradually less by the opinions given in advance.
Consider a game between team A and team B. In our model the number of goals scored by team A are Poisson distributed with parameter (a number) L(A,B). This means that we expect that team A will score about L(A,B) goals in a match against team B. Here
L(A,B) = "Normal number of goals"
x ("Strength of team A")/("Strength of team B").
"Normal number of goals" is a parameter (a number) which is interpreted as the average number of goals scored by one team in a match between to equally good teams. "Strength of team A" is a parameter (a number), which tells how good team A is, whereas "Strength of team B" tells how good team B is. The strength of Brazil is fixed to 100, and the strengths of all other teams are given relative to this. Further, the number of goals scored by team B are Poisson distributed with parameter L(B,A), independent of the number of goals scored by team A.
This means that if team A has a higher strength than team B, we will expect that team A wins, because L(A,B) will be greater than L(B,A). However, it will still be a possibility for a draw, or that team B wins.
Of course, our simple model is not able to cover all important aspects of a football match. However, in the Championship, few match results will be available for parameter estimation, so a simple model is needed to avoid over-fitting. However, in a national league, with many games over a season, one may consider several extensions to our model. These include
- Home-team advantage
- Offense strength and defence strength
- That the strengths of the teams vary over the season
- Number of goals of each team is correlated
In recent years several articles about this theme have been published in the statistical literature. A good introduction is Lee, A. (1997), "Modeling Scores in the Premier League: Is Manchester United Really the Best?", Chance, Vol 10, s. 15-19.
The parameters in the model are "Normal number of goals" and the strengths of each team. These parameters must be estimated before any probability calculations can be done. Before the start of the tournament, the estimation is based on evaluations from several Norwegian football experts. The experts have guessed the results of several hundred hypothetical games, and these results are transferred to number values of the parameters.
When the tournament starts the real games are taken into account as well. The information value of the hypothetical games (the expert guesses) is weighted versus the real games, such that the hypothetical games are equally important as the real games when all teams have played two matches. When all teams have played three or more matches the real matches are the most important part in the estimation of the parameters.
Estimating the parameters means to find the parameter values that fit the data (the match results) as good as possible. In our case, the parameters are estimated by maximising a modified Poisson likelihood. The difference from an ordinary Poisson likelihood is that it is robustified by down-weighting large victories and by adding a penalty term that shrinks the individual strength parameters towards a common mean.
This means we have three tuning parameters: 1) The weight of the expert guesses versus the real games, 2) the parameter that decides how large victories should be the down-weighted and 3) The penalty or shrinkage parameter that shrinks the strength parameters towards a common mean.
The value of these tuning parameters has been found by a forward validation experiment, similar to cross validation, on previous World and European Championships since 1998 and onwards. Here we start to predict the matches in the first round based on the model fitted to the expert guesses only. Then we re-estimate the method and predict the matches in the second round etc. until we predict the final. The set of tuning parameter values that is best on average over all tournaments has been chosen.
Estimated strength of all teams
The current estimate of "Normal number of goals" is 1.1.
The estimated strength values of each team are given in the table to the right (sorted), together with the FIFA ranking as per June 4, 2012. The discrepancy between the strength and the FIFA ranking is due to the fact that the expert opinions differ somewhat from the FIFA ranking. In addition, as the tournament progresses, teams with good results will obtain higher strength even though they may have a low FIFA ranking.
For more information and to follow the analysis visit the Norwegian Computing Center's website and keep an eye out for your favourite team - apologies to Austrians, Belgians, Turks and the Swiss - maybe at the next tournament in 2016 your team will be present!