Fig 1.
The general sketch of this study.
Fig 2.
The values of the attacking strengths, OA, and defensive vulnerabilities, VB, from the full dataset of 55 European teams, using matches between July 16, 2018 and May 28, 2021. Note that the parameters have been normalised so that the maximum defensive vulnerability is equal to 1.
Fig 3.
Parameter changes after removing San Marino.
A comparison of the estimates of the offensive strengths, OA, and defensive vulnerabilities, VB, depending on whether San Marino’s results are used. Note that the black lines join points corresponding to the same team, and that the variables have been normalised so that the defensive vulnerability of Gibraltar (0.79) is the same in both cases.
Fig 4.
Changes in offensive strength after removing San Marino.
The percentage change in the estimated offensive strength, OA, for different teams when San Marino is removed from the model. Note that only values that changed by at least 2% have been included.
Fig 5.
Changes in defensive vulnerability after removing San Marino.
The percentage change in the estimated defensive vulnerability, VB, for different teams when San Marino is removed from the model. Note that only values that changed by at least 2% have been included.
Fig 6.
Confidence intervals for the OA parameters.
Approximate 95% confidence intervals for the offensive strengths, OA of the teams playing in Euro 2020.
Fig 7.
Confidence intervals for the VB parameters.
Approximate 95% confidence intervals for the defensive vulnerabilities, VB of the teams playing in Euro 2020.
Fig 8.
Predicting goals scored by each team.
The number of goals scored by each team, alongside the predicted (mean) number of goals scored and an approximate 95% prediction interval.
Fig 9.
Predicting goals conceded by each team.
The number of goals conceded by each team, alongside the predicted (mean) number of goals conceded and an approximate 95% prediction interval.
Fig 10.
A Quantile-Quantile (QQ) plot for the re-generated uniform random variables.
A QQ Plot for the uniform random variable simulations described in (30), conditional on the results of Euro 2020. Note that for each result, 100 uniform random variables were sampled in order to minimise variance in the plot.
Fig 11.
The attacking strengths as a function of the start date of the dataset used.
This shows how the predicted attacking strengths, OA of the 24 teams that competed in Euro 2020 vary depending on the start date of the dataset of matches. Note that in all cases the end date was May 28, 2021—the date of submission to the competition.
Fig 12.
The log-likelihood of Euro 2020 as a function of the start date of the dataset used.
This shows how the log-likelihood of Euro 2020 according to the model (the metric for the RSS competition) depends on the start date of the dataset of matches. Note that in all cases the end date was May 28, 2021—the date of submission to the competition, and each international break was considered as a single unit (and thus it was assumed that one would not set the start date in the middle of such a break).
Fig 13.
A comparison of team rankings from the double Poisson model and those from the linear model.
This shows the rankings that are derived from each model for the 24 teams in Euro 2020. Note that the rankings have been normalised to ensure that they have the same sum, and to allow for easy comparison. This means that the Poisson rankings do not match the parameters considered in previous figures.