Statistical Soccer (Football) Predictions
Encyclopedia
Statistical Football prediction is a method used in sports betting
, to predict the outcome of Association football matches by means of statistical tools. The goal of statistical match prediction is to outperform the predictions of bookmakers, who use them to set odds on the outcome of football matches.
The most widely-used statistical approach to prediction is ranking
. Football ranking systems assign a rank to each team based on their past game results, so that the highest rank is assigned to the strongest team. The outcome of the match can be predicted by comparing the opponents’ ranks. Today, eight different football ranking systems exist: FIFA World Rankings
, World Football Elo Ratings
, AQB Sports Ratings, The Roon Ba, InternationalMark, Rsoccer, Mondfoot and Chance de Gol.
There are three main drawbacks to football match predictions that are based on ranking systems:
Another approach to football prediction is known as rating systems. While ranking refers only to team order, rating systems assign to each team a continuously scaled strength indicator. Moreover, rating can be assigned not only to a team but to its attacking and defensive strengths, home field advantage or even to the skills of each team player (according to Stern ).
and negative binomial distribution
provided an adequate fit to results of football games. The series of ball passing between players during football matches was successfully analyzed using negative binomial distribution by Reep and Benjamin in 1968. They improved this method in 1971, and in 1974 Hill indicated that soccer game results are to some degree predictable and not simply a matter of chance.
The first model predicting outcomes of football matches between teams with different skills was proposed by Michael Maher in 1982. According to his model, the goals, which the opponents score during the game, are drawn from the Poisson distribution
. The model parameters are defined by the difference between attacking and defensive skills, adjusted by the home field advantage factor. The methods for modeling the home field advantage factor were summarized in an article by Caurneya and Carron in 1992. Time-dependency of team strengths was analyzed by Knorr-Held in 1999. He used recursive Bayesian estimation
to rate football teams: this method was more realistic in comparison to soccer prediction based on common average statistics.
and Knockout competition. The methods for Knockout competition are summarized in an article by Diego Kuonen.
The table below summarizes the methods related to Round-robin tournament
.
Assume that the teams A, B, C and D are playing in a tournament and the match outcomes are as follows:
Though the ratings
, 
, 
and 
of teams A, B, C and D respectively are unknown, it may be assumed that the outcome of match #1 is proportional to the difference between the ranks of teams A and B: 
. In this way, 
corresponds to the score difference and 
is the noise observation. The same assumption can be made for all the matches in the tournament:
By introducing a selection matrix X, the equations above can be rewritten in a compact form:
Entries of the selection matrix can be either 1, 0 or -1, with 1 corresponding to home teams and -1 to away teams:
If the matrix
has full rank, the algebraic solution of the system may be found via the Least squares
method:
The final rating parameters are
In this case, the strongest team has the highest rating. The advantage of this rating method compared to the standard ranking systems is that the numbers are continuously scaled, defining the precise difference between the teams’ strengths.
and 
are the goals scored in the match where team i plays against team j, then:
and 
are independent random variables with means 
and 
. Thus, the joint probability of the home team scoring x goals and the away team scoring y goals is a product of the two independent probabilities:
while the generalized log-linear model for
and 
according to Kuonen and Lee is defined as: 
and 
, where 
refers to attacking and defensive strengths and to home field advantage respectively. 
and 
are correction factors which represent the means of goals scored during the season by home and away teams.
Assuming that C signifies the number of teams participating in a season and N stands for the number of matches played until now, the team strengths can be estimated by minimizing the negative log-likelihood function with respect to
and 
:
Given that
and 
are known, the team attacking and defensive strengths 
and home ground advantage 
that minimize the negative log-likelihood can be estimated by Expectation Maximization:
Improvements for this model were suggested by Mark Dixon (statistician) and Stuart Coles. They invented a correlation factor for low scores 0-0, 1-0, 0-1 and 1-1, where the independent Poisson model doesn't hold. Dimitris Karlis and Ioannis Ntzoufras built a Time-Independent Skellam distribution model. Unlike the Poisson model that fits the distribution of scores, the Skellam model fits the difference between home and away scores.
They suggested modifying the generalized linear model above for
and 
:
given that
corresponds to the strength difference between teams i and j. The parameter 
then represents the psychological effects caused by underestimation of the opposing teams’ strength.
According to the model, the attacking strength
of team A can be described by the standard equations of Brownian motion, 
, for time 
:
where
and 
refer to the loss of memory rate and to the prior attack variance respectively.
This model is based on the assumption that:
Assuming that three teams A, B and C are playing in the tournament and the matches are played in the following order:
: A-B; 
: A-C; 
: B-C, the joint probability density can be expressed as:
Since analytical estimation of the parameters is difficult in this case, the Monte Carlo method
is applied to estimate the parameters of the model.
Sports betting
Sports betting is the activity of predicting sports results and placing a wager on the outcome.-United States of America:Aside from simple wagers such as betting a friend that one's favorite baseball team will win its division or buying a football "square" for the Super Bowl, sports betting is...
, to predict the outcome of Association football matches by means of statistical tools. The goal of statistical match prediction is to outperform the predictions of bookmakers, who use them to set odds on the outcome of football matches.
The most widely-used statistical approach to prediction is ranking
Ranking
A ranking is a relationship between a set of items such that, for any two items, the first is either 'ranked higher than', 'ranked lower than' or 'ranked equal to' the second....
. Football ranking systems assign a rank to each team based on their past game results, so that the highest rank is assigned to the strongest team. The outcome of the match can be predicted by comparing the opponents’ ranks. Today, eight different football ranking systems exist: FIFA World Rankings
FIFA World Rankings
The FIFA World Rankings is a ranking system for men's national teams in association football, currently led by Spain. The teams of the member nations of FIFA , football's world governing body, are ranked based on their game results with the most successful teams being ranked highest...
, World Football Elo Ratings
World Football Elo Ratings
The World Football Elo Ratings is a ranking system for men's national teams in association football. The method used to rank teams is based upon the Elo rating system method but modified to take various football-specific variables into account...
, AQB Sports Ratings, The Roon Ba, InternationalMark, Rsoccer, Mondfoot and Chance de Gol.
There are three main drawbacks to football match predictions that are based on ranking systems:
- Ranks assigned to the teams do not differentiate between their attacking and defensive strengths.
- Ranks are accumulated averages which do not account for skill changes in football teams.
- The main goal of a ranking system is not to predict the results of football games, but to sort the teams according to their average strength.
Another approach to football prediction is known as rating systems. While ranking refers only to team order, rating systems assign to each team a continuously scaled strength indicator. Moreover, rating can be assigned not only to a team but to its attacking and defensive strengths, home field advantage or even to the skills of each team player (according to Stern ).
History
Publications about statistical models for football predictions started appearing from the 90s, but the first model was proposed much earlier by Moroney, who published his first statistical analysis of soccer match results in 1956. According to his analysis, both Poisson distributionPoisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
and negative binomial distribution
Negative binomial distribution
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified number of failures occur...
provided an adequate fit to results of football games. The series of ball passing between players during football matches was successfully analyzed using negative binomial distribution by Reep and Benjamin in 1968. They improved this method in 1971, and in 1974 Hill indicated that soccer game results are to some degree predictable and not simply a matter of chance.
The first model predicting outcomes of football matches between teams with different skills was proposed by Michael Maher in 1982. According to his model, the goals, which the opponents score during the game, are drawn from the Poisson distribution
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
. The model parameters are defined by the difference between attacking and defensive skills, adjusted by the home field advantage factor. The methods for modeling the home field advantage factor were summarized in an article by Caurneya and Carron in 1992. Time-dependency of team strengths was analyzed by Knorr-Held in 1999. He used recursive Bayesian estimation
Recursive Bayesian estimation
Recursive Bayesian estimation, also known as a Bayes filter, is a general probabilistic approach for estimating an unknown probability density function recursively over time using incoming measurements and a mathematical process model.-In robotics:...
to rate football teams: this method was more realistic in comparison to soccer prediction based on common average statistics.
Football Prediction Methods
All the prediction methods can be categorized according to tournament type, time-dependence and regression algorithm. Football prediction methods vary between Round-robin tournamentRound-robin tournament
A round-robin tournament is a competition "in which each contestant meets all other contestants in turn".-Terminology:...
and Knockout competition. The methods for Knockout competition are summarized in an article by Diego Kuonen.
The table below summarizes the methods related to Round-robin tournament
Round-robin tournament
A round-robin tournament is a competition "in which each contestant meets all other contestants in turn".-Terminology:...
.
| # | Code | Prediction Method | Regression Algorithm | Time Dependence | Performance |
|---|---|---|---|---|---|
| 1. | TILS | Time Independent Least Squares Rating | Linear Least Squares Regression | No | Poor |
| 2. | TIPR | Time Independent Poisson Regression | Maximum Likelihood | No | Medium |
| 3. | TISR | Time Independent Skellam Regression | Maximum Likelihood | No | Medium |
| 4. | TDPR | Time-Dependent Poisson Regression | Maximum Likelihood | Time dumping factor | High |
| 5. | TDMC | Time-Dependent Markov Chain | Monte-Carlo | Markov Chain Markov chain A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the... model |
High |
Time Independent Least Squares Rating
This method intends to assign to each team in the tournament a continuously scaled rating value, so that the strongest team will have the highest rating. The method is based on the assumption that the rating assigned to the rival teams is proportional to the outcome of each match.Assume that the teams A, B, C and D are playing in a tournament and the match outcomes are as follows:
| Match # | Home Team | Score | Away Team | Y |
|---|---|---|---|---|
| 1 | A | 3 - 1 | B |  |
| 2 | C | 2 - 1 | D |  |
| 3 | D | 1 - 4 | B |  |
| 4 | A | 3 - 1 | D |  |
| 5 | B | 2 - 0 | C |  |
Though the ratings
, , and of teams A, B, C and D respectively are unknown, it may be assumed that the outcome of match #1 is proportional to the difference between the ranks of teams A and B: . In this way, corresponds to the score difference and is the noise observation. The same assumption can be made for all the matches in the tournament:By introducing a selection matrix X, the equations above can be rewritten in a compact form:
Entries of the selection matrix can be either 1, 0 or -1, with 1 corresponding to home teams and -1 to away teams:
If the matrix
has full rank, the algebraic solution of the system may be found via the Least squaresLeast squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
method:
The final rating parameters are
In this case, the strongest team has the highest rating. The advantage of this rating method compared to the standard ranking systems is that the numbers are continuously scaled, defining the precise difference between the teams’ strengths.Time-Independent Poisson Regression
According to this model (Maher ), if and are the goals scored in the match where team i plays against team j, then: and are independent random variables with means and . Thus, the joint probability of the home team scoring x goals and the away team scoring y goals is a product of the two independent probabilities:while the generalized log-linear model for
and according to Kuonen and Lee is defined as: and , where refers to attacking and defensive strengths and to home field advantage respectively. and are correction factors which represent the means of goals scored during the season by home and away teams.Assuming that C signifies the number of teams participating in a season and N stands for the number of matches played until now, the team strengths can be estimated by minimizing the negative log-likelihood function with respect to
and :Given that
and are known, the team attacking and defensive strengths and home ground advantage that minimize the negative log-likelihood can be estimated by Expectation Maximization:Improvements for this model were suggested by Mark Dixon (statistician) and Stuart Coles. They invented a correlation factor for low scores 0-0, 1-0, 0-1 and 1-1, where the independent Poisson model doesn't hold. Dimitris Karlis and Ioannis Ntzoufras built a Time-Independent Skellam distribution model. Unlike the Poisson model that fits the distribution of scores, the Skellam model fits the difference between home and away scores.
Time-Dependent Markov Chain Monte Carlo
On the one hand, statistical models require a large number of observations to make an accurate estimation of its parameters. And when there are not enough observations available during a season (as is usually the situation), working with average statistics makes sense. On the other hand, it is well-known that team skills change during the season, making model parameters time-dependent. Mark Dixon (statistician) and Coles tried to solve this trade-off by assigning a larger weight to the latest match results. Rue and Salvesen introduced a novel time-dependent rating method using the Markov Chain model.They suggested modifying the generalized linear model above for
and :given that
corresponds to the strength difference between teams i and j. The parameter then represents the psychological effects caused by underestimation of the opposing teams’ strength.According to the model, the attacking strength
of team A can be described by the standard equations of Brownian motion, , for time :where
and refer to the loss of memory rate and to the prior attack variance respectively.This model is based on the assumption that:
Assuming that three teams A, B and C are playing in the tournament and the matches are played in the following order:
: A-B; : A-C; : B-C, the joint probability density can be expressed as:Since analytical estimation of the parameters is difficult in this case, the Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
is applied to estimate the parameters of the model.