The FIFA rankings for women’s national teams use a quite different methodology than the one used in ranking men’s national teams. The Women’s World Ranking (WWR) is based on the Elo rating system I wrote about in the previous post. The details for the men’s ranking can be found here, and the details the women’s ranking can be found here.

One thing that makes the WWR interesting is how goal differences are accounted for in the ratings. This is not something found in the ordinary chess-based Elo ratings. The method used in WWR is to let the ‘winning percentage’ change depending on two things: Goal difference and the number of goals the loosing team scored. This is in contrast to the original Elo ratings where the winning team wins 100% and the loosing team wins 0% (a draw is 50%). In WWR a team can never win 100%, the most it can win is 99%. This is the case when the goal difference is more than 6 and the loosing team haven’t scored any goals. The table below is from the pdf file linked to above and shows how many percent the loosing team win.

One strange thing about this table is the column for goal difference 0, implying a draw. I am guessing this is an error since it means that the winning percentage for the loosing team will be greater than the winning team. In the paper I mentioned in my last post where a number of different rating methods were compared (The predictive power of ranking systems in association football by J. Laset et. al), it was assumed that draw would yield both teams 50%, as in the original Elo-ratings. That paper also showed that the Women’s World Ranking was among the rating systems with best prediction.

Here is a plot showing the win percentage when the loosing team has scored 0 and 5 goals. We can see that there is not much gained for the loosing team to score one extra goal (assuming the goal difference stays the same, which of course is dubious), and most of the gain in winning percentage is when a team scores a goal such that the stance goes from a draw to a win.

The table above only goes up to 5 goals for the loosing team, but for the sake of implementation it is easy to generalize the rule about how much the loosing team gains by scoring an additional goal (with the goal difference is the same). For example will the loosing team gain 0.9 extra winning percentage points when the goal difference is 2. Similarly the gain is 0.6 percentage points when the goal difference is 5.

Below is a R function to compute the win percentage for football matches. It takes as input two vectors with the number of goals scored by the two opponents and returns a vector of win percentages (a number between 0 and 1) for the first team.

winPercentageWWR <- function(team1Goals, team2Goals){
  #calculates the win percentage for team 1.

  stopifnot(length(team1Goals) == length(team2Goals))
  
  perc <- c(0.01, 0.02, 0.03, 0.04, 0.08, 0.15, 0.50, 0.85, 0.92, 0.96, 0.97, 0.98, 0.99)
  add <- c(0, 0.01, 0.009, 0.008, 0.007, 0.006, 0.005)
  
  goalDifferences <- (team1Goals - team2Goals)
  goalDifferences[goalDifferences < -6] <- -6
  goalDifferences[goalDifferences > 6] <- 6

  team1WinPercentage <- numeric(length=length(goalDifferences))

  for (idx in 1:length(goalDifferences)){

    team1WinPercentage[idx] <- perc[goalDifferences[idx]+7] -  
      sign(goalDifferences[idx])*min(team1Goals[idx], team2Goals[idx])*add[abs(goalDifferences[idx])+1]
  }
  return(team1WinPercentage) 
}

I have previously written about some statistical methods for rating football teams and to predict the result of future matches. One was the last squares method and another was the Poisson regression method. None of these methods make good enough predictions. One problem with them is that they don’t incorporate a time perspective. Matches played a year ago is given equal importance as the most recent one. This could however be incorporated by weighing the the older matches less than newer matches. One other problem that I mentioned in the second post about Poisson regression is that teams are treated as categoricals which makes it hard to model the fact that a team’s ability changes over time.

One different kind of method that has been employed a lot in the recent years is the Elo rating system, which were originally developed for rating chess players. The method is rather simple, but I will not explain it in detail here since there are many good explanations of it elsewhere. Wikipedia has a very thorough coverage. The basic principle is that the difference in ratings between the two opposing teams provide a prediction for the result each game. The rating is then updated based on how the teams perform. If a team performs better than expected the rating increases, if they perform worse than expected the rating decrease. How much the rating changes depends on an update factor (often referred to as the K-factor).

Chess and football are of course different in many ways so the method for rating chess players is not directly suitable for rating football teams. The relative simplicity of the Elo system makes it easy to tweak and adjust to better fit football by incorporating things like home field advantage and goal difference. There are many sites around the Internet who provide different variants of Elo ratings, like the World Football Elo Ratings for national teams and Club Elo and Euro Club Index for club teams. FIFA even uses its own Elo system in its Womans World Ranking.

There has even been some research into different football rating systems. A paper titled The predictive power of ranking systems in association football (pdf) by Jan Lasek and others compared different rating systems. Their conclusion was that the different Elo type systems in general were better at predicting match outcomes than other types of rating systems.

I figured I wanted to implement a simple Elo rating system for rating football teams. There is already a package in R, PlayerRatings, which implements several different rating systems based on Elo. In my simple implementation there is no adjustment for goal difference, but I have support for home field advantage. All teams start with an initial rating of 1500. Here is what I got when I calculated the ratings for Premier League in November 2012 based on data going back to 1993. I used an update factor 24 without any home field advantage. There is no particular reason for this as I did this mostly as a proof of concept.

The table seems reasonable I think except for a couple of things. There is a problem related to relegation and promotion. Since I have used data back to 1993 every team who has played in the Premier League is given a rating. If a team is relegated to the Championship, their rating will no longer be updated. We can see that this creates some strange results. Take the two lowest rated teams for example. Derby has not been in the Premier League since the 2007-2008 season. Swindon, which is rated about 100 points higher than Derby, has not played in the Premier League since 1993-1994 season! Swindon now play in the fourth level of the English league system. So the ratings for the teams not in the Premier League should be considered invalid.

Relegation and promotion also creates a problem with inflated ratings. The Elo system is created so that the total number of points in the league should be constant. When a team is promoted they start with an initial rating of 1500, and if they later gets relegated they will probably have lost some of those points to the other teams in the league. In fact, we see that many of the teams with ratings less than 1500 no longer plays in the Premier League. The points they have lost are still in present in the league even though the team isn’t. This means that over time the average ratings of the teams in the league will increase.

The code I have written takes a data frame as input and works “out of the box” with data from football-data.co.uk. If you are going to use it yourself you have to make sure the data is sorted by date as the rating function just loops from top to bottom.

Here is how you can use it:

dta <- read.csv("yourdata.csv")
elo <- eloRating(data=dta)
print(elo)

And here is the code:

eloRating <- function(home="HomeTeam", away="AwayTeam", homeGoals="FTHG",
                      awayGoals="FTAG", data, kfactor=24, initialRating=1500,
                      homeAdvantage=0){
  
  #Make a list to hold ratings for all teams
  all.teams <- levels(as.factor(union(levels(as.factor(data[[home]])),
                                      levels(as.factor(data[[away]])))))
  
  ratings <- as.list(rep(initialRating, times=length(all.teams)))
  names(ratings) <- all.teams

  #Loop trough data and update ratings
  for (idx in 1:dim(data)[1]){
  
    #get current ratings
    homeTeamName <- data[[home]][idx]
    awayTeamName <- data[[away]][idx]
    homeTeamRating <- as.numeric(ratings[homeTeamName]) + homeAdvantage
    awayTeamRating <- as.numeric(ratings[awayTeamName])
    
    #calculate expected outcome 
    expectedHome <- 1 / (1 + 10^((awayTeamRating - homeTeamRating)/400))
    expectedAway <- 1 - expectedHome
    
    #Observed outcome
    goalDiff <- data[[homeGoals]][idx] - data[[awayGoals]][idx]
    if (goalDiff == 0){
      resultHome <- 0.5
      resultAway <- 0.5
    }
    else if (goalDiff < 0){
      resultHome <- 0
      resultAway <- 1
    }
    else if (goalDiff > 0){
      resultHome <- 1
      resultAway <- 0
    }
    
    #update ratings
    ratings[homeTeamName] <- as.numeric(ratings[homeTeamName]) + kfactor*(resultHome - expectedHome)
    ratings[awayTeamName] <- as.numeric(ratings[awayTeamName]) + kfactor*(resultAway - expectedAway)
  }
  
  #prepare output
  ratingsOut <- as.numeric(ratings)
  names(ratingsOut) <- names(ratings)
  ratingsOut <- sort(ratingsOut, decreasing=TRUE)

  return(ratingsOut) 
}