# The Dixon-Coles model, part 4: A trick to speed up estimation

In the previous installments in this series on implementing the Dixon-Coles model I complained a bit about the time it took to estimate the parameters. In the original implementation in part 1 it took about 40 seconds. Now 40 seconds is not much to complain about, there are a whole lot of other models and algorithms that takes much much longer time to fit (for my master’s I had some computations that took several months). Still, I wanted to make a few improvements.

The approach I described in part 3 is quite acceptable, I think, especially since it takes less than a second to fit the model. But still, I wanted to make some improvements to my original implementation.

There are several reasons for the estimation procedure being slow. I used a general purpose optimizer instead of a tailor-made algorithm, and I didn’t provide the optimizer with a function of the derivative of the model likelihood function, nor the function defining the constraint. This means that the optimizer have to estimate the derivatives by doing a lot of evaluations of the two functions with slight changes in the parameters. The most important speed bump, however, is probably due to how I implemented the constraint that all the average of the attack parameters should equal 1.

The alabama package I used relied on a technique called Lagrange multipliers, which is a very general method for constrained optimization. Instead of relying on general constrained optimization procedures, there is a trick commonly used in linear models with sum-to-zero constrained categorical parameters that we also can use here.

There has been some discussion and confusion in the comments about how categorical variables are coded and how R presents the results of the glm function. A thorough discussion of this is best left for another time, but let me explain how the sum-to-zero constraint is implemented in linear models. We will fit the model with this constraint and then make some adjustments later on to get the correct average-is-one constraint.

The sum-to-zero constraint basically says that the sum of all the parameters for a categorical variable must equal to zero:

$$\sum_{i=1} \theta_i = 0$$

If we for example have three levels, we can write out the equation like this:

$$\theta_1 + \theta_2 + \theta_3 = 0$$

If we subtract $$\theta_3$$ and multiply both sides of the equation by minus 1 we get

$$– \theta_1 – \theta_2 = \theta_3$$

Notice how we can write one of the parameters as a simple linear function of the other parameters. We can use this result to construct the design matrix for the categorical variable, incorporating the sum-to-zero constraint (exactly which parameter or level we chose to be a function of the others doesn’t matter, the end results does not differ). Suppose we have the following observations of a three-level categorical variable:

$$\begin{bmatrix} A & A & B & B & C & C \end{bmatrix}^T$$

We can then construct the following design matrix:

$$\begin{bmatrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ -1 & -1 & \\ -1 & -1 & \end{bmatrix}$$

Notice that we only need two columns (i.e. two variables) to encode the three levels. Since the last parameter is a function of the two other it is redundant. Also notice how the observations in the two last rows, corresponding to the $$C$$ observations, will influence the estimation of all the other parameters for this variable. When the two parameters are estimated, the last parameter can be obtained using the result from above relating the last parameter to all the other.

In the Dixon-Coles paper they used the constraint that the average of the attack parameters should be 1. This is not quite the same as the sum-to-zero constraint, but for prediction, it does not matter exactly which constraint we use. Anyway, I will explain later how we can fix this.

To use this trick in the Dixon-Coles implementation we need to make the following changes to our code from part 1. Obviously the first thing we need to change is how the design matrices in the DCmodelData function is computed. We need four matrices now, since the number of parameters estimated directly is different for the attack and defense parameters. Notice how I chose the last of team that appear last in the team.names vector. The teams get sorted alphabetically, so for the 2011-12 Premier League data this is is Wolves.

DCmodelData <- function(df){

team.names <- unique(c(levels(df$HomeTeam), levels(df$AwayTeam)))

# attack, with sum-to-zero constraint
## home
hm.a <- model.matrix(~ HomeTeam - 1, data=df)
hm.a[df$HomeTeam == team.names[length(team.names)], ] <- -1 hm.a <- hm.a[,1:(length(team.names)-1)] # away am.a <- model.matrix(~ AwayTeam -1, data=df) am.a[df$AwayTeam == team.names[length(team.names)], ] <- -1
am.a <- am.a[,1:(length(team.names)-1)]

# defence, same as before
hm.d <- model.matrix(~ HomeTeam - 1, data=df)
am.d <- model.matrix(~ AwayTeam -1, data=df)

return(list(homeTeamDMa=hm.a, homeTeamDMd=hm.d,
awayTeamDMa=am.a, awayTeamDMd=am.d,
homeGoals=df$FTHG, awayGoals=df$FTAG,
teams=team.names))
}

Some changes to the DCoptimFn function is also needed, so it properly handles the changes we made to the design matrices.

# I don't bother showing the rest of the function
nteams <- length(DCm$teams) attack.p <- matrix(params[3:(nteams+1)], ncol=1) #one column less defence.p <- matrix(params[(nteams+2):length(params)], ncol=1) # need to multiply with the correct matrices lambda <- exp(DCm$homeTeamDMa %*% attack.p + DCm$awayTeamDMd %*% defence.p + home.p) mu <- exp(DCm$awayTeamDMa %*% attack.p + DCm$homeTeamDMd %*% defence.p) We also need to make a the appropriate adjustments to the vectors with the initial parameter values, so that they have the correct lengths. dcm <- DCmodelData(data) nteams <- length(dcm$teams)

#initial parameter estimates
attack.params <- rep(.1, times=nteams-1) # one less parameter
defence.params <- rep(-0.8, times=nteams)
home.param <- 0.06
rho.init <- 0.03
par.inits <- c(home.param, rho.init, attack.params, defence.params)

#informative names
#skip the last team
names(par.inits) <- c('HOME', 'RHO',
paste('Attack', dcm$teams[1:(nteams-1)], sep='.'), paste('Defence', dcm$teams, sep='.'))

With these changes we can simply use the built-in optim function in R. There is no need for the DCattackConstr function anymore, or a third party package, since we built the constraint right into the design matrices.

res <- optim(par=par.inits, fn=DCoptimFn, DCm=dcm, method='BFGS')

This takes about 6-7 seconds on my laptop, a decent improvement to the 40 seconds it took before. If you take a look at the resulting parameter estimates in res$par you will see that the attack parameter for Wolves is missing. As I explained earlier, this parameter is easy to find. It is also easy to correct all the parameter estimates so that they become the same as if we had a mean-is-one constraint on the attack parameters. This is done by increasing the attack parameters by one, and decreasing the defense parameters by one. The reason it is that simple is that the sum-to-zero constraint is equivalent with a mean-is-zero constraint. parameters <- res$par

#compute Wolves attack parameter
missing.attack <- sum(parameters[3:(nteams+1)]) * -1

#put it in the parameters vector
parameters <- c(parameters[1:(nteams+1)], missing.attack, parameters[(nteams+2):length(parameters)])
names(parameters)[nteams+2] <- paste('Attack.', dcm\$teams[nteams], sep='')

#increase attack by one
parameters[3:(nteams+2)] <- parameters[3:(nteams+2)] + 1

#decrease defence by one
parameters[(nteams+3):length(parameters)] <- parameters[(nteams+3):length(parameters)] - 1

# Two Bayesian regression models for football results

Last fall I took a short introduction course in Bayesian modeling, and as part of the course we were going to analyze a data set of our own. I of course wanted to model football results. The inspiration came from a paper by Gianluca Baio and Marta A. Blangiardo Bayesian hierarchical model for the prediction of football results (link).

I used data from Premier League from 2012 and wanted to test the predictions on the last half of the 2012-23 season. With this data I fitted two models: One where the number of goals scored where modeled using th Poisson distribution, and one where I modeled the outcome directly (as home win, away win or draw) using an ordinal probit model. As predictors I used the teams as categorical predictors, meaning each team will be associated with two parameters.

The Poisson model was pretty much the same as the first and simplest model described in Baio and Blangiardo paper, but with slightly more informed priors. What makes this model interesting and different from the independent Poisson model I have written about before, apart from being estimated using Bayesian techniques, is that each match is not considered as two independent events when the parameters are estimated. Instead a correlation is implicitly modeled by specifying the priors in a smart way (see figure 1 in the paper, or here), thereby modeling the number of goals scored like a sort-of-bivariate Poisson.

Although I haven’t had time to look much into it yet, I should also mention that Baio and Blangiardo extended their model and used it this summer to model the World Cup. You can read more at Baio’s blog.

The ordinal probit model exploits the fact that the outcomes for a match can be thought to be on an ordinal scale, with a draw (D) considered to be ‘between’ a home win (H) and an away win (A). An ordinal probit model is in essence an ordinary linear regression model with a continuous response mu, that is coupled with a set of threshold parameters. For any value of mu the probabilities for any category is determined by the cumulative normal distribution and the threshold values. This is perhaps best explained with help from a figure:

Here we see an example where the predicted outcome is 0.9, and the threshold parameters has been estimated to 0 and 1.1. The area under the curve is then the probability of the different outcomes.

To model the match outcomes I use a model inspired by the structure in the predictors as the Poisson model above. Since the outcomes are given as Away, Draw and Home, the home field advantage is not needed as a separate term. This is instead implicit in the coefficients for each team. This gives the coefficients a different interpretation from the above model. The two coefficients here can be interpreted as the ability when playing at home and the ability when playing away.

To get this model to work I had to set the constrains that the threshold separating Away and Draw were below the Draw-Home threshold. This implies that a good team would be expected to have a negative Away coefficient and a positive Home coefficient. Also, the intercept parameter had to be fixed to an arbitrary value (I used 2).

To estimate the parameters and make predictions I used JAGS trough the rjags package.

For both models, I used the most credible match outcome as the prediction. How well were the last half of the 2012-13 season predictions? The results are shown in the confusion table below.

Confusion matrix for Poisson model

 actual/predicted A D H A 4 37 11 D 1 35 14 H 0 38 42

Confusion matrix for ordinal probit model

 actual/predicted A D H A 19 0 33 D 13 0 37 H 10 0 70

The Poisson got the result right in 44.5% of the matches while the ordinal probit got right in 48.9%. This was better than the Poisson model, but it completely failed to even consider draw as an outcome. Ordinal probit, however, does seem to be able to predict away wins, which the Poisson model was poor at.

Here is the JAGS model specification for the ordinal probit model.

model {

for( i in 1:Nmatches ) {

pr[i, 1] <- phi( thetaAD - mu[i]  )
pr[i, 2] <- max( 0 ,  phi( (thetaDH - mu[i]) ) - phi( (thetaAD - mu[i]) ) )
pr[i, 3] <- 1 - phi( (thetaDH - mu[i]) )

y[i] ~ dcat(pr[i, 1:3])

mu[i] <- b0 + homePerf[teamh[i]] + awayPerf[teama[i]]
}

for (j in 1:Nteams){
homePerf.p[j] ~ dnorm(muH, tauH)
awayPerf.p[j] ~ dnorm(muA, tauA)

#sum to zero constraint
homePerf[j] <- homePerf.p[j] - mean(homePerf.p[])
awayPerf[j] <- awayPerf.p[j] - mean(awayPerf.p[])
}

thetaAD ~ dnorm( 1.5 , 0.1 )
thetaDH ~ dnorm( 2.5 , 0.1 )

muH ~ dnorm(0, 0.01)
tauH ~ dgamma(0.1, 0.1)

muA ~ dnorm(0, 0.01)
tauA ~ dgamma(0.1, 0.1)

#predicting missing values
predictions <- y[392:573]
}

And here is the R code I used to run the above model in JAGS.

library('rjags')
library('coda')

#Remove the match outcomes that should be predicted
to.predict <- 392:573 #this is row numbers
observed.results <- dta[to.predict, 'FTR']
dta[to.predict, 'FTR'] <- NA

#list that is given to JAGS
data.list <- list(
teamh = as.numeric(dta[,'HomeTeam']),
teama = as.numeric(dta[,'AwayTeam']),
y = as.numeric(dta[, 'FTR']),
Nmatches = dim(dta)[1],
Nteams = length(unique(c(dta[,'HomeTeam'], dta[,'AwayTeam']))),
b0 = 2 #fixed
)

#MCMC settings
burnin <- 1000
nchains <- 1
steps <- 15000
thinsteps <- 5

#Fit the model
#script name is a string with the file name where the JAGS script is.
update(jagsmodel, n.iter=burnin)

samples <- coda.samples(jagsmodel, variable.names=parameters,
n.chains=nchains, thin=thinsteps,
n.iter=steps)

#Save the samples
save(samples, file='bayesProbit_20131030.RData')

#print summary
summary(samples)

# Predicting football results with Poisson regression pt. 1

I have been meaning to write about my take on using Poisson regression to predict football results for a while, so here we go. Poisson regression is one of the earliest statistical methods used for predicting football results. The goal here is to use available data to to say something about how many goals a team is expected to score and from that calculate the probabilities for different match outcomes.

The Poisson distribution
The Poisson distribution is a probability distribution that can be used to model data that can be counted (i.e something that can happen 0, 1, 2, 3, … times). If we know the number of times something is expected to happen, we can find the probabilities that it happens any number of times. For example if we know something is expected to happen 4 times, we can calculate the probabilities that it happens 0, 1, 2, … times.

It turns out that the number of goals a team scores in a football match are approximately Poisson distributed. This means we have a method of assigning probabilities to the number of goals in a match and from this we can find probabilities for different match results. Note that I write that goals are approximately Poisson. The Poisson distribution does not always perfectly describe the number of goals in a match. It sometimes over or under estimates the number of goals, and some football leagues seems fit the Poisson distribution better than others. Anyway, the Poisson distribution seems to be an OK approximation.

The regression model
To be able to find the probabilities for different number of goals we need to find the expected number of goals L (It is customary to denote the expectation in a Poisson distribution by the Greek letter lambda, but WordPress seem to have problems with greek letters so i call i L instead). This is where the regression method comes in. With regression we can estimate lambda conditioned on certain variables. The most obvious variable to look at is which team is playing. Manchester United obviously makes more goals than Wigan. The second thing we want to take into account is who the opponent is. Some teams are expected to concede fewer goals, while others are expected to let in more goals. The third thing we want to take into account is home field advantage.

Written in the language of regression models this becomes

log(L) = mu + home + teami + opponentj

The mu is the overall mean number of goals. The home is the effect on number of goals a team has by playing at home. Teami is the effect of team number i, opponentj is the effect of team j.

(Note: Some descriptions of the Poisson regression model on football data uses the terms offensive and defensive strength to describe what I have called team and opponent. The reason I prefer the terms I use here is because it makes it a bit easier to understand later when we look at the data set.)

The logarithm on the left hand side is called the link function. I will not dwell much on what a link function is, but the short story is that they ensure that the parameter we try to estimate don’t fall outside its domain. In this case it ensures us that we never get negative expected number of goals.

Data
In my example I will use data from football-data.co.uk. What data you would want to use is up to yourself. Typically you could choose to use data from the last year or the least season, but that is totally up to you to decide.

Each of the terms on the right hand side of the equation (except for mu) corresponds to a columns in a table, so we need to fix our data a bit before we proceed with fitting the model. Each match is essentially two observations, one for how many goals the home team scores, the second how many the away team scores. Basically, each match need two rows in our data set, not just one.

Doing the fix is an easy thing to do in excel or Libre Office Calc. We take the data rows (i.e. the matches) we want to use and duplicate them. Then we need to switch the away team and away goals columns so they become the same as the home team column. We also need a column to indicate the home team. Here is an example on how it will look like:

In the next part I will fit the actual model, calculate probabilities and describe how we can make predictions using R.

# Is goal difference the best way to rank and rate football teams?

In my previous post i compared the least squares rating of football teams to the ordinary three points for a win rating. In this post I will look closer at how these two systems rank teams differently. I briefly touched upon the subject in the last post, were we saw that the two systems generally ranked the teams in the same order, with a few exceptions. We saw that Sunderland and Newcastle were the two teams in the 2011-2012 Premier League season who differed most in their ranking in the two systems. The reason for this was of course because the least squares approach is based on goal difference, while the points system is based only on match outcome. This means that teams who win a match by many goals will benefit more on the least squares ranking than on the points system. For example, a 3-0 win will count more than a 2-1 win when we use goal difference, but they will give the same number of points based on match outcome. This also holds if wee look at the loosing team; a 2-1 loss is better than a 3-0 loss.

It seems more intuitive to rank teams on a system based on goal difference (using least squares or some other method) than the tree points for a win system, especially when we remind ourself that it lacks any theoretical justification. Awarding three points for a win instead of two was not used before the 1980’s and were not used in the World Cup until 1994. The reason for introducing the three points system was to give the teams more incentive to win. Also, as far as I know, even the two points for a win lacks a theoretical basis as a way to measure teams strength. But even if the points system lack an underlying mathematical theory, it still could be a better system than a system based on goal difference for deciding the true strength of a team. A paper titled Fitness, chance, and myths: an objective view on soccer results by the two German physicists A. Hauer and O. Rubner compares the two systems using data from the German Bundesliga. They looked at each team in each season from the late 1980’s and calculated how much the teams goal difference and points correlated between the first and second half of a season. A higher correlation means that there is less chance involved in how the measure reflects a teams real strength. What they found was that goal difference was more correlated between the half-seasons than the 3- and 2 points for a win system.

However, this does not mean that goal difference is the best way to measure team strength. I would like to see if there are some other measures that correlate better between season halves. What first comes to mind is to look at ball possession or shots at target.

As a last note, even if goal difference has a better theoretical foundation as a measure of “who is the best”, I do not think that leagues and tournaments should quit the points system. It may very well be that the points system makes a football competition more interesting since it adds more chance to it.

# Least squares rating of football teams

The Wikipedia article Statistical association football predictions mentions a method for least squares rating of football teams. The article does not give any source for this, but I found what I think may be the origin of this method. It appears to be from an undergrad thesis titled Statistical Models Applied to the Rating of Sports Teams by Kenneth Massey. It is not on football in particular, but on sports in general where two teams compete for points. A link to the thesis can be found here.

The basic method as described in Massey’s paper and the Wikipedia article is to use a n*k design matrix A where each of the k columns represents one team, and each of the n rows represents a match. In each match (or row) the home team is indicated by 1, and the away team by -1. Then we have a vector y indicating goal differences in each match, with respect to the home team (i.e. positive values for home wins, negative for away wins). Then the least squares solution to the system Ax = y is found, with the x vector now containing the rating values for each team.

When it comes to interpretation, the difference in least squares estimate for the rating of two teams can be seen as the expected goal difference between the teams in a game. The individual rating can be seen as how many goals a teams scores compared to the overall average.

Massey’s paper also discusses some extensions to this simple model that is not mentioned in the Wikipedia article. The most obvious is incorporation of home field advantage, but there is also a section on splitting the teams’ performances into offensive and defensive components. I am not going to go into these extensions here, you can read more about them i Massey’s paper, along with some other rating systems that are also discussed. What I will do, is to take a closer look at the simple least squares rating and compare it to the ordinary three points for a win rating used to determine the league winner.

I used the function I made earlier to compute the points for the 2011-2012 Premier League season, then I computed the least squares rating. Here you can see the result:

PTS LSR LSRrank RankDiff
Man City 89 1.600 1 0
Man United 89 1.400 2 0
Arsenal 70 0.625 3 0
Tottenham 69 0.625 4 0
Newcastle 65 0.125 8 3
Chelsea 64 0.475 5 -1
Everton 56 0.250 6 -1
Liverpool 52 0.175 7 -1
Fulham 52 -0.075 10 1
West Brom 47 -0.175 12 2
Swansea 47 -0.175 11 0
Norwich 47 -0.350 13 1
Sunderland 45 -0.025 9 -4
Stoke 45 -0.425 15 1
Wigan 43 -0.500 16 1
Aston Villa 38 -0.400 14 -2
QPR 37 -0.575 17 0
Bolton 36 -0.775 19 1
Blackburn 31 -0.750 18 -1
Wolves 25 -1.050 20 0

It looks like the Least squares approach gives similar results as the standard points system. It differentiates between the two top teams, Manchester City and Manchester United, even if they have the same number of points. This is perhaps not so surprising since City won the league because of greater goal difference than United, and this is what the least squares rating is based on. Another, perhaps more surprising thing is how relatively low least squares rating Newcastle has, compared to the other teams with approximately same number of points. If ranked according to the least squares rating, Newcastle should have been below Liverpool, instead they are three places above. This hints at Newcastle being better at winning, but with few goals, and Liverpool winning fewer times, but when they win, they win with more goals. We can also see that Sunderland comes poor out in the least squares rating, dropping four places.

If we now plot the number of points to the least squares rating we see that the two methods generally gives similar results. This is perhaps not so surprising, and despite some disparities like the ones I pointed out, there are no obvious outliers. I also calculated the correlation coefficient, 0.978, and I was actually a bit surprised of how big it was.