# Tuning the Elo ratings: The K-factor and home field advantage

The Elo rating system is quite simple, and therefore easy implement. In football, FIFA uses is in its womens rankings and the well respected website fivethirtyeight.com also uses Elo ratings to make predictions for NBA and NFL games. Another cool Elo rating site is clubelo.com.

Three year ago I posted some R code for calculating Elo ratings. Its simplicity also makes it easy to modify and extend to include more realistic aspects of the games and competitions that you want to make ratings for, for example home field advantage. I suggest reading the detailed description of the clubelo ratings to get a feel of how the system can be modified to get improved ratings. I have also discussed some ways to extend the Elo ratings here on this blog as well.

If you implement your own variant of the Elo ratings it is necessary to tune the underlying parameters to make the ratings as accurate as possible. For example, a too small K-factor will give ratings that update too slow. The ratings will not adapt well to more recent developments. Vice versa, a too large K-factor will put too much weight on the most recent results. The same goes for the extra points added to the home team rating to account for the home field advantage. If this is poorly tuned, you will get poor predictions.

In order to tune the rating system, we need a way to measure how accurate the ratings are. Luckily the formulation of the Elo system itself can be used for this. The Elo system updates the ratings by looking at the difference between the actual results and the results predicted by the rating difference between the two opposing teams. This difference can be used to tune the parameters of the system. The smaller this difference is, the more accurate are the predictions, so we want to tune the parameters so that this difference is as small as possible.

To formulate this more formally, we use the following criterion to assess the model accuracy:

$$\sum_i[ (exp_{hi} – obs_{hi})^2 + (exp_{ai} – obs_{ai})^2 ]$$

where $$exp_{hi}$$ and $$exp_{ai}$$ are the expected results of match i for the home team and the away team, respectively. These expectations are a number between 0 and 1, and is calculated based on the ratings of the two teams. $$obs_{hi}$$ and $$obs_{ai}$$ are the actual result of match i, encoded as 0 for loss, 0.5 for draw and 1 for a win. This criterion is called the squared error, but we will use the mean squared error.

With this criterion in hand, we can try to find the best K-factor. Using data from the English premier league as an example I applied the ratings on the match results from the January 1st 2010 to the end of the 2014-15 season, a total of 2048 matches. I tried it with different values of the K-factor between 7 and 25, in 0.1 increments. Then plotting the average squared error against the K-factor we see that 18.5 is the best K-factor.

The K-factor I have found here is, however, probably a bit too large. In this experiment I initialized the ratings for all teams to 1500. This includes the teams that was promoted from the Championship. A more realistic rating system would initialize these teams with a lower rating, perhaps be given the ratings from the relegated teams.

We can of course us this strategy to also find the best adjustment for the home field advantage. The simple way to add the home field advantage is to add some additional points to the ratings for the home team. Here I have used the same number of points in all matches across all season, but different strategies are possible. To find the optimal home field advantage I applied the Elo ratings with K=18.5, using different home field advantages.

From this plot we see that an additional 68.3 points is the optimal amount to add to the rating for the home team.

One might wonder if finding the best K-factor and home field advantage independent of each other is the best way to do it. When I tried to find the best K-factor with the home field advantage set to 68, I found that the best K was 19.5. This is a bit higher than when the home field advantage was 0. I tried to find the optimal pair of K and home field advantage by looking over a grid of possible values. Plotting the accuracy of the ratings against both K and the home field advantage in a contour we get the following:

The best K and home field advantage pair can be read from the plot, both of which is a bit higher than the first values I found.

Doing the grid search can take a bit of time, especially if you don’t narrow down the search space by doing some initial tests beforehand. I haven’t really tried it out, but alternating between finding the best K-factor and home field advantage and using the optimal value from the previous round is probably going to be a reasonable strategy here.

# My predictions for the 2016-17 Premier League

This year I am participating in Simon Gleave‘s Premier League prediction competition. It is an interesting initiative, as both statistical models and and more informal approaches are compared.

Last time I participated in something like this was midway trough the last Premier League season for statsbomb.com’s compilation. This time, however, the predictions are made before the first match has been played. To be honest, I think it is futile to try to model and predict an unplayed season since any model based only on previous results will necessarily reproduce what has already happened. This approach will work OK for predicting the result of the next couple of matches midway trough a season, but making predictions for the start of a season is really hard since the teams have brought inn some new players and gotten rid of other and perhaps also changed managers and so on. And not to forget that we also try predict results 9 months into the future.

When May comes and my predictions are completely wrong, I am not going to be embarrassed.

Last time I wanted to use the Conway-Maxwell-Poisson model, but I did not get it to work when I included data from several seasons plus data from the Championship. I still did not get it to work properly, but this time I tried a different approach to estimate the parameters. I ended up using a two-step approach, where I first estimate the attack and defense parameters with the independent Poisson model, and then, keeping those parameters fixed, I estimated the dispersion parameter by itself.

To fit the model I used Premier League data from the 2010-11 season to the 2015-16 season. I also included data from the 2015-16 season of the Championship (including the playoff) to be able to get some information on the promoted teams. I used the Dixon-Coles weighting scheme with $$\xi = 0.0019$$. I used a separate parameter for home field advantage for Premier League and the Championship. I also used separate dispersion parameters for the two divisions.

I estimated the dispersion parameter for the Premier League to be 1.103, about the same as I previously estimated in some individual Premier League seasons, indicating some underdispersion in the goals. Interestingly, the dispersion parameter for the Championship was only 1.015.

Anyway, here are my projected league table with expected (or average) point totals. This is completely based on the model, I have not done any adjustments to it.

Team Points
Manchester City 73.70
Arsenal 69.73
Leicester City 64.12
Manchester United 63.95
Chelsea 63.84
Tottenham 62.53
Southampton 60.51
Liverpool 60.37
Everton 51.48
West Ham 51.12
Middlesbrough 46.30
Swansea 44.59
Burnley 44.20
Stoke City 42.99
Hull 42.49
Crystal Palace 41.33
Watford 41.23
Sunderland 39.83
West Bromwich Albion 39.21
Bournemouth 36.37

# My predictions for the rest of the Premier League season

A couple of weeks ago Constantinos Chappas asked on twitter for predictions for the results of the remaining season of English Premier League:

I had been thinking about posting some predictions about the Premier League around new years, since this season is really exciting and it would be a great opportunity to see how well my models would cope with everything that is currently going on. I have never posted any predictions before, so this will surely be an interesting experience. And I thought Chappas’ initiative was really interesting, so that surely gave me a nice reason to come trough.

Today Chappas posted the combined results from all 15 participants so I thought I could share some of the details behind my contribution.

I originally wanted to use the Conway-Maxwell model I have written about recently, but I had some problems with the estimation procedure, so I instead used a classic Poisson model. I used data on Premier League and Championship results going back to the 2011-12 season. By including data from the Champoionship I hope to get better predictions, like I have demonstrated before. Since I used data from a long time back I used the Dixon-Coles weighting scheme, which make more recent games have a greater impact on the predictions. The weighting parameter $$\xi$$ was set to 0.0019, which gives a bit more weight on more recent games than the 0.0018 I found to be most optimal earlier.

I fitted the model and calculated the probabilities for the remaining games of the season. From these probabilities I simulated the rest of the season ten thousand times. From these simulations we can get the probabilities and expectations for the end of season results.

So how do I predict the league table will look like at the end of the season?

Team Points
Manchester City 75.7
Arsenal 75.2
Tottenham 65.6
Leicester City 64.8
Manchester United 64.3
Liverpool 58.2
West Ham 56.1
Chelsea 54.7
Everton 53.7
Crystal Palace 53.7
Stoke City 52.9
Watford 51.9
Southampton 50.6
West Bromwich Albion 45.8
Norwich City 43.7
Bournemouth 42.9
Swansea City 40.9
Newcastle 34.5
Sunderland 31.5
Aston Villa 23.1

Although I predict 0.2 points more for Manchester City than Arsenal, the probabilities for both of them to win is 47.0%. I also give Tottenham a 2.3% chance, Leicester 2.1% and Manchester United a 1.5%. At last, Liverpool have a 0.1% chance. The other teams have a chance less than 0.04%.

I will come back with an update with my entire table with probabilities for all positions for all teams.

# Underdispersed Poisson alternatives seem to be better at predicting football results

In the previous post I discussed some Poisson-like probability distributions that offer more flexibility than the Poisson distribution. They typically have an extra parameter that controls the variance, or dispersion. The reason I looked into these distributions was of course to see if they could be useful for modeling and predicting football results. I hoped in particular that the distributions that can be underdispersed would be most useful. If the underdispersed distributions describe the data well then the model should predict the outcome of a match better than the ordinary Poisson model.

The model I use is basically the same as the independent Poisson regression model, except that the part with the Poisson distribution is replaced by one of the alternative distributions. Let the $$Y_{ij}$$ be the number of goals scored in game i by team j

$$Y_{ij} \sim f(\mu_{ij}, \sigma)$$
$$log(\mu_{ij}) = \gamma + \alpha_j + \beta_k$$

where $$\alpha_j$$ is the attack parameter for team j, and $$\beta_k$$ is the defense parameter for opposing team k, and $$\gamma$$ is the home field advantage parameter that is applied only if team j plays at home. $$f(\mu_{ij}, \sigma)$$ is one of the probability distributions discussed in the last post, parameterized by the location parameter mu and dispersion parameter sigma.

To these models I fitted data from English Premier League from the 2010-11 season to the 2014-15 season. I also used Bundesliga data from the same seasons. The models were fitted separately for each season and compared to each other with AIC. I consider this only a preliminary analysis and I have therefore not done a full scale testing of the accuracy of predictions where I refit the model before each match day and use Dixon-Coles weighting.

The five probability distributions I used in the above model was the Poisson (PO), negative binomial (NBI), double Poisson (DPO), Conway-Maxwell Poisson (COM) and the Delaporte (DEL) which I did not mention in the last post. All of these, except the Conway-Maxwell Poisson, were easy to fit using the gamlss R package. I also tried two other gamlss-supported models, the Poisson inverse Gaussian and Waring distributions, but the fitting algorithm did not work properly. To fit the Conway-Maxwell Poisson model I used the CompGLM package. For good measure I also fitted the data to the Dixon-Coles bivariate Poisson model (DC). This model is a bit different from the rest of the models, but since I have written about it before and never really tested it I thought this was a nice opportunity to do just that.

The AIC calculated from each model fitted to the data is listed in the following table. A lower AIC indicates that the model is better. I have indicated the best model for each data set in red.

The first thing to notice is that the two models that only account for overdispersion, the Negative Binomial and Delaporte, are never better than the ordinary Poisson model. The other and more interesting thing to note, is that the Conway-Maxwell and Double Poisson models are almost always better than the ordinary Poisson model. The Dixon-Coles model is also the best model for three of the data sets.

It is of course necessary to take a look at the estimates of the parameters that extends the three models from the Poisson model, the $$\sigma$$ parameter for the Conway-Maxwell and double Poisson and the $$\rho$$ for the Dixon-Coles model. Remember that for the Conway-Maxwell a $$\sigma$$ greater than 1 indicates underdispersion, while for the Double Poisson model a $$\sigma$$ less than 1 is indicates underdispersion. For the Dixon-Coles model a $$\rho$$ less than 0 indicates an excess of 0-0 and 1-1 scores and fewer 0-1 and 1-0 scores, while it is the opposite for $$\rho$$ greater than 0.

It is interesting to see that the estimated dispersion parameters indicate underdispersion for all the data sets. It is also interesting to see that the data sets where the parameter estimates are most indicative of equidispersion is where the Poisson model is best according to AIC (Premier League 2013-14 and Bundesliga 2010-11 and 2014-15).

The parameter estimates for the Dixon-Coles model do not give a very consistent picture. The sign seem to change a lot from season to season for the Premier League data, and for the data sets where the Dixon-Coles model was found to be best, the signs were in the opposite direction of what where the motivation described in the original 1997 paper. Although it does not look so bad for the Bundesliga data, this makes me suspect that the Dixon-Coles model is prone to overfitting. Compared to the Conway-Maxwell and double Poisson models that can capture more general patterns in all of the data, the Dixon-Coles model extends the Poisson model to just parts of the data, the low scoring outcomes.

It would be interesting to do fuller tests of the prediction accuracy of these three models compared to the ordinary Poisson model.

# Some alternatives to the Poisson distribution

One important characteristic of the Poisson distribution is that both its expectation and the variance equals parameter $$\lambda$$. A consequence of this is that when we use the Poisson distribution, for example in a Poisson regression, we have to assume that the variance equals the expected value.

The equality assumption may of course not hold in practice and there are two ways in which this assumption can be wrong. Either the variance is less than the expectation or it is greater than the expectation. This is called under- and overdispersion, respectively. When the equality assumption holds, it is called equidispersion.

There are two main consequences if the assumption does not hold: The first is that standard errors of the parameter estimates, which are based on the Poisson, are wrong. This could lead to wrong conclusions when doing inference. The other consequence happens when you use the Poisson to make predictions, for example how many goals a football team will score. The probabilities assigned to each number of goals to be scored will be inaccurate.

(Under- and overdispersion should not be confused with heteroscedasticity in ordinary linear regression. Poisson regression models are naturally heteroscedastic because of the variance-expectation equality. Dispersion refers to what relationship there is between the variance and the expected value, in other words what form the heteroscedasticity takes.)

When it comes to modeling and predicting football results using the Poisson, a good thing would be if the data were actually underdispersed. That would mean that the probabilities for the predicted number of goals scored would be higher around the expectation, and it would be possible to make more precise predictions. The increase in precision would be greatest for the best teams. Even if the data were really overdispersed, we would still get probabilities that more accurately reflect the observed number of goals, although the predictions would be less precise.

This is the reason why I have looked into alternatives to the Poisson model that are suitable to model count data and that are capable of being over- and underdispersed. Except for the negative binomial model there seems to have been little focus on more flexible Poisson-like models in the literature, although there are a handful of papers from the last 15 years with some applied examples.

I should already mention the gamlss package, which is an extremely useful package that can fit a large number of regression type models in R. I like to think of it as the glm function on steroids. It can be used to create regression models for a large number of distributions (50+) and using different forms of dependent variables (for example random effects and splines) and doing regression on distribution parameters other than the usual expectation parameters.

The models that I have considered usually have two parameters. The two parameters are often not easy to interpret, but the distributions can be re-parameterized (which is done in the gamlss package) so that the parameters describe the location (denoted $$\mu$$, often the same as the expectation) and shape (denoted $$\sigma$$, often a dispersion parameter that modifies the association between the expectation and variance). Another typical property is that they equal the Poisson for certain values of the shape parameter.

As I have already mentioned, the kind of model that is most often put forward as an alternative to the Poisson is the Negative binomial distribution (NBI). The advantages of the negative binomial are that is well studied and good software packages exists for using it. The shape parameter $$\sigma > 0$$ determines the overdispersion (relative to the Poisson) so that the closer it is to 0, the more it resembles the Poisson. This is a disadvantage as it can not be used to model underdispersion (or equidispersion, although in practice it can come arbitrarily close to it). Another similar, but less studied, model is the Poisson-inverse Gaussian (PIG). It too has a parameter $$\sigma > 0$$ that determines the overdispersion.

A large class of distributions, called Weighted Poisson distributions, is capable of being both over- and underdispersed. (The terms Weighted in the name comes from a technique used to derive the distribution formulas, not that the data is weighted) A paper describing this class can be found here. The general form of the probability distribution is

$$P(x;\theta,\alpha)=\frac{e^{\mu x+\theta t(x)}}{x!C(\theta,\alpha)}$$

where $$t(x)$$ is one of a large number of possible functions, and $$C(\theta,\alpha)$$ is a normalizing constant which makes sure all probabilities in the distribution sum to 1. Note that I have denoted the two parameters using $$\theta$$ and $$\alpha$$ and not $$\mu$$ and $$\sigma$$ to indicate that these are not necessarily location and shape parameters. I think this and interesting class of distributions that I want to look more into, but since they are not generally implemented in any R package that I know of I will not consider them further now.

Another model that is capable of being over- and underdispersed is the Conway–Maxwell–Poisson distribution (COM), which incidentally is a special case of the class of Weighted Poisson distributions mentioned above (see this paper). The Poisson distribution is a special case of the COM when $$\sigma = 1$$, and is underdispersed when $$\sigma > 1$$ and overdispersed when $$\sigma$$ is between 0 and 1. One drawback with the COM model is that the expected value depends on both parameters $$\mu$$ and $$\sigma$$, although it is dominated by $$\mu$$. This makes the interpretation a bit difficult, but it may not be a problem when making predictions.

Unfortunately, the COM model is not supported by the gamlss package, but there are some other R packages that implements it. I have tried a few of them and the only one that I got to work is CompGLM, which for some reason does not use the location ($$\mu$$) and shape ($$\sigma$$) parameterization.

The Double Poisson (DP) is another interesting distribution which also equals the Poisson distribution when $$\sigma = 1$$, but is overdispersed when $$\sigma > 1$$ and underdispersed when $$\sigma$$ is between 0 and 1. The expectation does not depend on the shape parameter $$\sigma$$, and it is approximately equal to the location parameter $$\mu$$. Another interesting thing about the Double Poisson is that it is belongs to a larger group of distributions called double exponential families which also lets you derive a binomial-like distribution with an extra dispersion parameter which can be useful in a logistic regression setting (see this paper, or this preprint).

In a follow up post I will try to use these distributions in regression models similar to the independent Poisson model.

# A hectic schedule has some effect on the outcome of a football match

It may be that a football team who has had a hectic period with a lot of games will, because of lack of training and restitution, perform poorer. The Wikipedia page for the FA Cup mentions Manchester United’s absence from the cup as a reason for why they won the Premier League by 18 points in the 1999-2000 season. If this is indeed the case, then this is something we could try to exploit in a prediction model.

I basically used the same data and model as I have used before. I used data from the English Championship and the Premier League, and predicted the Premier League games from January 2007 until January 2015 using the independent Poisson model with the Dixon & Coles weighting method (more details on the setup here and here). In addition I constructed a new variable, the number of matches each team has played the last x number of days, were we can use and try different values of x. As a pretentious shorthand I will call this the Match Schedule Intensity Index (MSII). Matches from the FA Cup, Europa Cup and Champions League were also included in the calculations.

As usual I used the ranked probability score (RPS) to assess the prediction accuracy.

I tried four different number of days back in time: 21, 25, 28 and 31 days. I also varied the time weighing parameter $$\xi$$ a bit to see how these things varied together.

Plotting the RPS, number of days back in time and the different values of $$\xi$$ against each other gives the following:

We see that looking back 28 days in time gives the lowest RPS and this the most accurate predictions of the four alternatives. 25 days is almost as good as 28 days, while 21 and 31 days performs poorer than not having the MSII in the model at all. I am not sure how important the drop in RPS is, as the changes are around the 4th and 5th decimal place. It is probably not that much, but on the other hand, this is an average over 3000 matches, and the number of days back in time seems to be a more important parameter than the small changes in $$\xi$$ that I tried.

It is also interesting to see what effect the MSII has on the number of goals scored. I plotted the estimated multiplicative effect for each additional match for all the fitted models from 2007 to 2015 using the best model with 28 days and $$\xi=0.0020$$.

I expected the effect of additional matches to be negative, meaning the more games the team has recently played, the fewer goals will they be expected to score. This seems to be at least halfway true, except for a few dips over on the positive side around 2010 and 2013-2014, and a rather large positive effect from the start in 2007 until 2008. This was a bit surprising, and I don’t know why. It would be interesting to redo the analysis with data going further back in time to see how far back the positive effect goes.

Is the effect large? Not really. The most extreme values of the multiplicative effects for the MSII is around 0.97 and 1.04. These values means that for each match a team has played more in the last four weeks they are expected to score around 3-4% more or fewer goals. This effect is around 10% for a team that has played four matches in four weeks, which is a typical mid-season schedule. This is not that big of a deal for individual matches, but it seem to improve the predictions in the long run. But I also think it is necessary to keep in mind that the effect in seems to be mostly absent in some periods.

# Better prediction, not just for promoted teams

Ian posted an interesting question that had a lot to do with the post I posted last week:

I have implemented the model to make predictions with two different approaches. The first approach is the standard where I use all matches played in a league to predict a match between Team A and Team B. The second approach is to use just matches played by Team A and Team B to predict the outcome of when they both play each other.

Now would you say that the second approach should be more accurate? As surely the only results which matter for predicting the match between Team A and B is of those two teams?

My answer was that regression models use all the data to estimate the parameters, and that the parameter estimates for Team A and Team B probably will be more precise by including matches where neither team is playing. The intuition for this is that both teams play against a whole bunch of other teams during the season, and the more accurate parameter estimates we can get for these other teams, the more information are we going to get from the matches involving either Team A or Team B. One possible way of getting more accurate parameter estimates for all the other teams is to include data from more matches, if available. And at last, more precise parameter estimates should hopefully provide better predictions.

This is not exactly what I demonstrated in the last post. There I just demonstrated that more data, especially related to promoted teams, will give better predictions on average across the whole Premier League. I did not investigate exactly where these improved predictions occur. It could be that all that gain was just related to the improved parameter estimates of the promoted teams.

That is why, prompted by Ian’s comment, I took a closer look at the predictions. Using the model fitted with data from the Premier League and the Championship, with separate home field advantage for the two divisions, I decided to look at how well the predictions were for some Premier League Teams. Recall that this was the model that made the best predictions in the previous post. I decided to look at only the matches between Manchester United, Arsenal, Aston Villa, Chelsea, Liverpool, Everton and Tottenham since these teams have played in Premier League for a long time.

When only looking at these teams, and using Premier League data only, the RPS was 0.24462. When the Championship were included in the data, RPS were a bit smaller, 0.24436. So this means that including more data, not directly related to this group of teams, improved predictions within that group.

I also tried the model without separate home field advantage parameter for the two divisions, and the predictions got worse for this group of teams. This was not the case when looking at the predictions for all Premier League matches, were it got better on average. This demonstrates an important point that I did not mention in my reasoning above: More data is not necessarily a good thing if your model can’t properly handle it.

# Better prediction of Premier League matches using data from other competitions

In most of my football posts on this blog I have used data from the English Premier league to fit statistical models and make predictions. Only occasionally have I looked at other leagues, but always in isolation. That is, I have never combined data from different leagues and competitions into the same model. Using a league by itself works mostly fine, but I have experienced some issues. Model fitting and prediction making often simply does not work at the beginning of the season. The reason for this has mostly to do with newly promoted teams.

If only data from Premier League is used to fit a model, then no data on the new teams is available at the beginning of the season. This makes predicting the outcome of the first matches of the new teams impossible. In subsequent matches the information available is also very limited compared to the other teams, for which we can rely on data from the previous seasons. This uncertainty in the new teams also propagates into the estimates and predictions for the other teams.

This problem can be remedied by using data from outside the Premier League to help estimate the parameters for the promoted teams. The most obvious place to look for data related to the promoted teams is in the Championship, where the teams played before they were promoted. The FA Cup, where teams from the Championship and Premier League are automatically qualified, should also be a good place to use data from.

To test how much the extra data helps make predictions in the Premier League, I did something similar as I did in my post on the Dixon-Coles time weighting scheme. I used the independent Poisson model to make predictions for all the Premier League matches from 1st of January 2007 to 15th of January 2015. The predictions were made using a model fitted only with data from previous matches (going back to august 2005), thus emulating a realistic real-time prediction scenario. I weighted the data using the Dixon-Coles approach, with $$\xi=0.0018$$. This makes the scenario a bit unrealistic, since I estimated this parameter using the same Premier League matches I am going to predict here. I also experimented with using different home field advantage for each of the competitions.

To measure prediction quality I used the Ranked Probability Score (RPS), which goes from 0 to 1, with 0 being perfect prediction. RPS is calculated for each match, and the RPS I report here is the average RPS of all predictions made. Since this is over 3600 matches, I am going to report the RPS with quite a lot of decimal places.

Although the RPS goes from 0 to 1, using a RPS = 1 to mean worst possible prediction ability is unrealistic. To get a more realistic RPS to compare against I calculated the RPS using the probabilities of home, draw and away using the raw proportions of the outcome in my data. In statistical jargon this is often called the null model. The probabilities were 0.47, 0.25 and 0.28, respectively, and gave a RPS = 0.2249.

Using only Premier League data, skipping predictions for the first matches in a season involving newly promoted teams, gave a RPS of 0.19558.

Including data from the Championship in the model fitting, and assuming the home field advantage in both divisions were the same, gave a RPS of 0.19298. Adding a separate parameter for the home field advantage in the Championship gave an even better RPS of 0.19292.

Including data from the FA Cup (in addition to data from the Championship) were challenging. When data from the earliest round were included, the model fitting sometimes failed. I am not 100% sure of this, but I believe the reason for this is that some teams, or groups of teams, are mostly isolated from the rest of the teams. By that I mean that some group of teams have only played each other, but not any other team in the data. While this is not actually the case (it can not be) I nevertheless think the time weights makes this approximately true. Matches played a few years before the mathces that predictions are made for will have weights that are almost 0. It seems reasonable that this coupled with the incomplete design of the knockout format is where the trouble comes from.

Anyway, I got it to work by excluding matches played by a team not in the Championship or Premier League in the respective season. An additional parameter for home field advantage in the Cup were included in the model as well. Interestingly, this gave a somewhat poorer prediction ability that using additional data from the Championship only, with a RPS of 0.192972, but still better that using Premier League data only. With the same home overall field advantage for all the competitions, the prediction were unsurprisingly poorer with RPS = 0.1931.

I originally wanted to include data from Champions League and Europa League, as well as data from other European leagues, but the problems and results with the FA Cup made me dismiss the idea.

I am not sure why including the FA Cup didn’t give better predictions, but I have some theories. One is that a separate FA Cup home field advantage is unrealistic. Perhaps it would be better to assume that the home field advantage is the same as in the division the two opponents play in, if they play in the same division. If they played in different divisions, perhaps an overall average home field advantage could be used instead.

Another theory has to do with the time weighting scheme. The time weighting parameter I used was found by using data from the Premier League only. Since this gives uncertain estimates for the newly promoted teams, it will perhaps give more recent matches more weight to try to compensate. With more informative data from the previous season, this should probably be more influential. Perhaps the time weighting could be further refined with different weighting parameters for each division.

# Rain does not influence football results

I have often seen the weather mentioned as something that could influence football results, but I have yet to see anyone looking more into it. There are various ways in which the game could be influenced by the weather, and here I am going to look into the effects of precipitation (i.e. rain and snow). I have two hypotheses about what rain could do to the end result of a game.

The first is that rain makes the grass wet, which makes the the ball bounce less and makes running harder. This, I can imagine, should give make scoring goals harder, and thus we should see fewer goals scored in matches where it rains. Also, if it rains during the match the players also get wet, which of course is a burden that should influence the game.

My second hypothesis sort of follows from the first, and that is that rain should make draws more likely.

The obvious hindrance to test the two hypotheses is lack of data. It turns out that getting good historical weather data for a given location is not that simple. The Norwegian Meteorological Institute provides free data from Norwegian weather stations, but (for now at least) I didn’t want to test the hypotheses on Norwegian football results. Instead, I wanted to test it on data from England. What I ended up doing was scraping data from English weather stations from WeatherOnline. That site provides precipitation data from British weather stations in 6-hour intervals, in a window around 1400 o’clock.

Luckily, WeatherOnline provided the coordinates to the weather stations, and I used this together with the coordinates I have compiled in my football stadiums data set to figure out which weather station were nearest. Data from the weather station closest to the place where a match was played should hopefully serve as an adequate proxy for the conditions on the field.

As part of the work on this analysis I also updated the stadium data with some additional stadiums that I needed for this project.

Unfortunately, weather data from all match dates were not available, but all in all i ended up with precipitation data for 4826 matches from the Championship and 2702 matches from Premier League, going back to 2002.

How well can we expect the numbers from the weather stations to reflect the conditions on the stadiums where the mathces are played? After I had coupled the precipitation and match data I made a histogram of the distances from the stadium to the weather station. It reveals that some of the weather stations can be quite far away, some more than 300 kilometers.

This of course is a problem. The closer the station is to where the match is played, the more accurate is the data going to be. The usual way to deal with data points that are less accurate than others, is to weight them accordingly. That way they have less influence on the parameter estimation.

But how should we decide on how to weight the different matches? What we need is a way to relate the distance to accuracy. For this we need the precipitation levels at a specific location, and the precipitation at weather stations nearby. To do this, we can use the weather station themselves, and see how well the weather stations correlate with other weather stations.

I calculated the correlations between all pairs of weather stations, and plotted them against the distance between them:

Some of the weather stations are much farther away from each other than the farthest of the ones I have coupled to the matches. We see that there is a clear trend of diminishing correlation the farther away the stations are. Since the correlations are mostly positive (between 0 and 1), they can be used as weights.

The red line in the plot is an attempt to fit a function to the correlations that can be used to compute the weights for a given distance. I fitted (using least squares) the function

$$\lambda_0 e^{-\lambda d}$$

where d is the distance in kilometers, $$\lambda_0$$ is the value when d is 0, and $$\lambda$$ is the rate in which the function decreases. The estimated values of $$\lambda_0$$ and $$\lambda$$ that best describes the trend were found to be 0.75 and 0.0047, respectively. Judging from the line in the plot above, it reflects the trend quite well, although there are quite some variability around it.

To test the hypothesis of fewer overall goals scored I fitted a Poisson regression model of the total number of goals scored as response. As predictors I added an indicator for matches played in the Championship, and the amount of rain in millimeters.

Each millimeter rain is associated with 0.16% more goals, which is insignificantly different from 0% (p = 0.856).

To test whether rain makes draws more likely, I used the same predictors as in the Poisson model in a logistic regression model. The odds ratio associated with each millimeter rain were 0.952, insignificantly different from 1 (p=0.165).

To summarize: I found no evidence for any of my two hypotheses. Both were insignificantly different from the null hypotheses of no effect of rain on the number of goals and the probability of draws. The point estimates of the effects were both actually in the opposite direction of what I had thought. Rain was associated with more goals and fewer draws, but not more than we would expect to see if it all was due to chance.

# Some thoughts on goal differences in football matches without draws

In regular league matches, draws are a common occurrence. Modeling and predicting draws have some complications. Elo-type ratings allows for draws by simply treating them as half-wins for each team, but it does not allow for direct calculation of draw probabilities. Poisson regression models naturally lets you figure out the probability of a draw by calculating the probability of a goal difference of zero.

Poisson models have the additional strength over Elo-type systems in that they can be used to model and predict the number of goals scored, not only who wins (or loose, or draws). The models I have looked at all assumes that draws are possible, and that is the case in regular league matches. But what about the matches where draws are not allowed, such as in knockout tournaments? How could you calculate probabilities for different number of goals?

I haven’t really seen any discussion about this anywhere, but I have this one idea I just want to get out there. Bear in mind that this idea I present here is completely untested, so I can not say for sure if it is any good.

Matches where draws are impossible are a minority of the matches, so building and fitting a separate model for just those matches is not a good idea. Instead I propose an adjustment to be applied for just those matches. The adjustment can be motivated as follows: The game starts with 0-0 between the teams, so at least one goal has to be scored. This should increase the probabilities of 0-1 and 1-0 results. Similar argument can be given to a game that is in a 1-1 state, a 2-2 state, and so on; at least one goal has to be scored.

So the adjustment is to simply divide the probabilities for a draw and add them to the probabilities for a one-goal difference. This should of course be illustrated with an example.

Suppose you have a matrix with goal probabilities. This can be computed using a Poisson regression model, perhaps with the Dixon-Coles adjustment or some other bivariate structure, or perhaps it comes from a completely different kind of model. It doesn’t really matter.

Then it is just to divide the draw probabilities and add them to the appropriate cells in the matrix:

But how should the probabilities be divided? We could just divide them evenly between the two teams, but I think it is more appropriate to divide them based on the relative strengths of the two teams. There are may ways this could be done, but I think a reasonable method is to divide them based on the win-probabilities for the two teams (given that there is no draw). This does not rely on anything other than the goal probability matrix itself, and is easy to compute: Take the sum of the upper and lower triangle of the matrix, divided by the sum of the whole matrix except the diagonal. This also maintains the original win/lose probabilities.

This scheme is easy to implement in R. First we need a matrix of probabilities, which I here just compute using two Poisson distributions, then calculate the win probability of the team with goals on the vertical. After that we divide the diagonal with the win-probabilities.

# Matrix of goal probabilities
probability_matrix <- dpois(0:7, 1.1) %*% t(dpois(0:7, 1.6))

# Win probabilities, for dividing the draw probabilities
prop <- sum(mm[lower.tri(mm)]) / (1 - sum(diag(mm)))

# Diagonal values, split proportionally
divided_vertical <- (diag(probability_matrix) * prop)
divided_horizontal <- (diag(probability_matrix) * (1-prop))


Here we encounter a problem. The two vectors we are going to add to the two secondary diagonals are one element too long. If we have a big enough probability matrix, that last element is probably going to be so small that ignoring it should not matter too much.

# Increase the probabilities for one-goal wins.
diag(probability_matrix[-1,]) <- diag(probability_matrix[-1,]) + divided_vertical[-length(divided_vertical)]
diag(probability_matrix[,-1]) <- diag(probability_matrix[,-1]) + divided_horizontal[-length(divided_horizontal)]

# The main diagonal, with draw probabilities, should be 0.
diag(mm) <- 0


As always, it is nice to see how the probabilities of the goal differences are distributed. Here I have plotted the adjusted and unadjusted probability distributions:

We clearly see that one-goal wins are much more probable.

As I mentioned above, I haven’t really looked at any data, and it is quite possible that other adjustments are better. Perhaps boosting one-goal wins is a poor idea, and spreading the probabilities more out would be better.