Calculating Goal Expectancy

Way back in January, I did a post on how to understand Poisson (see entry HERE) and how it could be used to estimate various score lines between two teams. This was just a basic get-you-started guide, but it was generally well-received and generated a fair few comments. 


The post, however, while being all well and good, did rather leave everyone hanging because one of the corner-stones of being able to calculate Poisson to any degree of accuracy comes from having each team's attack and defence parameters to hand. When you have the attack and defence parameters, you can then calculate each teams scoring ability - in other words you will know the goal expectancy for each team.


Now that the Euros have come to an end, we have a natural, albeit brief, pause before all the European leagues kick-off again in August and September, so this seems like a good point to pop in some more analytical posts, rather than my normal "ooh", this is what I won/lost kind of entries.

Recap:
If you haven't read my post on Poisson and/or you haven't got a clue what I'm talking about, then do have a read of that post first. Briefly, however, given each team's scoring ability, (or goal expectancy) then we can use Poisson to generate probabilities for any scoreline. This is good for betting/trading in the correct score market, but can also be used for match odds, over/unders and asian bets.


But, as mentioned, the scoring ability bit was omitted from that post, so let's rectify that here. If you remember, in the horrible Poisson calculation we had this little symbol, "µ". This is from the Greek alphabet and is pronounced as "mu". In the Poisson calculation, this was used as a placeholder for the scoring rate of a particular team. So if we wanted to find out the actual scoring rate for Man City when playing at home last season, we would need to work out the actual value of µ. This is achieved thus:

µ = A1 x D2 x H

The value of µ is calculated as the attack rating of the home side (A1) times the defence rating of the away side (D2) times the home advantage value (H).


Oh dear, so we've now got to work-out three different values to get to our scoring rate? Well, we should do really, yes... but we don't absolutely have to do so.


It's at this point that I have a decision to make because there are different methods for calculating attack and defence parameters, and some are easier than others. The best method (in my view) is also the most complicated to explain and the most difficult to implement, and that's a method called Maximum Likelihood Estimation (or MLE). MLE basically uses best estimates for each attack and defence parameter, in addition to the home advantage parameter, by settling on a set of values that produce probabilities which actually reflect the real scorelines when looking at historical data. In other words, MLE is a "best fit" technique.


The problem is that calculating MLE would involve some detailed maths and some medium to advanced Excel, which personally I feel may be at odds with the rather simplistic approach I took when explaining Poisson in the first place. Afterall, what's the point in a dummies guide if the completion of that guide then moves on to advanced calculations?


You may strongly disagree with me here; you may think this is a cop-out or that I'm being outrageously patronising, but for now I'm going to leave MLE behind and show you a more "accessible" method for generating these values.

Before I do so, you may remember that I have already provided one method for calculating these values. I did it in my Fag Packet Calculations post, so have a look at that if you want a really rough-and-ready method.


Goal Expectancy Method:
Okay, let's get down to brass tacks:

• CityOverallRate = Man City scored 93 goals in 38 Premier League matches last season. If we divide one figure by the other ( 93 ÷ 38 ) we get 2.45
• WiganOverallRate = Wigan scored 42 goals in 38 Premier League matches last season. If we divide one figure by the other ( 42 ÷ 38 ) we get 1.10
• CityHomeRate = Man City scored 55 goals at home and 38 goals when away. Again, if we divide one figure by the other (55 ÷ 38 ), then we get a multiplication factor to show us how much better City are at home compared to when they are away. The result is 1.45.
• WiganAwayRate = Wigan scored 22 goals at home and 20 goals when away. This time we reverse the division (away divide home) because we want to know Wigan's multiplication factor when playing away. This is (20 ÷ 22 ) = 0.90.

Right the home side's scoring rate can now be calculated as:    CityOverallRate X CityHomeRate and the away side's scoring rate can be calculated as: WiganOverallRate X WiganAwayRate

         Man City Scoring Rate = 2.45 X 1.45  = 3.55
         Wigan Scoring Rate    = 1.10 X 0.90  = 0.99

So the goal count looks high. These scoring rate values can be plugged into Poisson now and used to calculate each individual scoreline. In Excel, Microsoft have kindly provided a Poisson function that we can use, so even though I explained the details behind how Poisson works, the truth is that you don't really need to know. Instead you can just plug the appropriate values into the function.

The POISSON() function in Excel takes three arguments; they are, "X", "Mean" and "Cumulative".

If you did revisit my Poisson post, then you'll recognise the "X" value, as that is the number of goals we're interested in. For example, if you want to know the likelihood of Man City scoring 2 goals, then "X" becomes 2.

The "Mean" figure is simply our scoring rate, which we calculated above - so for Man City that would be 3.55.

The "Cumulative" parameter is either TRUE or FALSE. If set to TRUE, then the result will return the cumulative probability between zero and "X", where as FALSE returns exactly "X". For our purpose, we should set it to FALSE. The POISSON function for finding out how likely Man City scoring two goals is then:

=POISSON( 2, 3.55, FALSE)

The answer is 0.181001136, or 18%.

Ideally you should complete a full set of POISSON calculations for all scores from 0 to 10 goals for both the home and away sides. You can then do things like add-up all the home win scorelines, draw scorelines and away win scorelines to produce match odds. The same can be done for overs and unders.

Outstanding Issue:
There is still one other issue that I mentioned in my original Poisson post back in January, and that's the weighting that needs to be applied to Poisson to prevent it from underforcasting score draws and low-score home wins. Judging from my normal output on this type of post, I'd say you only have six months or so to wait before I get round to talking about this. :-)