Soccermetrics

What I'm learning from the Soccer Pythagorean

I've had an opportunity to apply the soccer Pythagorean to various leagues and competitions, and I'm noticing some characteristics of the formula.

First of all, the Pythagorean exponent does vary significantly from league to league, and perhaps from season to season in a league, but I haven't checked that out yet.  The exponent does stay between 1.3 and 1.8, which indicates that the goal distribution of the teams is skewed toward the smaller numbers.  That result does make sense -- teams don't score five or more goals very often in soccer.

Second, the soccer Pythagorean consistently overestimates the number of wins and draws in a season.  I have yet to see a team finish above the estimated point total in the Pythagorean.  In general, the formula overestimates the point total by 12-16 points -- four or five games, which happens to be the stated accuracy of the baseball Pythagorean.  Now, I've applied this formula to the World Cup qualifying tournaments in CONCACAF and CONMEBOL, and while I still see the same overestimation of points, the point differences are much smaller.  

Third, the soccer Pythagorean estimates the relative places of the teams very well.  I think the ability of the Pythagorean to estimate final placement is perhaps the best measure of a team's ability to meet statistical expectations.  I am guessing that for predicting over/underperformance in a season, it is only useful to calculate the Pythagorean at the halfway point and compare results at the end of the season.  That test would be more difficult to carry out because I would need the goal scoring data up to the halfway point of the season, and that would require access to the match scores for each round of the league.

It looks like I'll have to think of a notion of "second-order" wins/draws in order to make the Pythagorean more accurate.  It ends up becoming a correction to the goals scored and allowed with respect to the expected goals scored and allowed in a league.  I need to give some more thought to how you would describe that mathematically.

Thanks to Dave, Eric, and the others who pointed me to the Pythagorean formula; it's been neat to think of the math foundations behind a deceptively simple formula, which turned out to be a little too simple for soccer.  Now to make it a little less intimidating, but I think we're well past that point!

Wilkommen!

I've been getting a lot of hits from Germany in the past few days.  I don't know if they're all from the same person or not, but welcome and thanks for stopping by!  I took a couple of days off after writing about my extended Pythagorean over the past week, but there are some posts on the latest club/country coefficients for CONCACAF, a paper review on "Football Fever" written by a group of German physicists, and a conference at MIT on Sports Analytics (and wondering why there aren't any football-related topics).  And I have started, with the help of a couple other contributors, a base design of a soccer results database suitable for statistical analysis.  I haven't forgotten that effort; it's gotten buried under some other efforts but I hope to have some updates on that end very soon.

Once again, welcome!

Pythagorean study -- the path forward

A few nights ago I posted a link to the MIT Sports Analytics Conference to be held in March and later included a link to a call for papers to the event.  I am thinking very seriously about submitting a paper to that conference.  It will likely be on the soccer Pythagorean formula, especially if I can demonstrate some kind of success in predicting points won at the end of a season. 

Here's where I could use some assistance. 

I'd like to study the ability of the soccer Pythagorean to predict team performance in the 2008-09 European season (2008 in the case of MLS or J-League).  To do this I will need goals scored and allowed data for all of the teams in the league during that season. Here are the leagues that I'd like to work with:

• Spanish Primera Liga
• German 1.Bundesliga
• Italian Serie A
• USA Major League Soccer
• Japanese J.League

The data aren't all that difficult to obtain, but it is not always arranged in column data format so it takes some time to collect.  The format should look something like this spreadsheet.  Don't worry about the histogram data; that command is very simple to write.  I am paying more attention to the raw data.  Once I get the raw data, processing it will be very quick now that I have my curve-fitting and Pythagorean codes written.

I plan on compiling most of the data myself, but if anyone is willing to help, I would be very appreciative.  You would also receive a mention in any publications I write -- everyone stands on the shoulders of giants, and I am a very strong believer in acknowledging assistance in every publication I write.  

The deadline for paper submission to this conference is the second week in December, so we're going to have to move quickly.  This isn't the only project that I'm working on, so I would like to make the most of my time.

ONE OTHER THING: I'll present a few of the results here, but I'm going to have to embargo most of the major findings until after the conference, assuming that my paper gets accepted.  If it's not accepted I'll publish all of the results here in mid-January.

Pythagorean results: 2008-09 English Premier League

I've reached the final step in my effort to test my extended Pythagorean formula, using the 2008-09 English Premier League as a test case.  There was a last-minute snag in my plans, however.  I was looking over my equations one more time, and I decided to re-derive everything from the beginning, which is a good thing to do after putting the original work aside for a couple of days.  I realized that I had made an error with distributing the exponent term in the exponential terms.  When I corrected it, I ended up with a much simpler expression for estimating draw probability, but the estimated point totals were worse!  It took me a while to convince myself that I had done everything okay this time.

I included the derivation in my writeup of the soccer Pythagorean and rewrote key parts of it so that it is easier to read and follow.  Your comments are always welcome.

Pythagorean exponents in the 2008-09 English Premier League

A couple of nights ago I presented goal distributions for all twenty teams in the 2008-09 English Premier League season, in an attempt to calculate the exponent that would be used for my expansion of the Pythagorean.  I realized after my calculation that I needed to consider the sum of the squares of both the error from the goals scored data and the goals allowed data, instead of separately like I had done before.  Fortunately that correction was easy to fix, so now I'd like to present the alpha and gamma terms for all 20 teams in last year's Premiership:

Goal distributions in the 2008-09 English Premier League

I'm using the 2008-09 English Premier League as a test case of the three-parameter Weibull distribution that is the basis of the Pythagorean formula.  Below I present the exponent values that I obtained from the least-squares estimation, as well as a few plots that show how the resulting Weibull distribution fit to the score data.

My 15 seconds of fame

This isn't really soccermetrics-related, but it is about me in a technical field, so I'm posting it here.

As a few people know, I work as an aerospace engineer in my day job.  I'm also an active member of the American Institute of Aeronautics and Astronautics (AIAA), which is the premier professional society in my field.  AIAA has a new public relations program called "When Did You Know?" which interviews society members and asks them about their interest in aviation/aerospace and how they came to pursuing it as a career.  I sat down for one of these interviews at the big winter meeting in 2008. 

I've been told that the interview is now on AIAA's website, including a blurb on the front page.  So please check it out if you're so inclined.

Test case for Soccer Pythagorean: English Premier League

As I said in my last post on my extended Pythagorean formula for soccer, I was in the process of compiling score data for the 2008-09 English Premier League season in order to test this formulation.  The first step was to fit the data to a Weibull probability distribution in order to estimate the alpha term (which is proportional to the goal average) and gamma term (which is the shape parameter of the distribution). 

I am using a nonlinear least-squares algorithm to estimate the two terms.  To do this, I had to form a Jacobian matrix, which involves expressions for the derivatives of the probability distribution function with respect to alpha and gamma.  The derivative with respect to alpha isn't all that complicated but it is tedious (Chain Rule has to be employed quite a bit).  The derivative with respect to gamma is more complicated because it appears as an exponent.  After some differentiation and then some algebra, I formed the Jacobian matrix and solved the problem.  The nonlinear least-squares algorithm converges very well and requires less than 10 iterations to three-digit accuracy, or less than a second of real time.  For a couple more seconds, it's possible to obtain an answer that's accurate to six digits.  If enough people are interested, I'll write a description of my algorithm and post it here.

I only have data from five clubs, but the alpha terms are reasonable, and the average gamma term is close to my wag that I made to someone in a separate email.  The spread in the gammas is fairly large, but that's to be expected with ten distributions.  It should tighten when I have all 40 (goals scored and goals allowed for all 20 teams in the Premier League). 

So hopefully by the end of the weekend I'll have an estimate of the exponent that best applies to the English Premier League, and then begin an estimate of the total points won over the 2009-2010 season.

UPDATE: I've completed my curve-fits for all 20 teams in the 2008-09 English Premier League season, goals scored and goals allowed.  I'm going to write a more complete summary in a day or so, but here are the main results:

• The mean exponent term in the English Premier League (2008-09 season) is 1.58.
• The standard deviation of that exponent term is large -- around 0.3.  That number reflects the number of heavily-skewed goal distributions in attack and defense.
• There is a gamma function that doesn't cancel out in the draw probability derivations, but with the current exponent, the function is close to 1.

The next step is to write a script that computes the extended Pythagorean for this season based on current goal data.

Updated CONCACAF club coefficients, 2009-10-24 version

I've updated my CONCACAF club/league coefficients with data from the now-completed group stage of the 2009-2010 CONCACAF Champions League.  My latest spreadsheet can be found below, in OpenOffice and Excel formats.

CCL_NatlCoefficients0910_20091024.ods

CCL_NatlCoefficients0910_20091024.xls

The Soccer Pythagorean document

I completed my Soccer Pythagorean document a couple of days ago, but for some reason I had the hardest time developing a PDF file (I used LyX).  I figured it out tonight and I'm linking it to the site now.  I greatly appreciate your comments.

Download PythagDerivation

MORE (23 October, 10:50pm): I'm compiling some score data from the 2008-09 English Premier League season to test this Pythagorean, and determine an estimate of the exponent.  I'm starting with Arsenal, Aston Villa, Blackburn, Bolton, and Chelsea; perhaps some other people would like to work with five other teams.