Thursday, May 19, 2011

Colley's Bias Free Matrix Rankings

(Note #1:  My original implementation of the Colley rating had an error.  See "Whoops!")
(Note #2: An interesting discussion about Colley can be found here.)

The next RPI alternative we'll look at comes from Wesley Colley.  Colley developed his rating system for rating NCAA football, and specifically to avoid bias towards conferences, traditions or regions.  Like RPI, the Colley Rating for a team has two components, one based on a team's performance and the other based upon the strength of its opponents.  Like the Iterative Strength Rating, and the Wilson Rating, the Colley Rating is calculated iteratively.

One unique feature of the Colley Rating is that, rather than use a team's winning percentage as a measure of its performance, Colley uses the "Laplace percentage."  This adjusts the winning percentage by adding a 1 in the numerator and a 2 in the denominator.  So a team with a 0-0 record will have a Laplace percentage of 1/2, and a team with a 5-2 record would have a percentage of 6/9.  This has several good effects. First, it eliminates problems that occur with undefeated teams and winless teams, which tend to introduce nasty divide-by-zero errors.  It also reduces the impact of wins and losses in the early season.  For example, with straight winning percentages, a team that is 1-0 is "infinitely" better than a team that is 0-1.  With the Laplace percentage, the 1-0 team is now 2/3, and the 0-1 team is 1/3, or half as good, which intuitively seems reasonable.

For each team, the Colley Rating calculates a "number of effective wins", determined as half the team's wins minus losses, plus the sum of all its opponents' ratings.  Then the ratings can be recalculated using Laplace percentage with the number of effective wins.  This is then repeated iteratively until the solution converges.

(Colley also provides a matrix solution to calculating the ratings, hence the "Colley Matrix" name for the rating.)

Testing this with our usual methodology provides this result:

Predictor    % Correct    MOV Error
Wilson77.7%10.33
Colley72.0%11.29

There is not much to "tweak" in the Colley Rating, and since this initial result is considerably worse than our current best, it seems unlikely that it can be tweaked to provide a new best result.