PyRate is a Python package for sports ratings. It is intended to provide a simple interface for testing and evaluating rating systems such as least-squares and maximum likelihood ratings. It is capable of extracting league score and schedule data from www.masseyratings.com, as well as from basic CSV file formats. Ratings, predictions, and other data can be written to a SQLite database. A Flask module is included that creates a website using data from the database.
Install:
python setup.py installRun the tests:
python -m unittestTo get started, see the example script
here. This script retrieves score and
schedule data from www.masseyratings.com, fits least-squares ratings,
and writes the ratings to a database file. To launch a website, copy
the database to pyrate/flaskr/static/pyrate.db and then run
FLASK_APP=flaskr flask run from the pyrate/ directory. See an
example website at www.mcfarlandratings.com.
Options for customizing the rating system (such as whether home court advantage is included, use of a custom game outcome measure, etc.) can be found within the API's for the specific rating system classes in leastsquares.py and mle.py.
The objective of the rating system is to assign quantitative ratings to a set of teams based on a given set of historical outcomes. Least squares sports ratings are attractive because they are simple to implement, easy to interpret, and allow for a variety of modifications such as inclusion of a home field advantage, custom game outcome measures, and game weighting schemes.
The least squares method begins with the concept of a "Game Outcome Measure" that is used to quantify the results of a particular game. A simple approach is to define the game outcome measure as the difference in points, although other measures are possible. The basic idea is to attempt to explain the game outcome measure for each game in terms of the difference in ratings between the two teams. As an example, suppose that Team A has a rating of 10, Team B has a rating of 6, and Team C has a rating of 0. Assuming that points are used as the game outcome measure, these ratings imply that we would expect Team A to beat Team B by 4 points, Team B to beat Team C by 6 points, and Team A to beat Team C by 10 points. Note that these are only the expected results, and that actual outcomes can vary.
The objective is to determine the rating values for each team based on a set of past outcomes. Of course, the ratings will not perfectly explain every outcome, so it is necessary to establish some definition of what constitutes the "best" ratings. Note that for any particular game, we can make a comparison between the actual outcome and what is predicted by the team ratings. Let yi denote the actual outcome for a game between teams A and B. The expected value of the outcome is equal to rA - rB. The difference between the actual and expected outcome is yi - (rA - rB). We seek ratings that minimize these differences between expected and actual outcomes. In least squares, this is done by minimizing the sum of the squares of the differences across all games.
Note that the predicted outcome of a game depends only on the difference in team ratings. Thus, a constant value c can be added to all ratings without affecting the model. This means that there is not a unique solution to the least squares ratings. This can be easily resolved by imposing an additional constraint that fixes the "location" of the overall set of rating values. Typically, this is done by constraining the sum (and hence the average) of the ratings to be zero. This is very convenient because it means that a rating value of zero denotes an average team. Taking the game outcomes together with this additional constraint produces a system of equations that can be solved directly using linear algebra: the result is the set of rating values for the teams.
It was already noted that the difference in the ratings between two teams can be interpreted as the expected value of the game outcome measure (e.g., point difference). Several other interpretations of the least squares approach are possible. For a given team, we can compute a "normalized score" for each game as the sum of the game outcome measure (expressed relative to that team, i.e., positive for a win and negative for a loss) and the opponent's rating. For example, a 10-point win over a team with a -5.0 rating corresponds to a normalized score of 5, whereas a 10-point win over a team with a 5.0 rating corresponds to a normalized score of 15. Thus, the normalized score measures the performance relative to the strength of the opponent. It turns out that due to the additive nature of the least squares model, each team's rating is equal to the average of the normalized scores from their games.
The ratings can also be used to evaluate strength of schedule. A simple metric for schedule strength is the average rating of a team's opponents. Using this metric, a strength of schedule of 0 indicates that on average, the strength of the team's opponents is average. Again, due to the additive nature of the system, the ratings can further be interpreted in terms of schedule strength: each team's rating is equal to the sum of their strength of schedule (average opponent rating) and their average game outcome measure (e.g., point difference). This is another view that indicates how each team's rating accounts for both the strength of the opposition and their performance against that opposition.
It is also possible to account for the effect of home field advantage within the least squares method. This can be done by adding one new parameter to the model, h, which represents the average home field advantage across the entire league. With this formulation, the expected outcome for a particular game is given by rA - rB + h · xh, where xh is an indicator that is assigned 1 for a home game and -1 for an away game. The equations are modified accordingly, so that the solution produces values for the team ratings and the universal home field advantage. Note that home field advantage is expressed in terms of the game outcome measure, which does not necessarily need to be defined as the point difference.