Recall from Problem Set 1 the Pythagorean Expectation formula was introduced by Bill James to estimate how many games a baseball team is ``expected’’ to win, based on the number of runs they scored and the number of runs they give up. For sports like baseball, basketball, and American football, win-loss records determine playoff seedings. In low-scoring sports like soccer (i.e. the other football) and ice hockey, more important than the number of games won is the number of points scored. Since tie games (or “draws”) are not that unusual, each team is awarded 1 point in the event of a tie and 3 points are awarded to a winner. In this project, we will try to devise a Pythagorean-style formula for the number of points scored in English football.
We will use data from the 1950 season all the way up to the 2015 season. The file “data/english_football_train.csv” contains a random subset of 75% of the full dataset. We will use this data to train several different models to forecast the number of points each team has scored. The file “data/english_football_test.csv” contains the remaining 25% of the data, which we will use to assess each of the models. Finally, the file “data/english_football_full.csv” contains the full dataset. We will only use this dataset to present our final results.
Exercise Load the training, testing, and full datasets into tbls names england_train, england_test, and england_full.
A naive attempt to get “Pythagorean Poitns” would be to take the Pythagorean expected win percentage, multiply it by the number of games (to get an ``expected number of games wons’’), and then multiply that by 3: \[ \text{Pythag. Points} = \text{Pythag. Win Percentage} \times \text{num. games} \times 3 \] where \[ \text{Pythag. Win Percentage} = \frac{\text{GF}^{2}}{\text{GF}^{2} + \text{GA}^{2}} \]
Exercise Add a column to england_train and england_test containing the forecast points from this formula. Call this column “points_0”.
One problem with this formula is that not every match awards a total of 3 points. Whenever teams tie (draw), the total number of points that are allocated is 2 (one to each team). Instead of multiplying by 3 points per match, we ought to multiply by a average number of points allocated per game: \[
\text{ppg} = 3 \times \frac{\text{Wins}}{\text{Num. Games}} + 2 \times \frac{\text{Draws}}{\text{Num. Games}}
\] To compute the average number of points per game, we should use all of the data (contained in the tbl england_full).
Exercise Add a column called “points_ppg” to england_train and england_test containing the new forecasts. Compute the training and testing RMSE for “points_ppg” and
On balance, there’s no reason for us to use the exponent 2 in the Pythagorean Expectation formula. Using our updated Pythagorean points formula (that uses ppg), try a few different exponents ranging from 0.5 to 2. Find the best exponent using the training – testing paradigm for predicting the number of points scored.
You can also consider a linear model that tries to predict the number of points scored using the ratio \(\frac{\text{GF - GA}}{{\text{GF + GA}}.\)