# Why does Random Forest Regression perform worse than autoregression

I have a dataset of NFL games. Each game has one row for each team in the game. Each team's row contains the team's statistics in that game (such as points scored, passing yards, red zone attempts, etc.), the team's statistics over the team's previous four games, and the team's opponent's statistics over their previous four games.

So, for example, the one row might contain a team's passing yards in game gt, that team's total passing yards over games gt - 4 through gt - 1, and the team's opponent's passing yards allowed over games go - 4 through go - 1.

What I have found is that, for some statistics, a team's previous four games' total (and possibly their opponent's previous four games' total) gives a better prediction of what they will do in a given game than a Random Forest Regressor trained on all the variables. Using the example from above:

>>> # keys beginning with 'tm_' represent the team's statistics in this game
>>> # keys beginning with '_tm_' represent the team's statistics over the previous 4 games
>>> # keys beginning with '_op_' represent the opponent's statistics (allowed) over the previous 4
>>>
>>> from scipy.stats.stats import pearsonr
>>> from sklearn.ensemble import RandomForestRegressor
>>>
>>> # Calculate r-squared between passing yards over the previous 4 and passing yards
>>> r, p = pearsonr(df['_tm_py'], df['tm_py'])
>>> r**2
0.10695998799573359
>>>
>>> # Use a random forest regressor to predict passing yards using all the features in the data set
>>> rgr = RandomForestRegressor(n_jobs=-1, max_features=None, n_estimators=500, oob_score=True)
>>> rgr.fit(df[features], df['tm_py'])
RandomForestRegressor(bootstrap=True, compute_importances=None,
criterion='mse', max_depth=None, max_features=None,
max_leaf_nodes=None, min_density=None, min_samples_leaf=1,
min_samples_split=2, n_estimators=500, n_jobs=-1,
oob_score=True, random_state=None, verbose=0)
>>> rgr.oob_score_
0.099712596456913757
>>>
>>>
>>> # print the top-10 most important features
>>> for f, i in sorted(zip(features, rgr.feature_importances_), key=lambda x: x, reverse=True)[:10]: print f, '  \t', round(i, 4)
...
_tm_py      0.0929
_tm_rzpy    0.0098
_op_py      0.008
_tm_pts     0.0075
_tm_pc      0.0071
_tm_sfpy    0.007
_tm_pfd     0.0068
_tm_td      0.0067
_op_p2y     0.0065
_tm_p2y     0.0065


As you can see, a team's passing yards over their previous four games is the most important feature for predicting their passing yards (followed by the team's red zone passing yards over their previous four games and their opponent's passing yards allowed over their opponent's previous four games). However, the oob_score_ for the Random Forest Regressor turns out to be lower than the r-squared between tm_py and _tm_py.

Why is this? Is it because all of the other features are somehow "polluting" the direct relationship between these two variables? Is it possible that using a linear correlation between a single feature and the target can be better than using a random forest regressor between many features and the target?

• By all the features in the model, do you also mean lagged outcome variables? – Brash Equilibrium Sep 7 '14 at 20:29