Interpreting the DecisionTreeRegressor score?

Question

I am trying to evaluate a relevance of features and I am using DecisionTreeRegressor()

The related part of the code is presented below:

# TODO: Make a copy of the DataFrame, using the 'drop' function to drop the given feature
new_data = data.drop(['Frozen'], axis = 1)

# TODO: Split the data into training and testing sets(0.25) using the given feature as the target
# TODO: Set a random state.

from sklearn.model_selection import train_test_split


X_train, X_test, y_train, y_test = train_test_split(new_data, data['Frozen'], test_size = 0.25, random_state = 1)

# TODO: Create a decision tree regressor and fit it to the training set

from sklearn.tree import DecisionTreeRegressor


regressor = DecisionTreeRegressor(random_state=1)
regressor.fit(X_train, y_train)

# TODO: Report the score of the prediction using the testing set

from sklearn.model_selection import cross_val_score


#score = cross_val_score(regressor, X_test, y_test)
score = regressor.score(X_test, y_test)

print score  # python 2.x 

When I run the print function, it returns the given score:

-0.649574327334

You can find the score function implementatioin and some explanation below here and below:

Returns the coefficient of determination R^2 of the prediction. ... The best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse).

I could not grasp the whole concept yet, so this explanation is not very helpful for me. For instance I could not understand why score could be negative and what exactly it indicates (if something is squared, I would expect it can only be positive).

---

What does this score indicates and why can it be negative?

If you know any article (for starters) it might be helpful as well!

Solution

R^2 can be negative from its definition (https://en.wikipedia.org/wiki/Coefficient_of_determination) if the model fits the data worse than a horizontal line. Basically

R^2 = 1 - SS_res/SS_tot

and SS_res and SS_tot are always positive. If SS_res >> SS_tot, you have a negative R^2. Look at this answer as well: https://stats.stackexchange.com/questions/12900/when-is-r-squared-negative