R Squared

Compute the coefficient of determination for a regression model.

R^2 = 1 - SS_res / SS_tot

SS_res = sum (y_true - y_pred)^2
SS_tot = sum (y_true - mean(y_true))^2

It is the fraction of the variance in the target that the model explains. 1.0 is a perfect fit and 0.0 means the model does no better than always predicting the mean. It can go negative, which just means the model is worse than that baseline, and that is a genuine result rather than an error.

If SS_tot is zero, every target is identical and the ratio is undefined. Return 1.0 if the predictions are also exact (within 1e-12), and 0.0 otherwise.

Function signature

def r_squared(y_true, y_pred):

Example

Input

y_true = np.array([3.0, -0.5, 2.0, 7.0])
y_pred = np.array([2.5,  0.0, 2.0, 8.0])
r_squared(y_true, y_pred)

Output

0.9486081370449679

Reasoning

SS_res = 0.25 + 0.25 + 0 + 1 = 1.5. The mean of y_true is 2.875, so SS_tot = 29.1875. R^2 = 1 - 1.5 / 29.1875, about 0.9486.

Constraints and notes

• The result can be negative.
• When SS_tot is 0, return 1.0 if the predictions are exact and 0.0 if not.