Python - Coefficient of Determination-R2 score

The Coefficient of determination, also called R² score, is used to evaluate the performance of a linear regression model. It is the amount of the variation in the output dependent attribute that is predictable from the input independent variable(s). It is used to check how well-observed results are reproduced by the model, depending on the ratio of the total deviation of results described by the model.

Mathematical Formula:

R^2= 1- \frac{SS_{res}}{SS_{tot}}

Where,

• SS_{res} is the sum of squares of the residual errors
• SS_{tot} is the total sum of the errors

Interpretation of R² score

Assume R² = 0.68. It can be inferred that 68% of the changeability of the dependent output attribute can be explained by the model, while the remaining 32 % of the variability is still unaccounted for. R² indicates the proportion of data points that lie within the line created by the regression equation. A higher value of R² is desirable as it indicates better results.

Examples

Case 1 (Model gives accurate results):

R^2 = 1 - \frac{0}{200} = 1

Case 2 (Model gives same results always):

R^2 = 1 - \frac{200}{200} = 0

Case 3 (Model gives ambiguous results):

R^2 = 1 - \frac{600}{200} = -2

We can import r2_score from sklearn.metrics in Python to compute R² score.

Python Implementation

Step 1: Import r2_score from sklearn.metrics

from sklearn.metrics import r2_score 

Step 2: Calculate R² score for all the above cases.

### Assume y is the actual value and f is the predicted values y =[10, 20, 30] f =[10, 20, 30] r2 = r2_score(y, f) print('r2 score for perfect model is', r2) 

Output:

r2 score for perfect model is 1.0

### Assume y is the actual value and f is the predicted values y =[10, 20, 30] f =[20, 20, 20] r2 = r2_score(y, f) print('r2 score for a model which predicts mean value always is', r2) 

Output:

r2 score for a model which predicts mean value always is 0.0

Code 3:

### Assume y is the actual value and f is the predicted values y = [10, 20, 30] f = [30, 10, 20] r2 = r2_score(y, f) print('r2 score for a worse model is', r2) 

Output:

r2 score for a worse model is -2.0

Conclusion

• The best possible score is 1 which is obtained when the predicted values are the same as the actual values.
• R² score of baseline model is 0.
• During the worse cases, R² score can even be negative.