Metrics: MAE, MSE, RMSE, R² Score

Mean Absolute Error (MAE)

MAE measures the average absolute difference between the actual and predicted values. It gives an idea of how much the predictions deviate from the true values, on average.

• MAE = (1/n) ∑ |yᵢ - ŷᵢ|
• Simple to interpret, same units as target variable
• Less sensitive to outliers than MSE and RMSE

Mean Squared Error (MSE)

MSE calculates the average squared difference between actual and predicted values. It penalizes larger errors more heavily because of squaring.

• MSE = (1/n) ∑ (yᵢ - ŷᵢ)²
• Useful for emphasizing larger errors
• Units are squared compared to the target variable, which can make interpretation harder

Root Mean Squared Error (RMSE)

RMSE is the square root of MSE and provides error in the same units as the target variable, making it more interpretable.

• RMSE = √MSE
• Heavily penalizes large errors
• Commonly used metric for regression problems

R² Score (Coefficient of Determination)

R² Score measures the proportion of variance in the dependent variable that is predictable from the independent variables.

• Ranges from 0 to 1 (or can be negative for poor models)
• R² = 1 - (SS_res / SS_tot), where SS_res is residual sum of squares, and SS_tot is total sum of squares
• Closer to 1 indicates a better fit

Python Code to Calculate MAE, MSE, RMSE, and R²

from sklearn.metrics import mean_absolute_error, mean_squared_error, r2_score
import numpy as np

# True and predicted values
y_true = np.array([3, -0.5, 2, 7])
y_pred = np.array([2.5, 0.0, 2, 8])

# Calculate MAE
mae = mean_absolute_error(y_true, y_pred)
print(f"MAE: {mae}")

# Calculate MSE
mse = mean_squared_error(y_true, y_pred)
print(f"MSE: {mse}")

# Calculate RMSE
rmse = np.sqrt(mse)
print(f"RMSE: {rmse}")

# Calculate R² Score
r2 = r2_score(y_true, y_pred)
print(f"R² Score: {r2}")