Mean Absolute Error (MAE)

1. Definition

MAE measures the average magnitude of errors between predictions and actual values, without considering direction (positive/negative).

$MAE = \frac{1}{N} \sum_{i=1}^N |y_i – \hat{y}_i|$

where:

• $N$ = number of predictions
• $y_i$ = actual value
• $\hat{y}_i$ = predicted value

2. Intuition

• It tells us “on average, how far off are our predictions?”
• Errors are in the same units as the target variable (e.g., dollars, kWh, degrees).
• Unlike RMSE, it does not penalize large errors disproportionately.

3. Example

Suppose actual vs predicted electricity demand (in MW):

$MAE = \frac{10 + 20 + 40}{3} = 23.3$

→ On average, predictions are 23.3 MW off.

4. Properties

• Range: 0 → ∞
• Interpretability: Directly understandable (same units as target).
• Robustness: Less sensitive to outliers than RMSE.
• Limitation: Treats all errors equally, so large mistakes are not penalized more heavily.

5. Comparison with Other Metrics

• MAE: Average absolute deviation (linear penalty).
• RMSE: Penalizes larger errors more (quadratic penalty).
• MAPE: Error expressed as a percentage (relative scale).
• R²: Proportion of variance explained, not an absolute error.

Use MAE when you want a straightforward, interpretable average error.
Use RMSE when large errors are especially costly.

6. Python Example

from sklearn.metrics import mean_absolute_error

y_true = [100, 200, 400]
y_pred = [90, 220, 360]

mae = mean_absolute_error(y_true, y_pred)
print("MAE:", mae)

from sklearn.metrics import mean_absolute_error

y_true = [100, 200, 400]
y_pred = [90, 220, 360]

mae = mean_absolute_error(y_true, y_pred)
print("MAE:", mae)

Output:

MAE: 23.3

MAE: 23.3

Summary

• MAE = average of absolute differences between actual and predicted values.
• Range: [0, ∞], lower = better.
• Same units as target → very interpretable.
• Less sensitive to outliers than RMSE.