MAPE & sMAPE

Last updated 2020-03-11 Read time 2 min

Summary

MAPE expresses errors as a percentage of actual values.
Compare MAPE and sMAPE on a sales-forecast example and observe instability near zero.
Review mitigation strategies when the series contains zeros, negatives, or very small numbers.

MAE & RMSE — understanding this concept first will make learning smoother

1. Definition #

$$ \mathrm{MAPE} = \frac{100}{n} \sum_{i=1}^n \left| \frac{y_i - \hat{y}_i}{y_i} \right| $$

Represents the mean percentage error relative to the actual value.
Lower is better.
Becomes unstable when any $y_i$ is zero or very close to zero.

2. Computing in Python #

1
2
3
4
5
6
7
8
import numpy as np
from sklearn.metrics import mean_absolute_percentage_error

y_true = np.array([120, 150, 80, 200])
y_pred = np.array([110, 160, 75, 210])

mape = mean_absolute_percentage_error(y_true, y_pred)
print(f"MAPE = {mape * 100:.2f}%")

mean_absolute_percentage_error returns a fraction; multiply by 100 for percentage format.

3. sMAPE (symmetrised MAPE) #

To reduce the blow-up at zero, sMAPE includes both actual and predicted values in the denominator:

$$ \mathrm{sMAPE} = \frac{100}{n} \sum_{i=1}^n \frac{|y_i - \hat{y}_i|}{(|y_i| + |\hat{y}_i|)/2} $$

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
import numpy as np

def smape(y_true: np.ndarray, y_pred: np.ndarray) -> float:
    numerator = np.abs(y_true - y_pred)
    denominator = (np.abs(y_true) + np.abs(y_pred)) / 2
    return float(np.mean(numerator / np.maximum(denominator, 1e-8)))

y_true = np.array([120, 150, 80, 200])
y_pred = np.array([110, 160, 75, 210])
print(f"sMAPE = {smape(y_true, y_pred) * 100:.2f}%")

The denominator can still be zero; add a small epsilon as needed.

4. Points to watch #

Zeros or negatives: MAPE explodes when actual values hit zero; use sMAPE or handle zero-demand periods separately.
Bias towards small values: percentage errors weight small-volume items more heavily. Communicate this effect to stakeholders.
Outliers: relative metrics downplay large-volume errors. Combine with MAE/RMSE for absolute impact.
Interpretation: business users like “average error is ±X%,” but pair it with monetary impact where possible.

5. Pairing with other metrics #

MAE / RMSE: report absolute errors alongside percentages to understand real-world loss.
RMSLE: emphasise underestimation for growth or demand forecasts.
Pinball loss: quantify upper/lower quantile forecasts when risk tolerances matter.

Summary #

MAPE delivers intuitive percentage error but needs careful handling near zero.
sMAPE mitigates instability by normalising with the sum of actual and predicted values.
Combine relative and absolute metrics to prioritise model improvements and communicate business impact.