WMAPE (Weighted Mean Absolute Percentage Error)

1. Definition

WMAPE is a weighted version of MAPE that gives more importance to data points with larger actual values.

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

where:

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

2. Intuition

• Standard MAPE = average of percentage errors for each observation → all points weighted equally.
• WMAPE = total absolute error divided by total actual demand → higher actual values contribute more weight.
• This makes WMAPE less distorted by small actual values (a major weakness of MAPE).

3. Example

Suppose actual vs predicted demand:

• Numerator = $10 + 20 + 40 = 70$
• Denominator = $100 + 200 + 400 = 700$

$WMAPE = \frac{70}{700} = 0.10 = 10\%$

4. Properties

• Range: 0 → ∞ (usually expressed as a percentage).
• Lower is better.
• Easy to interpret: “on average, the forecast was off by X% of actual demand.”
• More robust than MAPE when actuals can be close to zero.

5. Comparison with Other Metrics

• MAPE: Simple, but unstable if actuals are small.
• WMAPE: Weighted by actual demand → better for business forecasting (e.g., sales, inventory).
• sMAPE: Symmetric error measure → avoids asymmetry in over/under forecasting.
• RMSE/MAE: Absolute error measures (not percentages).

Use WMAPE when business cares about aggregate accuracy across all demand, not per-item fairness.

6. Python Example

import numpy as np

def wmape(y_true, y_pred):
    return np.sum(np.abs(y_true - y_pred)) / np.sum(np.abs(y_true))

y_true = np.array([100, 200, 400])
y_pred = np.array([90, 220, 360])

print("WMAPE:", wmape(y_true, y_pred))

import numpy as np

def wmape(y_true, y_pred):
    return np.sum(np.abs(y_true - y_pred)) / np.sum(np.abs(y_true))

y_true = np.array([100, 200, 400])
y_pred = np.array([90, 220, 360])

print("WMAPE:", wmape(y_true, y_pred))

Output:

WMAPE: 0.1  (10%)

WMAPE: 0.1  (10%)

Summary

• WMAPE = total absolute error ÷ total actual demand.
• More business-friendly than MAPE, avoids instability at low actuals.
• Range: 0 → ∞, lower = better.
• Common in forecasting accuracy for sales, demand, inventory, supply chain.