RMSPE (Root Mean Square Percentage Error)

Last updated 2020-07-29 Read time 2 min

Summary

RMSPE measures the root mean square of percentage errors between predictions and actuals.
In sales forecasting, it quantifies the average relative deviation between predicted and observed values.
This section explains how to handle zero values using ε and deal with low-demand cases.

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

1. Definition #

$$ \mathrm{RMSPE} = \sqrt{\frac{1}{n} \sum_{i=1}^n \left( \frac{y_i - \hat{y}_i}{y_i} \right)^2 } $$

In practice, it’s often multiplied by 100 to express it as a percentage.
Be cautious when actual values are close to zero, as this can cause divergence.

2. Computing in Python #

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import numpy as np

def rmspe(y_true: np.ndarray, y_pred: np.ndarray, eps: float = 1e-8) -> float:
    ratios = (y_true - y_pred) / np.maximum(np.abs(y_true), eps)
    return float(np.sqrt(np.mean(ratios**2)))

y_true = np.array([120, 150, 80, 200])
y_pred = np.array([118, 148, 79, 190])
print(f"RMSPE = {rmspe(y_true, y_pred) * 100:.2f}%")

Here, eps prevents division by zero.
If zero-demand samples are frequent, consider alternatives like RMSLE or WAPE.

3. When to Use #

Retail demand forecasting: Focus on proportional errors in sales volume.
Financial risk modeling: Evaluate prediction errors in returns or price movements as percentages.
Energy load prediction: Assess relative deviations even when magnitudes vary widely.

4. Comparison with Other Percentage Metrics #

Metric	Feature	Caution
MAPE	Mean percentage error; intuitive	Responds linearly to large errors
RMSPE	Penalizes large relative errors more strongly	Diverges when actuals are near zero
RMSLE	Logarithmic scale metric	Similar to ratio-based, but harder to interpret

RMSPE emphasizes large outliers more than MAPE, making it well-suited for risk-averse KPIs.

5. Practical Considerations #

RMSPE can be unstable with many zero or near-zero values. Exclude or adjust such samples, or complement with other metrics.
Report RMSPE together with MAE or WAPE to account for both absolute and relative performance.
When comparing across periods, calculate RMSPE on consistent sets (same products/time frames).

Summary #

RMSPE captures the root mean square of percentage errors, emphasizing large deviations.
It’s useful for evaluating proportional risk but requires care around zeros.
Use alongside MAPE, WAPE, or RMSLE for a balanced view of both absolute and relative errors.