📉 Continuous Ranked Probability Score (CRPS)

Continuous Ranked Probability Score (CRPS)#

A proper score comparing a full predicted distribution to the outcome.

Important

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What it is#

The Continuous Ranked Probability Score grades a full probabilistic forecast by comparing its predicted CDF to the observed outcome — the integrated squared gap between the forecast distribution and a step at the truth:

\[\text{CRPS}(F, y) = \int_{-\infty}^{\infty} \big(F(z) - \mathbb{1}\{y \le z\}\big)^2\, dz.\]

Lower is better, and it rewards mass placed near the observation.

Its key properties#

CRPS is a strictly proper scoring rule — it is minimized only by honest, well-calibrated distributions, penalizing overconfidence — and it reports in the target’s units (like MAE). It equals the integral of pinball loss over all quantiles, tying the whole family together.

How it relates to MAE#

For a point (degenerate) forecast, the predicted CDF becomes a step function and CRPS collapses to the MAE. So CRPS is literally MAE generalized to distributions — the natural score for weather, energy, and demand probabilistic forecasting, though it is unbounded.

Theme: Risk & Probabilistic Forecasting · All terminology

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Mind map — connected ideas

Pinball Loss (a.k.a. Quantile Loss) · Mean Absolute Error (MAE) · Strictly Proper Scoring Rules · Probabilistic Forecasts · Brier Score · Cumulative Distribution Function (CDF)

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More in Risk & Probabilistic Forecasting

Continuous Probabilistic Forecasts · Deterministic forecasts · Full Distribution · Pinball Loss (a.k.a. Quantile Loss) · Point Forecasts · Predicting Percentiles · Prediction Intervals (PI) · Probabilistic Forecasts · Probabilistic Scoring · Probability Forecasts · Quantile Forecasts · Quantile Level · Quantile Regression · Return Distribution