📉 Pinball Loss (a.k.a. Quantile Loss)

Pinball Loss (a.k.a. Quantile Loss)#

The loss minimised by an accurate quantile forecast.

Important

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What it is#

Pinball loss (a.k.a. quantile loss) scores a quantile forecast by penalizing errors asymmetrically — under- and over-prediction get different weights set by the target quantile \(\tau\):

\[L_\tau(y, \hat{y}) = \max\big(\tau(y - \hat{y}),\ (\tau - 1)(y - \hat{y})\big).\]

Minimizing it makes \(\hat{y}\) approach the true \(\tau\)-quantile.

Why asymmetry#

For a high quantile (say \(\tau = 0.9\)), under-predicting is penalized far more than over-predicting, pushing the forecast up to cover the upper tail — exactly what you want for a 90% prediction interval. At \(\tau = 0.5\) the two weights match and pinball loss reduces to (half) the MAE.

Where it’s used#

It trains and evaluates quantile regressors and probabilistic models that output intervals rather than points, without assuming any distribution. A caveat: fitting several quantiles independently can cause quantile crossing, where a lower quantile’s forecast exceeds a higher one’s.

Theme: Risk & Probabilistic Forecasting · All terminology

Hint

Mind map — connected ideas

Continuous Ranked Probability Score (CRPS) · Mean Absolute Error (MAE) · Probabilistic Forecasts · Strictly Proper Scoring Rules · Root Mean Squared Error (RMSE) · MASE (Mean Absolute Scaled Error)

Hint

More in Risk & Probabilistic Forecasting

Continuous Probabilistic Forecasts · Continuous Ranked Probability Score (CRPS) · Deterministic forecasts · Full Distribution · Point Forecasts · Predicting Percentiles · Prediction Intervals (PI) · Probabilistic Forecasts · Probabilistic Scoring · Probability Forecasts · Quantile Forecasts · Quantile Level · Quantile Regression · Return Distribution