1. Definition

Pinball loss measures the accuracy of predicted quantiles rather than the mean.
It’s used when we want to estimate, for example, the 90th percentile (quantile = 0.9) of a target variable.

For a quantile level $q \in (0,1)$:

$L_q(y, \hat{y}) = \begin{cases} q \cdot (y – \hat{y}), & \text{if } y \geq \hat{y} \\ (1-q) \cdot (\hat{y} – y), & \text{if } y < \hat{y} \end{cases}$

where:

  • $y$ = actual value
  • $\hat{y}$​ = predicted quantile value
  • $q$ = quantile (e.g., 0.5 = median, 0.9 = 90th percentile)

2. Intuition

  • If predicting the q-th quantile:
    • Under-predictions are penalized q times the error.
    • Over-predictions are penalized (1-q) times the error.
  • This asymmetry reflects the definition of a quantile.

Example:

  • If $q = 0.9$, underestimating the 90th percentile is worse than overestimating it.

3. Special Case – Median (q = 0.5)

When $q = 0.5$:

4. Example

Suppose we predict the 90th percentile of demand:

  • Quantile level: $q = 0.9$
  • Actual demand: $y = 100$
  • Predicted quantile: $\hat{y} = 80$

Since $y > \hat{y}$​:

$L_{0.9} = 0.9 \cdot (100 – 80) = 18$

If predicted $\hat{y} = 120$:

$L_{0.9} = (1 – 0.9) \cdot (120 – 100) = 0.1 \cdot 20 = 2$

Underestimating (loss = 18) is punished much more heavily than overestimating (loss = 2).

5. Why It Matters

6. Python Example

import numpy as np

def pinball_loss(y_true, y_pred, q=0.9):
    errors = y_true - y_pred
    return np.mean(np.where(errors >= 0, q * errors, (1 - q) * -errors))

# Example
y_true = np.array([100, 200, 300])
y_pred = np.array([90, 210, 280])

loss = pinball_loss(y_true, y_pred, q=0.9)
print("Pinball Loss (q=0.9):", loss)

Output:

Pinball Loss (q=0.9): 7.0

Summary

  • Pinball Loss = loss function for quantile regression.
  • Penalizes over/underestimation differently depending on quantile level $q$.
  • $q = 0.5$ → equivalent to MAE.
  • Useful in forecasting, risk management, inventory planning where asymmetric penalties matter.