1. Definition

  • A quantile forecast predicts a specific quantile (percentile) of the future outcome distribution, instead of just the mean (point forecast).
  • For quantile level $\alpha \in (0,1)$:

$\hat{q}_\alpha(t+h) \;\; \text{is the forecasted value such that} \;\; P(Y_{t+h} \leq \hat{q}_\alpha) = \alpha$

Example:

  • 0.5 quantile = median forecast.
  • 0.1 quantile = pessimistic lower bound.
  • 0.9 quantile = optimistic upper bound.

2. Why Quantile Forecasts Matter

  • Unlike point forecasts (single mean estimate), quantile forecasts give different possible scenarios.
  • Provide uncertainty information without requiring the full distribution.
  • Useful for prediction intervals:
    • A 90% prediction interval = [0.05 quantile forecast, 0.95 quantile forecast].

3. Examples

Weather

  • Deterministic: “Tomorrow = 25°C.”
  • Quantile forecasts:
    • 0.1 quantile = 22°C (10% chance temp ≤ 22).
    • 0.5 quantile = 25°C (median).
    • 0.9 quantile = 28°C (90% chance temp ≤ 28).

Retail Demand

  • 0.1 quantile forecast = 450 units (conservative, safe stock level).
  • 0.5 quantile forecast = 500 units (expected demand).
  • 0.9 quantile forecast = 560 units (upper bound for safety stock).

4. Methods for Quantile Forecasting

  • Quantile Regression: directly estimates conditional quantiles.
  • Quantile Regression Forests: ensemble tree methods for quantiles.
  • Gradient Boosting (LightGBM, XGBoost) with quantile loss.
  • Deep learning models (e.g., DeepAR, TFT) that predict quantile forecasts.

5. Loss Function: Pinball Loss

Quantile forecasts are typically trained using pinball loss:

$L_\alpha(y, \hat{q}_\alpha) = \begin{cases} \alpha \cdot (y – \hat{q}_\alpha) & \text{if } y \geq \hat{q}_\alpha \\[6pt] (1-\alpha) \cdot (\hat{q}_\alpha – y) & \text{if } y < \hat{q}_\alpha \end{cases}$

  • Asymmetric: penalizes underestimation and overestimation differently depending on quantile.
  • Ensures the model targets the correct quantile.

6. Relation to Other Forecasts

  • Point forecast = mean or median.
  • Quantile forecast = specific percentiles of the distribution.
  • Prediction interval = constructed from two quantile forecasts.
  • Probabilistic forecast = entire distribution (quantile forecasts approximate it).

Summary:
A quantile forecast predicts a chosen percentile of the outcome distribution (e.g., 10th, 50th, 90th). It’s useful for risk-aware decision-making, constructing prediction intervals, and understanding uncertainty. Trained using pinball loss, quantile forecasts are widely used in retail, energy, finance, and weather forecasting.