Quantile Regression

Unlike ordinary least squares which minimizes symmetric squared errors, quantile regression minimizes the pinball loss (tilted absolute value) function. This asymmetric loss penalizes over-prediction and under-prediction differently based on the target quantile τ.

• For the median (τ=0.5), the penalty is symmetric
• For the 95th percentile (τ=0.95), under-prediction is penalized 19x more than over-prediction
• This directly encodes the business cost of stockouts vs. overstocking into the model training objective

Quantile regression makes no parametric assumptions about the underlying error distribution. It does not require the residuals to be normally distributed, homoscedastic, or independent.

• Works robustly with heavy-tailed demand distributions common in intermittent spare parts
• Handles heteroscedasticity where forecast variance changes with the level of demand
• Provides valid inference without needing to specify a likelihood function

By fitting multiple quantile regressions simultaneously, you construct a prediction interval with a specified coverage probability. A 90% prediction interval is formed by the 5th and 95th conditional quantiles.

• The width of the interval directly reflects the model's uncertainty at each point
• Intervals can be asymmetric, capturing skewed demand distributions
• Service level targets (e.g., 98% fill rate) map directly to specific quantiles

Quantile regression reveals how covariates affect different parts of the demand distribution, not just the mean. A promotion might lift median demand by 10% but the 90th percentile by 40%.

• Identifies predictors that increase demand volatility without shifting the mean
• Detects heterogenous treatment effects across the distribution
• Provides richer operational insights than mean regression for inventory decisions

The quantile regression optimization problem can be expressed as a linear program, making it computationally tractable for large-scale supply chain datasets with thousands of SKUs.

• Solved efficiently using simplex or interior-point methods
• The dual formulation provides direct estimates of the conditional density
• Extends naturally to regularized variants (Lasso, Ridge) for high-dimensional feature spaces

While quantile regression minimizes absolute asymmetric loss, expectile regression minimizes asymmetric squared loss. Expectiles are more sensitive to extreme values and provide an alternative characterization of the distribution.

• Expectiles are coherent risk measures; quantiles are not
• Useful when the cost of extreme errors grows quadratically
• Both techniques can be combined for robust probabilistic forecasting ensembles

The primary output consumers of quantile regression in supply chain contexts. A prediction interval defines a range [L, U] expected to contain the actual future demand with a specified coverage probability.

• A 90% interval is constructed from the τ=0.05 and τ=0.95 quantiles
• Unlike confidence intervals, prediction intervals account for both model uncertainty and inherent noise
• Critical for setting safety stock levels at specific service level targets

The gold-standard metric for evaluating probabilistic forecasts, including those produced by quantile regression. CRPS measures the integrated squared difference between the predicted cumulative distribution function and the empirical observation.

• A strictly proper scoring rule that encourages honest probability estimates
• Generalizes the Mean Absolute Error to probabilistic predictions
• Penalizes both poor calibration and low sharpness in a single value

Quantile regression naturally captures aleatoric uncertainty—the irreducible noise in demand data. However, it does not automatically separate this from epistemic uncertainty, which stems from limited training data.

• Aleatoric: Random demand fluctuations that persist even with infinite data
• Epistemic: Model ignorance that shrinks as more training samples are collected
• Advanced approaches like Deep Ensembles or Bayesian quantile regression disentangle both sources for more honest prediction intervals

The downstream business application that directly consumes quantile regression outputs. Given a demand quantile forecast at a target service level (e.g., 95%), safety stock is calculated as the buffer above the mean forecast.

• Safety Stock = Q(τ) - Mean Forecast, where τ matches the desired fill rate
• Integrates with the newsvendor model to balance holding costs against stockout penalties
• Dynamic safety stock policies re-estimate quantiles daily as demand patterns shift