Quantile Regression

Ordinary least-squares regression targets a conditional mean. Quantile regression targets another point in the conditional outcome distribution, such as the median or the 90th percentile. The inputs might describe a parcel's route and dispatch time; the output could be a delivery-time percentile for parcels with those characteristics. The median is the 50th percentile. For a continuous outcome with a well-estimated conditional 90th percentile, roughly 90% of comparable outcomes should fall at or below that value. This is a statement about a distribution of outcomes. It is not a claim that the predicted value itself is correct with 90% probability. Quantile regression uses an asymmetric penalty called pinball loss. At a high quantile, underprediction costs more than an equally large overprediction. This pushes the fitted estimate upward relative to the median. Choosing a quantile therefore makes the planning objective explicit instead of hiding it inside an average. Separate lower and upper quantile estimates can form a prediction range. Estimates of the 10th and 90th percentiles suggest a nominal central 80% interval. The word nominal matters: a fitted model can be wrong. Evaluate the fraction of held-out outcomes covered by the interval and examine its width. Also inspect relevant groups rather than relying only on an overall average. Scikit-learn's QuantileRegressor provides a linear model with regularization. Quantile methods also exist for other model families. Predictions from separately fitted quantiles can cross, producing an upper estimate below a lower one. Check for that failure explicitly. Quantile regression gives a useful way to describe variation, but neither the training objective nor an attractive interval chart guarantees reliable uncertainty estimates.