Quantile Regression

Properties2

Type	Concept
Note created	Feb 17, 2025

A quantile regression is a type of regression that, instead of using the mean of the response variable (as per, for example, the least squares method would do), uses a given percentile to fit. That way, the prediction is biased by design: if $q=20\%$, then 80% of the samples are expected to fall above its value.

Quantile regressions are useful in econometrics and other predictions to define upper or lower bounds, and to be used as intervals.

Loss function

Quantile Loss

The quantile loss (often referred to as pinball loss) is an error metric that is able to quantify the error of a quantile regression prediction.

Contrary to the rest of the metrics, this metric does not aim to get the predictions as close to the real values as possible. Instead, there is a purposeful bias to match the prediction to a given percentile of the predictions. The result of the pinball loss could be understood as the “accuracy” of the quantile model.

Being $q$ the target quantile, $y$ the real value and $\hat{y}$ our quantile prediction, the loss function can be expressed as:

$$L_q(y,\hat{y})=\{\begin{array}{ll}(y-\hat{y})\cdot q& y\ge z\\ (y-\hat{y})\cdot (q-1)& y<z\end{array}$$Link to original