Quantile Regression

Can be used to give Quantile bounds for a Regression task. For example upper and lower bounds for the predicted value.

Uses a different Loss Function:

$$L=\{\begin{array}{l}\tau (y-X\theta ),\text{ if }y-X\theta \ge 0\\ (\tau -1)(y-X\theta ),\text{ if }y-X\theta <0\end{array}$$

The parameter $\tau$ ist the Quantile.

You have to train a model for each quantile that you want to obtain values for.

When the predicted values are smaller than the true values, you get a positive loss which is being penalized by the parameter $\tau$. When the predicted values are bigger than the true values, you get a negative loss which is being penalized by the inverse quantile.

You can use a Custom Loss Function for Linear Regression in Python to implement this in Python.

Example:

Ressources

• For non-linear Regression
  • http://ethen8181.github.io/machine-learning/ab_tests/quantile_regression/quantile_regression.html