  Conformalized quantile regression is a procedure that inherits the advantages
of conformal prediction and quantile regression. That is, we use quantile
regression to estimate the true conditional quantile and then apply a conformal
step on a calibration set to ensure marginal coverage. In this way, we get
adaptive prediction intervals that account for heteroscedasticity. However, the
aforementioned conformal step lacks adaptiveness as described in (Romano et
al., 2019). To overcome this limitation, instead of applying a single conformal
step after estimating conditional quantiles with quantile regression, we
propose to cluster the explanatory variables weighted by their permutation
importance with an optimized k-means and apply k conformal steps. To show that
this improved version outperforms the classic version of conformalized quantile
regression and is more adaptive to heteroscedasticity, we extensively compare
the prediction intervals of both in open datasets.
