  Tree-based ensemble methods, as Random Forests and Gradient Boosted Trees,
have been successfully used for regression in many applications and research
studies. Furthermore, these methods have been extended in order to deal with
uncertainty in the output variable, using for example a quantile loss in Random
Forests (Meinshausen, 2006). To the best of our knowledge, no extension has
been provided yet for dealing with uncertainties in the input variables, even
though such uncertainties are common in practical situations. We propose here
such an extension by showing how standard regression trees optimizing a
quadratic loss can be adapted and learned while taking into account the
uncertainties in the inputs. By doing so, one no longer assumes that an
observation lies into a single region of the regression tree, but rather that
it belongs to each region with a certain probability. Experiments conducted on
several data sets illustrate the good behavior of the proposed extension.
