  Spatio-temporal problems are ubiquitous and of vital importance in many
research fields. Despite the potential already demonstrated by deep learning
methods in modeling spatio-temporal data, typical approaches tend to focus
solely on conditional expectations of the output variables being modeled. In
this paper, we propose a multi-output multi-quantile deep learning approach for
jointly modeling several conditional quantiles together with the conditional
expectation as a way to provide a more complete "picture" of the predictive
density in spatio-temporal problems. Using two large-scale datasets from the
transportation domain, we empirically demonstrate that, by approaching the
quantile regression problem from a multi-task learning perspective, it is
possible to solve the embarrassing quantile crossings problem, while
simultaneously significantly outperforming state-of-the-art quantile regression
methods. Moreover, we show that jointly modeling the mean and several
conditional quantiles not only provides a rich description about the predictive
density that can capture heteroscedastic properties at a neglectable
computational overhead, but also leads to improved predictions of the
conditional expectation due to the extra information and a regularization
effect induced by the added quantiles.
