Beyond Expectation: Deep Joint Mean and Quantile Regression for Spatiotemporal Problems

Spatiotemporal problems are ubiquitous and of vital importance in many research fields. Despite the potential already demonstrated by deep learning methods in modeling spatiotemporal data, typical approaches tend to focus solely on conditional expectations of the output variables being modeled. In this article, we propose a multioutput multiquantile 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 spatiotemporal problems. Using two large-scale data sets from the transportation domain, we empirically demonstrate that, by approaching the quantile regression problem from a multitask 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 the regularization effect induced by the added quantiles.