Multiple-output quantile regression neural network

Quantile regression neural network (QRNN) model has received increasing attention in various fields to provide conditional quantiles of responses. However, almost all the available literature about QRNN is devoted to handling the case with one-dimensional responses, which presents a great limitation when we focus on the quantiles of multivariate responses. To deal with this issue, we propose a novel multiple-output quantile regression neural network (MOQRNN) model in this paper to estimate the conditional quantiles of multivariate data. The MOQRNN model is constructed by the following steps. Step 1 acquires the conditional distribution of multivariate responses by a nonparametric method. Step 2 obtains the optimal transport map that pushes the spherical uniform distribution forward to the conditional distribution through the input convex neural network (ICNN). Step 3 provides the conditional quantile contours and regions by the ICNN-based optimal transport map. In both simulation studies and real data application, comparative analyses with the existing method demonstrate that the proposed MOQRNN model is more appealing to yield excellent quantile contours, which are not only smoother but also closer to their theoretical counterparts.