  Quantile regression is an effective technique to quantify uncertainty, fit
challenging underlying distributions, and often provide full probabilistic
predictions through joint learnings over multiple quantile levels. A common
drawback of these joint quantile regressions, however, is \textit{quantile
crossing}, which violates the desirable monotone property of the conditional
quantile function. In this work, we propose the Incremental (Spline) Quantile
Functions I(S)QF, a flexible and efficient distribution-free quantile
estimation framework that resolves quantile crossing with a simple neural
network layer. Moreover, I(S)QF inter/extrapolate to predict arbitrary quantile
levels that differ from the underlying training ones. Equipped with the
analytical evaluation of the continuous ranked probability score of I(S)QF
representations, we apply our methods to NN-based times series forecasting
cases, where the savings of the expensive re-training costs for non-trained
quantile levels is particularly significant. We also provide a generalization
error analysis of our proposed approaches under the sequence-to-sequence
setting. Lastly, extensive experiments demonstrate the improvement of
consistency and accuracy errors over other baselines.
