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.
翻译:量化回归是量化不确定性的有效方法,符合具有挑战性的基本分布,而且往往通过对多个孔径水平的联合学习提供全面的概率预测。不过,这些共同的孔径回归的一个常见缺点是\textit{quantile 交叉点},这违反了有条件孔径函数的可取的单质属性。在这项工作中,我们建议采用递增(Spline)量函数I(S)QF,这是一个灵活而高效的无分布分数估计框架,它能用简单的神经网络层解决四分点交叉点问题。此外,I(S)QF间/外推法可以预测与基本培训不同的任意孔径位值。在对I(S)QF代表连续排序概率进行的分析评价后,我们运用了我们的方法对基于NN的时序预测案例进行计算,在这些案例中,为未受过训练的孔径水平节省的昂贵再培训费用尤其显著。我们还提供了在序列至序列基线设定的精确性下对拟议方法进行的一般性错误分析。最后,我们还展示了对其他测序至结果的精确性。