Uncertainty quantification for prediction is an intriguing problem with significant applications in various fields, such as biomedical science, economic studies, and weather forecasts. Numerous methods are available for constructing prediction intervals, such as quantile regression and conformal predictions, among others. Nevertheless, model misspecification (especially in high-dimension) or sub-optimal constructions can frequently result in biased or unnecessarily-wide prediction intervals. In this paper, we propose a novel and widely applicable technique for aggregating multiple prediction intervals to minimize the average width of the prediction band along with coverage guarantee, called Universally Trainable Optimal Predictive Intervals Aggregation (UTOPIA). The method also allows us to directly construct predictive bands based on elementary basis functions. Our approach is based on linear or convex programming which is easy to implement. All of our proposed methodologies are supported by theoretical guarantees on the coverage probability and optimal average length, which are detailed in this paper. The effectiveness of our approach is convincingly demonstrated by applying it to synthetic data and two real datasets on finance and macroeconomics.

翻译：预测的不确定性量化是一个引人入胜的问题，在生物医学、经济学研究和天气预报等多个领域具有重要应用。目前已有多种构建预测区间的方法，如分位数回归和共形预测等。然而，模型误设（特别是在高维场景中）或次优构造常常导致预测区间存在偏差或非必要地过宽。本文提出一种新颖且广泛适用的多预测区间聚合技术，旨在最小化预测带的平均宽度同时保证覆盖概率，称为通用可训练最优预测区间聚合（UTOPIA）。该方法还可基于基本基函数直接构建预测带。我们的方法基于线性规划或凸规划，易于实现。所有提出的方法均在覆盖概率和最优平均长度方面具有理论保证，详见本文。通过在合成数据及金融与宏观经济两个真实数据集上的应用，我们方法的有效性得到了令人信服的验证。