Isotonic Distributional Regression

TL;DR

Isotonic distributional regression (IDR) is a versatile nonparametric method for estimating conditional distributions with order restrictions, offering calibration, optimality, and broad applicability in probabilistic forecasting.

Contribution

The paper introduces IDR as a parameter-free, calibration-preserving distributional regression method that generalizes existing isotonic regression techniques and can be combined with subsample aggregation.

Findings

IDR performs well in simulation studies using CRPS and L2 error metrics.

IDR is competitive with state-of-the-art methods in weather forecast case studies.

The method benefits from smoother functions and improved computational efficiency when combined with subsample aggregation.

Abstract

Isotonic distributional regression (IDR) is a powerful nonparametric technique for the estimation of conditional distributions under order restrictions. In a nutshell, IDR learns conditional distributions that are calibrated, and simultaneously optimal relative to comprehensive classes of relevant loss functions, subject to isotonicity constraints in terms of a partial order on the covariate space. Nonparametric isotonic quantile regression and nonparametric isotonic binary regression emerge as special cases. For prediction, we propose an interpolation method that generalizes extant specifications under the pool adjacent violators algorithm. We recommend the use of IDR as a generic benchmark technique in probabilistic forecast problems, as it does not involve any parameter tuning nor implementation choices, except for the selection of a partial order on the covariate space. The method can be combined with subsample aggregation, with the benefits of smoother regression functions and gains in computational efficiency. In a simulation study, we compare methods for distributional regression in terms of the continuous ranked probability score (CRPS) and $L_2$ estimation error, which are closely linked. In a case study on raw and postprocessed quantitative precipitation forecasts from a leading numerical weather prediction system, IDR is competitive with state of the art techniques.