# Isotonic Distributional Regression

arXiv: https://arxiv.org/abs/1909.03725 · HTML: https://mlsanity.com/paper/1909.03725/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Sep 2019.

## Built on by (most cited first, 28 of 28)

- [Machine learning methods for postprocessing ensemble forecasts of wind gusts: A systematic comparison](https://mlsanity.com/paper/2106.09512/) (2021)
- [A review of predictive uncertainty estimation with machine learning](https://mlsanity.com/paper/2209.08307/) (2022)
- [Uncertainty quantification for data-driven weather models](https://mlsanity.com/paper/2403.13458/) (2024)
- [Precipitation nowcasting with generative diffusion models](https://mlsanity.com/paper/2308.06733/) (2023)
- [Proper scoring rules for estimation and forecast evaluation](https://mlsanity.com/paper/2504.01781/) (2025)
- [Conformal Frequency Estimation using Discrete Sketched Data with Coverage for Distinct Queries](https://mlsanity.com/paper/2211.04612/) (2022)
- [EnScale: Temporally-consistent multivariate generative downscaling via proper scoring rules](https://mlsanity.com/paper/2509.26258/) (2025)
- [Comparing Sequential Forecasters](https://mlsanity.com/paper/2110.00115/) (2021)
- [Isotonic Recalibration under a Low Signal-to-Noise Ratio](https://mlsanity.com/paper/2301.02692/) (2023)
- [Postprocessing of point predictions for probabilistic forecasting of day-ahead electricity prices: The benefits of using isotonic distributional regression](https://mlsanity.com/paper/2404.02270/) (2024)
- [Enforcing tail calibration when training probabilistic forecast models](https://mlsanity.com/paper/2506.13687/) (2025)
- [Conformal Frequency Estimation with Sketched Data](https://mlsanity.com/paper/2204.04270/) (2022)
- [Distributional regression and its evaluation with the CRPS: Bounds and convergence of the minimax risk](https://mlsanity.com/paper/2205.04360/) (2022)
- [Distributionally robust risk evaluation with an isotonic constraint](https://mlsanity.com/paper/2407.06867/) (2024)
- [Probabilistic measures afford fair comparisons of AIWP and NWP model output](https://mlsanity.com/paper/2506.03744/) (2025)
- [Distributional Regression with Tabular Foundation Models: Evaluating Probabilistic Predictions via Proper Scoring Rules](https://mlsanity.com/paper/2603.08206/) (2026)
- [Optimal and Provable Calibration in High-Dimensional Binary Classification: Angular Calibration and Platt Scaling](https://mlsanity.com/paper/2502.15131/) (2025)
- [Kernel Density Machines](https://mlsanity.com/paper/2504.21419/) (2025)
- [CRPS-Optimal Binning for Univariate Conformal Regression](https://mlsanity.com/paper/2603.22000/) (2026)
- [Hierarchies of Calibration: Classification meets Regression](https://mlsanity.com/paper/2606.03245/) (2026)
- [Machine learning models for daily rainfall forecasting in Northern Tropical Africa using tropical wave predictors](https://mlsanity.com/paper/2408.16349/) (2024)
- [Calibrated Multivariate Regression with Localized PIT Mappings](https://mlsanity.com/paper/2409.10855/) (2024)
- [Data-driven rainfall prediction at a regional scale: a case study with Ghana](https://mlsanity.com/paper/2410.14062/) (2024)
- [Statistical post-processing yields accurate probabilistic forecasts from Artificial Intelligence weather models](https://mlsanity.com/paper/2504.12672/) (2025)
- [Risk Bounds For Distributional Regression](https://mlsanity.com/paper/2505.09075/) (2025)
- [Residual Distribution Predictive Systems](https://mlsanity.com/paper/2510.26914/) (2025)
- [Isotonic Survival Regression: Calibrated Survival Distributions from Deep Cox Models](https://mlsanity.com/paper/2605.16571/) (2026)
- [Suitable Measures for the Potential Operational Utility of AI NWP Rainfall Forecasts Over Africa](https://mlsanity.com/paper/2609.31775/) (2026)

## Abstract (arXiv)

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 L2 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.