Calibrated Multivariate Distributional Regression with Pre-Rank Regularization

TL;DR

This paper introduces a regularization method using pre-rank functions, including a PCA-based approach, to improve multivariate calibration during training of distributional regression models, validated on simulations and real datasets.

Contribution

It presents a novel calibration technique that enforces multivariate calibration during training, incorporating a PCA-based pre-rank for better dependence structure assessment.

Findings

Significantly improves multivariate pre-rank calibration

Maintains predictive accuracy while enhancing calibration

Reveals dependence-structure misspecifications with PCA pre-rank

Abstract

The goal of probabilistic prediction is to issue predictive distributions that are as informative as possible, subject to being calibrated. Despite substantial progress in the univariate setting, achieving multivariate calibration remains challenging. Recent work has introduced pre-rank functions, scalar projections of multivariate forecasts and observations, as flexible diagnostics for assessing specific aspects of multivariate calibration, but their use has largely been limited to post-hoc evaluation. We propose a regularization-based calibration method that enforces multivariate calibration during training of multivariate distributional regression models using pre-rank functions. We further introduce a novel PCA-based pre-rank that projects predictions onto principal directions of the predictive distribution. Through simulation studies and experiments on 18 real-world multi-output regression datasets, we show that the proposed approach substantially improves multivariate pre-rank calibration without compromising predictive accuracy, and that the PCA pre-rank reveals dependence-structure misspecifications that are not detected by existing pre-ranks.