Calibrating Multivariate Regression with Localized PIT Mappings

Calibration ensures that predicted uncertainties align with observed uncertainties. While there is an extensive literature on recalibration methods for univariate probabilistic forecasts, work on calibration for multivariate forecasts is much more limited. This paper introduces a novel post-hoc recalibration approach that addresses multivariate calibration for potentially misspecified models. Our method involves constructing local mappings between vectors of marginal probability integral transform values and the space of observations, providing a flexible and model free solution applicable to continuous, discrete, and mixed responses. We present two versions of our approach: one using K-nearest neighbours, and the other using normalizing flows, each with distinct advantages depending on the problem. We demonstrate the effectiveness of our approach in two real data applications: recalibrating a deep neural network’s currency exchange rate forecast and improving a regression model for childhood malnutrition in India for which the multivariate response has both discrete and continuous components.