Calibrated Multivariate Regression with Localized PIT Mappings

TL;DR

This paper proposes a flexible post-hoc recalibration method for multivariate probabilistic forecasts, using local mappings with K-nearest neighbors or normalizing flows, applicable to various response types.

Contribution

It introduces a novel, model-free multivariate calibration technique that handles misspecified models and mixed data types using localized PIT mappings.

Findings

Effective recalibration of neural network currency forecasts.

Improved multivariate regression for childhood malnutrition.

Flexible approach applicable to continuous, discrete, and mixed responses.

Abstract

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 uses K-nearest neighbors, and the other uses normalizing flows. Each method has its own strengths in different situations. We demonstrate the effectiveness of our approach on 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.