Copula calibration

Abstract

We propose notions of calibration for probabilistic forecasts of general multivariate quantities. Probabilistic copula calibration is a natural analogue of probabilistic calibration in the univariate setting. It can be assessed empirically by checking for the uniformity of the copula probability integral transform (CopPIT), which is invariant under coordinate permutations and coordinatewise strictly monotone transformations of the predictive distribution and the outcome. The CopPIT histogram can be interpreted as a generalization and variant of the multivariate rank histogram, which has been used to check the calibration of ensemble forecasts. Kendall calibration is an analogue of marginal calibration in the univariate case. Methods and tools are illustrated in simulation studies and applied to compare raw numerical model and statistically postprocessed ensemble forecasts of bivariate wind vectors.