Multivariate ensemble Model Output Statistics using empirical copulas

Abstract

Statistical post-processing of ensemble forecasts usually is carried out independently for individual, scalar predictands. However, in some applications multivariate joint forecast distributions, which capture both the univariate marginal distributions of their constituent scalar predictands as well as the dependence structure among them, may be required. Copulas are functions that link multivariate distribution functions to their constituent univariate marginal distributions. Empirical copulas are non-parametric copula functions that are easy to implement. This article compares recently proposed variants of empirical copula coupling (ECC-Q and ECC-R), which take their dependence structures from raw forecast ensembles, and the Schaake shuffle, which is based on unconditional random samples from the historical climatology, in a four-dimensional multivariate ensemble post-processing setting. These alternatives were compared for probability forecasts of multi-day 'heat waves', based on the 11-member National Oceanic and Atmospheric Administration (NOAA) reforecast ensembles. Best forecast accuracy was achieved using the unconditional climatological dependence structures sampled by the Schaake shuffle, implying that any forecast improvements due to flow-specific dependencies that might be captured by the ensemble-based copulas are not sufficient to overcome errors in the ensemble's representation of those dependencies.

Variogram-Based Proper Scoring Rules for Probabilistic Forecasts of Multivariate Quantities*

“…Focusing on differences between components is probably the most natural, but by no means the only possible transformation of the multivariate quantity that leads to a multivariate score that is sensitive to correlations between components. In some applications, studying composite quantities like minima, maxima, or averages over several locations or lead times (Berrocal et al 2007;Feldmann et al 2015), or indexes that involve multiple quantities (Wilks 2014) is a natural way to turn multivariate quantities into univariate ones that can be evaluated by standard univariate scores. This way, specific (and practically relevant) aspects of the multivariate predictive distribution can be evaluated, and this sort of verification is another recommended supplement to general purpose multivariate scores like the ES or the VS-p presented here.…”

“…Moreover, simultaneous consideration of all locations in the river basin and several lead times may be required. A recent article by Wilks (2014) considers probabilistic forecasting of heat waves, which requires the simultaneous study of minimum temperature and dewpoint temperature at two consecutive days, and Feldmann et al (2015) study statistical postprocessing methods that yield calibrated temperature forecasts simultaneously at several locations. A number of multivariate generalizations of the verification rank histogram have been proposed (Smith and Hansen 2004;Wilks 2004;Gneiting et al 2008;Thorarinsdottir et al 2015;Ziegel and Gneiting 2014) that are sensitive to misrepresentations of both univariate characteristics and correlations between the different components of the multivariate quantity under consideration.…”

Maximum covariance analysis to identify intraseasonal oscillations over tropical Brazil

“…The MCA approach has been chosen because it is able to capture patterns of maximum covariance between two datasets; it has been found to reasonably capture atmospheric and oceanic processes (Wilks 2015). It is a robust method to investigate dominant modes of interaction, because it favors a better understanding of the relationship between groups of variables (Frankignoul et al 2011).…”

Ensemble reconstruction of spatio-temporal extreme low-flow events in France since 1871

“…Vrac and Friederichs (2015) also adapted it recently for multivariate bias correction of downscaled climate simulations. In the Schaake shuffle approach, which can be seen as an empirical copula on rank correlation (Wilks, 2014), the ensemble members are reordered so that their rank correlations across both space and variables match the ones from a randomly picked sample of observed multivariate fields. In the present application, rank correlations are considered across the 608 climatically homogeneous zones and across the three variables (precipitation, temperature, and reference evapotranspiration), and observed fields are taken from the Safran reanalysis.…”