  Critical decisions frequently rely on high-dimensional output from complex
computer simulation models that show intricate cross-variable, spatial and
temporal dependence structures, with weather and climate predictions being key
examples. There is a strongly increasing recognition of the need for
uncertainty quantification in such settings, for which we propose and review a
general multi-stage procedure called ensemble copula coupling (ECC), proceeding
as follows: 1. Generate a raw ensemble, consisting of multiple runs of the
computer model that differ in the inputs or model parameters in suitable ways.
2. Apply statistical postprocessing techniques, such as Bayesian model
averaging or nonhomogeneous regression, to correct for systematic errors in the
raw ensemble, to obtain calibrated and sharp predictive distributions for each
univariate output variable individually. 3. Draw a sample from each
postprocessed predictive distribution. 4. Rearrange the sampled values in the
rank order structure of the raw ensemble to obtain the ECC postprocessed
ensemble. The use of ensembles and statistical postprocessing have become
routine in weather forecasting over the past decade. We show that seemingly
unrelated, recent advances can be interpreted, fused and consolidated within
the framework of ECC, the common thread being the adoption of the empirical
copula of the raw ensemble. Depending on the use of Quantiles, Random draws or
Transformations at the sampling stage, we distinguish the ECC-Q, ECC-R and
ECC-T variants, respectively. We also describe relations to the Schaake shuffle
and extant copula-based techniques. In a case study, the ECC approach is
applied to predictions of temperature, pressure, precipitation and wind over
Germany, based on the 50-member European Centre for Medium-Range Weather
Forecasts (ECMWF) ensemble.
