# Uncertainty Quantification in Complex Simulation Models Using Ensemble Copula Coupling

arXiv: https://arxiv.org/abs/1302.7149 · HTML: https://mlsanity.com/paper/1302.7149/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2013.

## Built on by (most cited first, 22 of 22)

- [Increasing the accuracy and resolution of precipitation forecasts using deep generative models](https://mlsanity.com/paper/2203.12297/) (2022)
- [Machine learning methods for postprocessing ensemble forecasts of wind gusts: A systematic comparison](https://mlsanity.com/paper/2106.09512/) (2021)
- [Generative machine learning methods for multivariate ensemble post-processing](https://mlsanity.com/paper/2211.01345/) (2022)
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- [A two-step machine learning approach to statistical post-processing of weather forecasts for power generation](https://mlsanity.com/paper/2207.07589/) (2022)
- [Generating ensembles of spatially-coherent in-situ forecasts using flow matching](https://mlsanity.com/paper/2504.03463/) (2025)
- [A Composite-Loss Graph Neural Network for the Multivariate Post-Processing of Ensemble Weather Forecasts](https://mlsanity.com/paper/2509.02784/) (2025)
- [Enhancing multivariate post-processed visibility predictions utilizing CAMS forecasts](https://mlsanity.com/paper/2406.14159/) (2024)
- [Distributional Regression U-Nets for the Postprocessing of Precipitation Ensemble Forecasts](https://mlsanity.com/paper/2407.02125/) (2024)
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- [Improving ensemble extreme precipitation forecasts using generative artificial intelligence](https://mlsanity.com/paper/2407.04882/) (2024)
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- [A Weather Foundation Model for the Power Grid](https://mlsanity.com/paper/2509.25268/) (2025)
- [Joint distribution of upstream runoff governs downstream river-discharge prediction uncertainty in distributed ML models](https://mlsanity.com/paper/2607.03217/) (2026)
- [Efficient modeling of sub-kilometer surface wind with Gaussian processes and neural networks](https://mlsanity.com/paper/2405.12614/) (2024)
- [Foundation Model Forecasts: Form and Function](https://mlsanity.com/paper/2510.19345/) (2025)
- [STIPP: Space-time in situ postprocessing over the French Alps using proper scoring rules](https://mlsanity.com/paper/2601.02882/) (2026)
- [Evaluating Generative Time-Series Models on Data with Point Masses](https://mlsanity.com/paper/2608.09692/) (2026)
- [Interweaving Marginals into Multivariate Sample Paths: Training-Free Dependence Construction for Probabilistic Time Series Foundation Models](https://mlsanity.com/paper/2609.25980/) (2026)
- [A General Framework for Budgeted Threshold Incentives on Request](https://mlsanity.com/paper/2609.29724/) (2026)
- [When better traffic forecasts fail to improve signal control: a layered diagnostic study of forecast-to-decision value](https://mlsanity.com/paper/2610.06992/) (2026)

## Abstract (arXiv)

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.