Sklar's Theorem in Finite Settings

Mayor, Gaspar; Suñer, Jaume; Torrens, Joan

doi:10.1109/tfuzz.2006.882462

Cited by 25 publications

(9 citation statements)

References 5 publications

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“…The multivariate empirical cumulative distribution function R : R L → I M of the raw ensemble maps into I M , too. According to the discrete version of Sklar's theorem described by Mayor, Suñer and Torrens (2007) in the bivariate case, there exists a uniquely determined empirical copula E M : I L M → I M such that R(y 1 , . .…”

Section: Empirical Copula Interpretationmentioning

confidence: 99%

Uncertainty Quantification in Complex Simulation Models Using Ensemble Copula Coupling

Schefzik¹,

Thorarinsdottir²,

Gneiting³

2013

Statist. Sci.

237

292

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Critical decisions frequently rely on high-dimensional output from complex computer simulation models that show intricate crossvariable, 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.

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Section: Empirical Copula Interpretationmentioning

confidence: 99%

Uncertainty Quantification in Complex Simulation Models Using Ensemble Copula Coupling

Schefzik¹,

Thorarinsdottir²,

Gneiting³

2013

Statist. Sci.

237

292

View full text Add to dashboard Cite

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“…Our goal is now to prove a multivariate discrete version of Sklar's theorem, where the following extension lemma will play an essential role. A bivariate variant of this result has been shown by Mayor et al (2007).…”

Section: A Multivariate Discrete Version Of Sklar's Theoremmentioning

confidence: 68%

“…For a general overview of the mathematical theory of copulas, we refer to the textbooks by Joe (1997) and Nelsen (2006), as well as to the survey paper by Sempi (2011). A special type of copulas are the so-called discrete copulas, whose properties have been studied by Kolesárová et al (2006), Mayor et al (2005), Mayor et al (2007) and Mesiar (2005) in recent years. However, the discussion in the papers mentioned above focuses on the bivariate case, and it is natural to search for a treatment of the general multivariate situation.…”

Section: Introductionmentioning

confidence: 99%

Ensemble Copula Coupling as a Multivariate Discrete Copula Approach

Schefzik¹

2013

Preprint

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In probability and statistics, copulas play important roles theoretically as well as to address a wide range of problems in various application areas. In this paper, we introduce the concept of multivariate discrete copulas, discuss their equivalence to stochastic arrays, and provide a multivariate discrete version of Sklar's theorem. These results provide the theoretical frame for the ensemble copula coupling approach proposed by Schefzik et al. (2013) for the multivariate statistical postprocessing of weather forecasts made by ensemble systems.

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“…A very similar definition is given in Kolesárová et al (2006), who investigated such 'discrete copulas' in the case R = S. The 'copula pmf' here coincides essentially with the (rescaled) bistochastic matrix of their Proposition 2. See also Mayor et al (2005Mayor et al ( , 2007, Aguiló et al (2006), Kobayashi (2014) or de Amo et al (2017). Those papers investigate the analytical properties of such matrices, though, and show little overlap with what is discussed here.…”

Section: S×s Let Cl([p]mentioning

confidence: 90%

An essay on copula modelling for discrete random vectors; or how to pour new wine into old bottles

Geenens¹

2019

Preprint

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Copulas have now become ubiquitous statistical tools for describing, analysing and modelling dependence between random variables. Sklar's theorem, "the fundamental theorem of copulas", makes a clear distinction between the continuous case and the discrete case, though. In particular, the copula of a discrete random vector is not identifiable, which causes serious inconsistencies. In spite of this, downplaying statements are widespread in the related literature, and copula methods are used for modelling dependence between discrete variables. This paper calls to reconsidering the soundness of copula modelling for discrete data. It suggests a more fundamental construction which allows copula ideas to smoothly carry over to the discrete case. Actually it is an attempt at rejuvenating some century-old ideas of Udny Yule, who mentioned a similar construction a long time before copulas got in fashion.

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Sklar's Theorem in Finite Settings

Cited by 25 publications

References 5 publications

Uncertainty Quantification in Complex Simulation Models Using Ensemble Copula Coupling

Uncertainty Quantification in Complex Simulation Models Using Ensemble Copula Coupling

Ensemble Copula Coupling as a Multivariate Discrete Copula Approach

An essay on copula modelling for discrete random vectors; or how to pour new wine into old bottles

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