Copula analysis of mixture models

Abstract

Contemporary computers collect databases that can be too large for classical methods to handle. The present work takes data whose observations are distribution functions (rather than the single numerical point value of classical data) and presents the computational statistical approach of a new methodology to group the distributions into classes. The clustering method links the searched partition to the decomposition of mixture densities, through the notions of a function of distributions and of multidimensional copulas. The new clustering technique is illustrated by ascertaining distinct temperature and humidity regions for a global climate dataset and shows that the results compare favorably with those obtained from the standard EM algorithm method.

Multivariate bias adjustment of high-dimensional climate simulations: the Rank Resampling for Distributions and Dependences (R<sup>2</sup>D<sup>2</sup>) bias correction

“…Therefore, if the marginal (i.e., univariate) distributions are generally improved, that is closer to the reference ones -even when the BC is used as a preliminary step to downscaling (e.g., Colette et al, 2012;Vrac and Vaittinada Ayar, 2017) -the inter-site and inter-variable dependence structures are usually conserved from the climate model simulations to be corrected. Indeed, 1d-BC methods preserving the ranks of the simulations -as it is the case for quantile-mapping approaches -will not correct the copula functions characterizing the dependencies between sites and/or between variables (e.g., Nelsen, 2006;Schölzel and Friederichs, 2008;Vrac et al, 2011;Bevacqua et al, 2017). Such a preservation of the model dependence can obviously cause some deficiencies in the subsequent impact studies that will use the 1-dimensional bias corrected simulations if the model copula function is far from that of the references.…”

Model-Based Clustering

“…Some recent work on fractionally-supervised classification (Vrbik and McNicholas 2015) is sure to spawn further work in similar directions. The use of copulas in mixture model-based approaches has already received some attention (e.g., Jajuga and Papla 2006;Di Lascio and Giannerini 2012;Vrac et al 2012;Kosmidis and Karlis 2015;Marbac, Biernacki and Vandewalle 2015) and this sure to continue. Finally, there are some specific data types-both recently emerged and yet to emerge-that deserve their own special attention.…”

Multivariate bias adjustment of high-dimensional climate simulations: The Rank Resampling for Distributions and Dependences (R<sup>2</sup>D<sup>2</sup>) Bias Correction

“…So, if the marginal (i.e., univariate) distributions are generally improved, that is closer to the reference ones -even when the BC is used as a preliminary step to downscaling (e.g., Colette et al, 2012;Vrac and Vaittinada Ayar, 2017) -, the inter-sites and inter-variables dependence structures are usually conserved from the climate model simulations to 25 be corrected. Indeed, 1d-BC methods preserving the ranks of the simulations -as it is the case for quantile-mapping approaches -will not correct the copula functions characterizing the dependencies between sites and/or between variables (e.g., Nelsen, 2006;Schoelzel and Friederichs, 2008;Vrac et al, 2011;Bevacqua et al, 2017). Such a preservation of the model dependence can obviously cause some deficiencies in the subsequent impact studies that will use the 1-dimensional bias corrected simulations, if the model copula function is far from that of the references.…”