Selection of sparse vine copulas in high dimensions with the Lasso

  • Dominik Müller
  • Claudia Czado


We propose a novel structure selection method for high-dimensional (\(d > 100\)) sparse vine copulas. Current sequential greedy approaches for structure selection require calculating spanning trees in hundreds of dimensions and fitting the pair copulas and their parameters iteratively throughout the structure selection process. Our method uses a connection between the vine and structural equation models. The later can be estimated very fast using the Lasso, also in very high dimensions, to obtain sparse models. Thus, we obtain a structure estimate independently of the chosen pair copulas and parameters. Additionally, we define the novel concept of regularization paths for R-vine matrices. It relates sparsity of the vine copula model in terms of independence copulas to a penalization coefficient in the structural equation models. We illustrate our approach and provide many numerical examples. These include simulations and data applications in high dimensions, showing the superiority of our approach to other existing methods.


Dependence modeling Vine copula Lasso Sparsity 



The first author acknowledges financial support by a research stipend of the Technische Universität München. The second author is supported by the German Research Foundation (DFG Grant CZ 86/4-1). Numerical calculations were performed on a Linux cluster supported by DFG Grant INST 95/919-1 FUGG.

Supplementary material


  1. Aas, K.: Pair-copula constructions for financial applications: a review. Econometrics 4(4), 43 (2016)MathSciNetCrossRefGoogle Scholar
  2. Aas, K., Czado, C., Frigessi, A., Bakken, H.: Pair-copula constructions of multiple dependence. Insur. Math. Econ. 44, 182–198 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Akaike, H.: Information theory and an extension of the maximum likelihood principle. In: Petrov, B.N., Csaki, F. (eds.) Proceedings of the Second International Symposium on Information Theory Budapest, pp. 267–281. Akademiai Kiado, Budapest (1973)Google Scholar
  4. Andersson, S.A., Perlman, M.D.: Normal linear regression models with recursive graphical markov structure. J. Multivar. Anal. 66, 133–187 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bauer, A., Czado, C.: Pair-Copula Bayesian networks. J. Comput. Graph. Stat. 25(4), 1248–1271 (2016). MathSciNetCrossRefGoogle Scholar
  6. Bedford, T., Cooke, R.: Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intell. 32, 245–268 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bedford, T., Cooke, R.: Vines—a new graphical model for dependent random variables. Ann. Stat. 30(4), 1031–1068 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bollen, K.A.: Structural Equations with Latent Variables, 1st edn. Wiley, Chicester (1989)zbMATHGoogle Scholar
  9. Brechmann, E., Czado, C., Aas, K.: Truncated regular vines in high dimensions with application to financial data. Can. J. Stat. 40, 68–85 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Brechmann, E.C., Joe, H.: Parsimonious parameterization of correlation matrices using truncated vines and factor analysis. Comput. Stat. Data Anal. 77, 233–251 (2014)MathSciNetCrossRefGoogle Scholar
  11. Brechmann, E.C., Schepsmeier, U.: Modeling dependence with C- and D-vine copulas: the R package CDVine. J. Stat. Softw. 52(3), 1–27 (2013),
  12. Dißmann, J., Brechmann, E., Czado, C., Kurowicka, D.: Selecting and estimating regular vine copulae and application to financial returns. Comput. Stat. Data Anal. 52(1), 52–59 (2013)MathSciNetCrossRefGoogle Scholar
  13. Friedman, J., Hastie, T., Tibshirani, R.: Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9(3), 432 (2008).
  14. Friedman, J., Hastie, T., Tibshirani, R.: Regularization paths for generalized linear models via coordinate descent. J. Stat. Softw. 33(1), 1–22 (2010),
  15. Frommlet F, Chakrabarti A, Murawska M, Bogdan M (2011) Asymptotic Bayes optimality under sparsity for generally distributed effect sizes under the alternative. Technical report, arXiv:1005.4753
  16. Gruber L, Czado C (2015a) Bayesian model selection of regular vine copulas. Preprint
  17. Gruber, L., Czado, C.: Sequential bayesian model selection of regular vine copulas. Bayesian Anal. 10, 937–963 (2015b)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hastie, T., Tibshirani, R., Wainwright, M.: Statistical Learning with Sparsity The Lasso and Generalizations. CRC Press, Boca Raton (2015)zbMATHGoogle Scholar
  19. Hoyle, R.H.: Structural Equation Modeling, 1st edn. SAGE Publications, Thousand Oaks (1995)Google Scholar
  20. Joe, H.: Dependence Modeling with Copulas. Chapman & Hall/ CRC, London (2014)zbMATHGoogle Scholar
  21. Kaplan, D.: Structural Equation Modeling: Foundations and Extensions, 2nd edn. SAGE Publications, Thousand Oaks (2009)CrossRefzbMATHGoogle Scholar
  22. Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques, 1st edn. MIT Press, Cambridge, Massachusetts (2009)zbMATHGoogle Scholar
  23. Kurowicka, D., Cooke, R.: Uncertainty Analysis and High Dimensional Dependence Modelling, 1st edn. Wiley, Chicester (2006)CrossRefzbMATHGoogle Scholar
  24. Kurowicka, D., Joe, H.: Dependence Modeling—Handbook on Vine Copulae. World Scientific Publishing Co., Singapore (2011)Google Scholar
  25. Meinshausen, N., Bühlmann, P.: High-dimensional graphs and variable selection with the Lasso. Ann. Statist. 34(3), 1436–1462 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  26. Müller, D., Czado, C.: Representing sparse Gaussian DAGs as sparse R-vines allowing for non-Gaussian dependence. J. Comput. Graph. Stat. (2017).
  27. Schepsmeier U, Stöber J, Brechmann EC, Graeler B, Nagler T, Erhardt T (2016) VineCopula: Statistical Inference of Vine Copulas., r package version 2.0.6
  28. Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6(2), 461–464 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Sklar, A.: Fonctions dé repartition á n dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris 8, 229–231 (1959)zbMATHGoogle Scholar
  30. Stöber, J., Joe, H., Czado, C.: Simplified pair copula constructions-limitations and extensions. J. Multivar. Anal. 119, 101–118 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  31. Tibshirani, R.: Regression Shrinkage and Selection Via the Lasso. J. R. Stat. Soc. B 58, 267–288 (1994)MathSciNetzbMATHGoogle Scholar
  32. Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 67(2), 301–320 (2005). MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTechnical University of MunichGarchingGermany

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