Skip to main content
SpringerLink
Log in

The Relationship Between Graphical Representations of Regular Vine Copulas and Polytrees

  • Conference paper
  • First Online:
Information Processing and Management of Uncertainty in Knowledge-Based Systems. Applications (IPMU 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 855))

  • 1248 Accesses

Abstract

Graphical models (GMs) are powerful statistical tools for modeling the (in)dependencies among random variables. In this paper, we focus on two different types of graphical models: R-vines and polytrees. Regarding the graphical representation of these models, the former uses a sequence of undirected trees with edges representing pairwise dependencies, whereas the latter uses a directed graph without cycles to encode independence relationships among the variables. The research problem we deal with is whether it is possible to build an R-vine that represents the largest number of independencies found in a polytree and vice versa. Two algorithms are proposed to solve this problem. One algorithm is used to induce an R-vine that represents in each tree the largest number of graphical independencies existing in a polytree. The other one builds a polytree that represents all the independencies found in the R-vine. Through simple examples, both procedures are illustrated.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We assume that all multivariate, marginal and conditional distributions are absolutely continuous with corresponding densities.

References

  1. Aas, K., Czado, C., Frigessi, A., Bakken, H.: Pair-copula constructions of multiple dependence. Insur. Math. Econ. 44(2), 182–198 (2009)

    Article  MathSciNet  Google Scholar 

  2. Bauer, A., Czado, C.: Pair-copula Bayesian networks. arXiv:1211.5620 [stat.ME] (2012)

  3. Bedford, T., Cooke, R.M.: Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intell. 32(1), 245–268 (2001)

    Article  MathSciNet  Google Scholar 

  4. Bedford, T., Cooke, R.M.: Vines - a vew graphical model for dependent random variables. Ann. Stat. 30(4), 1031–1068 (2002)

    Article  Google Scholar 

  5. Brechmann, E.C., Czado, C., Aas, K.: Truncated regular vines in high dimensions with application to financial data. Can. J. Stat. 40(1), 68–85 (2012)

    Article  MathSciNet  Google Scholar 

  6. Castillo, E., Gutiérrez, J.M., Hadi, A.S.: Sistemas Expertos y Modelos de Redes Probabilísticas. Monografias de la Academia de Ingeniería (1996)

    Google Scholar 

  7. Cowell, G., Dawid, A.P., Lauritzen, S.L., Spiegelhalter, D.J.: Probabilistic Networks and Expert Systems: Exact Computational Methods for Bayesian Networks. Springer, New York (2003). https://doi.org/10.1007/b97670

    Book  MATH  Google Scholar 

  8. de Campos, L.M.: Independency relationships and learning algorithms for singly connected networks. J. Exper. Theor. Artif. Intell. 10(4), 511–549 (1998)

    Article  Google Scholar 

  9. Haff, I.H., Aas, K., Frigessi, A., Lacal, V.: Structure learning in Bayesian networks using regular vines. Comput. Stat. Data Anal. 101, 186–208 (2016)

    Article  MathSciNet  Google Scholar 

  10. Hanea, A.M.: Non-parameteric Bayesian belief nets versus vines. In: Joe, H., Kurowicka, D. (eds.) Dependence Modeling: Vine Copula Handbook, pp. 281–303. World Scientific Publishing (2011)

    Google Scholar 

  11. Joe, H.: Families of \(m\)-variate distributions with given margins and \(m(m-1)/2\) bivariate dependence parameters. In: Rüschendorf, L., Schweizer, B., Taylor, M.D. (eds.) Distributions with Fixed Marginals and Related Topics, pp. 120–141 (1996)

    Google Scholar 

  12. Kurowicka, D., Cooke, R.M.: Uncertainty Analysis with High Dimensional Dependence Modelling. Wiley, New York (2006)

    Book  Google Scholar 

  13. Lauritzen, L.: Graphical Models. Oxford University Press, Oxford (1996)

    MATH  Google Scholar 

  14. Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, New York (2006). https://doi.org/10.1007/0-387-28678-0

    Book  MATH  Google Scholar 

  15. Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan and Kaufmann, San Mateo (1998)

    MATH  Google Scholar 

  16. Prim, R.C.: Shortest connection networks and some generalizations. Bell Labs Tech. J. 36(6), 1389–1401 (1957)

    Article  Google Scholar 

  17. Sklar, A.: Fonctions de repartition à \(n\) dimensions et leurs marges. Publications de l’Institut de Statistique de l’Universite de Paris 8, 229–231 (1959)

    MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank Dr. Aritz Perez, of Basque Center for Applied Mathematics, BCAM, 48009 Bilbao, Spain, for valuable comments and suggestions. This work is partially supported by the Basque Government (IT609-13 and Elkartek), and Spanish Ministry of Science and Innovation (TIN2016-78365-R). Jose A. Lozano is also supported by BERC 2014–2017 and Elkartek programs (Basque government) and Severo Ochoa Program SEV-2013-0323 (Spanish Ministry of Economy and Competitiveness).

Author information

Authors and Affiliations

  1. Intelligent Systems Group, Department of Computer Science and Artificial Intelligence, University of the Basque Country (UPV/EHU), Manuel de Lardizabal, 1, 20018, Donostia, Gipuzkoa, Spain

    Diana Carrera, Roberto Santana & Jose A. Lozano

  2. Basque Center for Applied Mathematics, BCAM, 48009, Bilbao, Spain

    Jose A. Lozano

Authors
  1. Diana Carrera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roberto Santana
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jose A. Lozano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diana Carrera .

Editor information

Editors and Affiliations

  1. Universidad de Cádiz, Cádiz, Cadiz, Spain

    Jesús Medina

  2. Universidad de Málaga, Málaga, Málaga, Spain

    Manuel Ojeda-Aciego

  3. Universidad de Granada, Granada, Spain

    José Luis Verdegay

  4. Institute for Research and Applications, University of Ostrava, Ostrava, Czech Republic

    Irina Perfilieva

  5. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France

    Bernadette Bouchon-Meunier

  6. Iona College, New Rochelle, New York, USA

    Ronald R. Yager

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Carrera, D., Santana, R., Lozano, J.A. (2018). The Relationship Between Graphical Representations of Regular Vine Copulas and Polytrees. In: Medina, J., Ojeda-Aciego, M., Verdegay, J., Perfilieva, I., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. Applications. IPMU 2018. Communications in Computer and Information Science, vol 855. Springer, Cham. https://doi.org/10.1007/978-3-319-91479-4_56

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-91479-4_56

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-91478-7

  • Online ISBN: 978-3-319-91479-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation