# Abstract:

Vine copulas are a flexible way for modeling dependences using only pair-copulas as building blocks. However if the number of variables grows the problem gets fastly intractable. For dealing with this problem Brechmann at al. proposed the truncated R-vine copulas. The truncated R-vine copula has the very useful property that it can be constructed by using only pair-copulas and a lower number of conditional pair-copulas. In our earlier papers we introduced the concept of cherry-tree copulas. In this paper we characterize the relation between cherry-tree copulas and truncated R-vine copulas. It turns out that the concept of cherry-tree copula is more general than the concept of truncated R-vine copula. Although both contain in their expressions conditional independences between the variables, the truncated R-vines constructed in greedy way do not exploit the existing conditional independences in the data. We give a necessary and sufficient condition for a cherry-tree copula to be a truncated R-vine copula. We introduce a new method for truncated R-vine modeling. The new idea is that in the first step we construct the top tree by exploiting conditional independences for finding a good-fitting cherry-tree of order $k$. If this top tree is a tree in an R-vine structure then this will define a truncated R-vine at level $k$ and in the second step we construct a sequence of trees which leads to it. If this top tree is not a tree in an R-vine structure then we can transform it into such a tree at level $k+1$ and then we can again apply the second step. The second step is performed by a backward construction named Backward Algorithm. This way the cherry-tree copulas always can be expressed by pair-copulas and conditional pair-copulas.

# Keywords:

copula, conditional independences, Regular-vine, truncated vine, cherry-tree copula

60C05, 62H05

# References:

1. K. Aas, C. Czado, A. Frigessi and H. Bakken: Pair-copula constructions of multiple dependence. Insur. Math. Econom. 44 (2009), 182-198.   DOI:10.1016/j.insmatheco.2007.02.001
2. E. F. Acar, C. Genest and J. Nešlehová: Beyond simplified pair-copula constructions. J. Multivariate Anal. 110 (2012), 74-90.   DOI:10.1016/j.jmva.2012.02.001
3. A. Bauer, C. Czado and T. Klein: Pair-copula construction for non-Gaussian DAG models. Canad. J. Stat. 40 (2012), 1, 86-109.   DOI:10.1002/cjs.10131
4. T. Bedford and R. Cooke: Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intell. 32 (2001), 245-268.   DOI:10.1023/a:1016725902970
5. T. Bedford and R. Cooke: Vines - a new graphical model for dependent random variables. Ann. Statist. 30 (2002), 4, 1031-1068.   DOI:10.1214/aos/1031689016
6. E. C. Brechmann, C. Czado and K. Aas: Truncated regular vines in high dimensions with applications to financial data. Canad. J. Statist. 40 (2012), 1, 68-85.   DOI:10.1002/cjs.10141
7. J. Bukszár and A. Prékopa: Probability bounds with cherry trees. Math. Oper. Res. 26 (2001), 174-192.   DOI:10.1287/moor.26.1.174.10596
8. J. Bukszár and T. Szántai: Probability bounds given by hypercherry trees. Optim. Methods Software 17 (2002), 409-422.   DOI:10.1080/1055678021000033955
9. T. M. Cover and J. A. Thomas: Elements of Information Theory. Wiley Interscience, New York 1991.   DOI:10.1002/0471200611
10. C. Czado: Pair-copula constructions of multivariate copulas. In: Copula Theory and Its Applications (P. Jaworski, F. Durante, W. Härdle, and T. Rychlik, eds.), Springer, Berlin 2010.   DOI:10.1007/978-3-642-12465-5_4
11. J. Dissman, E. C. Brechmann, C. Czado and D. Kurowicka: Selecting and estimating regular vine copulae and application to financial returns. Comput. Statist. Data Anal. 59 (2013), 52-69.   DOI:10.1016/j.csda.2012.08.010
12. A. Hanea, D. Kurowicka and R. Cooke: Hybrid method for quantifying and analyzing Bayesian belief networks. Qual. Reliab. Engrg. 22 (2006), 708-729.   DOI:10.1002/qre.808
13. I. Hobaek Haff, K. Aas and A. Frigessi: On the simplified pair-copula construction - simply useful or too simplistic? J. Multivariate Anal. 101 (2010), 5, 1296-1310.   DOI:10.1016/j.jmva.2009.12.001
14. I. Hobaek Haff and J. Segers: Nonparametric estimation of pair-copula constructions with the empirical pair-copula.
15. I. Hobaek-Haff, K. Aas, A. Frigessi and V. Lacal: Structure learning in Bayesian Networks using regular vines. Computat. Statist. Data Anal. 101 (2016), 186-208.   DOI:10.1016/j.csda.2016.03.003
16. H. Joe: Multivariate Models and Dependence Concepts. Chapman and Hall, London 1997.   DOI:10.1201/b13150
17. E. Kovács and T. Szántai: On the approximation of discrete multivariate probability distribution using the new concept of $t$-cherry junction tree. Lect. Notes Economics Math. Systems 633, Proc. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty, Robust Solutions, 2008, IIASA, Laxenburg 2010, pp. 39-56.   DOI:10.1007/978-3-642-03735-1_3
18. E. Kovács and T. Szántai: Multivariate copula expressed by lower dimensional copulas. 2010.   arXiv:1009.2898
19. E. Kovács and T. Szántai: Hypergraphs in the characterization of regular-vine copula structures. In: Proc. 13th International Conference on Mathematics and its Applications, Timisoara 2012(a), pp. 335-344.   CrossRef
20. E. Kovács and T. Szántai: Vine copulas as a mean for the construction of high dimensional probability distribution associated to a Markov network. 2012(b).   arXiv:1105.1697
21. D. Kurowicka and R. Cooke: The vine copula method for representing high dimensional dependent distributions: Application to continuous belief nets. In: Proc. 2002 Winter Simulation Conference 2002, pp. 270-278.   DOI:10.1109/wsc.2002.1172895
22. D. Kurowicka and R. M. Cooke: Uncertainty Analysis with High Dimensional Dependence Modelling. John Wiley, Chichester 2006.   DOI:10.1002/0470863072
23. D. Kurowicka: Optimal truncation of vines. In: Dependence-Modeling - Handbook on Vine Copulas (D. Kurowicka and H. Joe, eds.), Word Scientific Publishing, Singapore 2011.   CrossRef
24. S. L. Lauritzen and D. J. Spiegelhalter: Local Computations with probabilites on graphical structures and their application to expert systems. J. Roy. Statist. Soc. B 50 (1988), 157-227.   CrossRef
25. S. L. Lauritzen: Graphical Models. Clarendon Press, Oxford 1996.   CrossRef
26. T. Szántai and E. Kovács: Hypergraphs as a mean of discovering the dependence structure of a discrete multivariate probability distribution. In: Proc. Conference Applied Mathematical Programming and Modelling (APMOD), Bratislava 2008, Ann. Oper. Res. 193 (2012), 1, 71-90.   DOI:10.1007/s10479-010-0814-y
27. J. Whittaker: Graphical Models in Applied Multivariate Statistics. John Wiley and Sons, 1990.   CrossRef