Article
Full entry |
PDF
(1.7 MB)
Feedback
PDF
(1.7 MB)
Feedback
Keywords:
copula; conditional independences; Regular-vine; truncated vine; cherry-tree copula
copula; conditional independences; Regular-vine; truncated vine; cherry-tree copula
Summary:
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.
References:
[1] Aas, K., Czado, C., Frigessi, A., Bakken, H.: Pair-copula constructions of multiple dependence. Insur. Math. Econom. 44 (2009), 182-198. DOI 10.1016/j.insmatheco.2007.02.001 | MR 2517884
[2] Acar, E. F., Genest, C., Nešlehová, J.: Beyond simplified pair-copula constructions. J. Multivariate Anal. 110 (2012), 74-90. DOI 10.1016/j.jmva.2012.02.001 | MR 2927510
[3] Bauer, A., Czado, C., Klein, T.: Pair-copula construction for non-Gaussian DAG models. Canad. J. Stat. 40 (2012), 1, 86-109. DOI 10.1002/cjs.10131 | MR 2896932
[4] Bedford, T., Cooke, R.: Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intell. 32 (2001), 245-268. DOI 10.1023/a:1016725902970 | MR 1859866
[5] Bedford, T., Cooke, R.: Vines - a new graphical model for dependent random variables. Ann. Statist. 30 (2002), 4, 1031-1068. DOI 10.1214/aos/1031689016 | MR 1926167
[6] Brechmann, E. C., Czado, C., Aas, K.: Truncated regular vines in high dimensions with applications to financial data. Canad. J. Statist. 40 (2012), 1, 68-85. DOI 10.1002/cjs.10141 | MR 2896931
[7] Bukszár, J., Prékopa, A.: Probability bounds with cherry trees. Math. Oper. Res. 26 (2001), 174-192. DOI 10.1287/moor.26.1.174.10596 | MR 1821836
[8] Bukszár, J., Szántai, T.: Probability bounds given by hypercherry trees. Optim. Methods Software 17 (2002), 409-422. DOI 10.1080/1055678021000033955 | MR 1944289
[9] Cover, T. M., Thomas, J. A.: Elements of Information Theory. Wiley Interscience, New York 1991. DOI 10.1002/0471200611 | MR 1122806
[10] Czado, C.: 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 | MR 3051264
[11] Dissman, J., Brechmann, E. C., Czado, C., Kurowicka, D.: 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 | MR 3000041
[12] Hanea, A., Kurowicka, D., Cooke, R.: Hybrid method for quantifying and analyzing Bayesian belief networks. Qual. Reliab. Engrg. 22 (2006), 708-729. DOI 10.1002/qre.808
[13] Haff, I. Hobaek, Aas, K., Frigessi, A.: 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 | MR 2595309
[14] Haff, I. Hobaek, Segers, J.: Nonparametric estimation of pair-copula constructions with the empirical pair-copula. 2010. DOI
[15] Hobaek-Haff, I., Aas, K., Frigessi, A., Lacal, V.: Structure learning in Bayesian Networks using regular vines. Computat. Statist. Data Anal. 101 (2016), 186-208. DOI 10.1016/j.csda.2016.03.003 | MR 3504845
[16] Joe, H.: Multivariate Models and Dependence Concepts. Chapman and Hall, London 1997. DOI 10.1201/b13150 | MR 1462613 | Zbl 0990.62517
[17] Kovács, E., Szántai, T.: 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 | MR 2681735
[18] Kovács, E., Szántai, T.: Multivariate copula expressed by lower dimensional copulas. 2010. DOI
[19] Kovács, E., Szántai, T.: 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.
[20] Kovács, E., Szántai, T.: Vine copulas as a mean for the construction of high dimensional probability distribution associated to a Markov network. 2012(b). DOI
[21] Kurowicka, D., Cooke, R.: 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] Kurowicka, D., Cooke, R. M.: Uncertainty Analysis with High Dimensional Dependence Modelling. John Wiley, Chichester 2006. DOI 10.1002/0470863072 | MR 2216540
[23] Kurowicka, D.: Optimal truncation of vines. In: Dependence-Modeling - Handbook on Vine Copulas (D. Kurowicka and H. Joe, eds.), Word Scientific Publishing, Singapore 2011. MR 2856976
[24] Lauritzen, S. L., Spiegelhalter, D. J.: Local Computations with probabilites on graphical structures and their application to expert systems. J. Roy. Statist. Soc. B 50 (1988), 157-227. MR 0964177
[25] Lauritzen, S. L.: Graphical Models. Clarendon Press, Oxford 1996. MR 1419991
[26] Szántai, T., Kovács, E.: 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 | MR 2874757
[27] Whittaker, J.: Graphical Models in Applied Multivariate Statistics. John Wiley and Sons, 1990. MR 1112133
Search
Partner of
