Contributions to Vine-Copula Modeling

Abstract

Regular vine-copula models (R-vines) are a powerful statistical tool for modeling thedependence structure of multivariate distribution functions. In particular, they allow modelingdierent types of dependencies among random variables independently of their marginaldistributions, which is deemed the most valued characteristic of these models. In this thesis, weinvestigate the theoretical properties of R-vines for representing dependencies and extend theiruse to solve supervised classication problems. We focus on three research directions.!In the rst line of research, the relationship between the graphical representations of R-vines!ÁREA LÍNEA1 2 0 3 0 4ÁREA LÍNEA1 2 0 3 1 7ÁREA LÍNEAÁREA LÍNEA!and Bayesian polytree networks is analyzed in terms of how conditional pairwise independence!relationships are represented by both models. In order to do that, we use an extended graphical!representation of R-vines in which the R-vine graph is endowed with further expressiveness,being possible to distinguish between edges representing independence and dependencerelationships. Using this representation, a separation criterion in the R-vine graph, called Rseparation,is dened. The proposed criterion is used in designing methods for building thegraphical structure of polytrees from that of R-vines, and vice versa. Moreover, possiblecorrespondences between the R-vine graph and the associated R-vine copula as well as dierentproperties of R-separation are analyzed. In the second research line, we design methods forlearning the graphical structure of R-vines from dependence lists. The main challenge of thistask lies in the extremely large size of the search space of all possible R-vine structures. Weprovide two strategies to solve the problem of learning R-vines that represent the largestnumber of dependencies in a list. The rst approach is a 0 -1 linear programming formulation forbuilding truncated R-vines with only two trees. The second approach is an evolutionaryalgorithm, which is able to learn complete and truncated R-vines. Experimental results show thesuccess of this strategy in solving the optimization problem posed. In the third research line, weintroduce a supervised classication approach where the dependence structure of the problemfeatures is modeled through R-vines. The ecacy of these classiers is validated in a mentaldecoding problem and in an image recognition task. While Rvines have been extensivelyapplied in elds such as economics, nance and statistics, only recently have they found theirplace in classication tasks. This contribution represents a step forward in understanding R-vinesand the prospect of extending their use to other machine learning tasks.