Multivariate statistical models can be simplified by assuming that a pattern of conditional independence is presented in the given data. A popular way of capturing the (conditional) independence is to use probabilistic graphical models. The relationship between strongly chordal graphs and m-saturated vines is proved. Moreover, an algorithm to construct an m-saturated vine structure corresponding to strongly chordal graph is provided. This allows the reduction of regular vine copula models complexity. When the underlying data is sparse our approach leads to model estimation improvement when compared with current heuristic methods. Furthermore, due to reduction of model complexity it is possible to evaluate all vine structures as well as to fit non-simplified vines. These advantages have been shown in the simulated and real data examples.¹

An extension of the D-vine based forward regression procedure to a R-vine forward regression is proposed. In this extension any R-vine structure can be taken into account. Moreover, a new heuristic is proposed to determine which R-vine structure is the most appropriate to model the conditional distribution of the response variable given the covariates. It is shown in the simulation that the performance of the heuristic is comparable to the D-vine based approach. Furthermore, it is explained how to extend the heuristic into a situation when more than one response variable are of interest. Finally, the proposed R-vine regression is applied to perform a stress analysis on the manufacturing sector which shows its impact on the whole economy.

The selection of vine structure to represent dependencies in a data set with a regular vine copula model is still an open question. Up to date, the most popular heuristic to choose the vine structure is to construct consecutive trees by capturing largest correlations in lower trees. However, this might not lead to the optimal vine structure. A new heuristic based on sampling orders implied by regular vines is investigated. The idea is to start with an initial vine structure, that can be chosen with any existing procedure and search for a regular vine copula representing the data better within vines having 2 common sampling orders with this structure. Several algorithms are proposed to support the new heuristic. Both in the simulation study and real data analysis, the potential of the new heuristic to find a structure fitting the data better than the initial vine copula model, is shown.