Optimal Truncation of Vines

The vine copulae representation is a very flexible model of n-dimensional random vectors. However when n becomes too large, some simplifying assumptions have to be made as fitting this model becomes too cumbersome. In this chapter truncations of vines are discussed. We could reduce a vine to a Markov tree structure but Markov trees allow n −1 copulae to be specified out of $(^{n}_{2})$ possible for vines. Hence trees may be too restrictive for a given set of data. Another possibility would be to model subsets of variables with vines and connect these smaller vines in a tree structure. We suggest one more strategy of choosing the “most suitable” vine for the correlation matrix. The “best vine” is the one whose nodes of top trees (tree with the most conditioning) correspond to the smallest absolute values of partial correlations. To search for the “best vine” we developed a new algorithm of generating a regular vine. We start building the vine from the top node (node in tree n−1) and progress to the lower trees, ensuring that the regularity condition is satisfied and that the partial correlations corresponding to these nodes are the smallest. If we assume that we can assign the independent copula to nodes of the vine with small absolute values of partial correlations, then this algorithm can be used to find an optimal truncation of a vine structure. We advocate using it as a preprocessing step of fitting a vine to the data.