*2.3. Development of C-Vine Copula-Based Quantile Regression (CVQR) Model*

In general, vine copulas are represented using a graph called R-vine, which consists of a series of trees (undirected acyclic graphs) [39]. Specially, the hierarchical structure, called a regular vine (R-vine), contains a series of connected trees *T* := (*T*1, *T*2, . . . , *Td*) along with the series of edges *<sup>E</sup>*(*T*) := *<sup>E</sup>*<sup>1</sup> ∪ *<sup>E</sup>*<sup>2</sup> ∪ . . . ∪ *<sup>E</sup>d*−<sup>1</sup> and the series of nodes *<sup>N</sup>*(*T*) := *N*<sup>1</sup> ∪ *N*<sup>2</sup> ∪ . . . ∪ *Nd*−<sup>1</sup> . However, regular vines in terms of pair-copulas are still very general and do not have unique decomposition. Thus, the canonical vine (C-vine) and the D-vine are two most common structures of regular vines [40]. C-vine has a stellar structure in their tree sequence, while D-vine has a path structure. In hydrological field in this study, the monthly streamflow is affected by various climatic and hydrological factors. Therefore, the runoff factor that has a strong dependence on all other variables is selected as the first root for C-vine construction instead of D-vines. Here, two five-dimensional examples of possible tree sequences are shown in Figure 2.
