2.3.1. Copula Function

The general expression of bivariate copulas can be written as follows:

$$H(\mathbf{x}, \mathbf{y}) = \mathbb{C}\left(\mathbf{u}\_{\mathbf{x}}, \mathbf{u}\_{\mathbf{y}}; \boldsymbol{\theta}\right) \tag{3}$$

where (*x*, *y*) are correlated random variables. *θ* can often be derived from Kendall's τ as a preliminary estimation, and (*ux*, *uy*) are the marginal cumulative distribution functions of *x* and *y*, respectively. Kendall's τ is the rank correlation coefficient proposed by Kendall [41]. Let (*x*1, *y*1), (*x*2, *y*2), ..., (*x*n, *y*n) be a set of observations of the joint random variables X and Y, respectively, and empirical Kendall's τ can be defined as *τ* = 2(*C<sup>n</sup>* − *Dn*)/*n*(*n* − 1), where *C<sup>n</sup>* and *D<sup>n</sup>* indicate the number of concordant and discordant pairs, respectively.

**Figure 2.** Examples of 5-dimensional C-vine (a) and D-vine (b). **Figure 2.** Examples of 5-dimensional C-vine (**a**) and D-vine (**b**).

2.3.1. Copula Function The general expression of bivariate copulas can be written as follows: *H xy Cu u* ( ) , ,; = ( ) *x y* θ (3) A d-dimensional copula C: [0, 1] d → [0, 1] with uniformly distributed marginals U (0, 1) on the interval [0, 1] was introduced by Sklar [42]. According to Sklar's theorem, every joint cumulative distribution function (CDF) *F* on *R <sup>d</sup>* with marginals *F*1(*x*1), *F*2(*x*2), . . . , *Fd*(*xd*) can be written as follows:

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.

$$F(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_d) = \mathbb{C}(F\_1(\mathbf{x}\_1), F\_2(\mathbf{x}\_2), \dots, F\_d(\mathbf{x}\_d)), \ \forall \mathbf{x} = (\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_d) \in \mathbb{R}^d \tag{4}$$

*x* and *y*, respectively. Kendall's τ is the rank correlation coefficient proposed by Kendall [41]. Let (*x*1, *y*1), (*x*2, *y*2), ..., (*x*n, *y*n) be a set of observations of the joint random variables X and Y, respectively, and empirical Kendall's τ can be defined as Similarly, the multivariate density *f*(*x*1, *x*2, . . . , *xd*) with marginal densities*f*1(*x*1), *f*2(*x*2), . . . , *fd*(*xd*) and join probability density of copula *c* (*u*1, *u*2, . . . , *u*d) can be written as follows:

preliminary estimation, and (*ux*, *uy*) are the marginal cumulative distribution functions of

$$f(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_d) = \left[ \prod\_{i=1}^d f\_i(\mathbf{x}\_i) \right] \mathbf{c}(\boldsymbol{\mu}\_1, \boldsymbol{\mu}\_2, \dots, \boldsymbol{\mu}\_d), \ \forall \mathbf{x} = (\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_d) \in \mathbb{R}^d \tag{5}$$

(0, 1) on the interval [0, 1] was introduced by Sklar [42]. According to Sklar's theorem, and vice versa:

τ

$$\mathbb{C}(\boldsymbol{u}\_1, \boldsymbol{u}\_2, \dots, \boldsymbol{u}\_d) = \mathbb{F}\left(\mathbb{F}\_1^{-1}(\boldsymbol{u}\_1), \mathbb{F}\_2^{-1}(\boldsymbol{u}\_2), \dots, \mathbb{F}\_d^{-1}(\boldsymbol{u}\_d)\right), \forall \boldsymbol{u} = (\boldsymbol{u}\_1, \boldsymbol{u}\_2, \dots, \boldsymbol{u}\_d) \in (0, 1) \tag{6}$$

*Fxx x CF x F x F x xx x* ( ) () () ( ) 12 1 1 2 2 12 , , ..., , , ..., , ,..., *d dd n* = ∀= ( ), *x* ( )∈ (4) where *u<sup>i</sup>* = *Fi*(*xi*), (*i* = 1, 2, . . . , *d*), and *F* −1 1 (*u*1), *F* −1 2 (*u*2), . . . , *F* −1 *d* (*ud*) are the inverse distribution functions of the marginals.
