*4.1. Marginal Probability Distribution Functions of C-Vine Model Variables*

A two-step approach that separately evaluates the dependence function and the marginals is of great advantage in stochastic modeling of multivariate data, since many manageable distribution models are available for simulating the marginal distributions. In this study, in order to build the CVQR model, firstly, after standardization, the data are fitted with some parametric distribution functions, including the gamma, lognormal, general extreme value (GEV), and Pearson type-III (P-III) distributions, which are commonly used parameter distributions to quantify the probability distribution characteristics of hydrometeorological variables in the hydrological process [62–64]. The expressions for the gamma, GEV, lognormal, P-III, and the associated parameter values for probability functions (PDFs) are shown in Table 2. The parameters of the above distributions were obtained through the Maximum Likelihood Estimation (MLE) method.


**Table 2.** Parameters of optimal marginal distribution functions.

Note: 32.12 \*, 32.34 \* indicate for −32.12 and −32.34, respectively; ∗∗ *m* = 1 + *ξ x*−*µ σ* −1/*<sup>ξ</sup>* ; ∗ ∗ ∗ Γ(*α*) = R∞ 0 *u α*−1 *e* <sup>−</sup>*udu*. 18

The goodness-of-fit (GOF) of each distribution was computed by using RMSE and AIC values to select the most appropriate distribution for fitting each individual variable. The results of GOF are presented in Table 3. The results demonstrate that all of the proposed four distribution models can be applied for processing the distributions of the variables (i.e., St-1, Pt-1, St-2, St-12, Tt, Pt, and St), except that the P-III distribution is not suitable for the average temperature (Tt). Specially, the P-III distribution are most suitable for the streamflow data series (i.e., St-1, St-2, St-12, and St), the Gamma distribution would perform best when fitting the distributions of precipitation data (Pt-1 and Pt), and the GEV method has advantages in quantifying the distributions of the average temperature (Tt).



Note: The RMSE and AIC values of the optimal fitting distribution are shown in bold.
