*4.4. Probabilistic and Interval Predictions Obtained by the CVQR Model*

As mentioned in Section 2.3, according to the C-vine copula-based quantile regression (CVQR) model, for any quantile *τ* ∈ (0, 1), the τth conditional quantile function of the predicted variable can be obtained. In this section, the relationships between the streamflow (St) abnormalities and other hydrometeorological indices at different levels of quantiles τ (i.e., τ = 0.05, 0.25, 0.50, 0.75, and 0.95) are explored.

The median prediction (i.e., α = 0.5) provides a general level about the monthly streamflow, while extreme values (e.g., flood, drought) in the upper tail (τ ≥ 0.75) or lower tail (τ ≤ 0.25) indicate the worst forecast scenarios. Table 6 describes the relative performance of the ANN model with respect to the CVQR model at different quantiles. It can be seen that the proposed CVQR model outperforms the ANN model at quantiles τ = 0.75 and 0.95 and that the ANN model performs better than the CVQR model at quantiles τ = 0.25 and 0.50, which indicate that the proposed CVQR model could perform better at upper extreme events (i.e., τ = 0.75 and 0.95 quantile levels) and that the ANN model provides good results in some cases of the mean and lower quantile values.


**Table 6.** The performance RRMSE and RMAE of the ANN model with respect to the CVQR model at different quantiles.

A scatter diagram of the simulated streamflow at different quantiles (τ = 0.05, 0.25, 0.5, 0.75, and 0.95) by the ANN and CVQR models with five-fold K cross-validations is depicted in Figure 8. The results also show that the proposed CVQR model performs a better fit in most cases, especially in the process of upper tail predictions, which are consistent with the earlier study of Kong in Xiangxi River basin [51]. While the ANN tends to overfit overestimated in the aspect of upper tail prediction. In general, the CVQR model shows a higher accuracy at upper tail levels while the ANN model provides overestimation predictions. The results indicate that the CVQR model can effectively capture upper tail dependences and has a relatively accurate assessment of the impact of upper extreme conditions (i.e., flood) in Xiangxi watershed.

models.

conditions (i.e., flood) in Xiangxi watershed.

**Figure 8.** Scatter diagram of predicted and observed monthly streamflow using the CVQR and ANN models and their corresponding fitting lines at different quantiles (*τ* = 0.05, 0.25, 0.5, 0.75, and 0.95) with the five-fold K cross-validation **Figure 8.** Scatter diagram of predicted and observed monthly streamflow using the CVQR and ANN models and their corresponding fitting lines at different quantiles (*τ* = 0.05, 0.25, 0.5, 0.75, and 0.95) with the five-fold K cross-validation models.

Figure 9 depicts the simulated streamflow with quantiles of 5% and 95% (90% uncertainty prediction intervals) using the ANN and CVQR models. The results indicate that the quantiles τ = 5% and 95% values of the predicted variable cover most of the observations and effectively reflect the fluctuation of the actual streamflow for the two models. Usually, hydrological forecasting in extreme cases can help policy makers make timely policy responses within the maximum risk range. The predicted 90% CI can reflect the fluctuation trend and abnormal value of the records well, whereas compared with the CVQR model, the ANN model often overestimates peaks in the prediction of flood events. Therefore, the CVQR model can effectively capture the complex nonlinear dependences among hydrological meteorological factors. This is of great significance to the practice of water resource management, for example, in rainy and dry seasons, managers Figure 9 depicts the simulated streamflow with quantiles of 5% and 95% (90% uncertainty prediction intervals) using the ANN and CVQR models. The results indicate that the quantiles τ = 5% and 95% values of the predicted variable cover most of the observations and effectively reflect the fluctuation of the actual streamflow for the two models. Usually, hydrological forecasting in extreme cases can help policy makers make timely policy responses within the maximum risk range. The predicted 90% CI can reflect the fluctuation trend and abnormal value of the records well, whereas compared with the CVQR model, the ANN model often overestimates peaks in the prediction of flood events. Therefore, the CVQR model can effectively capture the complex nonlinear dependences among hydrological meteorological factors. This is of great significance to the practice of water resource management, for example, in rainy and dry seasons, managers can well prevent and control the occurrence of flood and make timely corresponding countermeasures.

can well prevent and control the occurrence of flood and make timely corresponding

tail dependences and has a relatively accurate assessment of the impact of upper extreme

countermeasures.

**Figure 9.** Comparison of the predicted and observed monthly streamflow using the CVQR and ANN models with *τ* = 5% and 95% (90% uncertainty prediction intervals). **Figure 9.** Comparison of the predicted and observed monthly streamflow using the CVQR and ANN models with *τ* = 5% and 95% (90% uncertainty prediction intervals).

#### **5. Conclusions 5. Conclusions**

In this study, a C-vine copula-based quantile regression (CVQR) model was developed to model the relationship between streamflow and other hydrometeorological variables, such as temperature and precipitation. The proposed CVQR model couples vine copulas (known as pair copula constructions) with a quantile regression method, which was applied to monthly streamflow forecasting in the Xiangxi River basin. In this study, a C-vine copula-based quantile regression (CVQR) model was developed to model the relationship between streamflow and other hydrometeorological variables, such as temperature and precipitation. The proposed CVQR model couples vine copulas (known as pair copula constructions) with a quantile regression method, which was applied to monthly streamflow forecasting in the Xiangxi River basin.

Specifically, the CVQR model could process multidimensional data problems while satisfying the wide range of dependence. Meanwhile, the CVQR model can effectively capture the upper correlations between independent and dependent variables (i.e., flood events). In this paper, comparisons between the proposed CVQR model and the MLR and ANN models for monthly streamflow prediction are explored. The results indicate that the performance of the CVQR model is most effective for monthly streamflow forecasting in the calibration period. The performance of the MLR model in extreme quantile (flood events) and confidence intervals is the worst and is mainly determined by the inherent characteristics of the algorithm. Compared with the MLR model, the ANN model has good advantages in this aspect of flood events and confidence intervals, but it tends to be over-fit in the process of peaks prediction. Undeniably, the CVQR model can effectively capture both the linear and nonlinear dependence of these input variables and to perform best when dealing with upper tail correlation issues (i.e., flood events) in this Specifically, the CVQR model could process multidimensional data problems while satisfying the wide range of dependence. Meanwhile, the CVQR model can effectively capture the upper correlations between independent and dependent variables (i.e., flood events). In this paper, comparisons between the proposed CVQR model and the MLR and ANN models for monthly streamflow prediction are explored. The results indicate that the performance of the CVQR model is most effective for monthly streamflow forecasting in the calibration period. The performance of the MLR model in extreme quantile (flood events) and confidence intervals is the worst and is mainly determined by the inherent characteristics of the algorithm. Compared with the MLR model, the ANN model has good advantages in this aspect of flood events and confidence intervals, but it tends to be over-fit in the process of peaks prediction. Undeniably, the CVQR model can effectively capture both the linear and nonlinear dependence of these input variables and to perform best when dealing with upper tail correlation issues (i.e., flood events) in this study.

study. In summary, this proposed method can effectually depict the complicated dependencies between the hydrometeorological variables. However, there still remain some In summary, this proposed method can effectually depict the complicated dependencies between the hydrometeorological variables. However, there still remain some flaws in the process of model building. Pair-copula is joined by marginal distributions irrespective

of the conditional variables, which simplifies the construction of vine copulas [65]. The structure of PCCs is often not unique due to the flexibility of vine copulas [66]. Moreover, the proposed model can be used to explore temporal and spatial dependencies among hydrological series while spatial dependence is not considered in this study [67]. Consequently, the model will be explored further in the application process of future extensions.

**Author Contributions:** Conceptualization, H.L.; methodology, H.L.; software, H.L.; validation, P.G. and J.S.; formal analysis, H.L.; investigation, P.G.; resources, G.H. and Y.L.; data curation, J.S.; writing—original draft preparation, H.L.; writing—review and editing, J.S. and Y.L.; visualization, H.L.; supervision, G.H. and Y.L.; project administration, G.H. and Y.L.; funding acquisition, G.H. and Y.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant Number: XDA20060302).

**Data Availability Statement:** Publicly available datasets were analyzed in this study. The data presented in this study are available at http://data.cma.cn/, accessed on 20 January 2021.

**Acknowledgments:** The authors gratefully acknowledge all the reviewers and editors for their insightful comments.

**Conflicts of Interest:** The authors declare no conflict of interest.
