Predictive Uncertainty Estimation of Hydrological Multi-Model Ensembles Using Pair-Copula Construction

Abstract

Predictive uncertainty (PU) is defined as the probability of occurrence of an observed variable of interest, conditional on all available information. In this context, hydrological model predictions and forecasts are considered to be accessible but yet uncertain information. To estimate the PU of hydrological multi-model ensembles, we apply a method based on the use of copulas which enables modelling the dependency structures between variates independently of their marginal distributions. Given that the option to employ copula functions imposes certain limitations in the multivariate case, we model the multivariate distribution as a cascade of bivariate copulas by using the pair-copula construction. We apply a mixture of probability distributions to estimate the marginal densities and distributions of daily flow rates for various meteorological and hydrological situations. The proposed method is applied to a multi-model ensemble involving two hydrological and one statistical flow models at two gauge stations in the Moselle river basin. Verification and inter-comparison with other PU assessment methods show that copulas are well-suited for this scope and constitute a valid approach for predictive uncertainty estimation of hydrological multi-model predictions.

Statistical Analysis of Spatiotemporal Heterogeneity of the Distribution of Air Quality and Dominant Air Pollutants and the Effect Factors in Qingdao Urban Zones

“…Compared to linear models, the copula model can be applied to analyze nonlinear correlation between variables with the following advantages [52][53][54][55]: (i) The copula model captures abnormal information by visually displaying the tail features of the variable distribution, (ii) the copula model is suitable for variables obeying any type of distribution, and (iii) the copula model is powerful for analyzing the nonlinear correlation between variables. Recently, the copula model was applied with satisfactory results in geoscience, hydrology, finance, and other fields [56][57][58][59][60][61][62]. Therefore, the copula model was introduced to analyze the effect factors on the heterogeneity of dominant air pollutants in this study, and this analysis process is described below.…”

Conditional Copula-Based Spatial–Temporal Drought Characteristics Analysis—A Case Study over Turkey

“…This temporal and/or spatial dependency between different drought characteristics may immediately be exploited using copula functions, particularly using conditional probabilities. Even though spatial-temporal dependencies of drought characteristics have been investigated before using copula functions [22][23][24], such investigations have not been carried out in a conditional framework before.…”

Improving hydrological post-processing for assessing the conditional predictive uncertainty of monthly streamflows

“…Weerts et al (2011) proposed the quantile regression approach to avoid any assumptions in the regression. Currently, many methods were developed, e.g., a post-processing with error model (Evin et al, 2014;Woldemeskel et al, 2018), non-parametric post-processing methods (Brown and Seo, 2013), data-driven resampling techniques (Ehlers et al, 2019;Sikorska et al, 2015;Solomatine and Shrestha, 2009), copula post-processing methods (Klein et al, 2016;Madadgar and Moradkhani, 2014;Schefzik et al, 2013). This list of post-processing methods is not meant to be exhaustive but for a detailed review, see .…”

“…Schepen and Wang (2015) compared the performance of the Bayesian model averaging (BMA) and quantile model averaging (QMA) to merge statistical and dynamic forecasts for seasonal streamflows in 12 Australian catchments finding that both methods performed similarly. Klein et al (2016) compared the predictive skill of the copula uncertainty processor (COP) based on pair-copula construction, Bayesian model averaging (BMA), quantile regression and model conditional processor (MCP) using the multivariate truncated normal distribution for daily streamflows recommending the COP. Recently, Woldemeskel et al (2018) evaluated post-processing approaches using three transformations, namely logarithmic, log-sinh and Box-Cox over 300 Australian catchments for monthly and seasonal streamflow forecasts, concluding that the Box-Cox transformation with = 2 was the bestperforming post-processing method, especially in dry catchments.…”

“…However, the use of Gaussian mixture is not new in the hydrological post-processing community. Recently, Feng et al (2019) and Klein et al (2016) used Gaussian mixture for estimating the marginal distribution of hydrological post-processing approaches. On the other hand, the proposed GMM postprocessing method merges the BJP and GMM to model the joint probability distribution that describes the relationship between deterministic hydrological predictions (predictors) and corresponding observed streamflows (predictands).…”