Copulas for hydroclimatic applications – A practical note on common misconceptions and pitfalls

Abstract

Abstract. For most hydroclimatic applications, precipitation and temperature are of particular interest as they strongly affect the water cycle, can easily be measured and are often readily available from many meteorological stations worldwide. To account for precipitation and temperature variability, their co-dependence and their correlation, several multivariate analysis methods have been adopted in the hydroclimatic literature in recent years. In line with the steadily rising number of publications on this topic, the notion of copula-based probability distribution has also attracted tremendous interest to deal with the complexity of compound events in the multidimensional context. A copula is a function that connects a multivariate distribution to its one-dimensional margins, which allows for a joint distribution of random variables with great flexibility for the marginal distribution. However, there seems to be a lack of comprehensive understanding of the fundamental requirements of the copula concept such as the strength and significance of correlation between variables, autocorrelation effects and the choice of representative copula families, which potentially compromises the robustness of projections of future environmental processes and natural hazards. Therefore, by combining a systematic literature review with a specific hydroclimatic case study in Sweden, we illustrate a practical approach to copula-based modeling, which (1) provides end-users with an overview of necessary requirements, statistical assumptions and consequential limitations of copulas, (2) highlights possible pitfalls and misconceptions, and (3) offers a decision support framework for the application of copulas to support researchers and decision makers in addressing climatological hazards and sustainable development, thereby demystifying what is currently an area of great confusion.

Relationships among return and liquidity of cryptocurrencies

“…It should be noted that copulae are primarily applicable to stationary time series (Sadegh et al 2017). Autocorrelated time series can produce spurious dependencies between sets of variables, leading to inaccurate copula-dependency structures (Tootoonchi et al 2020). Therefore, a pre-processing step is necessary to ensure both mean and variance stationarity within the time series.…”

An approach based on multivariate distribution and Gaussian copulas to predict groundwater quality using DNN models in a data scarce environment

“…Meanwhile, the MVD-VSG method adopted elliptical copulas, as we are studying several groundwater quality parameters and they can generate random variables with high dimensions. These copulas, however, were widely applied in hydrological sciences and showed their utility for generating synthetic datasets [20] . The projection of the MVN produces the Gaussian Copula with a function density given by the following equation [21] : Where I is the identity matrix, r ∈ [−1, 1] m x m is the correlation matrix between variables with 1 in its diagonal.…”

Quantifying uncertainty in multivariate quantile estimation of hydrometeorological extremes via copula: A comparison between bootstrapping and Markov chain Monte Carlo

“…Based on the suggestion from Tootoonchi et al . (2020) that marginal variables for copula analysis should be correlated significantly, the three climate regions are excluded in our following analysis. Data in the rest six climate regions (Table 2) represent scenarios with weakly correlated variables.…”

“…The copula is a type of multivariate uniform distribution function, which could be used to simplify the representation of the multivariate cumulative distribution of two or more correlated variables. Copula has been widely applied in hydrometeorological studies to investigate the dependence structures of correlated variables, either strongly correlated (Zhang et al, 2013(Zhang et al, , 2015b or weakly correlated (Sharma and Mujumdar, 2019;Tootoonchi et al, 2020). Typically, the identification of copula function and parameters will be more difficult for weakly correlated variables (Huard et al, 2006).…”

“…It should be noted that the Spearman correlations between maximum temperature and precipitation deficit for three of the climate regions (NE, SE, and WNC) are not statistically significant (with a significance level of 0.1). Based on the suggestion from Tootoonchi et al (2020) that marginal variables for copula analysis should be correlated significantly, the three climate regions are excluded in our following analysis. Data in the rest six climate regions (Table 2) represent scenarios with weakly correlated variables.…”