Maximum entropy copula for bivariate drought analysis

Abstract

Copula functions are widely used to derive multivariate probability distributions in hydrometeorology. One of the key steps in the copula method is the derivation of marginal distributions of individual variables which can be accomplished using the principle of maximum entropy where the distribution parameters are estimated from the specified constraints. This study, investigated two drought variables (severity and duration) by coupling the principle of maximum entropy with parametric and empirical copulas. So, homogeneous climatic zones were first identified by applying the fuzzy clustering method to data from 39 synoptic stations in Iran and then drought severity and duration were determined with the standardized precipitation index. These two variables were scaled and their marginal probability distribution functions were derived using the principle of maximum entropy as well as empirically. Then, the joint probability distribution of drought severity and duration was determined using maximum entropy-copula, and parametric and empirical copulas. Thereafter, bivariate conditional return periods were determined for each homogeneous region. Results showed that 1) univariate and bivariate distributions can be obtained by maximizing entropy; 2) the dependence structure via Spearman's rho, which directly affects the Lagrange parameters of entropy copula, was a controlling factor to optimize the objective function; 3) for a given set of constraints, the maximum entropy copula is independent of the types of marginals; 4)The entropy-entropy copula (with entropy marginals) is considered a better method than alternatives, because it has a similar result to the parametric methods while it only needs to fit a single model.