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# Multivariate Copula Analysis Toolbox (MvCAT)

MvCAT is developed in Matlab as a user-friendly toolbox (software) to help scientists and researchers perform rigorous and comprehensive multivariate dependence analysis. It uses 26 copula families with 1 to 3 parameters to describe the dependence structure of two random variables. MvCAT uses local optimization and also Markov chain Monte Carlo simulation within a Bayesian framework to infer the parameter values of the copula families by contrasting them against available data. If Bayesian analysis with MCMC simulation is performed, an estimate of uncertainty for each copula family can be obtained from the posterior distribution of copula parameters. MCMC within Bayesian framework not only provide a robust estimate of the global optima, but also approximate the posterior distribution of the copula families which can be used to construct a prediction uncertainty range for the copulas. Local optimization methods are prone to getting trapped in local optima (see Sadegh et al., 2017 for more information).

The user can select any subset of the available 26 copulas and MvCAT will perform the analysis and rank the selected copula families based on their performance. Performance metrics used in this toolbox are Likelihood, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Nash-Sutcliffe Efficiency (NSE), and Root Mean Squared Error (RMSE). While Likelihood, NSE and RMSE only focus on minimizing the residuals between observations and model simulations, the other metrics take into consideration additional criteria. For example, AIC takes into account the model complexity and BIC account for model complexity and number of observations.

It is noteworthy that marginal distribution of each variable is modeled using best of the fitted (1) Beta, (2) Birnbaum-Saunders, (3) exponential,(4) extreme value, (5) Gamma, (6) generalized extreme value, (7) generalized Pareto, (8) inverse Gaussian,(9) logistic, (10) log-logistic, (11) lognormal, (12) Nakagami, (13) normal, (14) R ayleigh, (15) Rician, (16) t loca-tion scale, and (17) Weibull distributions (listed alphabetically). Best marginal is selected according to BIC.

**Reference**:

Sadegh, M., Ragno, E. and AghaKouchak, A.
(2017), Multivariate Copula Analysis Toolbox (MvCAT): Describing dependence and
underlying uncertainty using a Bayesian framework. *Water Resources Research*, 53, doi:10.1002/2016WR020242

Link: http://onlinelibrary.wiley.com/doi/10.1002/2016WR020242/epdf

Source codes of MvCAT in MATLAB:

Some data to play with MvCAT:

If you don’t have MATLAB installed on your MAC, use the following executable version of MvCAT:

If you don’t have MATLAB installed on your PC, use the following executable version of MvCAT: