🔗 Copulas

Sklar's theorem states that any multivariate joint distribution can be expressed in terms of univariate marginals and a copula that captures the dependency structure. Conversely, any copula combined with valid marginals yields a valid joint distribution. This powerful result allows the marginal distributions and dependencies to be modeled and estimated separately.

Common copula families include the Gaussian copula (which corresponds to linear correlations), the Clayton copula (which captures lower-tail dependence), the Gumbel copula (upper-tail dependence), and the Frank copula (symmetric dependence). My research has explored the use of different copula families in machine learning and evolutionary computation applications.

Research on the use of bivariate copulas for modeling pairwise statistical dependencies in multivariate datasets. Development of methods for selecting the best copula family and estimating its parameters from data, including both parametric (maximum likelihood) and rank-based (Kendall's tau, Spearman's rho) approaches.

Vine copulas (pair-copula constructions) extend bivariate copulas to the multivariate setting by decomposing the joint density into a product of bivariate copulas organized in a sequence of trees called a regular vine (R-vine). This provides a very flexible class of multivariate distributions that can capture asymmetric and tail dependencies that are beyond the reach of multivariate Gaussian models.

Development of algorithms for learning the graph structure of regular vine copulas from dependence data. The structure learning problem consists of finding the sequence of trees that best captures the statistical dependencies in the data. Research on efficient algorithms that scale to high-dimensional datasets and the application of vine copula structure learning within evolutionary optimization.

Research on special cases of vine copulas: C-vines (canonical vines) and D-vines (drawable vines), which impose structured constraints on the tree sequence. These constrained vine structures reduce the number of parameters and provide interpretable dependency models at the cost of reduced flexibility. Analysis of when C-vine and D-vine structures are appropriate for machine learning applications.

Development of classification algorithms based on vine copula models. By fitting a separate vine copula model to each class and using the learned class-conditional densities for Bayesian classification, it is possible to capture complex non-Gaussian dependencies in the features without making parametric assumptions about the feature distribution.

Application of vine copula classifiers to remote sensing problems, including the detection of sand dunes on Mars from orbital imagery. The approach outperforms standard Gaussian-based classifiers on these tasks because the distributions of spectral features are non-Gaussian and exhibit complex multivariate dependencies.

Development of estimation of distribution algorithms that use copula functions as the probabilistic model in the EDA loop. Copula EDAs separate the modeling of the marginal distributions of each variable from the modeling of their dependency structure, providing greater flexibility than standard Gaussian EDAs. Research on how different copula families and vine structures affect EDA performance on continuous benchmark problems.

Implementation and analysis of vine copula EDAs that use regular vine copulas as the probabilistic model for continuous EDAs. Research on how the quality of the learned vine structure affects the efficiency of the evolutionary search, and on methods for balancing the accuracy and computational cost of vine structure learning within the EDA loop.

Application of copula-based methods to the estimation of parameters for stochastic differential equations modeling financial assets and physical processes. Copulas provide a flexible framework for modeling the joint dynamics of multiple correlated time series, capturing dependencies that simpler models miss.

• Carrera D, Santana R and Lozano JA (2019). Detection of sand dunes on Mars using a regular vine-based classification approach. Knowledge-Based Systems.
• Carrera D, Santana R and Lozano JA (2018). The Relationship Between Graphical Representations of Regular Vine Copulas and Polytrees. PRICAI 2018.
• Carrera D, Santana R and Lozano JA (2016). Vine copula classifiers for the mind reading problem. Progress in Artificial Intelligence.
• Cheriet A and Santana R (2018). Modeling dependencies between decision variables and objectives with copula models. GECCO 2018.
• Cuesta-Infante A, Santana R, Hidalgo JI, Bielza C and Larrañaga P (2010). Bivariate empirical and n-variate Archimedean copulas in estimation of distribution algorithms. CEC 2010.
• Arenas ZG, Jimenez JC, Lozada-Chang L-V and Santana R (2021). Estimation of distribution algorithms for the computation of innovation estimators of diffusion processes. Mathematics and Computers in Simulation.