Copula Based Independent Component Analysis

We propose a parametric version of Independent Component Analysis (ICA) via Copulas families of multivariate distributions that join univariate margins to multivariate distributions. Our procedure exploits the role for copula models in information theory and in measures of association, specifically: the use of copulae densities as parametric mutual information, and as measures of association on the rank statistics. The copula approach offers a unified view of component analysis procedures, in particular, by parameterizing multivariate dependence. ICA then, via the copula, is a generalization of Principal Component Analysis (PCA) where the copula model may be non-Gaussian. Generally, the goal is to orthogonalize a measure of multivariate dispersion, yielding an orthogonal basis for a multivariate data set. The flexibility of the copula approach allows for parameterizations of non-gaussian, non-monotone dependence. Additionally, we note a possible use for the copula approach in generalized component extraction procedures (such as Canonical Correlation Analysis) and ultimately the broader class of Generalized Linear Models.