Nonparametric Copula Density Estimation in Sensor Networks

Statistical and machine learning is a fundamental task in sensor networks. Real world data almost always exhibit dependence among different features. Copulas are full measures of statistical dependence among random variables. Estimating the underlying copula density function from distributed data is an important aspect of statistical learning in sensor networks. With limited communication capacities or privacy concerns, centralization of the data is often impossible. By only collecting the ranks of the data observed by different sensors, we estimate and evaluate the copula density on an equally spaced grid after binning the standardized ranks at the fusion center. Without assuming any parametric forms of copula densities, we estimate them nonparametrically by maximum penalized likelihood estimation (MPLE) method with a Total Variation (TV) penalty. Linear equality and positivity constraints arise naturally as a consequence of marginal uniform densities of any copulas. Through local quadratic approximation to the likelihood function, the constrained TV-MPLE problem is cast as a sequence of corresponding quadratic optimization problems. A fast gradient based algorithm solves the constrained TV penalized quadratic optimization problem. Numerical experiments show that our algorithm can estimate the underlying copula density accurately.