A Generalized Cauchy Distribution Framework for Problems Requiring Robust Behavior

Rafael E. Carrillo, Tuncer C. Aysal, Kenneth E. Barner
2010 EURASIP Journal on Advances in Signal Processing  
Statistical modeling is at the heart of many engineering problems. The importance of statistical modeling emanates not only from the desire to accurately characterize stochastic events, but also from the fact that distributions are the central models utilized to derive sample processing theories and methods. The generalized Cauchy distribution (GCD) family has a closed-form pdf expression across the whole family as well as algebraic tails, which makes it suitable for modeling many real-life
more » ... many real-life impulsive processes. This paper develops a GCD theory-based approach that allows challenging problems to be formulated in a robust fashion. Notably, the proposed framework subsumes generalized Gaussian distribution (GGD) family-based developments, thereby guaranteeing performance improvements over traditional GCD-based problem formulation techniques. This robust framework can be adapted to a variety of applications in signal processing. As examples, we formulate four practical applications under this framework: (1) filtering for power line communications, (2) estimation in sensor networks with noisy channels, (3) reconstruction methods for compressed sensing, and (4) fuzzy clustering. Distributions, Optimal Filtering, and M-Estimation This section presents M-estimates, a generalization of maximum likelihood (ML) estimates, and discusses optimal filtering from an ML perspective. Specifically, it discusses statistical models of observed samples obeying generalized Gaussian statistics and relates the filtering problem to maximum likelihood estimation. Then, we present the generalized Cauchy distribution, and a relation between GGD and GCD random variables is introduced. The ML estimators for GCD statistics are also derived.
doi:10.1155/2010/312989 fatcat:ta5ajb7s5rhklfsifdbfasfszy