Penalized least squares estimation of Volterra filters and higher order statistics

Volterra filters (VFs) and higher order statistics (HOS) are important tools for nonlinear analysis, processing, and modeling. Despite their highly desirable properties, the transfer of VFs and HOS to real-world signal processing problems has been hindered by the requirement of very large data records needed to obtain reliable estimates. The identification of VFs and the estimation of HOS both fall into the category of ill-posed estimation problems. We develop penalized least squares (PLS) estimation methods for VFs and HOS. It is shown that PLS is a very effective way to incorporate prior information of the problem at hand without directly constraining the estimation procedure. Hence, PLS produces much more reliable estimates. The main contributions of this paper are the development of appropriate penalizing functionals and cross-validation procedures for PLS based VF identification and HOS estimation.