Polynomial Smoothing of Time Series With Additive Step Discontinuities

Abstract—This paper addresses the problem of estimating simultaneously a local polynomial signal and an approximately piecewise constant signal from a noisy additive mixture. The approach developed in this paper synthesizes the total variation filter and least-square polynomial signal smoothing into a unified problem formulation. The method is based on formulating an `1-norm regularized inverse problem. A computationally efficient algorithm, based on variable splitting and the alternating direc-tion method of multipliers (ADMM), is presented. Algorithms are derived for both unconstrained and constrained formulations. The method is illustrated on experimental data involving the detection of nano-particles with applications to real-time virus detection using a whispering-gallery mode detector. I.