Mixed ℓ2 and ℓ1-norm regularization for adaptive detrending with ARMA modeling

In this paper, the problem of detrending a time series and/or estimating a wandering baseline is addressed. We propose a new methodology that adaptively minimizes different regularized cost functions by introducing an ARMA model of the underlying trend. Mixed ℓ1/ℓ2-norm penalty functions are taken into consideration and novel RLS and LMS solutions are derived for the model parameters estimation. The proposed methods are applied to typical trend estimation/removal problems that can be found in the analysis of economic time series or biomedical signal acquisition. Comparisons with standard noncausal filtering techniques are also presented.