Normal priors with unknown variance (NUV) have long been known to promote sparsity and to blend well with parameter learning by expectation maximization (EM). In this paper, we advocate this approach for linear state space models for applications such as the estimation of impulsive signals, the detection of localized events, smoothing with occasional jumps in the state space, and the detection and removal of outliers.The actual computations boil down to multivariate-Gaussian message passing algorithms that are closely related to Kalman smoothing. We give improved tables of Gaussian-message computations from which such algorithms are easily synthesized, and we point out two preferred such algorithms.

Abstract-The paper addresses the problem of joint signal separation and estimation in a single-channel discrete-time signal composed of a wandering baseline and overlapping repetitions of unknown (or known) signal shapes. All signals are represented by a linear state space model (LSSM). The baseline model is driven by white Gaussian noise, but the other signal models are triggered by sparse inputs. Sparsity is achieved by normal priors with unknown variance (NUV) from sparse Bayesian learning. All signals and system parameters are jointly estimated with an efficient expectation maximization (EM) algorithm based on Gaussian message passing, which works both for known and unknown signal shapes. The proposed method outputs a sparse multi-channel representation of the given signal, which can be interpreted as a signal labeling.

This paper introduces a toolbox for model-based detection, separation, and reconstruction of signals that is especially suited for biomedical signals, such as electrocardiograms (ECGs) or electromyograms (EMGs). The modeling is based on autonomous linear state space models (LSSMs), which are localized with flexible windows. The models are fit to observations by minimizing the squared error while the use of LSSMs leads to efficient recursive error computations and minimizations. Multisection windows enable complex models, and per-sample weights enable multistage processing or adaptive smoothing. This paper is motivated by, and intended for, practical applications, for which several examples and tabulated cost computations are given.

Abstract-Surprisingly many signal processing problems can be approached by locally fitting autonomous deterministic linear state space models to the data. In this paper, we introduce local statistical models for such cases and discuss the computation both of the corresponding estimates and of local likelihoods for different models.

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