On Recursive State Estimation for Linear State-Space Models Having Quantized Output Data

In this paper, we study the problem of estimating the state of a dynamic state-space system where the output is subject to quantization. We compare some classical approaches and a new development in the literature to obtain the filtering and smoothing distributions of the state conditioned to quantized data. The classical approaches include the Extended Kalman filter/smoother in which we consider an approximation of the quantizer non-linearity based on the arctan function, the quantized Kalman filter/smoother, the Unscented Kalman filter/smoother, and the Sequential Monte Carlo sampling method also called particle filter/smoother. We consider a new approach based on the Gaussian sum filter/smoother where the probability mass function of the quantized data given the state is modeled as an integral equation and approximated using Gauss-Legendre quadrature. The Particle filter is addressed considering some resampling methods used to deal with the degeneracy problem. Also, the sample impoverishment caused by the resampling method is addressed by introducing diversity in the samples set using the Markov Chain Monte Carlo method. In this paper, we discuss the implementation of the aforementioned algorithms and the Particle filter/smoother implementation is studied by using different resampling methods combined with two Markov Chain algorithms. A numerical simulation is presented to analyze the accuracy of the estimation and the computational cost.

A MAP approach for $\ell_q$-norm regularized sparse parameter estimation using the EM algorithm

In this paper, Bayesian parameter estimation through the consideration of the Maximum A Posteriori (MAP) criterion is revisited under the prism of the Expectation-Maximization (EM) algorithm. By incorporating a sparsity-promoting penalty term in the cost function of the estimation problem through the use of an appropriate prior distribution, we show how the EM algorithm can be used to efficiently solve the corresponding optimization problem. To this end, we rely on variance-mean Gaussian mixtures (VMGM) to describe the prior distribution, while we incorporate many nice features of these mixtures to our estimation problem. The corresponding MAP estimation problem is completely expressed in terms of the EM algorithm, which allows for handling nonlinearities and hidden variables that cannot be easily handled with traditional methods. For comparison purposes, we also develop a Coordinate Descent algorithm for the $\ell_q$-norm penalized problem and present the performance results via simulations.