Smoothing Algorithms for State-Space Models

A prevalent problem in statistical signal processing, applied statistics, and time series analysis is the calculation of the smoothed posterior distribution, which describes the uncertainty associated with a state, or a sequence of states, conditional on data from the past, the present, and the future. The aim of this paper is to provide a rigorous foundation for the calculation, or approximation, of such smoothed distributions, to facilitate a robust and efficient implementation. Through a cohesive and generic exposition of the scientific literature we offer several novel extensions such that one can perform smoothing in the most general case. Experimental results for: a Jump Markov Linear System; a comparison of particle smoothing methods; and parameter estimation using a particle implementation of the EM algorithm, are provided. Index Terms State space smoothing, Hidden Markov Model, Kalman filter, Kalman smoother, Jump Markov Linear System, Particle filter, Particle smoother, Parameter estimation.