Efficient filtering and sampling for a class of time-varying linear systems
James K. Murphy, Simon J. Godsill
Abstract
This paper presents an O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ) time method for filtering and sampling of a time-varying n × n system matrix A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> in a restricted class of time-varying linear systems of the form X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> = A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t-1</sub> + C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> + ε <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> , via a matrix-variate normal formulation. This allows larger systems within this class to be inferred via Gibbs sampling in reasonable time than is possible with methods that rely on vectorization of the system matrix, followed by standard Kalman filtering, which run in O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sup> ) time. It is shown how to apply the method to vector autoregression problems with time-varying system matrices (TVP-VAR problems). Noisy observations of the underlying system state are also accommodated in a straightforward way.
BibTeX
@inproceedings{icassp2015_efficientfilteri,
title = {Efficient filtering and sampling for a class of time-varying linear systems},
author = {James K. Murphy and Simon J. Godsill},
booktitle = {ICASSP 2015},
year = {2015}
}