Efficient filtering and sampling for a class of time-varying linear systems

This paper presents an O(n⁴) time method for filtering and sampling of a time-varying n × n system matrix A_t in a restricted class of time-varying linear systems of the form X_t = A_tX_t-1 + C_t + ε_t, 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(n⁶) 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.