Linear filtering and state space representations of hidden Markov models

Sofia Andersson, Tobias Rydén and Rolf Johansson

Abstract:: The topic of this paper is linear filtering of hidden Markov models (HMMs) and linear innovation form representations of HMMs. The possibility to represent the widely used HMM as a state space model is interesting in its own respect, but our interest also comes from subspace estimation methods. To be able to fit the HMM into the framework of subspace methods the process needs to be formulated in state space form. This reformulation is complicated by the non-minimality within the state space representation of the HMM. The reformulation involves deriving solutions to algebraic Riccati equations which are usually treated under minimality assumptions.

Key words:: hidden Markov model, Kalman filter, non-minimality, prediction error representation, Riccati equation, state space model