Subspace estimation and prediction methods for hidden Markov models

Hidden Markov models (HMMs) are probabilistic functions of finite Markov chains, or, put in other words, state space models with finite state space. In this paper we examine subspace estimation methods for HMMs whose output lies a finite set as well. In particular we study the geometric structure arising from the non-minimality of the linear state space representation of HMMs, and consistency of a subspace algorithm arising from a certain factorisation of the singular value decomposition of the estimated linear prediction matrix. For this algorithm we show that the estimates of the transition and emission probability matrices are consistent up to a similarity transformation, and that the m-step linear predictor computed from the estimated system matrices is consistent, i.e. converges to the true optimal linear m-step predictor.