Approximation of stationary processes by hidden Markov models

Stochastic realization is still an open problem for the class of hidden Markov models (HMM): given the law Q of an HMM find a finite parametric description of it. Fifty years after the introduction of HMMs, no computationally effective realization algorithm has been proposed. In this paper we direct our attention to an approximate version of the stochastic realization problem for HMMs. We aim at the realization of an HMM of assigned complexity (number of states of the underlying Markov chain) which best approximates, in Kullback Leibler divergence rate, a given stationary law Q. In the special case of Q being the law of an HMM this corresponds to solving the approximate realization problem for HMMs. In general there is no closed form expression of the Kullback Leibler divergence rate, therefore we replace it, as approximation criterion, with the informational divergence between the Hankel matrices of the processes. This not only has the advantage of being easy to compute, while providing a good approximation of the divergence rate, but also makes the problem amenable to the use of nonnegative matrix factorization (NMF) techniques. We propose a three step algorithm, based on the NMF, which realizes an optimal HMM. The viability of the algorithm as a practical tool is tested on a few examples of HMM order reduction.