Recursive identification of chain dynamics in Hidden Markov Models using Non-Negative Matrix Factorization

Hidden Markov Models (HMMs) and associated Markov modulated time series are widely used for estimation and control in e.g. robotics, econometrics and bioinformatics. In this paper, we modify and extend a recently proposed approach in the machine learning literature that uses the method of moments and a Non-Negative Matrix Factorization (NNMF) to estimate the parameters of an HMM. In general, the method aims to solve a constrained non-convex optimization problem. In this paper, it is shown that if the observation probabilities of the HMM are known, then estimating the transition probabilities reduces to a convex optimization problem. Three recursive algorithms are proposed for estimating the transition probabilities of the underlying Markov chain, one of which employs a generalization of the Pythagorean trigonometric identity to recast the problem into a non-constrained optimization problem. Numerical examples are presented to illustrate how these algorithms can track slowly time-varying transition probabilities.