Learning Hidden Markov Models using Non-Negative Matrix Factorization

arXiv:0809.4086v2 [cs.LG] 8 Jan 2011 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2008 1 Learning Hidden Markov Models using Non-Negative Matrix Factorization George Cybenko, Fellow, IEEE, and Valentino Crespi, Member, IEEE Abstract—The Baum-Welch algorithm together with its deriva- tives and variations has been the main technique for learning Hidden Markov Models (HMM) from observational data. We present an HMM learning algorithm based on the non-negative matrix factorization (NMF) of higher order Markovian statistics that is structurally different from the Baum-Welch and its asso- ciated approaches. The described algorithm supports estimation of the number of recurrent states of an HMM and iterates the non-negative matrix factorization (NMF) algorithm to improve the learned HMM parameters. Numerical examples are provided as well. Index Terms—Hidden Markov Models, machine learning, non- negative matrix factorization. I. INTRODUCTION Hidden Markov Models (HMM) have been successfully used to model stochastic systems arising in a variety of appli- cations ranging from biology to engineering to finance [1], [2], [3], [4], [5], [6]. Following accepted notation for representing the parameters and structure of HMM’s (see [7], [8], [9], [1], [10] for example), we will use the following terminology and definitions: 1) N is the number of states of the Markov chain underly- ing the HMM. The state space is S = {S1, ..., SN} and the system’s state process at time t is denoted by xt; 2) M is the number of distinct observables or symbols generated by the HMM. The set of possible observables is V = {v1, ..., vM} and the observation process at time t is denoted by yt. We denote by yt2 t1 the subprocess yt1yt1+1 . . . yt2; 3) The joint probabilities aij(k) = P(xt+1 = Sj, yt+1 = vk|xt = Si); are the time-invariant probabilities of transitioning to state Sj at time t + 1 and emitting observation vk given that at time t the system was in state Si. Observation vk is emitted during the transition from state Si to state Sj. We use A(k) = (aij(k)) to denote the matrix of state transition probabilities that emit the same symbol vk. Note that A = P k A(k) is the stochastic matrix representing the Markov chain state process xt. 4) The initial state distribution, at time t = 1, is given by Γ = {γ1, ..., γN} where γi = P(x1 = Si) ≥0 and P i γi = 1. G. Cybenko is with the Thayer School of Engineering, Dartmouth College, Hanover, NH 03755 USA e-mail: gvc@dartmouth.edu. V. Crespi is with the Department of Computer Science, California State University at Los Angeles, LA, 90032 USA e-mail: vcrespi@calstatela.edu. Manuscript submitted September 2008 Collectively, matrices A(k) and Γ completely define the HMM and we say that a model for the HMM is λ = ({A(k) | 1 ≤ k ≤M}, Γ). We present an algorithm for learning an HMM from single or multiple observation sequences. The traditional approach for learning an HMM is the Baum-Welch Algorithm [1] which has been extended in a variety of ways by others [11], [12], [13]. Recently, a novel and promising approach to the HMM ap- proximation problem was proposed by Finesso et al. [14]. That approach is based on Anderson’s HMM stochastic realization technique [15] which demonstrates that a positive factorization of a certain Hankel matrix (consisting of observation string probabilities) can be used to recover the hidden Markov model’s probability matrices. Finesso and his coauthors used recently developed non-negative matrix factorization (NMF) algorithms [16] to express those stochastic realization tech- niques as an operational algorithm. Earlier ideas in that vein were anticipated by Upper in 1997 [17], although that work did not benefit from HMM stochastic realization techniques or NMF algorithms, both of which were developed after 1997. Methods based on stochastic realization techniques, includ- ing the one presented here, are fundamentally different from Baum-Welch based methods in that the algorithms use as input observation sequence probabilities as opposed to raw obser- vation sequences. Anderson’s and Finesso’s approaches use system realization methods while our algorithm is in the spirit of the Myhill-Nerode [18] construction for building automata from languages. In the Myhill-Nerode construction, states are defined as equivalence classes of pasts which produce the same futures. In an HMM, the “future” of a state is a probability distribution over future observations. Following this intuition we derive our result in a manner that appears comparatively more concise and elementary, in relation to the aforementioned approaches by Anderson and Finesso. At a conceptual level, our algorithm operates as follows. We first estimate the matrix of an observation sequence’s high order statistics. This matrix has a natural non-negative matrix factorization (NMF) [16] which can be interpreted in terms of the probability distribution of future observations given the current state of the underlyin

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