Continuously variable duration hidden Markov models for speech analysis

During the past decade, the applicability of hidden Markov models (HMM) to various facets of speech analysis had been demonstrated in several different experiments. These investigations all rest on the assumption that speech is a quasi-stationary process whose stationary intervals can be identified with the occupancy of a single state of an appropriate HMM. In the traditional form of the HMM, the probability of duration of a state decreases exponentially with time. This behavior does not provide an adequate representation of the temporal structure of speech. The solution proposed here is to replace the probability distributions of duration with continuous probability density functions to form a continuously variable duration hidden Markov model (CVDHMM). The gamma distribution is ideally suited to specification of the durational density since it is one-sided and has only two parameters which, together, define both mean and variance. The main result is a derivation and proof of convergence of reestimation formulae for all the parameters of the CVDHMM. It is interesting to note that if the state durations are gamma distributed, one of the formulae is nonalgebraic but, fortuitously, has properties such that it is easily and rapidly solved numerically to any desired degree of accuracy. Other results are presented including the performance of the formulae on simulated data.