Viterbi algorithm for acoustic vectors generated by a linear stochastic differential equation on each state

When using hidden Markov models for speech recognition, it is usually assumed that the probability that a particular acoustic vector is emitted at a given time only depends on the current state and the current acoustic vector observed. We introduce another idea, i.e., we assume that, in a given state, the acoustic vectors are generated by a linear stochastic differential equation. This work is motivated by the fact that the time evolution of the acoustic vector is inherently dynamic and continuous. So that the modelling could be performed in the continuous-time domain instead of the discrete-time domain. By the way, the links between the discrete-time model obtained after sampling, and the original continuous-time signal are not so trivial. In particular, the relationship between the coefficients of a continuous-time linear process and the coefficients of the discrete-time linear process obtained after sampling is nonlinear. We assign a probability density to the continuous-time trajectory of the acoustic vector inside the state, reflecting the probability that this particular path has been generated by the stochastic differential equation associated with this state. This allows us to compute the likelihood of the uttered word. Reestimation formulae for the parameters of the process, based on the maximization of the likelihood, can be derived for the Viterbi algorithm. As usual, the segmentation can be obtained by sampling the continuous process, and by applying dynamic programming to find the best path over all the possible sequences of states