Background and Motivation

The hidden Markov model (HMM), although having many benefits for modeling human speech, models the data using discrete states. It can only model continuous feature variations using a large number of states. This problem is generally solved by augmenting the features with time-derivatives [70]. Despite new probabilistic models that address the dynamic behavior of features such as segmental HMMs [71], and a wider class of graphical models [68], the derivative augmented feature (DAF) combined with hidden Markov model (DAF-HMM) remains the most widely-used method of modeling the dynamic behavior of features. Unfortunately, the DAF feature vector is of higher dimension with built-in redundancy. As a result, the assumption of conditional independence of the observations is violated. The probability density function (PDF), or likelihood function (LF) of DAF cannot be compared to the PDF of the original (un-augmented) features. Being able to do this could enable new quantitative means of evaluating dynamic models based on augmentation and comparing with those not based on augmentation and allow classifiers with ``mixed" models, taking advantage of DAF when necessary and using un-augmented features when not.

To this end, we derive an analytic expression for the integral of DAF-HMM model with respect to the un-differenced input data, allowing it to be normalized so that it integrates to one. The computational complexity of our method is order $ O(M^T)$ where $ M$ is the number of Markov states and $ T$ is the length of the feature stream. But, the correction term reaches a steady-state at low values of $ T$, allowing an efficient means to compensate PDFs for large $ T$.

Baggenstoss 2017-05-19