4 Hidden Dynamic Model Implemented Using Piece-wise Linear Approximation : In this section, we will describe a particular computational implementation of the hidden dynamic model in Eqs. 10.40 and 10.41, where a deterministic target vector is assumed and the general nonlinear function h[x(k)] in Eq. 10.41 is approximated by a set of

doi:10.1201/9781482276237-55

Chapter

4 Hidden Dynamic Model Implemented Using Piece-wise Linear Approximation

ABSTRACT

After the general nonlinear mapping in Eq. 10.41 from hidden dynamic variables to acoustic observational variables is approximated by a piecewise linear relationship, a new, discrete random variable m is introduced to provide the "region" (i.e., the mixturecomponent index) in the piecewise linear mapping. The conditional PDF for the output acoustic variables o(k) given the input hidden variables x(k) (for the fixed region m at time frame k) is:

P[o(k)lx(k), mk] = N[o(k); Ey]. (10.45) This is due to the Gaussian assumption for the observational noise vm(k). The conditional PDF related to the state equation for the fixed region m at time frame k is

P[x(k +1)1x(k), t(k), sk , mk] = N[x(k +1); (I'sk ,mk x(k)— (I-4'sk,mk)t(k), Ewl. (10.46)

This is due to the Gaussian assumption for the state noise wm(k).