Parameter Re-Estimation Formulae

For reference purposes, this section lists the various formulae employed within the HTK parameter estimation tools. All are standard, however, the use of non-emitting states and multiple data streams leads to various special cases which are usually not covered fully in the literature.

The following notation is used in this section

		 $N$ number of states 

$S$ number of streams 

$M_s$ number of mixture components in stream $s$

$T$ number of observations 

$Q$ number of models in an embedded training sequence 

$N_q$ number of states in the $q$'th model in a training sequence 


${\mbox{\boldmath $O$}}$ a sequence of observations 


${\mbox{\boldmath $o$}}_t$ the observation at time $t$, 
$1 \leq t \leq T$ 


${\mbox{\boldmath $o$}}_{st}$ the observation vector for stream $s$ at time $t$ 

$a_{ij}$ the probability of a transition from state $i$ to $j$ 

$c_{jsm}$ weight of mixture component $m$ in state $j$ stream $s$


${\mbox{\boldmath $\mu$}}_{jsm}$ vector of means for the mixture component $m$ of state $j$                        stream $s$


${\mbox{\boldmath $\Sigma$}}_{jsm}$ covariance matrix for the mixture component $m$                          of state $j$ stream $s$ 

$\lambda$ the set of all parameters defining a HMM