train.tex

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This can be avoided by using an incremental threshold.  For example,
executing 
\begin{verbatim}
    HERest -t 120.0 60.0 240.0 -S trainlist -I labs \
           -H dir1/hmacs -M dir2 hmmlist
\end{verbatim}
would cause \htool{HERest} to run normally
at a beam width\index{beam width} of 120.0.  However, if a pruning 
error\index{pruning errors} occurs, the
beam is increased by 60.0 and \htool{HERest} reprocesses the offending training
utterance.  Repeated errors cause the beam width to be increased
again and this continues until either the utterance is 
successfully processed or the upper beam limit is reached, in this
case 240.0.  Note that errors which occur at very high beam widths
are often caused by transcription errors, hence, it is best not to
set the upper limit too high.

\centrefig{parher}{90}{\htool{HERest} Parallel Operation}

\index{model training!in parallel}
The second way of speeding-up the operation of \htool{HERest} is to use more than
one computer in parallel.  The way that this is done is to divide the
training data amongst the available machines and then to run \htool{HERest} on each
machine such that each invocation of \htool{HERest} 
uses the same initial set of models but has its own private set of data.
By setting the option {\tt -p N} where {\tt N} is an integer, \htool{HERest} will
dump the contents of all its accumulators\index{accumulators}  into a file called {\tt HERN.acc}
rather than updating and outputing a new set of models.  These dumped
files are collected together and input to a new invocation of \htool{HERest} with
the option {\tt -p 0} set.  \htool{HERest} then reloads the accumulators from
all of the dump files and updates the models in the normal way.
This process is illustrated in Figure~\href{f:parher}.

To give a concrete example, suppose that four networked workstations
were available to execute the \htool{HERest} command given earlier. The training files 
listed previously in \texttt{trainlist} would be split 
into four equal sets and a list
of the files in each set stored in {\tt trlist1}, 
{\tt trlist2}, {\tt trlist3}, and {\tt trlist4}.
On the first workstation, the command
\begin{verbatim}
    HERest -S trlist1 -I labs -H dir1/hmacs -M dir2 -p 1 hmmlist
\end{verbatim}
would be executed.  This will load in the HMM definitions in 
{\tt dir1/hmacs}, process the files listed in {\tt trlist1} and finally
dump its accumulators into a file called {\tt HER1.acc} in the output
directory {\tt dir2}.  At the same time, the command
\begin{verbatim}
    HERest -S trlist2 -I labs -H dir1/hmacs -M dir2 -p 2 hmmlist
\end{verbatim}
would be executed on the second workstation, and so on.  When 
\htool{HERest} has finished on all four
workstations,  the following command will be executed on just one of them
\begin{verbatim}
    HERest -H dir1/hmacs -M dir2 -p 0 hmmlist dir2/*.acc
\end{verbatim}
where the list of training files has been replaced by the dumped accumulator
files.  This will cause the accumulated
statistics to be reloaded and merged so that the model parameters can
be reestimated and the new model set output to \texttt{dir2}
The time to perform this last phase of the operation is very small, hence
the whole process will be around four times quicker than for the
straightforward sequential case. 

\mysect{Single-Pass Retraining}{singlepass}

In addition to re-estimating the parameters of a HMM set, \htool{HERest}
also provides a mechanism for mapping a set of models trained using
one parameterisation into another set based on a different parameterisation.
This facility allows the front-end of a HMM-based recogniser to be 
modified without having to rebuild the models from scratch.

This facility is known as single-pass retraining\index{single-pass retraining}.
Given one set of well-trained models, a new set matching a
different training data parameterisation can be generated in a single
re-estimation pass. This is done by computing the forward and backward
probabilities using the original models together with the original
training data, but then switching to the new training data to compute
the parameter estimates for the new set of models.

Single-pass retraining is enabled in \htool{HERest} by setting the
\texttt{-r} switch.  This causes the input training files to be read
in pairs.  The first of each pair is used to compute the
forward/backward probabilities and the second is used to estimate the
parameters for the new models.  Very often, of course, data input to
\HTK\ is modified by the \htool{HParm} module in accordance with
parameters set in a configuration file.  In single-pass retraining mode,
configuration parameters can be prefixed by the pseudo-module names
\texttt{HPARM1} and \texttt{HPARM2}.  Then when reading in the first
file of each pair, only the \texttt{HPARM1} parameters are used and
when reading the second file of each pair, only the \texttt{HPARM2}
parameters are used.\index{configuration parameters!switching}

As an example, suppose that a set of models has been trained on data
with \texttt{MFCC\_E\_D} parameterisation and a new set of models using
Cepstral Mean Normalisation (\texttt{\_Z}) is required. These two data
parameterisations are specified in a configuration file
(\texttt{config}) as two separate instances of the configuration
variable \texttt{TARGETKIND} i.e.
\begin{verbatim}
    # Single pass retraining
    HPARM1: TARGETKIND = MFCC_E_D
    HPARM2: TARGETKIND = MFCC_E_D_Z
\end{verbatim}
\htool{HERest} would then be invoked with the \texttt{-r} option set to enable 
single-pass retraining. For example,
\begin{verbatim}
    HERest -r -C config -S trainList -I labs -H dir1/hmacs -M dir2 hmmList
\end{verbatim}
The script file  \texttt{trainlist} contains a list of data file
pairs. For each pair, the first file should match the parameterisation of 
the original model set and the second file should match that of the
required new set.
This will cause the model parameter estimates to be performed using the 
new set of training data and a new set of models matching this data will  
be output to \texttt{dir2}. This process of single-pass retraining is
a significantly faster route to a new set of models than training a fresh 
set from scratch.

\mysect{Two-model Re-Estimation}{twomodel}

Another method for initialisation of model parameters implemented in
\htool{HERest} is two-model re-estimation. HMM sets often use the same
basic units such as triphones but differ in the way the underlying HMM
parameters are tied. In these cases two-model re-estimation can be
used to obtain the state-level alignment using one model set which is
used to update the parameters of a second model set. This is helpful
when the model set to be updated is less well trained.

A typical use of two-model re-estimation\index{two-model
  re-estimation} is the initialisation of state clustered triphone
models. In the standard case triphone models are obtained by cloning
of monophone models and subsequent clustering of triphone states.
However, the unclustered triphone models are considerably less
powerful than state clustered triphone HMMs using mixtures of
Gaussians. The consequence is poor state level alignment and thus poor
parameter estimates, prior to clustering.  This can be ameliorated by
the use of well-trained \textit{alignment models} for computing the
forward-backward probabilities. In the maximisation stage of the
Baum-Welch algorithm the state level posteriors are used to
re-restimate the parameters of the \textit{update model set}. Note that
the corresponding models in the two sets must have the same number of
states.

As an example, suppose that we would like to update a set of cloned
single Gaussian monophone models in {\tt dir1/hmacs} using the well
trained state-clustered triphones in {\tt dir2/hmacs} as alignment
models. Associated with each model set are the model lists {\tt
  hmmlist1} and {\tt hmmlist2} respectively. In order to use the
second model set for alignment a configuration file {\tt
  config.2model} containing
\begin{verbatim}
   # alignment model set for two-model re-estimation
   ALIGNMODELMMF = dir2/hmacs
   ALIGNHMMLIST  = hmmlist2
\end{verbatim}
is necessary. \htool{HERest} only needs to be invoked using that
configuration file.
\begin{verbatim}
   HERest -C config -C config.2model -S trainlist -I labs -H dir1/hmacs -M dir3 hmmlist1
\end{verbatim}
The models in directory {\tt dir1} are updated using the alignment
models stored in directory {\tt dir2} and the result is written to
directory {\tt dir3}. Note that {\tt trainlist} is a standard \HTK\ 
script and that the above command uses the capability of HERest to
accept multiple configuration files on the command line. If each HMM
is stored in a separate file, the configuration variables {\tt
  ALIGNMODELDIR} and {\tt ALIGNMODELEXT} can be used.

Only the state level alignment is obtained using the alignment models.
In the exceptional case that the update model set contains mixtures of
Gaussians, component level posterior probabilities are obtained from
the update models themselves.

\mysect{Parameter Re-Estimation Formulae}{bwformulae}

\index{model training!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
\begin{tabbing}
++ \= ++++++++ \= \kill
\> $N$ \> 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 \\
\> $\bm{O}$      \> a sequence of observations \\
\> $\bm{o}_t$    \> the observation at time $t$, $1 \leq t \leq T $ \\
\> $\bm{o}_{st}$ \> 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 $s$\\
\> $\bm{\mu}_{jsm}$  \> vector of means for the mixture component $m$ of state $j$
                        stream $s$\\ 
\> $\bm{\Sigma}_{jsm}$  \> covariance matrix for the mixture component $m$ 
                         of state $j$  stream $s$ \\
\> $\lambda$ \> the set of all parameters defining a HMM
\end{tabbing}

\subsection{Viterbi Training (\htool{HInit})}

\index{model training!Viterbi formulae}
In this style of model training, a set of training observations
$\bm{O}^r, \;\; 1 \leq r \leq R$ is used to estimate the 
parameters of a single HMM by iteratively computing Viterbi alignments.
When used to initialise a new HMM, the Viterbi segmentation is
replaced by a uniform segmentation (i.e.\ each training
observation is divided into $N$ equal segments) 
for the first iteration.

Apart from the first iteration on a new model, 
each training sequence $\bm{O}$ is segmented using a state alignment procedure
which results from maximising
\[
    \phi_N(T) = \max_i \phi_i(T) a_{iN}
\]
for $1<i<N$ where
\[
  \phi_j(t) = \left[ \max_i \phi_i(t-1) a_{ij} \right] b_j(\bm{o}_t)
\]
with initial conditions given by 
\[
    \phi_1(1) = 1
\]
\[
    \phi_j(1) = a_{1j} b_j(\bm{o}_1)
\]
for $1<j<N$. 
In this and all subsequent cases, the output  probability $b_j(\cdot)$ is as defined in
equations~\ref{e:cdpdf} and \ref{e:gnorm} in section~\ref{s:HMMparm}.

If $A_{ij}$ represents the total number of transitions from state $i$ to state $j$
in performing the above maximisations, then the transition probabilities can
be estimated from the relative frequencies
\[
  \hat{a}_{ij} = \frac{A_{ij}}{\sum_{k=2}^{N}A_{ik}}
\]

The sequence of states which maximises $\phi_N(T)$ implies an alignment of
training data observations with states.  Within each state, a further alignment
of observations to mixture components is made.  The tool \htool{HInit} provides
two mechanisms for this:  for each state and each stream
\begin{enumerate}
\item use clustering to allocate each observation $\bm{o}_{st}$ to one of $M_s$ clusters, or
\item associate each observation $\bm{o}_{st}$ with the mixture component with the
       highest probability
\end{enumerate}
In either case, the net result is that every observation is associated with a single
unique mixture component.  This association can be
represented by the indicator function $\psi^r_{jsm}(t)$ which is 1
if $\bm{o}^r_{st}$ is associated with mixture component $m$ of stream $s$ of 
state $j$ and is zero otherwise.

The means and variances are then estimated via simple averages
\newcommand{\vitsum}[2]{
                  \sum_{r=1}^R  \sum_{t=1}^{T_r} #1 \psi^r_{js#2}(t)
}

\[
   \hat{\bm{\mu}}_{jsm} = \frac{
                \vitsum{}{m}\bm{o}^r_{st}}{\vitsum{}{m}}
\]

\[
   \hat{\bm{\Sigma}}_{jsm} = \frac{
        \vitsum{}{m}(\bm{o}^r_{st} - \hat{\bm{\mu}}_{jsm})
                                        (\bm{o}^r_{st} - \hat{\bm{\mu}}_{jsm})'
                }{\vitsum{}{m}}
\]

Finally, the mixture  weights are based on the number of
observations allocated to each component
\[
   \bm{c}_{jsm} = \frac{\vitsum{}{m}}{
        \vitsum{\sum_{l=1}^{M_s}}{l} }
\]

\subsection{Forward/Backward Probabilities}

\index{model training!forward/backward formulae}
Baum-Welch training is similar to the Viterbi training described
in the previous section except that the \textit{hard} boundary implied
by the $\psi$ function is replaced by a \textit{soft} boundary
function $L$ which represents the probability of an observation being
associated any given Gaussian mixture component.  
This \textit{occupation} probability is computed from the \textit{forward}
and \textit{backward} probabilities.

For the isolated-unit style of training, the forward 
probability $\alpha_j(t)$ for $1<j<N$ and
$1<t \leq T$ is calculated by the forward recursion