.#models.tex.1.3

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% HTKBook - Steve Young 15/11/95
%

\mychap{HMM Definition Files}{HMMDefs}

\sidepic{Tool.model}{80}{}
The principle function of \HTK\ is to manipulate sets of hidden
Markov  models (HMMs). The definition of a HMM must specify the
model topology, the transition parameters and the output
distribution parameters. The HMM observation vectors can be
divided into multiple independent data streams and each stream can
have its own weight. In addition, a HMM can have ancillary
information such as duration parameters.
\HTK\ supports both continuous mixture densities and discrete
distributions. \HTK\ also provides a generalised tying mechanism which
allows parameters to be shared within and between models.  

\index{HMM!definitions}
In order to encompass this rich variety of HMM types within
a single framework, \HTK\ uses a formal language 
to define HMMs.  The interpretation of this language is handled
by the library module \htool{HModel} which is responsible for converting
between the external and internal representations of HMMs.  In addition,
it provides all the basic probability function calculations.
A second module \htool{HUtil} provides various additional facilities for
manipulating HMMs once they have been loaded into memory.

The purpose of this chapter is to describe
the HMM definition language in some detail.  The chapter begins by 
describing how to write
individual HMM definitions.  \HTK\ macros are
then explained and the mechanisms for defining a complete
model set are presented.  The various flavours of
HMM are then described and the use of binary files discussed.
Finally, a formal description of the \HTK\ HMM definition
language is given.

As will be seen, the definition of a large 
HMM system can involve considerable complexity.  However, in
practice, HMM systems are built incremently.  The usual 
starting point is a single HMM definition which is then
repeatedly cloned and refined using the various \HTK\ tools
(in particular, \htool{HERest} and \htool{HHEd}).
Hence, in practice, the \HTK\ user rarely has to 
generate complex HMM definition files directly.  

\mysect{The HMM Parameters}{HMMparm}

A HMM consists of a number of states.  Each state $j$ has an associated
observation probability distribution $b_{j}(\bm{o}_t)$  which
determines the probability of generating observation $\bm{o}_t$ at
time $t$ and each pair of states $i$ and $j$ has an associated
transition probability $a_{ij}$.  In \HTK\,  the entry state $1$ and
the exit state $N$ of an $N$ state HMM are non-emitting.  

\sidefig{hmm1}{70}{Simple Left-Right HMM}{-4}{
Fig.~\href{f:hmm1} shows a simple left-right HMM with five states in
total.  Three of these are emitting states and have output probability
distributions associated with them.   The transition matrix for
this model will have 5 rows and 5 columns.  Each row will sum to one
except for the final row which is always all zero since no
transitions are allowed out of the final state.

\HTK\ is principally concerned with continuous\index{HMM!parameters}
density models in which each observation probability distribution
is represented by a mixture Gaussian density.  In this case,
for state $j$
the probability $b_{j}(\bm{o}_t)$ of generating 
observation $\bm{o}_t$ is given by
}

\hequation{
  b_{j}(\bm{o}_t) = \prod_{s=1}^S \left[
     \sum_{m=1}^{M_{js}} c_{jsm} {\cal N}(\bm{o}_{st};
                    \bm{\mu}_{jsm}, \bm{\Sigma}_{jsm})
  \right]^{\gamma_s}
}{cdpdf}
where $M_{js}$ is the number of mixture components\index{mixture component} 
in state $j$ for stream $s$,
$c_{jsm}$ is the weight of the $m$'th  component and 
${\cal N}(\cdot; \bm{\mu}, \bm{\Sigma})$ is a multivariate Gaussian
with mean vector\index{mean vector} $\bm{\mu}$ and 
covariance matrix\index{covariance matrix} $\bm{\Sigma}$, that
is\index{output probability!continuous case}
\hequation{
{\cal N}(\bm{o}; \bm{\mu}, \bm{\Sigma}) =
       \frac{1}{\sqrt{(2 \pi)^n | \bm{\Sigma} |}} 
       e^{- \frac{1}{2}(\bm{o}-\bm{\mu})' \bm{\Sigma}^{-1}(\bm{o}-\bm{\mu})}
}{gnorm}
where $n$ is the dimensionality of $\bm{o}$.  The exponent $\gamma_s$ is
a stream weight\index{stream weight} and its 
default value is one.  Other values can be
used to emphasise particular streams, however, none of the standard
\HTK\ tools manipulate it.

\HTK\ also supports discrete probability 
distributions\index{discrete probability} in which
case \index{output probability!discrete case}
\hequation{
  b_{j}(\bm{o}_t) = \prod_{s=1}^S \left\{
     P_{js}[v_s(\bm{o}_{st})]
  \right\}^{\gamma_s}
}{ddpdf}
where $v_s(\bm{o}_{st})$ is the output of the vector 
quantiser for stream $s$
given input vector $\bm{o}_{st}$ and $P_{js}[v]$ is the 
probability of state $j$ generating symbol $v$ in stream $s$.

In addition to the above, any model or state can have an
associated vector of duration parameters
\index{duration parameters} $\{d_k\}$\footnote{
No current \HTK\ tool can estimate or use these.
}.  
Also,
it is necessary to specify the kind of the observation
vectors, and the width of the observation vector in each stream.
Thus, the total information needed to define a single HMM is 
as follows
   \begin{itemize}
    \item type of observation vector
    \item number and width of each data stream
    \item optional model duration parameter vector
    \item number of states
    \item for each emitting state and each stream
        \begin{itemize}
           \item mixture component weights or discrete probabilities
           \item if continuous density, then means and covariances
           \item optional stream weight vector    
           \item optional duration parameter vector
         \end{itemize}  
    \item transition matrix
\end{itemize}
The following sections explain how these are defined.

\mysect{Basic HMM Definitions}{OneHMM}

Some \HTK\ tools require a single HMM to be defined.  For example, the
isolated-unit re-estimation tool \htool{HRest} would be invoked as
\begin{verbatim}
    HRest hmmdef s1 s2 s3 ....
\end{verbatim}

\noindent
This would cause the model defined in the file \texttt{hmmdef}
to be input and its parameters re-estimated using the speech data
files \texttt{s1}, \texttt{s2}, etc.\index{HMM definition!basic form}

\sideprog{hmm1def}{60}{Definition for Simple L-R HMM}{
\hmmc{h}{hmm1} \\
\hmkw{BeginHMM} \\
\> \hmkw{VecSize} 4 \hmkw{MFCC}  \\
\> \hmkw{NumStates} 5  \\
\> \hmkw{State} 2 \\
\>\>    \hmkw{Mean} 4 \\
\>\>\>       0.2 0.1 0.1 0.9  \\
\>\>    \hmkw{Variance} 4 \\
\>\>\>       1.0 1.0 1.0 1.0  \\
\> \hmkw{State} 3 \\
\>\>    \hmkw{Mean} 4 \\
\>\>\>       0.4 0.9 0.2 0.1  \\
\>\>    \hmkw{Variance} 4 \\
\>\>\>       1.0 2.0 2.0 0.5  \\
\> \hmkw{State} 4 \\
\>\>    \hmkw{Mean} 4 \\
\>\>\>       1.2 3.1 0.5 0.9  \\
\>\>    \hmkw{Variance} 4 \\
\>\>\>       5.0 5.0 5.0 5.0  \\
\> \hmkw{TransP} 5 \\
\>\>   0.0 0.5 0.5 0.0 0.0 \\
\>\>   0.0 0.4 0.4 0.2 0.0 \\
\>\>   0.0 0.0 0.6 0.4 0.0 \\
\>\>   0.0 0.0 0.0 0.7 0.3 \\
\>\>   0.0 0.0 0.0 0.0 0.0  \\
\hmkw{EndHMM}
}{}
HMM definition files consist of a sequence of symbols representing
the elements of a simple language.  These symbols are mainly
keywords written within angle brackets and integer and
floating point numbers. 
The full \HTK\ definition language is presented
more formally later in section~\ref{s:hmmdef}.  For now, the
main features of the language will be described by some
examples.\index{HMM definition!symbols in}

Fig~\href{f:hmm1def} shows a HMM definition corresponding to the simple
left-right HMM illustrated in Fig~\href{f:hmm1}.  It is a continuous density
HMM with 5 states in total, 3 of which are emitting.  The first symbol in the
file \hmmt{h} indicates that the following string is the name of a macro of
type \textsf{h} which means that it is a HMM definition (macros are explained
in detail later).  Thus, this definition describes a HMM called ``hmm1''.  
Note that HMM names should be composed of alphanumeric characters only and must
not consist solely of numbers. The HMM definition itself is bracketed by the
symbols \hmkw{BeginHMM}\index{beginhmm@$<$BeginHMM$>$} and
\hmkw{EndHMM}\index{endhmm@$<$EndHMM$>$}.\index{HMM name}

The first 
line of the definition proper specifies\index{HMM definition!global features}
the \textit{global} features of the HMM.  In any system
consisting of many HMMs, these
features will be the same for all of them.
In this case, the global definitions indicate that
the observation vectors have 4 components
(\hmkw{VecSize}\index{vecsize@$<$VecSize$>$} 4) and that they denote
MFCC coefficients\index{MFCC coefficients} (\hmkw{MFCC}).

The next line specifies the number of states in the HMM. There
then follows a definition for each emitting state $j$, each of which
has a single mean 
vector $\bm{\mu}_j$ introduced by the keyword \hmkw{Mean}
\index{mean@$<$Mean$>$} 
and a diagonal variance vector $\bm{\Sigma}_j$
introduced by the keyword \hmkw{Variance}.
\index{variance@$<$Variance$>$}
The definition ends with the transition matrix $\{a_{ij}\}$
introduced by the keyword 
\hmkw{TransP}\index{transp@$<$TransP$>$}.  
\index{HMM definition!mean vector}
\index{HMM definition!covariance matrix}
\index{HMM definition!transition matrix}

Notice that
the dimension of each vector or matrix is specified
explicitly before listing the component values.  These
dimensions must be consistent with the corresponding observation 
width (in the case of output distribution parameters) or
number of states (in the case of transition matrices).
Although in this example they could be inferred, 
\HTK\ requires that 
they are included explicitly since, as will
be described shortly, they can be detached from the HMM definition
and stored elsewhere as a macro.\index{vector dimensions}\index{matrix dimensions}


The definition for \textsf{hmm1} makes use of many defaults.
In particular, there is no definition for the number of
input data streams or for the number of 
mixture components per output distribution.  Hence, in both
cases, a default of 1 is assumed.

Fig~\href{f:hmm2def} shows a HMM definition in which
the emitting states are 2 component mixture Gaussians.
The number of mixture components in each state $j$ is indicated by the keyword
\hmkw{NumMixes}\index{nummixes@$<$NumMixes$>$} and each mixture component 
is prefixed by the keyword \hmkw{Mixture}\index{mixture@$<$Mixture$>$}  followed by the 
component index $m$ and component weight $c_{jm}$.  Note
that there is no requirement for the number of mixture components
to be the same in each distribution.\index{HMM definition!mixture components}


State definitions and the mixture components within them may be
listed in any order.  When a HMM definition is loaded, a check is made
that all the required components have been defined.  In addition,
checks are made that the mixture component weights and each row
of the transition matrix sum to one.
If very rapid loading is required, this consistency checking can be inhibited
by setting the Boolean configuration variable 
\texttt{CHKHMMDEFS}\index{chkhmmdefs@\texttt{CHKHMMDEFS}} to
false.

As an alternative to diagonal variance vectors, a Gaussian distribution
can have a full rank covariance\index{full rank covariance} matrix.   An example of
this is shown in the definition for \textsf{hmm3} shown in 
Fig~\href{f:hmm3def}.  Since covariance matrices are symmetric,
they are stored in upper triangular form\index{upper triangular form}  
i.e. each row of the matrix
starts at the diagonal element\footnote{
Covariance matrices are actually stored internally in lower triangular
form}.  Also, covariance matrices are stored
in their inverse form i.e.\ HMM definitions contain $\bm{\Sigma}^{-1}$
rather than  $\bm{\Sigma}$.  To reflect this, the keyword chosen to
introduce a full covariance matrix is \hmkw{InvCovar}\index{invcovar@$<$InvCovar$>$}.  


\sideprog{hmm2def}{60}{Simple Mixture Gaussian HMM}{
\hmmc{h}{hmm2} \\
\hmkw{BeginHMM} \\
\>\hmkw{VecSize} 4 \hmkw{MFCC} \\