.#adapt.tex.1.2

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\mychap{HMM Adaptation}{Adapt}

\sidepic{headapt}{80}{
Chapter~\ref{c:Training} described how the parameters are estimated
for plain continuous density HMMs within \HTK, primarily using the
embedded training tool \htool{HERest}. Using the training strategy
depicted in figure~\ref{f:subword}, together with other techniques can
produce high performance speaker independent acoustic models for a 
large vocabulary recognition system. However it is possible to build
improved acoustic models by tailoring a model set to a specific
speaker. By collecting data from a speaker and training
a model set on this speaker's data alone, the speaker's characteristics
can be modelled more accurately. Such systems are commonly 
known as \textit{speaker dependent} systems, and on a typical word
recognition task, may have half the errors of a speaker
independent system. The drawback of speaker dependent systems is that
a large amount of data (typically hours) must be collected in order to
obtain sufficient model accuracy.
}

Rather than training speaker dependent models, 
\textit{adaptation} techniques can be applied. In this case, by using
only a small amount of data from a new speaker, a good speaker
independent system model set can be adapted to better fit the
characteristics of this new speaker.

Speaker adaptation techniques can be used in various different
modes\index{adaptation!adaptation modes}. If
the true transcription of the adaptation data is known then
it is termed 
\textit{supervised adaptation}\index{adaptation!supervised adaptation}, 
whereas if the adaptation
data is unlabelled then it is termed 
\textit{unsupervised adaptation}\index{adaptation!unsupervised
adaptation}.
In the case where all the adaptation data is available in one block,
e.g. from a speaker enrollment session, then this termed \textit{static
adaptation}. Alternatively adaptation can proceed incrementally as
adaptation data becomes available, and this is termed 
\textit{incremental adaptation}.  
% \htool{HVite} can provide unsupervised incremental adaptation.

\HTK\ provides two tools to adapt continuous density HMMs. 
\htool{HERest}\index{headapt@\htool{HERest}} 
performs offline supervised adaptation using various forms of linear
transformation and/or maximum a-posteriori (MAP) adaptation, while
unsupervised adaptation is supported by \htool{HVite} (using only
linear transformations).  In this case \htool{HVite} not only performs
recognition, but simultaneously adapts the model set as the data
becomes available through recognition. Currently, linear
transformation adaptation can be applied in both incremental and
static modes while MAP supports only static adaptation.

This chapter describes the operation of supervised adaptation with the
\htool{HERest} tools.  The first sections of the chapter give an
overview of linear transformation schemes and MAP adaptation and this
is followed by a section describing the general usages of
\htool{HERest} to build simple and more complex adapted systems. The
chapter concludes with a section detailing the various formulae used
by the adaptation tool.  The use of \htool{HVite} to perform
unsupervised adaptation is discussed in section~\ref{s:unsup_adapt}.

\mysect{Model Adaptation using Linear Transformations}{mllr}

\mysubsect{Linear Transformations}{whatismllr}

This section briefly discusses the forms of transform available. Note
that this form of adaptation is only available with diagonal
continuous density HMMs.

The transformation matrices are all obtained by solving a maximisation
problem using the \textit{Expectation-Maximisation} (EM)
technique. Using EM results in the maximisation of a standard
\textit{auxiliary function}. (Full details are available in
section~\ref{s:mllrformulae}.)

\mysubsub{Maximum Likelihood Linear Regression ({\tt MLLRMEAN})}{mumllr}

Maximum likelihood linear regression or MLLR\index{adaptation!MLLR}
computes a set of transformations that will reduce the mismatch
between an initial model set and the adaptation data\footnote{
MLLR can also be used to perform environmental compensation by
reducing the mismatch due to channel or additive noise effects.}.
More specifically MLLR is a model adaptation technique
that estimates a set of linear transformations for the mean and
variance parameters of a Gaussian mixture HMM system. 
%The set of
%transformations are estimated so to as to maximise the likelihood of the
%adaptation data. 
The effect of these transformations is to shift the
component means and alter the variances in the initial system 
so that each state in the HMM system is more likely to generate the 
adaptation data.

The transformation matrix used to give a new estimate of the adapted mean is
given by
\hequation{ 
        \hat{\bm{\mu}} = \bm{W}\bm{\xi}, 
}{mtrans}
where $\bm{W}$ is the $n \times \left( n + 1 \right)$
transformation matrix (where $n$ is the dimensionality of the data)
and $\bm{\xi}$ is the extended mean vector,
\[
        \bm{\xi} = \left[\mbox{ }w\mbox{ }\mu_1\mbox{ }\mu_2\mbox{ }\dots\mbox{ }\mu_n\mbox{ }\right]^T
\]
where $w$ represents a bias offset whose value is fixed (within \HTK) at 1.\\
Hence $\bm{W}$ can be decomposed into
\hequation{
        \bm{W} = \left[\mbox{ }\bm{b}\mbox{ }\bm{A}\mbox{ }\right]
}{decompmtrans}
where $\bm{A}$ represents an $n \times n$
transformation matrix and $\bm{b}$ represents a bias vector. This
form of transform is referred to in the code as {\tt MLLRMEAN}.

\mysubsub{Variance MLLR ({\tt MLLRVAR} and {\tt MLLRCOV})}{vmllr}

There are two standard forms of linear adaptation of the variances. The first
is of the form
\[
        \hat{\bm{\Sigma}}_{m} = \bm{B}_m^T\bm{H}_m\bm{B}_m
\]
where $\bm{H}_m$ is the linear transformation to be estimated and
$\bm{B}_m$ is the inverse of the Choleski factor of $\bm{\Sigma}_{m}^{-1}$,
so
\[ 
        \bm{\Sigma}_{m}^{-1} = \bm{C}_m\bm{C}_m^T
\]
and
\[
        \bm{B}_m = \bm{C}_m^{-1}
\]
This form of transform results in an effective full covariance matrix if
the transform matrix $\bm{H}_m$ is full. This makes likelihood calculations
highly inefficient. This form of transform is only available with a 
diagonal transform and in conjunction with estimating an MLLR transform. The
MLLR transform is used as a parent transform for estimating $\bm{H}_m$.
This form of transform is referred to in the code as {\tt MLLRVAR}.

An alternative more efficient form of variance transformation is also available.
Here, the transformation of the covariance matrix is  of the form
\hequation{ 
        \hat{\bm{\Sigma}} = \bm{H}\bm{\Sigma}\bm{H}, 
}{covtrans}
where $\bm{H}$ is the $n\times n$ covariance transformation matrix.
This form of transformation, referred to in the code as {\tt MLLRCOV}
can be efficiently implemented as a transformation of the means and 
the features.
\hequation{ 
       {\cal N}(\bm{o};\bm{\mu},\bm{H}\bm{\Sigma}\bm{H}) = 
       \frac{1}{|\bm{H}|^2}{\cal N}(\bm{H}^{-1}\bm{o};\bm{H}^{-1}\bm{\mu},\bm{\Sigma}) = 
       {|\bm{A}|^2}{\cal N}(\bm{A}\bm{o};\bm{A}\bm{\mu},\bm{\Sigma})
} {covlike}
where $\bm{A}=\bm{H}^{-1}$.
Using this form it is possible to estimate and efficiently apply full transformations.
{\tt MLLRCOV} transformations are normally estimated using {\tt MLLRMEAN} transformations
as the parent transform.

\mysubsub{Constrained MLLR ({\tt CMLLR})}{cmllr}

Constrained maximum likelihood linear regression or CMLLR\index{adaptation!CMLLR}
computes a set of transformations that will reduce the mismatch
between an initial model set and the adaptation data\footnote{
MLLR can also be used to perform environmental compensation by
reducing the mismatch due to channel or additive noise effects.}.
More specifically CMLLR is a feature adaptation technique
that estimates a set of linear transformations for the features. 
The effect of these transformations is to shift the
feature vector in the initial system 
so that each state in the HMM system is more likely to generate the 
adaptation data.
Note that due to computational reasons, CMLLR is only implemented
within \HTK\ for diagonal covariance, continuous density
HMMs.

The transformation matrix used to give a new estimate of the adapted mean is
given by
\hequation{ 
        \hat{\bm{o}} = \bm{W}\bm{\zeta}, 
}{mtrans2}
where $\bm{W}$ is the $n \times \left( n + 1 \right)$
transformation matrix (where $n$ is the dimensionality of the data)
and $\bm{\zeta}$ is the extended observation vector,
\[
        \bm{\zeta} = \left[\mbox{ }w\mbox{ }o_1\mbox{ }o_2\mbox{ }\dots\mbox{ }o_n\mbox{ }\right]^T
\]
where $w$ represents a bias offset whose value is fixed (within \HTK) at 1.\\
Hence $\bm{W}$ can be decomposed into
\hequation{
        \bm{W} = \left[\mbox{ }\bm{b}\mbox{ }\bm{A}\mbox{ }\right]
}{decompmtrans2}
where $\bm{A}$ represents an $n \times n$
transformation matrix and $\bm{b}$ represents a bias vector.
This form of transform is referred to in the code as {\tt CMLLR}.

\mysubsect{Input/Output/Parent Transformations}{whattransform}
There are three types of linear transform that may be used with 
the HTKTools.
\begin{itemize}

\item {\it Input transform}: the input transform is used to determine
the forward-backward probabilities, hence the component posteriors,
for estimating model and transform parameters. MLLR transforms can be
iteratively estimated by refining the posteriors using a newly
estimated transform.

\item {\it Output transform}: the output transform is the transform
that is generated. The form of the transform is specified using the 
appropriate configuration options.

\item {\it Parent transform}: the parent transform determines the 
model, or features, on which the model set or transform is to be 
generated. For transform estimation this allows {\em cascades} of transforms
to be used to adapt the model parameters. For model estimation this 
supports {\em speaker adaptive training}. Note the current implementation 
only supports adaptive training with CMLLR. Any parent transform can be
used when generating transforms.
\end{itemize}

There is no difference in the storage of the transform parameters depending
on whether it is to be a parent transform or an input transform. There is
also no restrictions on the base classes, or regression classes, that 
are used for each transform.

\mysubsect{Base Class Definitions}{base_classes}
The first requirement to allow adaptation is to specify the set of
the components that share the same transform. This is achieved using a
baseclass.  The baseclass definition files uses the same syntax for
defining components as the \htool{HHEd} command. However, for
baseclass definitions the components must always be specified.

\begin{figure}[htbp]
\begin{verbatim}
  ~b ``global''
  <MMFIDMASK> CUED_WSJ* 
  <PARAMETERS> MIXBASE
  <NUMCLASSES> 1
    <CLASS> 1  {*.state[2-4].mix[1-12]}      
\end{verbatim}
\caption{Global base class definition}
\label{fig:globbase}
\end{figure}
The simplest form of transform uses a global transformation for all
components.  Figure~\ref{fig:globbase} shows a global transformation
for a system where there are upto 3 emitting states and upto 12
Gaussian components per state.

\begin{figure}[htbp]
\begin{verbatim}
  ~b ``baseclass_4.base''
  <MMFIDMASK> CUED_WSJ*
  <PARAMETERS> MIXBASE
  <NUMCLASSES> 4
    <CLASS> 1  {(one,sil).state[2-4].mix[1-12]}
    <CLASS> 2  {two.state[2-4].mix[1-12]}
    <CLASS> 3  {three.state[2-4].mix[1-12]}
    <CLASS> 4  {four.state[2-4].mix[1-12]}
\end{verbatim}
\caption{Four base classes definition}
\end{figure}

These baseclasses may be directly used to determine which components
share a particular transform. However a more general approach
is to use a regression class tree.

\mysubsect{Regression Class Trees}{reg_classes}
\index{adaptation!regression tree}
To improve the flexibility of the adaptation process it is possible to
determine the appropriate set of baseclasses depending on the amount
of adaptation data that is available. If a small amount of data is
available then a \textit{global} adaptation transform
\index{adaptation!global transforms} can be generated. A global transform 
(as its name suggests) is applied
to every Gaussian component in the model set. However as more
adaptation data becomes available, improved adaptation is possible by
increasing the number of transformations. Each transformation is now
more specific and applied to certain groupings of Gaussian components.
For instance the Gaussian components could be grouped into the broad 
phone classes: silence, vowels, stops, glides, nasals, fricatives, etc.
The adaptation data could now be used to construct more specific broad