Digital Signal Processing Reference
In-Depth Information
The new estimated parameters in the M-Step, involved in the Baum-Welch
algorithm are calculated as:
T
γ
t
(
j, m
)
t
=1
c
jm
=
(2.63)
T
M
γ
t
(
j, m
)
t
=1
m
=1
T
γ
t
(
j, m
)
x
t
t
=1
μ
jm
=
(2.64)
T
γ
t
(
j, m
)
t
=1
T
γ
t
(
j, m
)(
x
t
−
μ
jm
)
T
μ
jm
)(
x
t
−
Σ
jm
=
t
=1
(2.65)
T
γ
t
(
j, m
)
t
=1
Several issues are of high importance while using HMM such as the topology
of the model, the initial estimates and the number of parameters. For the case
of GMM, it has been mentioned that they can model any distribution with
a sucient number of normal mixtures. However, there is a compromise be-
tween the amount of training data and the number of components for obtaining
probability distributions robustly estimated. On the other hand, for reducing
number of parameters, the features can be decorrelated by means of features
transformation such as those described in Section 2.2.2. Thus, the use of diag-
onal covariance matrices is suitable in the transformed feature space.
2.3.4 Hybrid HMM/ANN
In the previous section, a HMM based on GMM has been presented. In fact,
the GMM is a generative model where the parameters are estimated so that
the likelihood of the training data given the model is maximized. In contrary
to generative models, discriminative models are also widely used in some
ASR application. In Section 2.2.3 a discriminative model based on multi-
layered perceptrons has been explained in detail. As it was mentioned, the
training procedure is based on minimizing an error function. In addition, the
parameters are estimated so that the model discriminates among the output
classes, mainly because a label or target vector at a particular time instance
corresponds to a one and zeros.
We have also seen that by using a softmax activation function as output
layer in a MLP, the output units have probability properties such as they are
between zero and one, and all of them sum to one. In fact, by considering each
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