MACHINE LEARNING TECHNIQUES - LASA

167
7.2.4 Decoding an HMM
There are two ways to use a HMM once built. It can be used either for classification or for
prediction.
Classification: assume that you have trained two different HMM-s on the basis of two classes of
time series (say you trained a HMM to recognize a waving gesture and another HMM to
recognize a pointing gesture on the basis of ten examples each). One can use these two HMM-s
classify the observation of a new times series (a new movement) to determine whether it
belonged more to the first or second type of time series (i.e. whether the person was waving or
rather pointing). This can be done by computing the likelihood of the new times series given the
model and determining simply which model is most likely to explain this time series. If the new
times series is X’, the set of parameters of each model are λ 1 = ( A 1 , B
1 , π 1 ), λ 2 = ( A 2 , B
2 , π
2 )
respectively, then the likelihoods of X’ to have been generated by each of these two models
1 2
are p( X'| ), p( X'|
)
backward procedure.
λ λ . These likelihoods can be computed using either the forward or
One can then decide which of model 1 and 2 is most likely to have explained the time series by
directly comparing the two likelihoods. That is, model 1 is more likely to have generated X’ if:
( '| λ
1 ) p( X'|
λ
2
)
p X
> (7.11)
Such a comparison is however difficult as the value of the likelihood is unbounded (recall that pdf
are positive functions with no upper bound). It can take very large values just as an effect of one
model having much more parameters than the other. Hence, in practice, one sometimes
normalizes the likelihood by the number of parameters of each model. Since one only compares
two values, one has no idea how well any of the two models actually explain the data. It could be
that both likelihoods are very low and that hence the observed time series belongs to neither of
these two models. In speech processing, people avoid this problem by creating a “garbage”
model that ensures that, if a times series is not well represented by any of the meaningful model,
then it will be automatically classified in the garbage model. Building a garbage model requires
training the model on a large variety of things that cannot be the pattern one is looking for (e.g.
noise from cars, doors shutting down, etc). Generating such a large set of non-class samples is
not very practical. An alternative is to retain a value of the average likelihood obtained with the
training examples for each model and use this to ensure that the likelihood of the new time series
is at least as big as that seen during training (or within some bounds).
Prediction: Once built a HMM can be used to make prediction about the evolution of a given time
series if provided with part of the time series. Clearly the farther back in time one can go, the
better the prediction.
Say that you have built a HMM to represent the temporal evolution of the weather day after day
taking into account the seasons and the particular geographical area, one can then use the model
to predict the weather for the following couple of days. The farther in the future the prediction, the
less reliable it may be.
To perform prediction one uses the Viterbi algorithm. The viterbi algorithm aims at generating the
most likely path. Given a HMM model with parameters λ = ( AB , , π)
, one looks for the most
likely state sequence { }
Q= q , , 1
… q T
for T time steps in the future, i.e. one optimizes for:
Q arg max P( Q' | O, λ)
= (7.12)
Q'
© A.G.Billard 2004 – Last Update March 2011