Single channel (univariate) signal - Practical Biomedical Signal Analysis Using MATLAB - page 30

2.2 Probabilistic models
2.2.1 Hidden Markov model
An excellent introduction to the Hidden Markov models (HMM) concept, estima-
tion and illustrative applications can be found in [Rabiner, 1989]. Here we review
the basic facts only.
Let us consider a system which can be in a number of states . In discrete time mo-
ments the state of the system may change. The transitions between states are prob-
abilistic. The probability distribution of the next state depends only on the current
state and it doesn't depend on the way the system reached the current state. If the
states of the system can be observed directly then it can be mathematically described
as a regular Markov model. The Markov model is specified by a set of states and the
probabilities of transitions between each pair of states. An extension of the Markov
model allowing for much wider applications is a Hidden Markov model (HMM).
It is hidden in the sense that the states of the system are not observed directly, but
the observations of the model depend in the probabilistic way on the states of the
model. Although the states are not observed directly, often some physical sense can
be attached to them. As the system evolves in time the state transitions occur and the
system passes through a sequence of states S

( t ) . This sequence is reflected in a se-

quence of observations Y . An important property of HMM is that the conditional

probability distribution of the hidden state S

( t )

( t ) at time t , depends only on the hid-

den state S ( t − 1 ) . Similarly, the observation Y ( t ) depends only on the hidden state

S —both occurring at the same time t . A simple example of such a model is shown

in Figure 2.2.
The HMM is specified by the pair of parameters: number of states ( N ) and number
of possible observations per state ( M ), and additionally by three probability distri-
butions governing: state transitions ( A ), observations ( B ), initial conditions (π). The
HMMs can be used to generate possible sequences of observations. But more inter-
esting applications of HMMs rely on:

( t )

• Estimating the model parameters given the sequence of obser-

vations. This can be done with maximum likelihood parameter estimation us-
ing expectation-maximization Baum-Welch algorithm [Baum et al., 1970].

( N , M , A , B , π )

• Given a sequence of observations one can evaluate probability that the se-

quence was generated by a specific model

( N , M , A , B , π ) .

Imagine that we have a set of observations known to be generated by a number of
different models. We can estimate parameters of each of the models. Later, when
a new sequence of observations is obtained, we could identify which model most
probably generated that sequence. In this sense HMMs can be used as classifiers
with supervised learning.
Another problem that can be addressed with HMM is the question of the most
probable sequence of states given the model and the sequence of observations. The