Biomedical Engineering Reference
In-Depth Information
a
1
,
1
a
2
,
2
a
1
,
2
S
1
S
2
a
2
,
1
b
2
,
1
b
1
,
2
b
1
,
1
b
2
,
2
Y
1
Y
2
(a)
S
(
t
)
:
S
1
S
2
S
2
S
1
S
2
Y
(
t
)
:
Y
2
Y
1
Y
2
Y
1
Y
1
.
.
.
.
.
time
(b)
FIGURE 2.2:
(a) An example of two state Hidden Markov model.
S
1
and
S
2
are
states of the model,
a
i
,
j
are probabilities of transition from state
i
to
j
.
Y
1
and
Y
2
are
observations,
b
i
,
j
are probabilities of observation
Y
j
if the system is in state
S
i
.(b)A
possible sequence of states and observations. Arrows indicate the dependence.
solution to this problem is usually obtained with the Viterbi algorithm [Viterbi,
1967].
Hidden Markov Model (HMM) Toolbox for MATLAB written by Kevin Mur-
phy supports maximum likelihood parameter estimation, sequence classification and
computation of the most probable sequence. The toolbox can be downloaded from
2.2.2 Kalman filters
AKalmanfilter is a method for estimation of the state of a linear dynamic system
(LDS) discretized in time domain. At time
k
the system is in a
hidden
state
x
k
.It
is hidden since it can't be directly observed. The observer's knowledge about the
state of the system comes from measurements
z
k
which are distorted by noise
w
k
.
Formally it can be expressed as:
z
k
=
M
k
x
k
+
w
k
(2.19)
where
M
k
is the matrix describing the linear operation of taking the observation. The
measurement noise
w
k
is assumed to come from a zero mean normal distribution
with covariance matrix
R
k
.
The current state of the system
x
k
is assumed to depend only on the previous state
x
k
−
1
, on the current value of a control vector
u
k
, and current value of a random
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