Biomedical Engineering Reference
In-Depth Information
select the estimated feature metrics, i.e., objective function J(
) using within-class
scatter (S
W
) and between-class scatter (S
B
)[
38
,
39
]. Here we define the S
W
as
follows:
2
;
S
W
¼
z
P
S
i
¼
P
u
i
¼
n
i
P
G
n
i
n
i
S
i
;
x
ij
u
i
x
ij
;
ð
6
:
4
Þ
i
¼
1
j
¼
1
j
¼
1
where z is the element number of a feature combination vector in S
W
, G is the total
number of class in the given datasets and n
i
is the data number of the feature
combination vector in the ith class. We define the S
B
as follows:
S
B
¼
P
u
¼
n
P
G
n
Þ
2
;
n
i
u
i
u
ð
x
i
;
ð
6
:
5
Þ
i
¼
1
i
¼
1
where n is the total data number of the feature combination vector. The objective
function J to select the optimal feature combination vector can be written as
follows:
;
S
W
S
B
J
ð
x
Þ¼
arg min
x
ð
6
:
6
Þ
where x can be the estimated feature vector for the rest of the modules for
breathing patterns classification.
6.3.2 Clustering of Respiratory Patterns Based on EM
After extracting the estimated feature vector (x) for the breathing feature, we can
model the joint probability density that consists of the mixture of Gaussians
/ x
j
l
m
;
P
m
for the breathing feature as follows [
31
,
32
]:
p
ð
x
;
H
Þ¼
P
a
m
u
ð
x
j
l
m
;
P
m
Þ;
a
m
0
;
P
M
M
a
m
¼
1
;
ð
6
:
7
Þ
m
¼
1
m
¼
1
where x is the d-dimensional feature vector, a
m
is the prior probability, l
m
is the
mean vector, R
m
is the covariance matrix of the mth component data, and the
parameter H
a
m
;
l
m
;
P
m
M
m
¼
1
is a set of finite mixture model parameter vec-
tors. For the solution of the joint distribution p x
; ð Þ
, we assume that the training
feature vector sets x
k
are independent and identically distributed, and our purpose
of the M components
that maximize the log-likelihood function as follows [
32
,
33
]:
of this section is to estimate the parameters a
m
;
l
m
;
P
m
L
ðÞ¼
X
K
log p x
k
;
H
ð
Þ;
ð
6
:
8
Þ
k
¼
1
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