Biomedical Engineering Reference
In-Depth Information
changed depending on the applications from time to time. To make the collabo-
rative grouping system, we introduce some background knowledge that can be
presented as a hypothetical prior probability b
y
that we call hyper-parameter
[
41
,
42
]. Suppose that a
y
ðÞ
is an initial prior probability at time k. The initial
hyper-parameter b
y
can be found as follows:
k
!1
a
y
ðÞ ð
3
:
4
Þ
In practice, we can get the hyper-parameter b
y
using sample training data
instead of the infinite training data with respect to time. After calculating the hyper-
parameter with sample training data using (
3.4
), the hyper-parameter b
y
should be
adaptive with respect to time k. Please note that the hyper-parameter can be selected
based on the global information of sample data. This parameter is selected for
corresponding to the steady state. It can be accomplished using the switching
probability that will be explained in detail in
Sect. 3.4.3
. The adaptive hyper-
parameter can be increased or decreased based on the current switching probability
comparing to the previous switching probability, and can be calculated as follows:
b
y
ðÞ¼
lim
Þþ
Dl
y
;
b
y
ðÞ¼
b
y
k
1
ð
ð
3
:
5
Þ
where Dl
y
is the difference between the current switching probability and the
previous one. Dl
y
can be calculated using a switching probability at time k, i.e.,
l
y
ðÞ
indicating the switching weight of group y. We will describe how to select
the difference Dl
ð Þ
in detail in
Sect. 3.4.4
. After calculating the adaptive hyper-
parameter, the adaptive (ADT) posterior probability p
ADT
(y|z
j
) is calculated at time
k in E-step as follows:
þ
b
y
ðÞ
a
ð
t
y
u z
j
; m
ð
t
y
;
P
ð
t
Þ
p
ADT
y
j
z
j
ðÞ¼
y
ð
3
:
6
Þ
þ
P
P
a
ð
t
y
u z
j
; m
ð
t
y
;
P
ð
t
Þ
G
G
b
y
ðÞ
y
y
¼
1
y
¼
1
Using the modified one, we can proceed to the M-step at time k as follows:
p
ADT
y
j
z
j
ðÞ;
ðÞ¼
L
P
L
a
t
þ
1
ð
Þ
y
j
¼
1
P
L
y
j
z
ð
z
j
p
ADT
p
ADT
y
j
z
j
z
j
;
¼
a
y
L
P
L
m
t
þ
1
ð
Þ
ðÞ¼
j
¼
1
ð
3
:
7
Þ
P
y
L
y
j
z
ðÞ
j
¼
1
p
ADT
j
¼
1
T
p
ADT
y
j
z
j
z
j
m
t
þ
1
P
t
þ
1
ðÞ¼
a
y
L
P
L
ð
Þ
z
j
m
ð
t
þ
1
Þ
ð
Þ
y
y
y
j
¼
1
We can estimate the tth iteration result of the adaptive posterior probability
p
ADT
y
j
z
j
at time k from (
3.6
). Based on the modified result we can calculate the
prior probability a
y
, the mean (m
y
), and the covariance
P
y
in the (t ? 1)th
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