Biomedical Engineering Reference
In-Depth Information
Table 1. Average clustering results on 10 runs of these algorithms, where
n
indicates the number of
c
p
is the correct cassification percentage (that is, the percentage of simulations in
obtained clusters,
n
=
n
) and Err. is the average error percentage.
which
a
n
=3
n
=4
K
RL
IRL
RL
IRL
n
p
Err.
n
p
Err.
n
p
Err.
n
p
Err.
cc
cc
cc
cc
75
3.1
90
0.53
3.1
90
0.53
4.7
60
3.46
4.1
90
0.26
150
3.3
70
0.33
3.1
90
0.13
4.8
70
1.06
4.4
80
0.73
300
3.7
70
0.60
3.3
80
0.43
5.5
30
3.06
4.6
60
4.80
450
3.5
70
0.37
3.1
90
0.17
4.9
60
0.55
4.4
80
0.31
600
3.2
80
0.25
3.2
80
0.25
5.4
50
0.61
4.6
50
0.31
750
3.3
80
0.06
3.0
100
0.00
5.4
30
3.49
4.6
60
3.12
900
3.2
90
3.23
3.0
100
0.00
5.5
20
0.56
4.8
40
0.42
Av.
3.3
78
0.76
3.1
90
0.21
5.2
46
1.82
4.5
66
1.42
concept. This concept is thus formed and learned.
So, the process of formation and learning concepts
can be seen in Figure 3.
SIMULATIONS
a.
In order to show the ability of RL and IRL
to perform a clustering task, as mentioned
above, several simulations have been made
whose purpose is the clustering of discrete
data.
Iterative relationships learning method:
RL
can be improved in many ways. In this sense, an
iterative approach, IRL, to enhance the solution
given by RL, is presented.
Several datasets have been created, each of
them formed by
K
50-dimensional patterns ran-
domly generated around
n
centroids, whose com-
ponents were integers in the interval [1,10]. That
is, the
n
centroids were first generated and input
patterns were formed from them by introducing
some random noise modifying one component
of the centroid with probability 0.018. So, the
Hamming distance between input patterns and
the corresponding centroids is a binomial distribu-
tion B(50,0.018). Patterns are equally distributed
among the
n
clusters. It must be noted that pat-
terns may have Hamming distance even 5 or 6
from their respective centroid, and new clusters
can be formed by this kind of patterns.
So, a network with
N =
50 neurons taking
value in the set
Suppose that, by using equation (5.1) of RL,
matrix
W
X
related to pattern set
X X k
∈
has been learned and denote by
Y
(
k
)
the stable state
reached by the network (with weight matrix
W
X
)
when beginning from the initial state given by
V
=
X
(
k
)
.
Then, the cardinal of
(
k
)
= {
:
}
k
Y k
∈
is (no
multiplicities) the number of classes that RL
finds,
n
.
(
)
{
:
}
k
Y Y k
∈
can be considered (with
all multiplicities included) as a new pattern set,
formed by a noiseless version of patterns in
X
.
So, if applying a second time RL to
Y
, by using
equation (5.1) to build a new matrix
W
Y
, better
results are expected than in the first iteration,
since the algorithm is working with a more refined
pattern set.
This whole process can be repeated iteratively
until a given stop criterion is satisfied. For example,
when two consecutive classifications assign each
pattern to the same cluster.
(
)
:= {
:
}
=
has been consid-
ered. The parameter of learning reinforcement
has been chosen
1.=
. It has been observed
that similar results are obtained for a wide range
of values of b.
The results obtained in our experiments are
shown in Table 1. It can be observed not only the
{1,
,10}
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