Information Technology Reference
In-Depth Information
Let
d
=
P
−
P
be Euclidean distance between cells
V
i
and
V
j
. Let
h
be given
ij
i
j
E
threshold.
Definition 3
. Cell
V
i
recognizes cell
V
k
if the following conditions are satisfied:
k
c
=
c
,
d
ik
<
h
,
d
<
d
,
∀
V
j
∈
W
,
j
≠
i
,
k
≠
j
.
i
ik
ij
Let us define the behavior of SFIN by the following two rules.
Rule 1 (Apoptosis)
. If cell
V
i
∈
W
recognizes cell
V
k
∈
W
then remove
V
i
from
SFIN.
Rule 2 (Immunization)
. If cell
V
k
∈
W
is nearest to cell
V
i
∈
W
\
W
among all cells
0
of SFIN:
d
<
d
,
∀
V
j
∈
W
,
whereas
c
≠
c
,
then add
V
i
to SFIN.
ik
ij
i
k
2.2 Pattern Recognition
Definition 4
. Pattern is any
n
-dimensional column-vector
Z
=
[
1
z
,...,
z
]'
, where
n
z
,...,
1
z
are real values and (') is symbol of matrix transposing.
n
q
Definition 5
. Pattern recognition is mapping
Z
→
R
and assigning to
Z
a class
c
of
nearest cell of SFIN.
2.3 Training
Let
A
,...,
1
A
be
n
-dimensional training patterns with known classes
c
,...,
1
c
.
m
Definition 6
. Training is mapping of training patterns to cells of SFIN
W
:
A
→
V
,...,
A
→
V
, and application of the rules of Apoptosis and Immunization to
1
1
m
m
all cells of
W
.
Let
A
=
[
A
,...,
A
]'
be training matrix of dimension
m
×
n
. Consider singular
1
value decomposition (SVD: see, e.g., [11]) of this matrix:
'
1
'
2
'
3
'
A
=
s
Y
X
+
s
Y
X
+
s
Y
X
+
...
+
s
Y
X
, (1)
1
1
2
2
3
3
r
r
r
where
r
is the rank of matrix
A
,
s
are singular values and
Y
,
X
are left and right
k
k
k
'
'
'
singular vectors with the following properties:
Y
k
Y
=
1
,
X
k
X
=
1
,
Y
k
Y
=
0
,
k
k
i
'
X
k
X
.
Consider the following mapping
=
0
,
i
≠
k
,
k
=
1
,...,
r
i
q
Z
→
P
∈
R
of any
n
-dimensional pattern
Z
:
1
p
=
Z
'
X
,
k
=
1
,...,
q
. (2)
k
k
s
k
