Biomedical Engineering Reference
In-Depth Information
Let vector
U
have
N
components.
K
components of the vector have values “1.”
Let us introduce the two vectors
U
a
*
and
U
b
*
. Vector
U
a
*
contains
R
(
X
)·
N
components, obtained from the initial vector
U
using permutations
E
(
X
) times.
All the rest of the components of
U
a
*
are zero. Vector
U
b
*
contains (1
—R
(
X
)) ·
N
components, obtained from the initial vector
U
using permutations (
E
(
X
) + 1) times,
and the rest of the components are 0.
The probability
p
a
of “1” in an arbitrary component of vector
U
a
*
is:
:
K
N
p
a
¼
RX
ðÞ
(4.13)
In the vector
U
b
, the corresponding probability
p
b
is:
:
K
N
p
b
¼ð
1
RX
ðÞÞ
(4.14)
Absorption occurs if both vectors
U
a
*
and
U
b
*
have “1” in the same place. The
probability of this event for one component is:
:
K
2
N
2
p
¼
p
a
p
b
¼
RX
ðÞð
1
RX
ðÞÞ
(4.15)
It is easy to show that we have the maximum of
p
when
R
(
X
)
¼
(1 -
R
(
X
))
¼
0.5.
Thus:
K
2
P
N
2
:
(4.16)
4
For all the
N
components of the vector
U
, at least one absorption probability
shall be:
N
K
2
K
2
4
N
p
¼
e
1
ð
1
p
Þ
1
¼
:
(4.17)
N
4
N
2
If we have, for example,
K
¼
16,
N
¼
128,000,
1
2000
p
¼
1
e
¼
0
:
0005
:
(4.18)
Thus, for a large number of neurons
N
and a small number of active neurons
K
,
absorption has a very low probability and has practically no influence on the coding
process. The code vector
V
is composed from all code vectors
U
*
s
of the detected
features:
