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Fig. 10.4
Partition of the fuzzy universe of discourse
0
@
1
A
0:8141 0:1481 0:0369
0:1481 0:8141 0:0369
0:0369 0:0369 0:9247
W D
(10.13)
It can be easily shown that W
u
1
D
u
1
, W
u
2
D
u
2
, W
u
3
D
u
3
, i.e.
u
1
,
u
2
,
u
3
are stable states (attractors) of the network. The eigenvalues of matrix W are
1
D 0:667,
2
D 0:888, and
3
D 1:0. The associated eigenvectors of W
are
v
1
D Œ0:7071; 0:7071;0
T
,
v
2
D Œ0:4066; 0:4066;0:8181
T
, and
v
3
D
Œ0:5785;0:5785;0:5750
T
. It can be observed that
v
1
is collinear to
u
3
,
v
2
is collinear
to
u
2
, and
v
3
is collinear to
u
1
. That was expected from Lemma
1
, in Sect.
10.1
.
Next, the elements of the weight matrix W are considered to be stochastic
variables, with p.d.f. (possibility distribution) as the one depicted in Fig.
7.1
b and
thus matrix W , given by Eq. (
10.13
), can be decomposed into a superposition
of weight matrices W
i
. Assume that only the non-diagonal elements of W are
considered and that the possibility distribution of the stochastic variables
w
ij
is
depicted in Fig.
10.4
.
Then, the weight matrix W is decomposed into a superposition of weight
matrices
W
i
, i D 1; ;8:
8
<
9
=
0
1
0
1
0:405 0:155
0 0:14 0:02
0:14 0 0:02
0:02 0:02 0
@
A
;
@
A
W D
0:405
0:155
:
;
0:155 0:155
8
<
9
=
0
1
0
1
0:405 0:155
0 0:14 0:02
0:14 0 0:04
0:02 0:04 0
@
A
;
@
A
C
0:405
0:845
:
;
0:155 0:845
