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symmetric, but not transitive. In order to obtain a transitive similarity relation
S
i
,
the max-min transitive closure
C
T
i
, of
C
i
is calculated (Klir and Yuan, 1995):
x
C
c
max
C
C
D
T C
.
i
i
i
i
x If
CC
c z
,
set
CC
c
,
and go to previous step.
i
i
i
i
x Stop:
C
c
,
set
SC
.
Ti
i
i
Ti
Here, the
t
-
norm
used is the
min
-
operator
and “o
T
” is the
sup
-
t
composition. The
lm
th
element of the fuzzy similarity relation
> @
, of size
M
u , gives the
transitive similarity between the concepts represented by the fuzzy sets
A
li
and
A
mi
.
The merging of similar fuzzy sets takes place by applying a threshold
Ss
i
ilm
O to
the similarity relation. Therefore, the similar fuzzy sets are merged, when their
similarities are greater than a threshold O, to produce a fuzzy set representing
generalization of the individual concepts represented by the similar fuzzy sets.
Thereafter, updated rule base is checked for any fuzzy set which is similar to the
universal fuzzy set. The approach is illustrated in Algorithm 7.2 and Example 7.1.
0,1
Algorithm 7.2.
Algorithm of similarity relations
Given a fuzzy rule base
^
`
l
RRl
with the
l
th
rule
R
l
: If
x
1
is
G
l
1
and, ...,
"
1, 2,
,
M
,
and
x
n
is
G
l
n
Then
y
l
=
f
(
x
1
, ...,
x
n
), where
G
l
i
,
i
= 1, 2,...,
n,
are fuzzy sets with
>@
OJ
.
,
0,1
membership functions
P
:
x
o
0,1 ,
select
l
i
i
G
Repeat for inputs i =1, 2, ..., n;
Step 1
.
Calculate similarity relation:
> @
;,
Cc l m
"
,2, , ;
M
i
ilm
>@
,
Ss
C
i
ilm
Ti
where the elements of the MM
u
fuzzy compatibility relation C
i
are given by
> @
c
S
l
x
,
m
x
.
GG
ilm
i
i
Step 2
.
Aggregate similar fuzzy sets
for l =1,2, ..., M
^
`
^
`
l
m
S
!
O
,
m
1, 2,
"
,
M
GG
Merge
i
i
ilm
^`
l
l
,
G
c
G
i
i
end
Step 3
to
Step-6
.
The steps 3-6 are same as in iterative merging algorithm.
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