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alternatives to previous proposals. Detailed comparison of our approach with
these related works is ongoing. Implementation and experimental testing of
our approach will be covered by future research.
A Definitions of Similarity Functions
Here, we review some basic definitions of the similarities between fuzzy sets
over finite domains [3]. Given two membership functions,
µ
1
and
µ
2
, for fuzzy
subsets
F
1
and
F
2
, respectively, of a finite domain
V
,
sim
(
µ
1
,µ
2
) can be
defined in several ways.
One way is based on the cardinality of the intersection and union of two
sets. This is a generalization of the Jaccard index (also called the coe
cient of
similarity or index of commonality) between crisp sets [5]. For the crisp case,
the Jaccard index between two crisp sets
F
and
G
is defined as
|
F
∩
G
|
|F∪G|
,where
|·|
denotes the cardinality of a set. Analogously, for fuzzy sets,
sim
(
µ
1
,µ
2
)=
v
∈
V
µ
1
(
v
)
⊗
µ
2
(
v
)
µ
2
(
v
)
=
|
F
1
∩
F
2
|
v∈V
µ
1
(
v
)
,
(10)
⊕
|
F
1
∪
F
2
|
where
denotes the sigma count of a fuzzy set.
Another definition of similarity based on cardinality is called the simple
matching coe
cient [18], defined as
|·|
|
F
1
∩
F
2
|
+
|
F
1
∩
F
2
|
,
(11)
|
V
|
where
F
i
is the complement set of
F
i
in
V
,for
i
=1
,
2.
Yet another definition measures the degree of mutual inclusion of two fuzzy
sets.
sim
(
µ
1
,µ
2
)=
v
∈
V
µ
1
(
v
)
↔
⊗
µ
2
(
v
)
,
(12)
|
V
|
where
a
↔
⊗
b
=(
a
→
⊗
b
)
⊗
(
b
→
⊗
a
) is the equivalence function with respect
to a t-norm
⊗
.
=
k
,
µ
1
and
µ
2
can be seen as two
k
-dimensional vectors, then the
similarity can be measured with the cosine of the angle between two vectors:
As
|
V
|
µ
1
◦
µ
2
sim
(
µ
1
,µ
2
)=
,
(13)
µ
1
·
µ
2
where
µ
1
◦
µ
2
is the inner product of
µ
1
and
µ
2
,and
µ
i
is the length of
µ
i
,
for
i
=1
,
2.
