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In-Depth Information
The classification can be the movie genre that the viewer would like to
watch, such as
C
=
.
All the attributes are vague by definition. For example, people's feelings
of happiness, indifference, sadness, sourness and grumpiness are vague
without any crisp boundaries between them. Although the vagueness of
“Age Group” or “Time of Day” can be avoided by indicating the exact age
or exact time, a rule induced with a crisp decision tree may then have an
artificial crisp boundary, such as “IF Age
<
16 THEN action movie”. But
how about someone who is 17 years of age? Should this viewer definitely
not watch an action movie? The viewer preferred genre may still be vague.
For example, the viewer may be in a mood for both comedy and drama
movies. Moreover, the association of movies into genres may also be vague.
For instance, the movie “Lethal Weapon” (starring Mel Gibson and Danny
Glover) is considered to be both comedy and action movie.
Fuzzy concept can be introduced into a classical problem if at least
one of the input attributes is fuzzy or if the target attribute is fuzzy. In
the example described above, both input and target attributes are fuzzy.
Formally, the problem is defined as following
[
Yuan and Shaw (1995)
]
:
Each class
c
j
is defined as a fuzzy set on the universe of objects
U
.
The membership function
µ
c
j
(
u
) indicates the degree to which object
u
belongs to class
c
j
. Each attribute
a
i
is defined as a linguistic attribute
which takes linguistic values from
dom
(
a
i
)=
{
Action
,
Comedy
,
Drama
}
.
Each linguistic value
v
i,k
is also a fuzzy set defined on
U
.Themembership
µ
v
i,k
(
u
) specifies the degree to which object
u
's attribute
a
i
is
v
i,k
. Recall
that the membership of a linguistic value can be subjectively assigned or
transferred from numerical values by a membership function defined on the
range of the numerical value.
{
v
i,
1
,v
i,
2
,...,v
i,|dom
(
a
i
)
|
}
14.4 Fuzzy Set Operations
Like classical set theory, fuzzy set theory includes such operations as union,
intersection, complement, and inclusion, but also includes operations that
have no classical counterpart, such as the modifiers concentration and
dilation, and the connective fuzzy aggregation. Definitions of fuzzy set
operations are provided in this section.
Definition 14.2.
The membership function of the union of two fuzzy sets
A
and
B
with membership functions
µ
A
and
µ
B
respectively is defined
as the maximum of the two individual membership functions
µ
A∪B
(
u
)=
max
{
µ
A
(
u
)
,µ
B
(
u
)
}
.
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