Biomedical Engineering Reference
In-Depth Information
3.5.2 Fuzzy Membership Functions
Membership functions are used to describe the membership value given to
fuzzy set elements. Their use depends on the meaning or linguistic variable
attached to the fuzzy set, for example, if the values are taken to denote the
linguistic variable “high,” then it must be defined with respect to a quan-
tity such as temperature. Examples of membership functions are given in the
following:
a.
Triangular membership function
.
⎧
⎨
0
if
x
≤
a
x
−
a
if
x ∈
[
a, d
]
d
−
a
m
(
x
)=
(3.62)
⎩
b
x
b − d
−
if
x
∈
[
d, b
]
0
if
x
≥
b
where
d
is the modal value and
a
and
b
are the upper and lower bounds
of
m
(
x
), respectively.
b.
Γ-
membership function
.
⎧
⎨
0
if
x
≤
a
m
(
x
)=
(3.63)
⎩
e
−k
(
x−a
)
2
1
−
if
x>a
or
⎧
⎨
0
if
x
≤
a
m
(
x
)=
(3.64)
a
)
2
k
(
x
−
⎩
if
x>a
1+
k
(
x
−
a
)
2
where
k>
0.
c.
S-membership function
.
⎧
⎨
if
x
≤
a
0
2
x
2
−
a
if
x
∈
[
a, d
]
b
−
a
m
(
x
)=
(3.65)
2
x
2
⎩
−
a
1
−
if
x
∈
[
d, b
]
b
−
a
if
x
≥
b
1
Here the point
d
=(
a
+
b
)
/
2 is the crossover of the
S
-function.
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