Biomedical Engineering Reference
In-Depth Information
d.
Trapezoidal-membership function
.
⎧
⎨
0
if
x<a
x
−
a
if
x
∈
[
a, d
]
d
−
a
m
(
x
)=
1
if
x
∈
[
d, e
]
(3.66)
⎩
b
−
x
if
x
∈
[
e, b
]
b
−
e
0
if
x>b
e.
Gaussian-membership function
.
m
(
x
)=e
−k
(
x−d
)
2
(3.67)
for
k>
0.
f.
Exponential-membership functions
.
1
1+
k
(
x
m
(
x
)=
(3.68)
−
d
)
2
for
k>
1or
d
)
2
k
(
x
−
m
(
x
)=
(3.69)
1+
k
(
x
−
d
)
2
The membership value assigned to an element in the fuzzy set depends on
the application at hand, for example, high temperature in an ore smelting fur-
nace is clearly different from the meaning of high temperature in a house with
gas heating. Several methods are used to estimate the proper membership val-
ues such as horizontal estimation, vertical estimation, pairwise comparison,
and inference. Horizontal estimation uses statistical distributions to estimate
the membership value based on responses from a group of observers. Verti-
cal estimation discretely separates the responses of the observer group and
sections the elements with respect to the importance placed by the group.
In pairwise comparisons, elements are compared with other elements in the
fuzzy set
A
and valued according to a suitable ratio scale. Problem inference
involves constructing approximating functions to a problem that can be used
as membership functions.
3.5.3 Fuzzy Operations
Since a fuzzy set is characterized by its membership function, operations
on fuzzy sets manipulate these functions in a similar manner as standard
functional analysis and set theory. Some of the more common mathematical
notions are outlined in the following.
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