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In-Depth Information
Take into account for such an example that the input linguistic variable,
X
,as-
sumes values within the standardized domain
X
=[
,
]
0
10
, whose associated linguis-
=
{
,
,
}
tic terms are
T
Light
Mild
Severe
. Consider yet the membership functions
∈
→
[
,
]
(
)=
{
∈
|
μ
(
)=
}
of the term
M
T
be
μ
M
:
X
0
1
, such that
c
M
x
0
X
x
0
1
M
and
s
denote the core and the support, respectively.
The support of
M
is the set of points in
X
in which
(
M
)=
{
x
0
∈
X
|
μ
M
(
x
0
)
>
0
}
μ
M
(
x
)
is positive while the
core is the set of point in which
μ
M
(
x
)
is unitary.
The linguistic terms
Light
and
Severe
∈
T
assumes the trapezoidal membership function being represented,
respectively, by
s
1
light
,
s
2
light
, such that
s
c
1
light
,
c
2
light
,
(
Light
)=[
s
1
light
,
s
2
light
]
and
c
(
Light
)=[
c
1
light
,
c
2
light
]
and by
s
1
severe
,
c
1
severe
,
c
2
severe
,
s
2
severe
, such that
s
∈
T
is defined as a triangular membership function and represented by the triple
(
Severe
)=[
s
1
severe
,
s
2
severe
]
and
c
(
Severe
)=[
c
1
severe
,
c
2
severe
]
.Theterm
Mild
.
Here, the fuzzy sets in
T
are defined by the following set of supports and cores:
Light
s
1
mild
,
c
1
mild
,
s
2
mild
, such that
s
(
Mild
)=[
s
1
mold
,
s
2
mold
]
and
c
(
Mild
)=[
c
1
mild
]
. Likewise the
input linguistic variable, the output linguistic variable,
Y
, assumes values within
the domain
Y
=
0
,
0
,
1
,
4
,
Mild
=
1
,
4
,
8
and
Severe
=
4
,
8
,
10
,
10
=[
0
,
10
]
. The linguistic terms
T
=
{
Reduced
,
Moderate
,
Strong
}
can
also be defined by their supports and cores as given by
Reduced
=
0
,
0
,
4
,
6
,
Moderate
=
4
,
6
,
9
and
Strong
=
6
,
9
,
10
,
10
. The linguistic terms and their
associated fuzzy sets are depicted in Fig. 16.5.
The mapping,
f
,of
X
=
{
x
}
into
Y
=
{
y
}
, shown in (16.9) is obtained by em-
M
represents
the input measure or observation (cause), and
y
the output (consequence) that can
be associated to decision, diagnosis, assessment, therapeutic conduct, and so on.
Consider also a fuzzy restriction characterizing a signal measurement or a patient
cognition concerning the symptom as depicted in Fig. 16.6b. A Gaussian function
is employed to represent such a subjective measure:
M
ploying the compositional operation,
◦
R
, in such a way that
x
=
ex p
2
−
(
e
−
μ
)
g
(
e
)=
,
(16.10)
2
2
σ
where
μ
is the mean value and
σ
, the standard deviation. Assume, for instance,
that
4. Employing fuzzy sets to stand for the fifth vital sign
of medical condition was first presented in [3] to represent the inherent impreci-
sion, uncertainty and vagueness presented in the pain report and assessment. The
unidimensional fuzzy pain intensity scales therein proposed extend the accepted
classical unidimensional pain scales to fuzzy set theory obtaining the fuzzy visual
analog scale (FVAS), fuzzy numerical rating scale (FNRS), fuzzy qualitative pain
scale (FQPS), and fuzzy face pain scale (FFPS). The input measure as illustrated in
Fig. 16.6b can immediately be associated to any of the FVAS, FNRS, FFPS by using
traditional measurement systems or by a computer program [9]. When interested in
representing physiological, behavioral, psychological, and cultural aspects of in-
dividual life experience encompassing pain-related disabilities, the tridimensional
Fuzzy Professional-Social-Sexual Pain Assessment also furnishes both singleton
or fuzzy pain measurement [6], as portrayed in Fig. 16.6b. Another source of fuzzy
pain is the Fuzzy Musculoskeletal Pain Scale (FUMPS). Such a multi-criteria fuzzy
μ
=
5
.
5and
σ
=
0
.
