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where
R
is a fuzzy relation (16.3) and
M
is the input measure; fuzzy or singleton
ones. The expression in (16.7) is decomposed into:
pro j
con j
cyl
M
)
,
)
N
(
y
)=
(
x
R
(
x
,
y
sup
x
M
)
=
(
x
)
∧
R
(
x
,
y
(16.8)
sup
x
M
)
,
N
=
(
x
)
∧∨
i
=
1
R
i
(
x
,
u
since the operation
is related to
pro j
,
con j
,and
cyl
which refers, respectively,
to the principle of projection, principle of conjunction, and cylindrical extension.
The composition of the fuzzy relations is important for building up a fuzzy mapping
by rules and directly related to a fuzzy graph - as presented by Zadeh and not by
Rosenfeld [34].
The importance of the compositional rule of inference is due to the fact that
it is the key mechanism used for accomplishing the linguistic (Mamdani, Larsen
etc.) fuzzy inference system and the interpolative (Takagi-Sugeno, Tsukamoto etc.)
fuzzy inference system.
◦
16.3
Fuzziness in Medical Therapeutic Conduct
and Measurement
Suppose there is a fuzzy diagnostic support system in which there is just one input
linguistic variable related to a
signal
or
symptom
as well as one output linguistic
variable associated to an
action
, for instance, a
prescription, classification etc.
that
a professional should take. Consider, for example, the simplified and hypothetical
set of medical fuzzy IF-THEN rules in the form:
:IF
x
is Light THEN
y
is Reduced
f
∗
:IF
x
is Mild THEN
y
is Moderate
:IF
x
is Severe THEN
y
is Strong
(16.9)
.
The set of inference rules can be comprehended as the knowledge base to a certain
domain of problem. Consider that the input linguistic variable,
X
, can be assigned,
for instance, to
pain
meanwhile the output linguistic variable,
Y
, can be related to
opioid
,orthe
dose of opioid
to be applied to the patient. In so doing, this set of rule
works as a meta mental modeling representing the manner a professional would de-
termine the dose of opioid to be administered according to the pain reported by the
patient or measured by instruments. The input linguistic terms assigned
light
,
mild
,
and
severe
have their respective membership functions shown in Fig. 16.5a. The
output linguistic terms assigned
reduced
,
moderate
,and
strong
have their respec-
tive membership functions shown in Fig. 16.5b. These membership functions part,
respectively, the universes of discourse associated to the input linguistic variable,
X
=
pain, and the output linguistic variable,
Y
=
opioid.
