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In-Depth Information
The important new idea in Sanchez's work was his suggestion to use Zadeh's
max-min composition rule as an inference rule to develop diagnoses. Given symp-
tom and diagnosis sets
S
and
D
and an existing fuzzy relation
R
⊆
×
D
between
them, the max-min composition can serve as an “inference rule”, which makes it
possible to deduce imprecise descriptions of a patient's illnesses (fuzzy sets of
D
)
from imprecise symptom descriptions (fuzzy sets of
S
). With this inference rule,
medical diagnoses
D
j
about a patient's disease can be derived by fuzzy logic from
symptoms
S
i
with the help of the medical knowledge represented by the fuzzy rela-
tion
R
. The membership function is then computed as follows:
S
μ
D
i
(
d
)=
max
s
∈
S
min
{
μ
S
i
(
s
)
;
μ
R
(
s
,
d
)
} ,
s
∈
S
,
d
∈
D
.
(3.10)
By taking into account a set
P
of all patients considered and a fuzzy relation
Q
between
P
and the symptom set
S
, it was now possible with the aid of the max-min
composition rule to obtain a fuzzy relation
T
=
Q
◦
R
with the membership function
T
(
p
,
d
)
:
μ
T
(
p
,
d
)=
max
s
∈
S
min
{
μ
Q
(
p
,
s
)
;
μ
R
(
s
,
d
)
},
s
∈
S
,
d
∈
D
,
p
∈
P
.
(3.11)
(
,
)
The membership function of the fuzzy relation
R
is denoted with
. The fuzzy
relation
R
can be expressed as a matrix, the entries of which can be made after inter-
viewing doctors about their diagnostic experiences. This expert medical knowledge
must additionally be translated into
degrees of association
between symptoms and
diagnoses.
Sanchez interpreted this equation in this way: If the condition of a patient
p
is
described with the help of a fuzzy set
A
of symptoms from
S
, then diagnoses from
D
can be associated with this patient
p
with the help of a fuzzy set
B
, specifically by
means of the fuzzy relation
R
between
S
and
D
. Given fuzzy subsets
A
of
S
and
B
of
D
, the max-min composition
B
μ
s
d
R
R
describes the condition of the patient with
respect to the symptoms he is experiencing and the diseases from which he may be
suffering. The membership function below defined the fuzzy subset
B
in
D
.
=
A
◦
μ
B
(
d
)=
max
s
∈
S
(
μ
A
(
s
)
;
μ
R
(
s
,
d
))
,
d
∈
D
.
(3.12)
Simultaneously studying an entire set
P
of patients
p
led Sanchez to the definition of
the fuzzy relation
Q
S
to characterize the relationship between these patients
and their possible symptoms. Finally, the newly composed fuzzy relation
TonP
⊆
P
×
×
D
can be composed from the fuzzy relations
Q
and
R
:
T
=
Q
◦
R
with the membership
function
μ
T
(
p
,
d
)=
max
s
∈
S
min
{
μ
Q
(
p
,
s
)
;
μ
R
(
s
,
d
)
},
s
∈
S
,
d
∈
D
,
p
∈
P
.
(3.13)
Zadeh had devised the max-min rule in 1965 as a composition rule for fuzzy rela-
tions and - as we said already - Assilian and Mamdani used it in 1972 to calculate
inference rule relationships once they had implemented the fuzzy IF-THEN rules
for their fuzzy algorithm to control their steam engine. Sanchez now interpreted it
directly as a “fuzzy inference rule”:
B
=
A
◦
R
:IF
A
THEN
B
by
R
.
